e:/_/clanky/phillips/bruck/11.dviConsider the following set of axioms:
Axiom 1: x1 = mult(x1, unit()) Axiom 2: x1 = mult(i(x2), mult(x2, x1))
Axiom 3: x1 = mult(unit(), x1) Axiom 4: x1 = mult(rd(x1, x2), x2) Axiom 5: x1 = rd(mult(x1, x2), x2)
Axiom 6: mult(x1, mult(x2, mult(x1, x3))) = mult(mult(x1, mult(x2, x1)), x3) Axiom 7: mult(x1, i(x1)) = unit() Axiom 8: mult(mult(x1, x2), x3) = mult(mult(x1, mult(x2, x3)), asoc(x1, x2, x3)) Axiom 9: mult(i(x1), x1) = unit()
Axiom 10: mult(i(x1), mult(x2, x1)) = opt(x2, x1) Axiom 11: mult(i(x1), i(x2)) = i(mult(x1, x2)) Axiom 12: mult(i(mult(x1, x2)), mult(x1, mult(x2, x3))) = opl(x3, x2, x1) Axiom 13: opt(opt(x1, x2), x3) = opt(opt(x1, x3), x2)
Axiom 14: opt(opr(x1, x2, x3), x4) = opr(opt(x1, x4), x2, x3) Axiom 15: opt(opl(x1, x2, x3), x4) = opl(opt(x1, x4), x2, x3) Axiom 16: opr(x1, x2, x3) = rd(mult(mult(x1, x2), x3), mult(x2, x3))
Axiom 17: opr(opr(x1, x2, x3), x4, x5) = opr(opr(x1, x4, x5), x2, x3) Axiom 18: opr(opl(x1, x2, x3), x4, x5) = opl(opr(x1, x4, x5), x2, x3) Axiom 19: opl(opl(x1, x2, x3), x4, x5) = opl(opl(x1, x4, x5), x2, x3)
This theorem holds true:
Proof:
rd(x1, unit())


rd( x1
rd(


mult(i(i(x1)),

opt(x1, x1)


opr(x1, x2, unit())

rd(
rd(mult(x1, x2),

opt(i(x1), x1)

mult(i(x1),
unit())

x1


opt(
rd( x1
, mult(x2, x1))
rd(

i(rd(x1, x2))

rd(x2,
x1 )
opt(opt(x1, x2), x1)

opt(
mult( x1
, mult(i(x1), x2))
mult(

mult(x1, opt(x2, x1)
mult(x1,

opt(rd(x1, x2), x2)

mult(i(x2),
x1 )
4
opt(i(x1), i(x2))

mult(i(i(x2)),

i(
i(mult(i(x1), x2))

mult(
i(mult(x1, i(x2)))

mult(i(x1),
x2 )
i(opt(i(x1), x2))

opt(
i(opt(x1, i(x2)))

opt(i(x1),
x2 )
5
mult(x1, i(mult(x1, x2))
mult(x1,

mult(rd(i(x1), x2)
mult(

i(
x2 )
mult(rd(x1, x2), i( x1
mult(rd(x1, x2), i(
mult(rd(x1, x2), x2)))

rd(i(x1)
, i(x2))
rd(

mult(mult(x1, x1
mult(mult(x1,
mult(unit(), x1)), x2)

mult(x1,
opl(x1, x2, x2)

)
mult(i(mult(x2, x2)),

opr(x1, x1, x2)

rd(

mult(opt(x1, x2), x1)

mult(x1,
opr(opt(x1, x2), x1, x3)

opt(
rd(mult(x1, x2), opt(x1, x2)
rd(mult(x1, x2),


mult( x1
, mult(x2, i(x1)))
mult(

mult(x1, i(mult(x2, x1)))

i(
opt(rd(i(x2), x1)
opt(

i(
opt(rd(i(x1), x2), x1)



opt(rd(x1, x2), i(x1))

mult(i(i(x1)),
i(x2))


opt(i(opt(x1, x2)), x1)


opt(
opt(i(x1), i(x2)), i(i(x1)))

opt(

rd(opt(x1, x2)
, mult(x1, x2))
rd(

rd(i(x1), x2)

rd(
rd(x2, i(x1))


rd(x1,
rd(rd(x1, x2)
rd(

rd(i(x1), rd(x2, x3)
rd(i(x1),

rd(mult(x1, mult(x1, x2))
rd(

rd( x1
rd(
rd(mult(x1,

i(mult(x1, x1))


rd(mult(i(x1),
unit())
, x1)
rd(
i(mult(rd(x1, x2), x3))

mult(
i(mult(x1, rd(x2, x3)))

mult(i(x1),
i(opt(rd(x1, x2), x3))

opt(
i(opt(x1, rd(x2, x3)))

opt(i(x1),
opt(x1, i(opt(x1, x2)))


, x2))))
mult(opt(x1, x2), i(mult(i(x1), opt(
)))
= by Axiom 13 LR with {x3 ← x2, x2 ← i(x1), x1 ← x1}
mult(opt(x1, x2), i(mult(i(x1),
opt(opt(x1, x2), i(x1)))
= by Lemma 13 LR with {x2 ← opt(x1, x2), x1 ← i(x1)}
mult(opt(x1, x2), i(
mult(opt(x1, x2), i(x1))))


rd(i(x1), x1)


rd(x2,
mult(x1, opt(opt(x2, x1), x3)
)
= by Axiom 13 RL with {x3 ← x1, x2 ← x3, x1 ← x2}
mult(x1,

opr(rd(x1, x2), rd(x2, x1), x3)

rd(mult(mult(rd(x1, x2),
i(rd(x1, x2)))
rd(mult(
rd(


opr(x1, i(x1)
opr( x1
opr(
rd(mult(x1, x2), x2), rd(x2, mult(x1, x2)), x2)


rd(x1, x2)

))
opt(rd(x1, x2), i(
opt(rd(x1, x2),
mult(i(x1), x2))
Lemma 54: mult(x1, mult(rd(x2, x1), mult(x1, x3))) = mult(mult(x1, x2), x3)
mult(x1, mult(rd(x2, x1), mult(x1, x3)))

mult(mult(x1,
mult(mult(x1, x2), i(x1))

))
mult(x1, mult(rd(x2, x1),
mult(x1,
mult(i(mult(x1, x2)), x1)


mult(

mult(i(x1),
rd(x1, mult(i(x1), rd(x1, x2))
)
= by Lemma 45 RL with {x3 ← x1, x2 ← x2, x1 ← x1}
rd(x1,
i(mult(x1, rd(x2, x1))))

rd(

rd(x1, i(mult(x1, x2))
rd(x1,
rd(x1, mult(i(x1),

i(mult(x1, mult(x2, x1)))

mult(i(x1),

rd(i(x1), i(
i(mult(x1, x2))))


mult(x1, mult(i(x2), x1))

rd(x1,
rd(mult(x1, i(x2))
rd(

rd(i(x1), mult(x2, i(x3))
rd(i(x1),

rd(mult(i(x1), x2)
rd(

rd(i(x1), mult(i(x2), x3)
rd(i(x1),

rd(opt(x1, i(x2))
rd(

rd(i(x1), opt(x2, i(x3))
rd(i(x1),

rd(opt(i(x1), x2)
rd(

rd(i(x1), opt(i(x2), x3)
rd(i(x1),

mult(i(x1), rd(x2, i(x1))
mult(i(x1),
mult(i(x1), mult(rd(x2, i(x1)),
mult(i(x1), x1)))

mult( x1
mult(
mult(i(i(x1)),

mult(i(x1), rd(x1, i(x2)))

mult(
mult(x1, mult(opt(x2, x1), x1))

rd(x1, i(
mult(x2, x1)))
mult(i(x1), mult(x2, i(x1)))


mult(i(x2), x1)


opt(
mult( x1
mult(

opt(mult(
Lemma 76: rd(mult(x1, mult(x2, mult(x1, x3))), x3) = mult(x1, mult(x2, x1))
rd(mult(x1, mult(x2, mult(x1, x3)))
, x3)
= by Axiom 6 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
rd(
mult(mult(x1, mult(x2, x1)), x3), x3)

rd( x1
rd(

opr(i(x1), x2, mult(x1, i(x2)))

rd(mult(mult(i(x1), x2),
i(mult(i(x1), x2)))
rd(

i(

opr( x1
opr(
opr(i(i(x1)), x2,

opt(
opt(rd(x2, i(i(x1)))
opt(

)
opr(i(rd(i(i(x1)), x2)), x1,
mult(i(x1), x2)

opt(rd(x2,

mult(i(x1), x2)

opr(
opr(rd(x2,
rd(i(x1), mult(x1, rd(x2, x1))
rd(i(x1),

mult(x1, rd(mult(i(x1), x2), x1))

mult(
rd(mult(i(x1), rd(x1, x2))
rd(

mult(x1, mult(mult(i(x1), x2), x1))

rd(x1, i(
i(mult(x1, mult(i(x2), x3)))

mult(i(x1),
i(mult(mult(x1, i(x2))
i(mult(

i(opt(mult(x1, i(x2)), x3))

opt(
i(opt(x1, opt(i(x2), x3)))

opt(i(x1),
Lemma 91: mult(opr(x1, x2, x3), mult(x2, x3)) = mult(mult(x1, x2), x3)
mult(opr(x1, x2, x3)
, mult(x2, x3))
= by Axiom 16 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(
rd(mult(mult(x1, x2), x3), mult(x2, x3)), mult(x2, x3))

mult(mult(x1, x2), i(x2))

)
mult(opr(x1, x2, i(x2)),

mult(mult(x1, i(x2)), x2)

)
mult(opr(x1, i(x2), x2),

opt(x1, x2)


mult(
mult(x1, rd(opt(x1, x2), x1))


mult(i(x1), opt(x1, x2)
mult(i(x1),
mult(x1, rd(opt(x1, x2), x1)))

mult(opt(x1, x2), i(x1))

)
= by Axiom 13 RL with {x3 ← x2, x2 ← i(x1), x1 ← x1}
mult(i(x1),
mult(i(x1), opt(
x1 , x2))

rd(x1, opt(x1, x2))

i(rd(opt(x1, x2), x1))

))
mult(x1, i(
opt(x1, x2)))
mult(i(opt(x1, x2)), x1)

mult(x1,

mult(i(mult(x1, x2)), x2)


mult(

opr(i(x1), i(x2),
opr(i(x1), i(x2), x2)


i(
rd(x1, opt(x1, i(x2)))

rd(opt(i(x1), x2), i(x1))

)
mult(x1,
mult(x1, opt(x1, i(x2))
mult(x1,
= by Lemma 21 RL with {x2 ← opt(i(x1), x2), x1 ← x1}
mult(x1,
mult(x1, mult(

rd(x1,
opr(rd(x1, x2), x2, i(x2))

mult(
opr(rd(x1, x2), i(x1), x1
opr(rd(x1, x2), i(x1),

mult(
mult(i(x1), x2)


rd(x2,
mult( x1
mult(
mult(mult(x1,
mult(x2, i(x2))), asoc(x1, x2, i(x2)))


Lemma 108: mult(i(mult(x1, mult(x2, x3))), mult(mult(x1, x2), x3)) = asoc(x1, x2, x3)
mult(i(mult(x1, mult(x2, x3))), mult(mult(x1, x2), x3)
)
= by Axiom 8 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(mult(x1, mult(x2, x3))),
mult(mult(x1, mult(x2, x3)), asoc(x1, x2, x3)))

mult(x1, rd(opt(x2, x1), x1))

mult(

mult( x1
mult(
mult(i(i(x1)),

mult(i(x1), opt(i(x1), x2)
= by Axiom 4 LR with {x2 ← i(x1), x1 ← opt(i(x1), x2)}
mult(i(x1),
mult(rd(opt(i(x1), x2), i(x1)), i(x1)))

))
rd(i(x1), i(
opt(i(x1), x2))
rd(i(x1),
opt(x1, i(x2)))

Lemma 112: mult(opr(x1, x2, x3), i(x1)) = rd(opr(x1, x2, x3), x1)
mult(opr(x1, x2, x3), i(x1))

, x1)
= by Lemma 93 LR with {x2 ← x1, x1 ← opr(x1, x2, x3)}
rd(
, x1)
= by Axiom 17 RL with {x5 ← x3, x4 ← x2, x3 ← x1, x2 ← i(x1), x1 ← x1}
rd(
rd(opr(
Lemma 113: mult(i(opr(x1, x2, x3)), x1) = rd(x1, opr(x1, x2, x3))
mult(i(opr(x1, x2, x3)), x1)

)
= by Axiom 17 RL with {x5 ← x3, x4 ← x2, x3 ← x1, x2 ← i(x1), x1 ← x1}
rd(x1,
rd(x1, opr(
Lemma 114: rd(x1, i(opr(x1, x2, x3))) = mult(opr(x1, x2, x3), x1)
rd(x1, i(opr(x1, x2, x3)))

rd(opr(x1, x2, x3), i(x1))

, opr(x1, x2, x3)))
= by Lemma 113 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(opr(x1, x2, x3), mult(
)
= by Axiom 4 RL with {x2 ← opr(x1, x2, x3), x1 ← x1}
mult(opr(x1, x2, x3),
Lemma 115: mult(x1, i(opr(x1, x2, x3))) = rd(x1, opr(x1, x2, x3))
mult(x1, i(opr(x1, x2, x3)))

))
= by Lemma 114 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
rd(x1, mult(i(x1),
)
= by Axiom 10 LR with {x2 ← opr(x1, x2, x3), x1 ← x1}
rd(x1,
)
= by Axiom 14 LR with {x4 ← x1, x3 ← x3, x2 ← x2, x1 ← x1}
rd(x1,
rd(x1, opr(
Lemma 116: mult(i(x1), opr(x1, x2, x3)) = rd(mult(i(x1), mult(opr(x1, x2, x3), x1)), x1)
mult(i(x1), opr(x1, x2, x3))

i(mult(x1, i(opr(x1, x2, x3))))

), x1)
= by Lemma 114 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
rd(mult(i(x1),
30
Lemma 117: mult(i(x1), opr(x1, x2, x3)) = rd(opr(x1, x2, x3), x1)
mult(i(x1), opr(x1, x2, x3))

, x1)
= by Axiom 10 LR with {x2 ← opr(x1, x2, x3), x1 ← x1}
rd(
, x1)
= by Axiom 14 LR with {x4 ← x1, x3 ← x3, x2 ← x2, x1 ← x1}
rd(
rd(opr(
x1 , x2, x3), x1)
Lemma 118: mult(opr(x1, x2, x3), x1) = mult(x1, mult(rd(opr(x1, x2, x3), x1), x1))
mult(opr(x1, x2, x3), x1)

rd(x1, i(opr(x1, x2, x3)))

, x1))
= by Lemma 117 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(x1, mult(
Lemma 119: mult(opr(x1, x2, x3), x1) = mult(x1, opr(x1, x2, x3))
mult(opr(x1, x2, x3), x1)

)
= by Axiom 4 RL with {x2 ← x1, x1 ← opr(x1, x2, x3)}
mult(x1,
opt(x1, opr(x1, x2, x3))

)
= by Lemma 119 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(opr(x1, x2, x3)),
mult(opr(x1, x2, x3), x1))

Lemma 121: rd(x1, i(opr(x1, x2, x3))) = mult(x1, opr(x1, x2, x3))
rd(x1, i(opr(x1, x2, x3)))

mult(opr(x1, x2, x3), x1)

rd(i(x1), opr(x2, x1, i(x1))
rd(i(x1),

Lemma 123: mult(x1, rd(opr(x1, x2, x3), x1)) = opr(x1, x2, x3)
mult( x1
mult(
)
= by Lemma 112 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(i(x1)),
mult(opr(x1, x2, x3), i(x1)))

opt(opr(x1, x2, x3), i(x1))

opr(
Lemma 124: mult(mult(x1, opt(x1, x2)), x3) = mult(opt(x1, x2), mult(x1, x3))
mult(mult(x1, opt(x1, x2))
mult(
mult(opt(x1, x2), x1), x3)

, mult(x1, x3))
= by Lemma 28 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(
Lemma 125: opr(mult(x1, x2), i(x1), x1) = mult(mult(x1, rd(x2, x1)), i(i(x1)))
opr(mult(x1, x2), i(x1), x1
opr(mult(x1, x2), i(x1),

mult(
Lemma 126: opr(mult(x1, x2), i(x2), x2) = mult(opr(x1, x2, i(x2)), i(i(x2)))
opr(mult(x1, x2), i(x2), x2
opr(mult(x1, x2), i(x2),

mult(
Lemma 127: mult(opr(x1, x2, i(x2)), x2) = opr(mult(x1, x2), i(x2), x2)
mult(opr(x1, x2, i(x2)), x2
mult(opr(x1, x2, i(x2)),

rd(mult(x1, i(opt(x1, x2))), i(x1))


i(mult(i(x1),
Lemma 129: rd(x1, rd(opt(x1, x2), x1)) = i(mult(i(x1), rd(opt(x1, x2), x1)))
rd(x1, rd(opt(x1, x2), x1)
rd(x1,
mult(i(x1), opt(x1, x2)))


Lemma 130: rd(x1, rd(opt(x1, x2), x1)) = mult(x1, rd(x1, opt(x1, x2)))
rd(x1, rd(opt(x1, x2), x1))

i(mult(i(x1), rd(opt(x1, x2), x1)))

= by Lemma 10 LR with {x2 ← x1, x1 ← opt(x1, x2)}
mult(x1,
Lemma 131: rd(i(x1), opr(x1, x2, x3)) = i(mult(x1, opr(x1, x2, x3)))
rd(i(x1), opr(x1, x2, x3))

rd(i(opr(x1, x2, x3)), x1)

)
= by Lemma 121 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
i(
Lemma 132: mult(mult(x1, x2), opl(x3, x2, x1)) = mult(x1, mult(x2, x3))
mult(mult(x1, x2), opl(x3, x2, x1)
)
= by Axiom 12 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(mult(x1, x2),
mult(i(mult(x1, x2)), mult(x1, mult(x2, x3))))

mult(i(opt(x1, x2)), mult(x1, x2)
mult(i(opt(x1, x2)),

mult(x1, mult(x1, x2)
mult(x1,

opt(
mult(mult(x1, x2), i(x1))


))
rd(i(i(mult(x1, x2))), mult(i(mult(x1, x2)),
mult(x1, rd(x2, x1))


rd(i(i(mult(x1, x2))), mult(i(mult(x1, x2)), mult(x1, mult(x2, x1))))

35
Lemma 137: mult(x1, mult(x2, opl(x3, x1, x2))) = mult(mult(x1, x2), x3)
mult(x1, mult(x2, opl(x3, x1, x2)
))
= by Axiom 12 RL with {x3 ← x3, x2 ← x1, x1 ← x2}
mult(x1, mult(x2,
)
= by Axiom 6 LR with {x3 ← mult(x1, x3), x2 ← i(mult(x2, x1)), x1 ← x2}
mult(x1,
mult(mult(x2, mult(i(mult(x2, x1)), x2)), mult(x1, x3)))

), x3)
= by Axiom 6 RL with {x3 ← x1, x2 ← i(mult(x2, x1)), x1 ← x2}
mult(mult(x1,
)), x3)
mult(mult(x1, mult(x2,
mult(mult(x1,
opl(opl(x1, x2, x3), x3, x2)

)
= by Lemma 137 LR with {x3 ← x1, x2 ← x3, x1 ← x2}
mult(i(mult(x2, x3)),

mult(i(x1), opr(x2, x1, i(x1))
mult(i(x1),

rd(
rd(x1, rd(opt(i(x2), x1), x2))

rd(
rd(mult(x2,
mult(i(i(x1)), mult(x2, i(x1)))), i(x1))


rd(x2,

rd(opr(x1, x2, i(x2)), x1)

mult(i(x1),
mult(x1, asoc(x1, x2, i(x2))))

asoc(x1, i(x2), x2
asoc(x1, i(x2),

rd(opr(x1, i(x2),
x2 ), x1)
rd(x1, opr( x1
rd(x1, opr(
rd(x1,
i(opr(i(x1), x2, i(x2))))


asoc(i(x1), i(x2), x2
asoc(i(x1), i(x2),

rd(x1, opr(x1,
x2 , i(x2)))
asoc(i(x1), i(x2), x2)

rd(x1, opr(x1, x2, i(x2)))

i(
Lemma 146: rd(x1, mult(x1, opr(x1, x2, x3))) = i(opr(x1, x2, x3))
rd(x1, mult(x1, opr(x1, x2, x3))
)
= by Lemma 119 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
rd(x1,
mult(opr(x1, x2, x3), x1))

opr(mult(x1, x2)
opr(
opr(i(
, x3)
opr(i(opt(rd(i(x1), x2), x1))
= by Lemma 15 RL with {x2 ← x1, x1 ← rd(i(x1), x2)}
opr(
opt(i(rd(i(x1), x2)), i(x1)), i(i(rd(i(x1), x2))), x3)

, i(x1))
= by Lemma 52 LR with {x2 ← x3, x1 ← i(rd(i(x1), x2))}
opt(

i(opt(rd(i(x1), x2), x1))


mult(
i(x2)

mult(x1, i(mult(x1, x2)))

opr(
39
i(x2)

mult(rd(x1, x2), i(x1))

opr(
Lemma 150: i(mult(i(x1), rd(mult(x2, x1), x2))) = rd(i(rd(mult(x2, x1), x2)), i(x1))
i(mult(i(x1), rd(mult(x2, x1), x2)))

, i(rd(mult(x2, x1), x2))))
= by Lemma 23 RL with {x2 ← mult(x2, x1), x1 ← x2}
rd(i(rd(mult(x2, x1), x2)), opr(i(x1),
)
= by Lemma 148 RL with {x3 ← i(rd(mult(x2, x1), x2)), x1 ← x2, x2 ← x1}
rd(i(rd(mult(x2, x1), x2)),
Lemma 151: mult(x1, rd(x2, mult(x2, x1))) = rd(i(rd(mult(x2, x1), x2)), i(x1))
mult(x1, rd(x2, mult(x2, x1))
= by Lemma 10 RL with {x2 ← x2, x1 ← mult(x2, x1)}
mult(x1,
i(rd(mult(x2, x1), x2)))


Lemma 152: i(mult(i(x1), rd(x2, rd(x2, x1)))) = rd(i(rd(x2, rd(x2, x1))), i(x1))
i(mult(i(x1), rd(x2, rd(x2, x1))))

, i(rd(x2, rd(x2, x1)))))
= by Lemma 23 RL with {x2 ← x2, x1 ← rd(x2, x1)}
rd(i(rd(x2, rd(x2, x1))), opr(i(x1),
)
= by Lemma 149 RL with {x3 ← i(rd(x2, rd(x2, x1))), x1 ← x2, x2 ← x1}
rd(i(rd(x2, rd(x2, x1))),
Lemma 153: mult(x1, rd(rd(x2, x1), x2)) = rd(i(rd(x2, rd(x2, x1))), i(x1))
mult(x1, rd(rd(x2, x1), x2)
= by Lemma 10 RL with {x2 ← rd(x2, x1), x1 ← x2}
mult(x1,
i(rd(x2, rd(x2, x1))))


opt(x1, rd(i(x1), opr(x1, x2, x3))
)
= by Lemma 131 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
opt(x1,
i(mult(x1, opr(x1, x2, x3))))

))
= by Lemma 119 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
i(opt(i(x1),
)
= by Axiom 10 RL with {x2 ← i(x1), x1 ← mult(opr(x1, x2, x3), x1)}
i(
))
= by Axiom 10 LR with {x2 ← opr(x1, x2, x3), x1 ← x1}
i(mult(i(mult(opr(x1, x2, x3), x1)
opt(opr(x1, x2, x3), x1)))
= by Lemma 119 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
i(mult(i(
mult(x1, opr(x1, x2, x3)))
, opt(opr(x1, x2, x3), x1)))
= by Lemma 131 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
i(mult(
))
= by Axiom 14 LR with {x4 ← x1, x3 ← x3, x2 ← x2, x1 ← x1}
i(mult(rd(i(x1), opr(x1, x2, x3)),
i(mult(rd(i(x1), opr(x1, x2, x3)), opr(
x1 , x2, x3))
)
= by Axiom 4 RL with {x2 ← opr(x1, x2, x3), x1 ← i(x1)}
i(
i(x1))

x1
41
Lemma 155: mult(x1, opr(opt(x2, x1), x3, x4)) = mult(opr(x2, x3, x4), x1)
mult(x1, opr(opt(x2, x1), x3, x4)
)
= by Axiom 14 RL with {x4 ← x1, x3 ← x4, x2 ← x3, x1 ← x2}
mult(x1,
opt(opr(x2, x3, x4), x1))

mult(opr(x1, opt(x1, x2), x3), x2)

)
= by Lemma 26 LR with {x2 ← x3, x1 ← opt(x1, x2)}
mult(x2,

opr(x1, opt(x1, x2), x3)

, x2)
= by Lemma 156 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
rd(

mult(x1, asoc(x1, x2, x1))

, x1, x2)
= by Axiom 12 RL with {x3 ← mult(x1, asoc(x1, x2, x1)), x2 ← x2, x1 ← x1}
opl(
), x1, x2)
= by Axiom 6 LR with {x3 ← asoc(x1, x2, x1), x2 ← x2, x1 ← x1}
opl(mult(i(mult(x1, x2)),
), x1, x2)
= by Axiom 8 RL with {x3 ← x1, x2 ← x2, x1 ← x1}
opl(mult(i(mult(x1, x2)),
= by Axiom 2 RL with {x2 ← mult(x1, x2), x1 ← x1}
opl(
Lemma 159: mult(opr(i(x1), x2, x3), x1) = mult(x1, opr(i(x1), x2, x3))
mult(opr(i(x1), x2, x3), x1)

mult(x1, opr(
opt(x1, i(opr( x1
opt(x1, i(opr(
i(i(x1)), x2, x3)))

)
= by Axiom 10 RL with {x2 ← i(x1), x1 ← opr(i(i(x1)), x2, x3)}
i(
))
= by Lemma 159 RL with {x3 ← x3, x2 ← x2, x1 ← i(x1)}
i(mult(i(opr(i(i(x1)), x2, x3)),
)
= by Axiom 2 RL with {x2 ← opr(i(i(x1)), x2, x3), x1 ← i(x1)}
i(
i(x1))

Lemma 161: mult(opt(x1, x2), mult(x1, x3)) = mult(x1, mult(opt(x1, x2), x3))
mult(opt(x1, x2), mult(x1, x3))

mult(mult(x1, opt(x1, x2)), x3)

, mult(opt(x1, x2), x3))
= by Lemma 157 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(
Lemma 162: mult(mult(x1, opt(x1, x2)), x3) = mult(x1, mult(opt(x1, x2), x3))
mult(mult(x1, opt(x1, x2)), x3)

mult(opt(x1, x2), mult(x1, x3))

opl(x1, opt(x2, x3), x2)

)
= by Lemma 162 RL with {x3 ← x1, x2 ← x3, x1 ← x2}
mult(i(mult(x2, opt(x2, x3))),
mult(mult(x2, opt(x2, x3)), x1))

opl(x1, mult(i(x2), x3)
opl(x1,
opt(rd(x3, x2), x2), rd(x3, x2))

opl( x1
, x2, opr(x2, x3, i(x3)))
= by Lemma 164 RL with {x3 ← mult(rd(i(x3), opr(x2, x3, i(x3))), opr(x2, x3, i(x3))), x2 ← rd(i(x3), opr(x2, x3, i(x3))), x1 ←
opl(
, rd(mult(rd(i(x3), op
= by Axiom 2 RL with {x2 ← rd(i(x3), opr(x2, x3, i(x3))), x1 ← opr(x2, x3, i(x3))}
opl(opl(x1,
opr(x2, x3, i(x3)), rd(mult(rd(i(x3), opr(x2, x3, i(x3))), opr(x2, x3, i(x3)))
, rd(i(x3), opr(x2, x3, i(x3))))), x2, op
= by Axiom 4 RL with {x2 ← opr(x2, x3, i(x3)), x1 ← i(x3)}
opl(opl(x1, opr(x2, x3, i(x3)), rd(
i(x3), rd(i(x3), opr(x2, x3, i(x3)))
)), x2, opr(x2, x3, i(x3)))
opl(opl(x1, opr(x2, x3, i(x3)), rd(i(x3),
i(mult(x2, x3)))
= by Lemma 23 LR with {x2 ← mult(x2, x3), x1 ← x3}
opl(opl(x1, opr(x2, x3, i(x3)),
opl(opl(x1, opr(x2, x3, i(x3)),
x2 ), x2, opr(x2, x3, i(x3)))

mult(opr(x1, i(opt(x1, x2)), x3), x2)

)
= by Lemma 52 LR with {x2 ← x3, x1 ← opt(x1, x2)}
mult(x2,

mult(x1, x2)
Lemma 167: mult(x1, opl(opt(x2, x1), x3, x4)) = mult(opl(x2, x3, x4), x1)
mult(x1, opl(opt(x2, x1), x3, x4)
)
= by Axiom 15 RL with {x4 ← x1, x3 ← x4, x2 ← x3, x1 ← x2}
mult(x1,
opt(opl(x2, x3, x4), x1))

Lemma 168: mult(opl(x1, x2, x3), x1) = mult(x1, opl(x1, x2, x3))
mult(opl(x1, x2, x3), x1)

mult(x1, opl(
opt(x1, opl(x1, x2, x3))

)
= by Lemma 168 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(opl(x1, x2, x3)),
mult(opl(x1, x2, x3), x1))

Lemma 170: mult(x1, rd(opl(x1, x2, x3), x1)) = opl(x1, x2, x3)
mult(x1, rd(opl(x1, x2, x3), x1))

, i(x1))
= by Lemma 168 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(
mult(opl(x1, x2, x3), x1), i(x1))

opr(opl(x1, x2, x3), x1, i(x1))

opl(
Lemma 171: mult(i(x1), opl(x1, x2, x3)) = rd(opl(x1, x2, x3), x1)
mult(i(x1), opl(x1, x2, x3)
)
= by Lemma 170 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(x1),
mult(x1, rd(opl(x1, x2, x3), x1)))

rd(opl(x1, x1, x2), x1)

mult(i(x1),
mult(x1, asoc(x1, x2, x1)))

mult(asoc(x1, x2, x1)
mult(
rd(opl(x1, x1, x2), x1), x1)

rd(x1, opl(x1, x1, x2))

i(
opt(opt(x1, x2), asoc(x1, x3, x1))

opt(opt(x1,
, x2)
= by Axiom 10 RL with {x2 ← x1, x1 ← rd(opl(x1, x1, x3), x1)}
opt(
), x2)
= by Lemma 170 LR with {x3 ← x3, x2 ← x1, x1 ← x1}
opt(mult(i(rd(opl(x1, x1, x3), x1))
opl(x1, x1, x3)), x2)
= by Lemma 10 LR with {x2 ← x1, x1 ← opl(x1, x1, x3)}
opt(mult(
, x2)
= by Axiom 4 RL with {x2 ← opl(x1, x1, x3), x1 ← x1}
opt(
Lemma 176: rd(x1, opl(x1, x2, x3)) = mult(x1, i(opl(x1, x2, x3)))
rd(x1, opl(x1, x2, x3))

i(rd(opl(x1, x2, x3), x1))

))
= by Lemma 170 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(x1, i(
Lemma 177: rd(x1, opl(x1, x2, x3)) = mult(i(opl(x1, x2, x3)), x1)
rd( x1
, opl(x1, x2, x3))
= by Lemma 138 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
rd(
opl(opl(x1, x2, x3), x3, x2), opl(x1, x2, x3))

)
= by Lemma 138 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(opl(x1, x2, x3)),
Lemma 178: mult(opl(x1, x2, x3), i(x1)) = rd(opl(x1, x2, x3), x1)
mult(opl(x1, x2, x3), i( x1
))
= by Lemma 138 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(opl(x1, x2, x3), i(
opl(opl(x1, x2, x3), x3, x2)))

)
= by Lemma 138 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
rd(opl(x1, x2, x3),
Lemma 179: opt(asoc(x1, x2, x1), opt(x1, x3)) = mult(i(opt(x1, x3)), mult(opt(x1, x3), asoc(x1, x2, x1)))
opt(asoc(x1, x2, x1), opt(x1, x3)
)
= by Lemma 175 RL with {x3 ← x2, x2 ← x3, x1 ← x1}
opt(asoc(x1, x2, x1),
opt(opt(x1, x3), asoc(x1, x2, x1)))

mult(i(opt(opt(x1, x3), asoc(x1, x2, x1))
), mult(opt(x1, x3), asoc(x1, x2, x1)))
= by Lemma 175 LR with {x3 ← x2, x2 ← x3, x1 ← x1}
mult(i(
Lemma 180: opl(x1, x2, x3) = rd(x1, rd(x1, opl(x1, x2, x3)))
opl(x1, x2, x3)

i(i(opl(x1, x2, x3)))

)
= by Lemma 177 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
rd(x1,
rd(x1, i(asoc(x1, x2, x1))
rd(x1,
rd(x1, opl(x1, x1, x2)))

rd(i(x1), asoc(x1, x2, x1))

rd(i(asoc(x1, x2, x1)), x1)

i(
opl(x1, x1, x2))
Lemma 183: mult(x1, i(asoc(x1, x2, x1))) = rd(x1, rd(opl(x1, x1, x2), x1))
mult(x1, i(asoc(x1, x2, x1)))

))
rd(x1, mult(i(x1),
)
= by Lemma 171 LR with {x3 ← x2, x2 ← x1, x1 ← x1}
rd(x1,
rd(opl(x1, x1, x2), x1))
Lemma 184: mult(i(x1), asoc(x1, x2, x1)) = rd(mult(i(x1), opl(x1, x1, x2)), x1)
mult(i(x1), asoc(x1, x2, x1))

i(mult(x1, i(asoc(x1, x2, x1))))

), x1)
rd(mult(i(x1),
Lemma 185: mult(i(x1), asoc(x1, x2, x1)) = rd(asoc(x1, x2, x1), x1)
mult(i(x1), asoc(x1, x2, x1))

, x1)
= by Lemma 171 LR with {x3 ← x2, x2 ← x1, x1 ← x1}
rd(
rd(
Lemma 186: i(i(asoc(x1, x2, x1))) = mult(x1, i(rd(x1, asoc(x1, x2, x1))))
i(i(asoc(x1, x2, x1)))

))
mult(x1, i(
)))
mult(x1, i(rd(x1,
Lemma 187: asoc(x1, x2, x1) = mult(x1, rd(asoc(x1, x2, x1), x1))
asoc(x1, x2, x1)


)
= by Lemma 10 LR with {x2 ← asoc(x1, x2, x1), x1 ← x1}
mult(x1,
Lemma 188: mult(opl(i(x1), x2, x3), x1) = mult(x1, opl(i(x1), x2, x3))
mult(opl(i(x1), x2, x3), x1)

mult(x1, opl(
opt(x1, opl(i(x1), x2, x3))

)
= by Lemma 188 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(opl(i(x1), x2, x3)),
mult(opl(i(x1), x2, x3), x1))

opt(x1, i(opl( x1
opt(x1, i(opl(
i(i(x1)), x2, x3)))

)
= by Lemma 189 LR with {x3 ← x3, x2 ← x2, x1 ← i(x1)}
i(
i(x1))
opt(x1, i(opl(x1, x2, x3)))


x1
Lemma 192: opt(i(opl(x1, x2, x3)), x1) = mult(i(x1), mult(x1, i(opl(x1, x2, x3))))
opt(i(opl(x1, x2, x3)), x1
)
= by Lemma 191 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
opt(i(opl(x1, x2, x3)),
opt(x1, i(opl(x1, x2, x3))))

mult(i(opt(x1, i(opl(x1, x2, x3)))
), mult(x1, i(opl(x1, x2, x3))))
= by Lemma 191 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(
51
opt(i(opl(x1, x2, x3)), x1)

mult(i(x1), mult(x1, i(opl(x1, x2, x3))))

opt(opt(rd(x1, x2), x3), i(x1))

opt(
opl(x1, i(x2), mult(x2, i(x1)))

, mult(i(x2), x1)))
mult(i(mult(mult(x2, i(x1)), i(x2))), mult(
i(mult(i(x2), x1)), mult(i(x2), x1))
mult(i(mult(mult(x2, i(x1)), i(x2))),


mult(
opl(i(mult(x1, x2)), x1, x2
opl(i(mult(x1, x2)), x1
opl(i(mult(x1, x2))
opl(
opl(mult(i(x1), i(x2)), i(i(x1)),
mult(i(x1), i(mult(i(x1), i(x2)))))

mult(

opl(mult(i(x1), x2), x1, i( x2
))
opl(mult(i(x1), x2), x1
opl(mult(i(x1), x2),
= by Axiom 11 RL with {x2 ← mult(i(x1), x2), x1 ← x1}
opl(mult(i(x1), x2), i(i(x1)),
mult(i(x1), i(mult(i(x1), x2))))

mult(mult(


Lemma 198: mult(mult(x1, x2), rd(i(x1), x1)) = mult(x1, mult(rd(x2, x1), i(x1)))
mult(mult(x1, x2), rd(i(x1), x1))

)))
mult(x1, mult(rd(x2, x1), mult(x1,
mult(x1, mult(rd(x2, x1),
mult(mult(x1, x2), rd(i(x1), x1))


opt(mult(x1, mult(x2, x1))
opt(
= by Lemma 38 RL with {x1 ← x1, x2 ← mult(x1, x2)}
opt(
opt(rd(mult(x1, x2), i(x1)),

mult(
mult(x2,
mult(x2,
mult(rd(x1, x2), mult(x2, x2))

opt(mult(x2,
rd(x1, i(x1))

i(rd(i(x1), x1))

))
mult(mult(x1, x2), i(
opt(rd(x2, x1), i(x1))))
mult(x1, x1)


)
= by Lemma 46 LR with {x3 ← i(x1), x2 ← x1, x1 ← x2}
mult(mult(x1, x2),
mult(mult(x1, x2), opt(rd(x1, x2), x1
))
mult(mult(x1, x2), opt(rd(x1, x2),

rd(opt( x1
rd(opt(
= by Lemma 201 RL with {x2 ← x2, x1 ← mult(i(x2), x1)}
rd(
mult(rd(mult(i(x2), x1), x2), mult(x2, x2)), mult(x2, x2))

Lemma 206: rd(rd(opt(x1, x2), x1), x1) = rd(opt(x1, x2), mult(x1, x1))
rd(rd(opt(x1, x2), x1)
rd(

rd(
mult(i(x1), rd(x1, opt(x2, x1)))

mult(i(

opr(i(x1), i(x2), x2)


opr(
opr(i(x1), i(x2), x2)

opr(i(x1), mult(x1, x2),
mult(x1, i(mult(x1, x2))))

mult(x1, opr(i(x1), x2, i(x2))
mult(x1,
mult(i(x1), asoc(i(x1), x2, i(x2))))

Lemma 211: opr(i(x1), x2, i(x2)) = rd(asoc(i(x1), x2, i(x2)), opt(x1, opr(i(x1), x2, i(x2))))
opr(i(x1), x2, i(x2))

mult(i(x1), asoc(i(x1), x2, i(x2)))

))
rd(asoc(i(x1), x2, i(x2)), opt(x1,
opt(rd(x1, i(x2)), x2
opt(rd(x1, i(x2)),

mult(mult(i(x2), x1), rd(
mult(mult(i(x1), x2), mult(x1, x1)
mult(mult(i(x1), x2),
rd(x1, i(x1)))

opt(mult(x1, i(mult(x2, i(x1))))
= by Lemma 16 RL with {x2 ← mult(x2, i(x1)), x1 ← x1}
opt(
i(mult(i(x1), mult(x2, i(x1)))), x1)

i(
opt(rd(x1, mult(i(x1), x2))
opt(
opt(mult(x1,

opt(rd(x1, mult(i(x1), x2)), x1)

i(mult(x2, mult(i(x1), i(x1))))

mult(i(x2), mult(x1,
mult(i(x1), rd(x1, mult(i(x1), x2)))

mult(i(
rd( x1
rd(
)
= by Axiom 12 RL with {x3 ← x2, x2 ← mult(i(x2), x1), x1 ← x2}
rd(mult(x2, mult(i(x2), x1)),
mult(i(mult(x2, mult(i(x2), x1))), mult(x2, mult(mult(i(x2), x1), x2))))



rd(opl(x1, mult(i(x1), x2), x1), x2)

)
i(

opl(mult(x1, x2), x2, i(mult(x2, x1)))

))
mult(i(mult(i(mult(x2, x1)), x2)), mult(i(mult(x2, x1)),
rd(x2, i(mult(x2, x1))))
= by Lemma 55 RL with {x2 ← x2, x1 ← i(mult(x2, x1))}
mult(i(mult(i(mult(x2, x1)), x2)),
mult(mult(i(mult(x2, x1)), x2), i(i(mult(x2, x1)))))


i(x2)


))
opl(mult(i(mult(x1, x2)), x1), x1, i(
i(x2)))
i(x2)

opl(
)
opl(mult(i(x1), rd(x1, x2)), x1,
opl(mult(x1, rd(x2, x1))
opl(
))
opl(mult(mult(x1, x2), i(x1)), i(x1), i(
mult(i(x1), mult(x1, x2))))


opl( x1
, i(x1), opl(x1, x2, x3))
= by Lemma 223 RL with {x2 ← x1, x1 ← i(opl(x1, x2, x3))}
opl(
))), i(i(opl(x1, x2, x3))), i(x1)), i(x1), opl(x1, x2, x3))
= by Lemma 170 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
opl(opl(mult(i(opl(x1, x2, x3)), rd(x1, i(
mult(x1, rd(opl(x1, x2, x3), x1))))
), i(i(opl(x1, x2, x3))), i(x1)), i(x1), opl(x1, x2, x3))
= by Lemma 58 LR with {x2 ← rd(opl(x1, x2, x3), x1), x1 ← x1}
opl(opl(mult(i(opl(x1, x2, x3)),
)), i(i(opl(x1, x2, x3))), i(x1)), i(x1), opl(x1, x2, x3))
= by Axiom 4 RL with {x2 ← x1, x1 ← opl(x1, x2, x3)}
opl(opl(mult(i(opl(x1, x2, x3)), mult(x1,
, i(i(opl(x1, x2, x3))), i(x1)), i(x1), opl(x1, x2, x3))
= by Axiom 10 LR with {x2 ← x1, x1 ← opl(x1, x2, x3)}
opl(opl(
, i(i(opl(x1, x2, x3))), i(x1)), i(x1), opl(x1, x2, x3))
= by Lemma 169 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
opl(opl(
= by Lemma 3 LR with {x1 ← opl(x1, x2, x3)}
opl(opl(x1,
opl(x1, x2, x3), i(x1)), i(x1), opl(x1, x2, x3))

opl(i(x1), x1
opl(i(x1),
i(i(x1)), opl(i(x1), x2, x3))

x2


))
opl(mult(mult(i(x1), x2), x1), x1, i(
x2 ))
opl( x1
= by Lemma 223 RL with {x2 ← x1, x1 ← i(mult(x1, x2))}
opl(
), i(i(mult(x1, x2))), i(x1)), i(x1), mult(x1, x2))
= by Lemma 58 LR with {x2 ← x2, x1 ← x1}
opl(opl(mult(i(mult(x1, x2)),
, i(i(mult(x1, x2))), i(x1)), i(x1), mult(x1, x2))
= by Axiom 12 LR with {x3 ← x1, x2 ← x2, x1 ← x1}
opl(opl(
, i(x1)), i(x1), mult(x1, x2))
opl(opl(opl(x1, x2, x1),
mult(x1, x2), i(x1)), i(x1), mult(x1, x2))

opl(x1, asoc(x1, x2, x1), x1)

)
opl(x1, i(x1),
opl(x1, x1, x2))

opl(x1, mult(i(x1), x2), x1)

)
opl(x1, i(x1),
rd(x1, opl(x2, i(x2), x1)
rd(x1,

mult(x1, x2
mult(x1,

rd(x1, opl(i(x2),
x2 , x1))
mult(x1, opl(i(x2), x2, x1))

))
rd(x1, mult(i(x1),
mult(x1, x2))
rd(x1,
x2 )
rd(opl(x1, i(x1), x2)
rd(
= by Lemma 219 LR with {x2 ← x2, x1 ← x1}

opl(x1, rd(opl(x1, x2, x3), x1), x1)

)
= by Lemma 170 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
opl(x1, i(x1),
opl(x1, x2, x3))

i(i(x1))

)
rd(opl(i(i(x2)), i(x2), i(x1)),
rd(opl(i(i(x2)), i(x2), i(x1)),
x1


rd(opl(
64
Lemma 237: mult(x1, opr(x1, x2, x3)) = mult(rd(opr(x1, x2, x3), x1), mult(x1, x1))
mult(x1, opr(x1, x2, x3))

, x1)
= by Axiom 4 LR with {x2 ← x1, x1 ← opr(x1, x2, x3)}
mult(
mult(rd(opr(x1, x2, x3), x1), x1), x1)

))
= by Lemma 227 RL with {x2 ← rd(opr(x1, x2, x3), x1), x1 ← x1}
mult(rd(opr(x1, x2, x3), x1), mult(x1,

)))
= by Lemma 123 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(rd(opr(x1, x2, x3), x1), mult(x1, opl( x1
, i(x1),
opr(x1, x2, x3))))
= by Lemma 223 RL with {x2 ← x1, x1 ← i(opr(x1, x2, x3))}
mult(rd(opr(x1, x2, x3), x1), mult(x1, opl(
opl(mult(i(opr(x1, x2, x3)), rd(x1, i(opr(x1, x2, x3)))
), i(i(opr(x1, x2, x3))), i(x1))
= by Lemma 121 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(rd(opr(x1, x2, x3), x1), mult(x1, opl(opl(mult(i(opr(x1, x2, x3)),
mult(x1, opr(x1, x2, x3)))
, i(i(opr(x1, x2, x3))), i(x1))
= by Axiom 10 LR with {x2 ← x1, x1 ← opr(x1, x2, x3)}
mult(rd(opr(x1, x2, x3), x1), mult(x1, opl(opl(
opt(x1, opr(x1, x2, x3))
, i(i(opr(x1, x2, x3))), i(x1)), i(x1), opr(x1, x2, x3))))
= by Lemma 120 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(rd(opr(x1, x2, x3), x1), mult(x1, opl(opl(
x1 , i(i(opr(x1, x2, x3)))
= by Lemma 3 LR with {x1 ← opr(x1, x2, x3)}
mult(rd(opr(x1, x2, x3), x1), mult(x1, opl(opl(x1,

))
= by Lemma 138 LR with {x3 ← i(x1), x2 ← opr(x1, x2, x3), x1 ← x1}
mult(rd(opr(x1, x2, x3), x1), mult(x1,
x1 ))
Lemma 238: opt(mult(x1, opr(x1, x2, x3)), x1) = mult(x1, opr(x1, x2, x3))
opt(mult(x1, opr(x1, x2, x3)), x1)

mult(rd(opr(x1, x2, x3), x1), mult(x1, x1))

65
Lemma 239: opt(x1, mult(x1, opr(x1, x2, x3))) = mult(i(mult(x1, opr(x1, x2, x3))), mult(mult(x1, opr(x1, x2, x3)), x1))
opt(x1, mult(x1, opr(x1, x2, x3))
)
= by Lemma 238 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
opt(x1,
opt(mult(x1, opr(x1, x2, x3)), x1))

mult(i(opt(mult(x1, opr(x1, x2, x3)), x1)
), mult(mult(x1, opr(x1, x2, x3)), x1))
= by Lemma 238 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(
mult(x1, opr(x1, x2, x3))), mult(mult(x1, opr(x1, x2, x3)), x1))
Lemma 240: opt(x1, mult(x1, opl(x1, x2, x3))) = mult(i(mult(x1, opl(x1, x2, x3))), mult(mult(x1, opl(x1, x2, x3)), x1))
opt(x1, mult(x1, opl(x1, x2, x3))
)
= by Lemma 168 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
opt(x1,
, x1))
= by Axiom 4 LR with {x2 ← x1, x1 ← opl(x1, x2, x3)}
opt(x1, mult(
)
= by Lemma 137 RL with {x3 ← x1, x2 ← x1, x1 ← rd(opl(x1, x2, x3), x1)}
opt(x1,

)))
= by Lemma 234 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
opt(x1, mult(rd(opl(x1, x2, x3), x1), mult(x1,
x1 ))
)
= by Lemma 201 LR with {x2 ← x1, x1 ← opl(x1, x2, x3)}
opt(x1,
opt(mult(x1, opl(x1, x2, x3)), x1))

mult(i(opt(mult(x1, opl(x1, x2, x3)), x1)
), mult(mult(x1, opl(x1, x2, x3)), x1))
= by Lemma 201 RL with {x2 ← x1, x1 ← opl(x1, x2, x3)}
mult(i(
))), mult(mult(x1, opl(x1, x2, x3)), x1))
= by Lemma 234 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(mult(rd(opl(x1, x2, x3), x1), mult(x1,

), mult(mult(x1, opl(x1, x2, x3)), x1))
= by Lemma 137 LR with {x3 ← x1, x2 ← x1, x1 ← rd(opl(x1, x2, x3), x1)}
mult(i(
, x1)), mult(mult(x1, opl(x1, x2, x3)), x1))
= by Axiom 4 RL with {x2 ← x1, x1 ← opl(x1, x2, x3)}
mult(i(mult(
), mult(mult(x1, opl(x1, x2, x3)), x1))
= by Lemma 168 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(
66
opl( x1
opl(
= by Axiom 2 LR with {x2 ← x2, x1 ← mult(x3, x1)}
opl(mult(i(x3),
)), i(x2), mult(x2, x3))
= by Lemma 132 RL with {x3 ← x1, x2 ← x3, x1 ← x2}
opl(mult(i( x3
= by Axiom 2 LR with {x2 ← x2, x1 ← x3}
opl(mult(i(
, i(x2), mult(x2, x3))
= by Axiom 12 LR with {x3 ← opl(x1, x3, x2), x2 ← mult(x2, x3), x1 ← i(x2)}
opl(
opl(opl(x1, x3, x2), mult(x2, x3), i(x2)), i(x2), mult(x2, x3))

opl( x1
, mult(x2, x3), i(x2))
= by Lemma 138 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
opl(
opl(opl(x1, x2, x3), x3, x2), mult(x2, x3), i(x2))

= by Axiom 2 RL with {x2 ← x2, x1 ← x3}
mult(i(
))
= by Lemma 132 LR with {x3 ← opl(x1, x2, x3), x2 ← x3, x1 ← x2}
mult(i(x3), mult(i(x2),
)
= by Axiom 2 RL with {x2 ← x2, x1 ← mult(x3, opl(x1, x2, x3))}
mult(i(x3),
mult(x3, opl(x1, x2, x3)))

opl(x1, x2
, mult(i(x2), x3))
opl(x1,
i(i(x2)), mult(i(x2), x3))

opl(x1, mult(i(x2), x3), x2)

)
opl(x1, i(x2),
opl(x1, x2
, i(mult(x2, x3)))
opl(x1,
opl(x1, i(i(x2)),
mult(i(x2), i(x3)))

Lemma 246: opl(x1, i(x2), mult(x3, x2)) = opl(x1, opt(x3, x2), x2)
opl(x1, i(x2), mult(x3, x2)
opl(x1, i(x2),
mult(x2, opt(x3, x2)))

opl(x1, i(mult(x2, x3)), x2)

)
opl(x1, i(x2),
Lemma 248: rd(opl(x1, x2, i(x1)), x1) = asoc(x1, mult(i(x1), x2), x1)
rd(opl(x1, x2, i(x1))
, x1)
= by Lemma 243 RL with {x3 ← x2, x2 ← x1, x1 ← x1}
rd(

opl(mult(i(x1), x2), x1
opl(mult(i(x1), x2),
i(i(x1)), x2)

))
mult(i(mult(x2, i(i(x1))))
mult(

opt(
opl(opt(rd(x1, x2), x1)
opl(
opl(mult(i(x2), x1), x2, x1), x1, x2)

opl(opt(x1, x2), x2
opl(opt(x1, x2
opl(opt(x1,
opl(
opl(
mult(i(x2), i(mult(i(x1), i(x2)))), i(i(x2)), x1)

))
= by Axiom 2 RL with {x2 ← i(x2), x1 ← i(mult(i(x1), i(x2)))}
mult(i(mult(x1, i(i(x2))))
mult(

opt(x1,
opt(x1, mult(x1,
opl( x1
, x1, opl(x1, x2, x3))
= by Axiom 2 LR with {x2 ← mult(x1, opl(x1, x2, x3)), x1 ← x1}
opl(
, x1, opl(x1, x2, x3))
= by Lemma 240 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
opl(
, x1, opl(x1, x2, x3))
= by Lemma 251 RL with {x2 ← opl(x1, x2, x3), x1 ← x1}
opl(
, opl(x1, x2, x3), x1), x1, opl(x1, x2, x3))
= by Lemma 169 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
opl(opl(
x1 , opl(x1, x2, x3), x1), x1, opl(x1, x2, x3))

asoc(x1, opl(x1, x2, x3), x1)

rd(opl(x1, x1, opl(x1, x2, x3)), x1)

)
= by Lemma 252 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(x1),
x1 )
opl(opt(x1, mult(x1, x2))
opl(
opl(opt(x1, x2), x2, x1), x1, x2)

Lemma 255: mult(x1, opt(x2, mult(x2, x1))) = mult(opl(x2, x1, x2), x1)
mult(x1, opt(x2, mult(x2, x1))
mult(x1,
opl(opt(x2, x1), x1, x2))

mult(opl(i(x1), x2, x1), x2)

)
= by Lemma 241 RL with {x3 ← x2, x2 ← x1, x1 ← opt(i(x1), x2)}
mult(x2,
mult(x2, opl(opt(i(x1),
)
= by Lemma 254 LR with {x2 ← mult(x1, x2), x1 ← i(x1)}
mult(x2,
opt(i(x1), mult(x1, x2)))

rd(x2, mult(i(x2), rd(
))
= by Lemma 29 LR with {x2 ← mult(x1, x2), x1 ← i(x1)}
rd(x2, mult(i(x2),
mult(x1, x2))
rd(x2,
opt(x1, mult(i(x1), x2))

opl(opt(x1,
x2 ), x1, mult(i(x1), x2))

mult(mult( x1
mult(mult(
, opt(x2, x1))
= by Lemma 44 RL with {x3 ← x2, x2 ← mult(x2, x1), x1 ← opt(x2, x1)}
mult(
)
= by Axiom 4 LR with {x2 ← mult(x2, x1), x1 ← opt(x2, x1)}
mult(i(mult(rd(opt(x2, x1), mult(x2, x1)), x2)),
mult(rd(opt(x2, x1), mult(x2, x1)), mult(x2, x1)))

)
opl(x1, x2,
Lemma 259: opl(x1, x1, mult(x2, x1)) = mult(mult(x1, i(mult(x1, mult(x2, x1)))), mult(x2, mult(x1, x1)))
opl(x1, x1, mult(x2, x1))

opl(x1, mult(x1, mult(x2, x1)), i(x1))

)
mult(mult(x1, i(mult(x1, mult(x2, x1)))),
opl(x1, x1, mult(x2, x1))

= by Lemma 20 LR with {x2 ← mult(x2, x1), x1 ← x1}
mult(
i(mult(x2, x1)), mult(x2, mult(x1, x1)))

Lemma 261: asoc(x1, mult(x2, x1), x1) = mult(i(x1), opl(x1, x1, x2))
asoc(x1, mult(x2, x1), x1)

rd(opl(x1, x1, mult(x2, x1)), x1)

)
mult(i(x1),
asoc(x1, mult(x2, x1), x1)

mult(i(x1), opl(x1, x1, x2))


asoc(x1, rd(x2, x1), x1)

rd(opl(x1, x1, rd(x2, x1)), x1)

)
= by Lemma 260 RL with {x2 ← rd(x2, x1), x1 ← x1}
mult(i(x1),
))
mult(i(x1), opl(x1, x1,
x2 ))


mult( x1
mult(
))
mult(mult(rd(x1, x2), x2), opl(x2, x2,

= by Lemma 260 LR with {x2 ← rd(x1, x2), x1 ← x2}
mult(mult(rd(x1, x2), x2),
opl(x2, x2, rd(x1, x2)))


mult(i(mult(i(x1), i(x2))), opt(i(x2), x1))

i(mult(mult(i(x1), i(x2)),
opt(x2, i(x1)))
i(
opl(x1, i(x2), i(x1))

mult(

i(opl(i(x1), x2, x1
i(opl(i(x1), x2,

opl(i(x1), i(x2), x1
opl(i(x1), i(x2),

i(opl(
opl(i(x1), i(x1), x2)

opl(i(x1),

)
= by Lemma 247 LR with {x3 ← x2, x2 ← i(x1), x1 ← x1}
i(
i(opl(x1, x1
i(opl(x1,

i(asoc(x1, i(x2), x1))

rd(x1, opl(x1, x1, i(x2)))

mult(x1,
mult(x1,
mult(i(x1), asoc(i(x1), x2, i(x1))))

opl(x1, x1, opl(i(x1), x2, x3)
)
opl(x1, x1,
)
= by Lemma 21 RL with {x2 ← i(opl(i(x1), x2, x3)), x1 ← x1}
opl(x1, x1,
mult(rd(i(x1), i(opl(i(x1), x2, x3))), x1))

)))
= by Lemma 170 RL with {x3 ← x3, x2 ← x2, x1 ← i(x1)}
opl(x1, x1, rd(i(x1), i(
)
= by Lemma 58 LR with {x2 ← rd(opl(i(x1), x2, x3), i(x1)), x1 ← i(x1)}
opl(x1, x1,
))
= by Axiom 4 RL with {x2 ← i(x1), x1 ← opl(i(x1), x2, x3)}
opl(x1, x1, mult(i(x1),
opl(i(x1), x2, x3)))

)
= by Lemma 225 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
opl(x1, opl(i(x1), x2, x3),
opl(i(x1), x1, opl(i(x1), x2, x3)))

mult(i(mult(opl(i(x1), x1, opl(i(x1), x2, x3)), opl(i(x1), x2, x3))), mult(opl(i(x1), x1, opl(i(x1), x2, x3)), mult(opl(i(x1), x2,
= by Lemma 231 LR with {x2 ← x1, x1 ← opl(i(x1), x2, x3)}
mult(i(mult(opl(i(x1), x1, opl(i(x1), x2, x3)), opl(i(x1), x2, x3))), mult(opl(i(x1), x1, opl(i(x1), x2, x3)),
rd(opl(i(x1), x2, x3)
= by Lemma 55 RL with {x2 ← opl(i(x1), x2, x3), x1 ← opl(i(x1), x1, opl(i(x1), x2, x3))}
mult(i(mult(opl(i(x1), x1, opl(i(x1), x2, x3)), opl(i(x1), x2, x3))),
mult(mult(opl(i(x1), x1, opl(i(x1), x2, x3)), opl(i(x1), x2,

)
= by Lemma 225 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
i(
i(x1))

Lemma 273: opl(i(x1), x2, x1) = opl(i(x1), i(x1), mult(x1, rd(x2, x1)))
opl(i(x1), x2, x1)


)
opl(i(x1), i(x1),
asoc(x1, i(opr(x1, x2, x3))
, x1)
= by Lemma 21 RL with {x2 ← opr(x1, x2, x3), x1 ← x1}
asoc(x1,
mult(rd(i(x1), opr(x1, x2, x3)), x1), x1)

, x1)
= by Lemma 131 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
asoc(x1,
, x1)
= by Axiom 11 RL with {x2 ← opr(x1, x2, x3), x1 ← x1}
asoc(x1,
mult(i(x1), i(opr(x1, x2, x3))), x1)

rd(opl(x1, i(opr(x1, x2, x3)), i(x1)), x1)

, i(opr(x1, x2, x3)), i(x1)))
= by Lemma 160 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(x1), opl(
)
= by Lemma 257 RL with {x2 ← i(opr(x1, x2, x3)), x1 ← x1}
mult(i(x1),
))
= by Axiom 11 LR with {x2 ← opr(x1, x2, x3), x1 ← x1}
mult(i(x1), opt(x1,
))
= by Lemma 131 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(x1), opt(x1,
)
= by Lemma 154 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(x1),
x1 )

opl(i(x1), i(mult(x1, x2)), x1
opl(i(x1), i(mult(x1, x2)),

)
= by Lemma 242 LR with {x3 ← x2, x2 ← x1, x1 ← i(i(x1))}
i(
opl(i(x1), i(x1), i(x2))


i(opl(
opl(x1, i(i(x1)), i(x2))


)
= by Lemma 244 LR with {x3 ← x2, x2 ← x1, x1 ← i(x1)}
i(
opl(x1, x1
opl(x1,
i(i(x1)), i(opl(i(i(x1)), x2, x3)))

)
= by Lemma 272 LR with {x3 ← x3, x2 ← x2, x1 ← i(x1)}
i(
i(x1))
80
opl(x1, x1, i(opl( x1
opl(x1, x1, i(opl(
i(i(x1)), x2, x3)))


x1
Lemma 280: opl(x1, rd(x2, x1), x3) = mult(mult(i(x3), rd(x1, x2)), mult(x3, x2))
opl(x1, rd(x2, x1), x3)

))
mult(i(mult(x3, rd(x2, x1)))
, mult(x3,
x2 ))
= by Lemma 45 LR with {x3 ← x1, x2 ← x2, x1 ← x3}
mult(
Lemma 281: opl(x1, x1, x2) = mult(i(x2), opt(mult(x1, x2), x1))
opl(x1, x1, x2)


mult(
Lemma 282: asoc(i(x1), rd(x1, x2), i(x1)) = mult(x1, i(opl(x1, x1, x2)))
asoc(i(x1), rd(x1, x2), i(x1))

rd(opl(i(x1), i(x1), rd(x1, x2)), i(x1))

)
= by Lemma 244 RL with {x3 ← rd(x1, x2), x2 ← x1, x1 ← i(x1)}
mult(i(i(x1)),
mult(i(i(x1)), opl(i(x1),
)
= by Lemma 273 LR with {x2 ← mult(i(mult(x1, x2)), x1), x1 ← x1}
mult(i(i(x1)),
)
= by Lemma 241 LR with {x3 ← rd(mult(i(mult(x1, x2)), x1), x1), x2 ← x1, x1 ← i(x1)}
mult(i(i(x1)),
, x1))
= by Axiom 5 RL with {x2 ← x1, x1 ← i(mult(x1, x2))}
mult(i(i(x1)), opl(i(x1),
)
= by Lemma 247 LR with {x3 ← x2, x2 ← x1, x1 ← i(x1)}
mult(i(i(x1)),
mult(i(i(x1)),
i(opl(x1, x1, x2)))

i(mult(i(x1), opl(x1, x1, x2)))

rd(opt(mult(x1, x2), x1)
, opl(x1, x1, x2))
rd(
mult(x2, opl(x1, x1, x2)), opl(x1, x1, x2))

Lemma 284: mult(opl(rd(x1, x2), x1, x2), x1) = rd(x1, mult(i(x1), x2))
mult(opl(rd(x1, x2), x1, x2), x1)

)
mult(x1,

opl(x1, mult(x2, x3), x3)

opl(x1, i(x3), mult(x3, mult(x2, x3)))

), mult(mult(x3, mult(x2, x3)), mult(i(x3), x1)))
= by Axiom 6 RL with {x3 ← i(x3), x2 ← x2, x1 ← x3}
mult(i(
)
= by Axiom 6 RL with {x3 ← mult(i(x3), x1), x2 ← x2, x1 ← x3}
mult(i(mult(x3, mult(x2, mult(x3, i(x3))
= by Axiom 7 LR with {x1 ← x3}
mult(i(mult(x3, mult(x2,
= by Axiom 1 RL with {x1 ← x2}
mult(i(mult(x3,
x2 )), mult(x3, mult(x2, mult(x3, mult(i(x3), x1)))))

opl(
opl(x1, x2, rd(x3, x2
opl(x1, x2, rd(x3,
= by Lemma 60 RL with {x2 ← mult(x3, x2), x1 ← x3}
opl(x1, x2,
mult(x3, mult(i(mult(x3, x2)), x3)))

), mult(mult(x3, mult(i(mult(x3, x2)), x3)), mult(x2, x1)))
= by Axiom 6 RL with {x3 ← x2, x2 ← i(mult(x3, x2)), x1 ← x3}
mult(i(
)
= by Axiom 6 RL with {x3 ← mult(x2, x1), x2 ← i(mult(x3, x2)), x1 ← x3}
mult(i(mult(x3, mult(i(mult(x3, x2)), mult(x3, x2))
)),
= by Axiom 9 LR with {x1 ← mult(x3, x2)}
mult(i(mult(x3,
= by Axiom 1 RL with {x1 ← x3}
mult(i(
x3 ), mult(x3, mult(i(mult(x3, x2)), mult(x3, mult(x2, x1)))))

mult(i(mult(x3, x2)), mult(x3, mult(x2, x1)))

opl(x1, x2, opt(i(x2), x3))

= by Axiom 2 LR with {x2 ← opt(i(x2), x3), x1 ← x1}
opl(
), x2, rd(opt(i(x2), x3), x2))
= by Axiom 2 LR with {x2 ← i(x2), x1 ← mult(opt(i(x2), x3), x1)}
opl(mult(i(opt(i(x2), x3)),
)), x2, rd(opt(i(x2), x3), x2))
= by Lemma 162 RL with {x3 ← x1, x2 ← x3, x1 ← i(x2)}
opl(mult(i(opt(i(x2), x3)
= by Axiom 2 LR with {x2 ← i(x2), x1 ← opt(i(x2), x3)}
opl(mult(i(
, x2, rd(opt(i(x
= by Axiom 12 LR with {x3 ← x1, x2 ← mult(i(x2), opt(i(x2), x3)), x1 ← i(i(x2))}
opl(
), x2, rd(opt(i(x2), x3), x2))
opl(opl(x1, mult(i(x2), opt(i(x2), x3))
= by Lemma 111 LR with {x2 ← x3, x1 ← x2}
opl(opl(x1,
rd(opt(i(x2), x3), x2), x2), x2, rd(opt(i(x2), x3), x2))

opl(x1, i(opt(x2, x3)), x2)

)))
opl(x1, i(opt(x2, x3)), rd(x2, i(
mult(x2, rd(opt(x2, x3), x2))))
)
= by Lemma 58 LR with {x2 ← rd(opt(x2, x3), x2), x1 ← x2}
opl(x1, i(opt(x2, x3)),
))
= by Axiom 4 RL with {x2 ← x2, x1 ← opt(x2, x3)}
opl(x1, i(opt(x2, x3)), mult(x2,
opt(x2, x3)))

= by Axiom 10 RL with {x2 ← x2, x1 ← opt(x2, x3)}
opl(x1,
), opt(x2, x3))
opl(x1, mult(i(opt(x2, x3)),
mult(opt(x2, x3), x2))
= by Axiom 2 RL with {x2 ← opt(x2, x3), x1 ← x2}
opl( x1
x2 , opt(x2, x3))
= by Lemma 163 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
opl(
opl(x1, opt(x2, x3), x2), x2, opt(x2, x3))

opl(x1, x2, i(opt(x2, x3))
opl(x1, x2,
opt(i(x2), i(x3)))

x1

opl(x1, i(
i(opt(x2, x3)))
opl(x1,
opl(x1, x2
, mult(x3, x2))
opl(x1,
mult(i(x3), mult(x3, x2)), mult(x3, x2))

opl(x1, i(x3), mult(x3, x2))

opl(x1, rd(x2, x3), x3)

opl(x1,
x1

opl(x1, x2, opr(x2, x3, i(x3)))

)
opl(x1, x2,
x1

opl(x1, x2, i(opt(x2, x3)))

)
opl(x1, x2,
opl(x1, rd(i(x2), x3), x2)

opl(x1,
opl(x1, x2, rd(i(x2), x3)
opl(x1, x2,
rd(i(x3), x2))

x1

))
opl(x1, i(x2), rd(i(x2),
opr(i(x2), i(x3), x3))
opl(x1, i(x2),
opl(x1, i(x2), asoc(
x2 , x3, i(x3)))
x1

, i(x2))
= by Lemma 29 RL with {x2 ← i(opt(i(x2), mult(x2, x3))), x1 ← i(x2)}
opl(x1,
), i(x2))
= by Lemma 48 LR with {x2 ← mult(x2, x3), x1 ← i(x2)}
opl(x1, rd(mult(i(x2), i(opt(i(x2), mult(x2, x3))))
,

= by Lemma 98 RL with {x2 ← mult(x2, x3), x1 ← i(x2)}
opl(x1, rd(
rd(i(x2), opt(i(x2), mult(x2, x3))), i(x2)), i(x2))

), i(x2))
opl(x1, rd(i(x2),
opl(x1,
opl(x1, asoc(
opl(x1, rd(x2, x3), i(x2))

opl(x1,
Lemma 300: opl(x1, i(x2), i(x3)) = opl(x1, mult(i(x2), rd(x2, x3)), x2)
opl(x1, i(x2), i(x3))

opl(x1, i(mult(x2, x3)), x2)

opl(x1,
opl(x1, i(x2), rd(x2, x3))

opl(x1, mult(i(x2), rd(x2, x3)), x2)

Lemma 302: opl(x1, x2, opl(x3, i(x3), x2)) = opl(x1, x3, i(x2))
opl(x1, x2, opl(x3, i(x3), x2))

)
opl(x1, x2,
mult(i(x2), x3))

i(opl(x1, i(x1), x2))

opl(mult(i(x2), rd(x2, opl(x1, i(x1), x2))), x2, opl(x1, i(x1), x2))

), x1, i(x2))
opl(mult(i(x2),
opl(
opl( x1
, i(x1), i(x2))
opl(

i(opl(i(x1),
opl(i(x1), x1, mult(i(x1), x2)
opl(i(x1), x1,

)
= by Lemma 245 LR with {x3 ← x2, x2 ← i(x1), x1 ← x1}
i(
opl(i(x1), x2, i(x1))


i(opl(x1, i(x2),
opl(i(x1), i(x2), i(x1))


)
= by Lemma 241 LR with {x3 ← x2, x2 ← x1, x1 ← x1}
i(
Lemma 308: rd(x1, rd(x2, x1)) = mult(x1, opl(rd(x1, x2), x2, x1))
rd(x1, rd(x2, x1))

)
= by Lemma 292 LR with {x3 ← x1, x2 ← x2, x1 ← i(rd(x2, x1))}
mult(x1,
mult(x1, opl(
Lemma 309: mult(mult(x1, x2), rd(x2, x1)) = mult(x1, rd(x2, rd(x1, x2)))
mult(mult(x1, x2), rd(x2, x1))

)
mult(x1,
opl(i(x1), x1, x2)

))
mult(i(mult(x2, x1)), mult(x2,
mult(i(mult(x2, x1)),

mult(i(x2), rd(x2, x1))
Lemma 311: opt(rd(x1, opl(x1, x2, x3)), x1) = rd(x1, opl(x1, x2, x3))
opt(rd(x1, opl(x1, x2, x3))
, x1)
= by Lemma 176 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
opt(
mult(x1, i(opl(x1, x2, x3))), x1)

)
= by Lemma 279 LR with {x3 ← x3, x2 ← x2, x1 ← x1}
mult(i(opl(x1, x2, x3)),
x1 )

opl(x1, i(x3), i(x2))

opl(x1, x2, i(mult(x2, x3)))

opl(x1,
92
Lemma 313: opl(x1, x2, mult(x3, mult(x3, x2))) = opl(x1, x2, mult(x3, x3))
opl(x1, x2, mult(x3, mult(x3, x2)))

)
opl(x1, x2,
Lemma 314: mult(i(x1), opt(x2, rd(x2, x1))) = opl(rd(x2, x1), x2, x1)
mult(i(x1), opt( x2
mult(i(x1), opt(
)
= by Lemma 264 RL with {x2 ← rd(x2, x1), x1 ← x1}
mult(i(x1),
))
= by Lemma 292 LR with {x3 ← x1, x2 ← x2, x1 ← rd(x2, x1)}
mult(i(x1), mult(x1,
opl(rd(x2, x1), x2, x1)))

Lemma 315: opl(opt(x1, x2), i(x1), i(x2)) = opl(opr(opl(x1, i(x1), i(x2)), i(x2), x2), i(x2), i(x1))
opl(opt(x1, x2), i(x1), i(x2))

opt(
)
= by Axiom 2 LR with {x2 ← opl(i(x1), x1, x2), x1 ← x2}
opt(i(opl(i(x1), x1, x2)),
mult(i(opl(i(x1), x1, x2)), mult(opl(i(x1), x1, x2), x2)))

opl(opt(i(opl(i(x1), x1, x2)), mult(opl(i(x1), x1, x2), x2))
, mult(opl(i(x1), x1, x2), x2), i(opl(i(x1), x1, x2)))
= by Lemma 209 RL with {x2 ← x2, x1 ← opl(i(x1), x1, x2)}
opl(
opr(i(opl(i(x1), x1, x2)), i(x2), x2), mult(opl(i(x1), x1, x2), x2), i(opl(i(x1), x1, x2)))

opl(opr(i(opl(i(x1), x1, x2)), i(x2), x2), opl(i(x1), x1, x2), x2)

, x2)
= by Lemma 10 RL with {x2 ← opl(i(x1), x1, x2), x1 ← x2}
opl(opr(i(opl(i(x1), x1, x2)), i(x2), x2),
i(rd(x2, opl(i(x1), x1, x2))
opl(opr(i(opl(i(x1), x1, x2)), i(x2), x2), i(
mult(x2, x1)), x2)

opl(opr(
94
mult(opl(x1, i(x1), x2), i(x2))

))
mult(i(x2), opl(opt(x1, i(x2)), i(x1),
mult(i(x2),
, i(i(x2)), i(x1)))
= by Axiom 18 LR with {x5 ← i(x2), x4 ← i(i(x2)), x3 ← i(i(x2)), x2 ← i(x1), x1 ← x1}
mult(i(x2), opl(
)
= by Lemma 138 LR with {x3 ← i(i(x2)), x2 ← i(x1), x1 ← opr(x1, i(i(x2)), i(x2))}
mult(i(x2),
mult(i(x2), opr(x1,
x2 , i(x2)))

rd(opt(mult(x1, x2), x1), x1)

, i(x1))
= by Lemma 247 LR with {x3 ← x2, x2 ← x1, x1 ← mult(x1, x2)}
mult(
mult(opl(mult(x1, x2), i(x1),
mult(x1, i(mult(x1, x2))))
= by Lemma 195 LR with {x2 ← x1, x1 ← mult(x1, x2)}
mult(
mult(mult(

mult(opr(x1, x2, i(x2))
mult(
rd(opt(mult(x2, x1), x2), x2), x2)

Lemma 319: rd(opt(mult(x1, x2), x2), x2) = opr(opt(x1, x2), x2, i(x2))
rd(opt(mult(x1, x2)
, x2), x2)
rd(opt(

opr(mult(x1, x2), x2, i(x2))

)
mult(x2,
opr(opt(x1, x2), x2, i(x2)))


opr( x1
, i(x2), x2)
opr(

= by Lemma 317 RL with {x2 ← rd(x1, x2), x1 ← x2}
mult(
rd(opt(mult(x2, rd(x1, x2)), x2), x2), x2)


opr(
asoc(x1, x2, i(x2))

rd(

mult(mult(x1, i(x2)), x2)


opt(mult(x1, i(x2)), x2)

mult(i(x2),
mult(i(x2),

mult(i(mult(x1, x2)), x2)


opt(mult(x1, x2), i(x2))

mult(x2,
opr(x1, i(x2), x2))

))
rd(x2, mult(i(x2),
mult(i(x1), x2))
rd(x2,
rd(i(x1), opt(x2, i(x1))
rd(i(x1),

i(mult(x1,
rd(opt(i(x2), x1), x1)


i(
opt(i(mult(x1, x2)), x2)


opr(rd(x1, x2), x1, i(x1))




mult(rd(x2, x1), i(x1))


)
rd(i(x1),
mult(rd(x1, x2), i(x2))

rd(i(x2), mult(i(x1), x2))

mult(rd(x2, x1)
mult(

i(
rd(x1, mult(x2, i(x1)))


rd(x2, mult(i(x1), i(x2)))


mult(rd(x1, i(x2)),
mult(rd(x1, i(x2)), x2)

rd(x2,
opt(rd(x1, i(x2)), x2)

mult(i(x2),

mult(
opt(rd(i(i(x2)), x1)
opt(

i(
opt(rd( x2
, x1), x2)
opt(rd(

i(mult(

mult(opt(i(x1), x2), x2
mult(opt(i(x1), x2),

opt(rd(x1, i(x2))
opt(

rd(opr(x1, x2, i(x2))
rd(
rd(

rd(x1, opr(x2, x1, i(x1))
rd(x1,

rd(x1, opt(rd(x1, x2)
rd(x1, opt(
i(rd(x2, x1)), x1))

= by Axiom 2 LR with {x2 ← x1, x1 ← rd(x2, x1)}
rd(opt(
mult(i(x1), mult(x1, rd(x2, x1))), i(x1)), i(x1))

opr(

opt(
rd(opt(rd( x1
rd(opt(rd(

opt(
mult(rd(x1, rd( x1
mult(rd(x1, rd(

rd(x1,
rd(x1, rd(x1, i(x2))
rd(x1,
mult(rd(x1, rd(x1, x2)), x1))


rd(x1, rd(x2, i(x1)))


rd(i(x1),
opt(mult(x1, rd(x2, x1)), x1)

, x1)
= by Lemma 317 LR with {x2 ← rd(x2, x1), x1 ← x1}
mult(

opr(

mult(opt(mult(x1, x2), x1)
mult(


opr(opr(x1, x2, x2), x2, i(x2))

, x2)
= by Axiom 5 LR with {x2 ← x2, x1 ← opt(mult(x2, opr(x1, x2, x2)), x2)}
rd(
, x2), x2)
= by Lemma 350 LR with {x2 ← opr(x1, x2, x2), x1 ← x2}
rd(rd(
, x2), x2)
= by Lemma 91 LR with {x3 ← x2, x2 ← x2, x1 ← x1}
rd(rd(
= by Axiom 5 RL with {x2 ← x2, x1 ← mult(x1, x2)}
rd(

mult(opr(x1, x2, x2), i(x2))

, x2)
rd(
rd(x1, rd(x1, x2))

rd(
mult(opr(x2, x1, x1), i(x1)), i(x1))

rd(rd(x1, x2), x1)


i(opr(x1, x2, x2))

)
rd(i(x2),
rd(x1, x2))

i(opr(x1, x2, x2))


mult(opr(x1, x2, x2)
mult(

rd(rd(x1, i(x2)), x2)

rd(i(x2), rd(i(x2), x1))

rd(
rd(
rd(x1, x2), i(x2))


opr(mult(x1, x2), x2, x2)

rd(x2,
opr( x1
, i(x2), i(x2))
opr(

rd(
opr(x1, i(x2), i(x2))

rd(rd(x1, x2), i(x2))


rd(x1, rd(x2, i(x1)))

rd(i(x1), rd(x2, x1))


opr(mult(x1, i(x2)), x2, x2)

rd(x2,

opr(mult(x1, i(x2)), x2, x2)


x1

i(i(x1))

))
mult(opr(x2, x1, x1), i(
mult(opr(x1, x2, x2), rd(x2, x1)
)
mult(opr(x1, x2, x2),

Lemma 367: mult(rd(x2, x1), x1) = opt(opr(x2, x1, x1), rd(x1, x2))
mult(rd(x2, x1)
mult(

opt(
109
opr(opt(x1, rd(x2, x1)), x2, x2)

= by Lemma 367 RL with {x1 ← x2, x2 ← x1}


Lemma 369: rd(x1, opr(x2, x1, x1)) = opr(rd(x1, x2), x1, x1)
rd(x1, opr(x2, x1, x1)
rd(x1,

Lemma 370: opr(rd(x2, x1), x1, x1) = rd(opr(x2, x1, x1), x1)
opr(rd(x2, x1)
, x1, x1)
opr(

rd(
opr(i(mult(x1, x2)), x2, x2)

rd(
rd(opt(x1, rd(x2, x1)), x2)

, i(x2))
mult(
rd(opt(x1, x2
), mult(x2, x1))
rd(opt(x1,


opt(x1, opt(x2, x1))

mult(
rd(opt(x1, x2), mult(x2, x1)), mult(x2, x1))

opt(x1, mult(i(x1), x2)
opt(x1,

rd(mult(x1, x2), opt(x2, x1)
rd(mult(x1, x2)
rd(

mult(x2, x1)


)
mult(opt(x2, x1),
opt(x1, x2))
opt(x1, rd(mult(x1, x2), x1))

opt(x1,
x2 )
opt(x1, i(mult(x1, x2))
opt(x1,

rd(x1, opt(x2, rd(x1, x2))
rd(x1,

opr(opt(x1, i(x2)), x2, x2)

opt(opr(x1, x2, x2), i(x2))

)
mult(i(i(x2))
mult(
Lemma 382: mult(x1, opr(x2, x1, x1)) = opr(mult(i(i(x1)), x2), x1, x1)
mult(x1, opr(x2, x1, x1)
mult(x1,

opr(
Lemma 383: mult(x1, opr(x2, x1, x1)) = opr(mult(x1, x2), x1, x1)
mult(x1, opr(x2, x1, x1))

opr(mult(
mult(i(opt(x1, x2)), mult(x1, x2))


opr(opt(mult(x1, x2), x1), x1, x1)

opt(
mult(x1, opr(x2, x1, x1)), x1)

, x1)
= by Lemma 350 LR with {x2 ← opr(x2, x1, x1), x1 ← x1}
rd(
, x1)
= by Lemma 91 LR with {x3 ← x1, x2 ← x1, x1 ← x2}
rd(

rd(x1, opt(i(x2), rd(x2, x1))
)
= by Lemma 47 RL with {x3 ← x2, x2 ← x1, x1 ← x2}
rd(x1,
i(opt(x2, rd(x1, x2))))

, x1)
mult(
mult(rd(opt(x1, x2), x1)
mult(
mult(mult(
, opt(x2, x1))
= by Lemma 44 RL with {x3 ← x1, x2 ← mult(x1, x2), x1 ← opt(x2, x1)}
mult(
)
= by Axiom 4 LR with {x2 ← mult(x1, x2), x1 ← opt(x2, x1)}
mult(i(mult(rd(opt(x2, x1), mult(x1, x2)), x1)),
mult(rd(opt(x2, x1), mult(x1, x2)), mult(x1, x2)))

)
opl(x2, x1,
i(opt(x1, x2)))

mult(rd(x1, opt(x1, x2))
= by Lemma 10 RL with {x2 ← x1, x1 ← opt(x1, x2)}
mult(
mult(i(rd(opt(x1, x2), x1)),
mult(rd(opt(x1, x2), x1), opt(x2, x1)))

Lemma 389: rd(x1, rd(opt(x2, x1), x2)) = opr(opt(x1, x2), x1, x1)
rd(x1, rd(opt(x2, x1), x2)
= by Lemma 10 RL with {x2 ← opt(x2, x1), x1 ← x2}
rd(x1,
i(rd(x2, opt(x2, x1))))

opr(
rd(x1, rd(opt(x2, x1), x2))

opr(opt(x1, x2), x1, x1)

mult(x1, rd(x2, opt(x2, x1))
= by Lemma 10 RL with {x2 ← x2, x1 ← opt(x2, x1)}
mult(x1,
i(rd(opt(x2, x1), x2)))

opt(
opl(x1, rd(x2, opt(x2, x3)), x3)

, x3)
opl(x1,
opt(x3, x2), x3)

Lemma 393: rd(mult(x1, x2), x1) = mult(rd(opt(x1, x2), x1), mult(i(x1), mult(x1, x2)))
rd(mult(x1, x2), x1)

= by Lemma 378 LR with {x2 ← x2, x1 ← x1}
mult(rd(
)
= by Lemma 14 LR with {x2 ← x1, x1 ← mult(x1, x2)}
mult(rd(opt(x1, x2), x1),
mult(i(x1), mult(x1, x2)))
mult(rd(opt(x1, x2), x1), x2
mult(rd(opt(x1, x2), x1),
mult(i(x1), mult(x1, x2)))

rd(opt(x2, x1), x2)

mult(i(x2),
mult(rd(x1, opt(x1, x2)), x2))

opt(rd(x1, opt(x1, x2)), x2)

)
= by Axiom 10 RL with {x2 ← x2, x1 ← rd(x1, opt(x1, x2))}
rd(mult(rd(x1, opt(x1, x2)), x2),
))
rd(mult(rd(x1, opt(x1, x2)), x2), mult(i(rd(x1, opt(x1, x2))),
opt(opt(x2, x1), i(x2))
))
= by Axiom 13 RL with {x3 ← x1, x2 ← i(x2), x1 ← x2}
rd(mult(rd(x1, opt(x1, x2)), x2), mult(i(rd(x1, opt(x1, x2))),
opt(opt(x2, i(x2))
, opt(
= by Lemma 10 LR with {x2 ← opt(x1, x2), x1 ← x1}
rd(mult(rd(x1, opt(x1, x2)), x2), mult(
rd(opt(x1, x2), x1), opt(x2, x1))
)
rd(mult(rd(x1, opt(x1, x2)), x2),
x2 )

rd(x1, opt(x1, x2))

i(rd(opt(x1, x2), x1))

)
rd(x2,
rd(mult(x1, x2), x1))
Lemma 397: rd(x1, rd(x2, opt(x2, x1))) = opr(rd(mult(x2, x1), x2), x1, x1)
rd(x1, rd(x2, opt(x2, x1))
= by Lemma 10 RL with {x2 ← x2, x1 ← opt(x2, x1)}
rd(x1,
i(rd(opt(x2, x1), x2)))

opr(
Lemma 398: rd(x1, rd(x2, opt(x2, x1))) = rd(mult(x2, x1), x2)
rd(x1, rd(x2, opt(x2, x1)))

opr(rd(mult(x2, x1), x2),
mult(i(x2), mult(x2, x1))
= by Lemma 14 RL with {x2 ← x2, x1 ← mult(x2, x1)}
opr(rd(mult(x2, x1), x2),
opt(rd(mult(x2, x1), x2), x2), x1)

Lemma 399: mult(i(x1), mult(x1, x2)) = opt(rd(mult(x1, x2), x1), opt(x1, x2))
mult(i(x1), mult(x1, x2))

opt(rd(mult(x1, x2), x1), x1)

)
opt(rd(mult(x1, x2), x1),
opt(rd(mult(x1, x2), x1), opt(x1, x2))
= by Lemma 399 RL with {x2 ← x2, x1 ← x1}


opt(rd(x1, rd(opt(x1, x2), x1)
opt(rd(x1,
= by Lemma 60 RL with {x2 ← opt(x1, x2), x1 ← x1}
opt(
mult(x1, mult(i(opt(x1, x2)), x1)), x2)

)
= by Axiom 6 RL with {x3 ← x2, x2 ← i(opt(x1, x2)), x1 ← x1}
mult(i(x2),
))
mult(i(x2), mult(x1,
mult(i(x2),

mult(x1, opr(x2, x1, i(x1)))


Lemma 403: mult(x1, rd(rd(x2, x1), x2)) = rd(x1, opr(x1, x2, x2))
mult(x1, rd(rd(x2, x1), x2))

rd(i(rd(x2, rd(x2, x1))), i(x1))

rd(x1,
rd(x1, opt(mult(x1, x2), x1
rd(x1, opt(mult(x1, x2),
i(i(x1))))

rd(opt(

opr(i(x2), i(x1),

opr(mult(x1, x2), x1, i(x1))

)
mult(x1,


opt(mult(x1, rd(x1, x2)), x1)

mult(
mult(mult(x1, i(x2))
mult(
rd(x1, opl(x2, i(x2), x1)), x1)

)
rd(x1,
opt(rd(i(x1), x2), i(x2))

i(
Lemma 409: i(opt(mult(x1, x2), x1)) = rd(x1, opt(mult(x1, mult(x2, x1)), x1))
i(opt(mult(x1, x2), x1))


rd(x1, opt(
rd(x1, mult(x2, mult(x1, x1))
rd(x1,

opt(i(opr(x1, x2, x2)), x2)

i(opt(opr(x1, x2, x2), i(x2)))

))
mult(i(x2), i(
rd(x1, x2)))
Lemma 412: opr(mult(x1, x2), x1, x1) = rd(mult(x1, mult(x2, x1)), x1)
opr(mult(x1, x2)
, x1, x1)
opr(

rd(
opr(i(opt(x1, x2)), x2, x2)

)
= by Axiom 14 RL with {x4 ← x2, x3 ← x2, x2 ← x2, x1 ← x1}
i(
opt(opr(x1, x2, x2), x2))

))
mult(x2, i(
rd(x2, i(x1)))
mult(x2,
opt(i(x1), mult(x1, i(x2))
opt(i(x1),

opt(i(x1), rd(x2, i(x1)))

opt(i(x1), mult(
x1 , x2))
Lemma 416: mult(x1, rd(rd(x2, x1), x1)) = opr(mult(opt(x2, i(x1)), i(x1)), x1, x1)
mult(x1, rd(rd(x2, x1
mult(x1, rd(rd(x2,

opr(
Lemma 417: mult(x1, rd(rd(x2, x1), x1)) = rd(opt(x2, i(x1)), x1)
mult(x1, rd(rd(x2, x1), x1))


opl(i(mult(x1, x2))
, i(x2), x1)
opl(
mult(i(x1)
opl(mult(
= by Lemma 92 LR with {x2 ← x2, x1 ← rd(i(x2), x1)}
opl(
opr(rd(i(x2), x1), x2, i(x2)), i(x2), x1)

opr(opl(rd(i(x2), x1), i(x2), x1), x2, i(x2))

, x2)
mult(



mult(mult(x1, x2), i(x2))


opl(i(
124
Lemma 420: opr(x1, x2, i(x2)) = opl(opt(x1, x2), i(i(mult(x1, x2))), x2)
opr(x1, x2, i(x2))


opl(
Lemma 421: opr(x1, x2, i(x2)) = opl(opt(x1, x2), x1, x2)
opr(x1, x2, i(x2))

opl(opt(x1, x2),
mult(x1, x2), x2)

opl(mult(x1, x2), x2
opl(mult(x1, x2)
opl(
opl(i(


mult(x2,
x1 )
125
opl(mult(x1, x2)
, i(x2), x1)
opl(
opl(mult(x2, x1), x1, i(x2)), i(x2), x1)

mult(opl(x1, x1, x2), x2)

)
mult(x2,

mult(x1, i(x2))


opl(i(mult(x2, i(x1))),
x1 , x2)
opl(mult(i(x1), x2)
, x2, x1)
opl(

opl(x1, x1, x2)

rd(


mult(x2, rd(
x1 , x2))
Lemma 428: opl(x1, i(x1), x2) = mult(opl(x1, x1, i(x2)), mult(x2, i(x2)))
opl(x1, i(x1), x2)

opl(x1, rd(i(x2), x1), x2)

mult(
Lemma 429: opl(x1, i(x1), x2) = opl(x1, x1, i(x2))
opl(x1, i(x1), x2)

)
mult(opl(x1, x1, i(x2)),

i(opl(x1, x1, x2))


opl(i(x1),
opl(x1, x1, mult(i(x1), x2)
opl(x1, x1,

opl(x1, i(x1), i(mult(i(x1), x2)))

opl(x1, x2, i(x1))


opl(x1, i(x2),
i(opl(x1, x2, x1))


opl(i(x1), x2,
Lemma 434: mult(mult(i(x1), mult(x1, x2)), x1) = opl(mult(x1, x2), x1, x2)
mult(mult(i(x1), mult(x1, x2)), x1)


opl(mult(x1, x2), mult(x1, x2), i(x1))

opl(mult(x1, x2), x1, x2)

mult(
i(opl(i(x1), x1, x2))


opl(x1, x1,
Lemma 437: opl(x1, i(x2), i(x1)) = mult(i(mult(x1, x2)), mult(x1, mult(x2, x1)))
opl(x1, i(x2), i(x1))


)
mult(i(mult(x1, x2)),
opl(x1, i(x2), i(x1))

mult(i(mult(x1, x2)), mult(x1, mult(x2, x1)))

opl(x1, x1, i(x2))

opl(x1, i(x1), x2)

))
mult(i(mult(x2, i(x1))), mult(x2,
mult(i(mult(x2, i(x1))),


mult(i(x1), opl(x2, x2, x1)
mult(i(x1),

rd(opl(x1, x1, x2
rd(opl(x1, x1,
rd(
= by Lemma 236 RL with {x2 ← x1, x1 ← x2}

opl(mult(x1, rd(x2, x1))
opl(
opl(x2, x2, x1), x1, x2)

opl(i(x1), i(x1), x2)


opl(i(x1), i(x2), x1)

i(

rd(x1, opl(x2, x2, i(x1))
rd(x1,

rd(opl(x1, x1, i(x2))
rd(

rd(i(x1), opl(x2, x2, x1)
rd(i(x1),

rd(mult(x1, rd(x2, x1))
rd(

rd(rd(x1, x2), opl(x1, x1, x2)
)
rd(rd(x1, x2),

rd(i(x1), opt(i(x2), i(x1)))

mult(opl(
mult(
rd(opt(x2, x1
rd(opt(x2,
i(i(x1))), x1)


Lemma 452: rd(mult(x1, x2), x1) = opr(opl(x2, x2, i(i(x1))), x1, x1)
rd(mult( x1
, x2), x1)
rd(mult(

opr(
Lemma 453: opl(opr(x1, x2, x2), x1, x2) = rd(mult(x2, x1), x2)
opl(opr(x1, x2, x2), x1, x2
)
opl(opr(x1, x2, x2), x1,
i(i(x2)))

= by Lemma 452 RL with {x2 ← x1, x1 ← x2}

mult(i(x1), rd(x1, x2))

opl(
opl(i(x2),
opl(opt(x1, rd(x2, x1)
opl(opt(x1,
= by Lemma 18 RL with {x2 ← rd(x1, x2), x1 ← x1}
opl(
i(opt(i(x1), rd(x1, x2))), x2, x1)

)
opl(i(opt(i(x1), rd(x1, x2))), x2,
rd(x2, opt(i(x1), rd(x1, x2))))

)
mult(i(x2),

opt(opt(mult(x1, x2), x1)
opt(
mult(x2, opl(x1, x1, x2)), x2)

opr(

mult(
rd(opt(x2, i(x1)), x1)


mult(
Lemma 458: asoc(x1, mult(i(x1), i(x2)), x1) = mult(i(x1), opl(x1, x2, x1))
asoc(x1, mult(i(x1), i(x2)), x1)

rd(opl(x1, i(x2), i(x1)), x1)

mult(i(x1),
rd(opt(x1, mult(x1, x2))
rd(
, x2)
= by Axiom 15 RL with {x4 ← x2, x3 ← x1, x2 ← x2, x1 ← x1}
rd(
opt(opl(x1, x2, x1), x2), x2)

opt(

Lemma 460: mult(opl(x2, x1, x2), x1) = rd(x1, mult(i(x1), mult(x1, i(opt(x2, x1)))))
mult(opl(x2, x1, x2), x1)

mult(x1, opt(x2, mult(x2, x1)))

)))
rd(x1, mult(i(x1), rd(x1,
rd(x1, mult(i(x1), rd(x1, opl(opt(x2,
)))
= by Axiom 15 RL with {x4 ← i(i(x1)), x3 ← x2, x2 ← x1, x1 ← x2}
rd(x1, mult(i(x1), rd(x1,
))
= by Lemma 67 RL with {x3 ← x1, x2 ← i(x1), x1 ← opl(x2, x1, x2)}
rd(x1, mult(i(x1),
))
= by Lemma 329 RL with {x2 ← i(x1), x1 ← opl(x2, x1, x2)}
rd(x1, mult(i(x1),
rd(x1, mult(i(x1), opt(i(
))
= by Lemma 15 LR with {x2 ← x1, x1 ← rd(opt(x2, x1), x1)}
rd(x1, mult(i(x1),
))
= by Lemma 46 LR with {x3 ← x1, x2 ← x1, x1 ← opt(x2, x1)}
rd(x1, mult(i(x1),
= by Lemma 34 LR with {x2 ← opt(x2, x1), x1 ← x1}
rd(x1, mult(i(x1),
mult(x1, opt(x2, mult(x2, x1)))


)
= by Axiom 2 RL with {x2 ← x1, x1 ← i(opt(x2, x1))}
rd(x1,
Lemma 462: mult(x1, opt(rd(x1, x2), i(x1))) = rd(opt(rd(x1, x2), i(x2)), i(x1))
mult( x1
mult(

), i(x1))
rd(opt(rd(x1, x2),
i(x2)), i(x1))
mult(x1, mult(x1, i(x2))
mult(x1,

rd(opt(rd(x1, x2), i(x2)), i(x1))

rd(x1, opt(rd(x2, x1)
rd(x1, opt(

mult(x1, opt(rd(x2, x1), x2
mult(x1, opt(rd(x2, x1),
mult(rd(x2, x1), x1)))

)
= by Lemma 46 LR with {x3 ← x1, x2 ← x1, x1 ← x2}
rd(x1,
rd(x1,
mult(mult(x1, x2
), opt(i(x2), x1))
mult(mult(x1,


opr(rd(opt(x1, x2), x2), x2, x2)

, x2)
= by Axiom 14 RL with {x4 ← x2, x3 ← x2, x2 ← x2, x1 ← x1}
rd(
opt(opr(x1, x2, x2), x2), x2)

)
mult(i(x2),
x1 )
Lemma 468: rd(x2, opt(opr(x1, x2, x2), i(i(x2)))) = opt(i(rd(x1, x2)), i(x2))
rd(x2, opt(opr(x1, x2, x2), i(i(x2))))

rd(opt(i(opr(x1, x2, x2)), i(x2)), i(x2))

opt(i(
opr(rd(x1, opt(x2, x1)), x1, x1)

rd(x1, opr(opt(x2,
)
= by Axiom 14 RL with {x4 ← i(i(x1)), x3 ← x1, x2 ← x1, x1 ← x2}
rd(x1,


i(opt(rd(x2, x1), x1))


opt(rd(x1, x2), opr(x2, x1, x1)
)
opt(rd(x1, x2),
rd( x1
, rd(x1, x2)))
opt(rd(x1, x2), rd(


mult(rd(opt(x1, i(x2)), x1)
= by Lemma 10 RL with {x2 ← opt(x1, i(x2)), x1 ← x1}
mult(
i(rd(x1, opt(x1, i(x2)))), x2)

i(
Lemma 472: i(opt(rd(x2, x1), x1)) = rd(x1, opl(x2, rd(x1, x2), x1))
i(opt(rd(x2, x1)
= by Lemma 12 RL with {x2 ← rd(x2, x1), x1 ← x1}
i(opt(
mult(x1, mult(i(x1), rd(x2, x1))), x1))

)
= by Lemma 280 RL with {x3 ← x1, x2 ← x1, x1 ← x2}
rd(x1,
mult(x1, i(x2))

i(

Lemma 474: i(opl(x1, rd(x2, x1), x2)) = mult(x2, rd(i(x1), x2))
i(opl(x1, rd(x2, x1), x2))

, i(x2))
mult(

opl(opr(x1, x2, i(x2)), x1, x2)

opr(opl(x1, x1, x2), x2, i(x2))

mult(


opt(rd(x1, x2), i(x2))

opl(
mult(x1, i(x2)), rd(x1, x2), x2)

opt(rd(x1, i(x2)), x2
opt(rd(x1, i(x2)),

opl(mult(x1,
Lemma 478: opt(i(x2), i(mult(x2, x1))) = opl(i(opt(x2, x1)), x1, mult(x2, x1))
opt(i(x2)
opt(

opl(
Lemma 479: i(opt(x2, mult(x2, x1))) = opl(i(opt(x2, x1)), x1, x2)
i(opt(x2, mult(x2, x1)))


opl(i(opt(x2, x1)), x1, mult(x2, x1))

opl(i(opt(x2, x1)), x1, x2)
Lemma 480: rd(x1, opt(rd(x1, x2), x1)) = mult(opl(rd(x2, x1), x2, x1), x1)
rd(x1, opt(rd(x1, x2)
rd(x1, opt(

, x1)
= by Lemma 292 LR with {x3 ← x1, x2 ← x2, x1 ← rd(x2, x1)}
mult(
Lemma 481: mult(opl(rd(x1, x2), x1, x2), x2) = opt(x1, i(x2))
mult(opl(rd(x1, x2), x1, x2), x2)
= by Lemma 480 RL with {x2 ← x1, x1 ← x2}


asoc(x1, x2, x1)

rd(opl(x1, x1, x2), x1)

mult(
Lemma 483: opl(x1, x2, x3) = opl(x1, i(x3), opl(x2, x2, x3))
opl(x1, x2, x3)

opl(x1, rd(x2, x3), x3)

)
opl(x1, i(x3),
Lemma 484: opl(x1, x2, x3) = opl(x1, opl(x3, x3, x2), i(x2))
opl(x1, x2, x3)

opl(x1, x2, rd(x3, x2))

opl(x1,
Lemma 485: opl(rd(mult(x1, x2), x1), x1, x2) = opr(x2, x1, x1)
opl(rd(mult(x1, x2), x1)
opl(
opl(opr(x2, x1, x1), x2, x1), x1, x2)

mult(x1, asoc(i(x2), x1, i(x1))
mult(x1,
mult(x1, mult(x2,
= by Lemma 13 RL with {x2 ← i(mult(x2, x1)), x1 ← x1}
mult(x1, mult(x2,
mult(x1, opt(i(mult(x2, x1)), x1))))


opl(x1, mult(x2, x1), x1)

rd(x1, opl(x1, x2, x1))

mult(x1, i(
mult(x1, asoc(i(x2), x1, i(x1)))))



asoc(
Lemma 488: asoc(mult(x1, x2), i(x1), x1) = rd(i(x1), opl(i(x1), x1, x2))
asoc(mult(x1, x2), i(x1), x1
asoc(mult(x1, x2), i(x1),

)
= by Lemma 242 LR with {x3 ← x2, x2 ← x1, x1 ← i(x1)}
rd(i(x1),
asoc(mult(x1, x2), x1, i(x1))


rd(i(x1), opl(i(x1), x1, x2))

mult(i(x1),
opl(x1, x1, x2))


Lemma 490: rd(x1, asoc(x1, x2, x1)) = opl(x1, mult(x1, x2), x1)
rd(x1, asoc(x1, x2, x1)
rd(x1,
= by Lemma 487 RL with {x2 ← mult(x1, x2), x1 ← x1}
rd(x1,
rd(x1, opl(x1, mult(x1, x2), x1)))

opr(opl(x1, mult(x1, x2), x1), x1, x1)

opl(
mult(x1, asoc( x2
mult(x1, asoc(


asoc(mult(x1, x2), x2, i(x2))

)
= by Lemma 285 LR with {x3 ← x2, x2 ← x1, x1 ← x2}
rd(x2,

opl(x1, x1
, opt(x2, x1))
opl(x1,
opl(x1, i(i(x1)),
mult(i(x1), mult(x2, x1)))


)
mult(x1,

opl(x1, rd(x1, x2)
opl(x1,


)
opl(x1, x1,
mult(i(x1), x2))

opl(x1, x2, i(x1))
Lemma 495: opl(x1, x1, rd(x1, x2)) = opl(x1, x1, mult(mult(i(x2), x1), x1))
opl(x1, x1, rd(x1, x2))

opl(x1, mult(x1, rd(x1, x2)), i(x1))

)
opl(x1, x1,
opl(x1, x1, rd(x1, x2))

opl(x1, x1, mult(mult(i(x2), x1), x1))

opl(x1, x1, mult(i(x2), x1))

opl(x1, opt(x2, x1), x1)



)
opl(x1, x1,
asoc(x1, rd(x1, x2), x1)

rd(opl(x1, x1, rd(x1, x2)), x1)

)
mult(i(x1),
opl(x1, x1, i(x2)))


opl(x1, rd(x2, i(x1))
opl(x1,


)
opl(x1, x1,
i(mult(x1, x2)))


Lemma 500: opl(x1, i(x2), i(x1)) = opl(x1, opr(x2, x1, x1), x1)
opl(x1, i(x2), i(x1))


opl(x1,
Lemma 501: opl(x1, x1, opr(x2, x1, x1)) = opl(x1, x1, x2)
opl(x1, x1, opr(x2, x1, x1)
)
opl(x1, x1,

opl(x1, x1,
rd(x2, x1))

Lemma 502: asoc(x1, opr(x2, x1, x1), x1) = asoc(x1, x2, x1)
asoc(x1, opr(x2, x1, x1)
asoc(x1,

asoc(x1,

asoc(x1, x2, x1)
Lemma 503: opr(opl(i(x1), x2, x1), x1, x1) = rd(asoc(x2, x1, i(x1)), x1)
opr(opl(i(x1), x2, x1)
opr(
i(opl(x1, x2, x1)), x1, x1)

rd(
asoc(x1, mult(i(x1), x2), x1)

rd(
opl(x1, i(x2), x1), x1)

mult(i(x1),
mult(x1, asoc(i(i(x2)), x1, i(x1))))

asoc(
Lemma 505: asoc(rd(x1, x2), i(x1), x1) = rd(i(x1), opl(i(x1), i(x2), i(x1)))
asoc(rd(x1, x2), i(x1), x1
asoc(rd(x1, x2), i(x1),

)
= by Lemma 299 LR with {x3 ← x2, x2 ← x1, x1 ← i(x1)}
rd(i(x1),
Lemma 506: rd(asoc(x1, x2, x1), x1) = opl(i(x1), mult(x1, x2), x1)
rd(asoc(x1, x2, x1), x1)

i(

Lemma 507: opl(i(x1), rd(x1, x2), x1) = opt(i(opl(x1, x2, i(x1))), x1)
opl(i(x1), rd(x1, x2), x1)

i(opl(x1, rd(x1, x2), x1))

opt(i(
opl(i(x1), rd(x1, x2), x1)

opt(i(opl(x1, x2, i(x1))), x1)



opl(i(x1), x1, i(i(i(x2))))


opl(i(x1),
rd(i(x1), opt(x2, rd(x1, x2)
rd(i(x1), opt( x2
rd(i(x1), opt(


opl(opt(x1, rd(x2, x1)), x1
opl(opt(x1, rd(x2, x1))
opr(opt(x1, rd(x2, x1)), x2, x2), x2)
= by Lemma 351 RL with {x2 ← x2, x1 ← opt(x1, rd(x2, x1))}
opl(
opr(opr(opt(x1, rd(x2, x1)), x2, x2), x2, i(x2)), opr(opt(x1, rd(x2, x1)), x2, x2), x2)

opt(opr(opt(x1, rd(x2, x1)), x2, x2), i(x2))

opr(opt(opt(x1, rd(x2, x1)), i(x2)), x2, x2)

)
mult(x2,

152
Lemma 512: opl(x1, x1, mult(x2, x1)) = opl(x1, x1, mult(x2, mult(x1, x1)))
opl(x1, x1, mult(x2, x1))

opl(x1, mult(x1, mult(x2, x1)), i(x1))

)
opl(x1, x1,
mult(x2, mult(x1, x1)))
Lemma 513: opl(x1, x1, rd(x2, x1)) = opl(x1, x1, opr(x2, x1, i(x1)))
opl(x1, x1, rd(x2, x1))

opl(x1, mult(x1, rd(x2, x1)), i(x1))

)
opl(x1, x1,
Lemma 514: opl(x1, x1, opt(x2, i(x1))) = opl(x1, x1, mult(x1, x2))
opl(x1, x1, opt(x2, i(x1))
opl(x1, x1,

)
opl(x1, x1,

opl(x1, x1, rd(mult(x1, x2), x1))

153
Lemma 515: asoc(x1, opt(x2, i(x1)), x1) = mult(i(x1), opl(x1, x1, mult(x1, x2)))
asoc(x1, opt(x2, i(x1)), x1)

rd(opl(x1, x1, opt(x2, i(x1))), x1)

)
mult(i(x1),
Lemma 516: asoc(x1, opt(x2, i(x1)), x1) = asoc(x1, mult(x1, x2), x1)
asoc(x1, opt(x2, i(x1)), x1)

mult(i(x1), opl(x1, x1, mult(x1, x2)))


i(opl(x1, i(x2), i(x3))
)
= by Axiom 12 RL with {x3 ← x1, x2 ← i(x2), x1 ← i(x3)}
i(
i(mult(i(
i(mult(
mult(x3, x2), mult(i(x3), mult(i(x2), x1))))

))
mult(i(mult(x3, x2)), mult(x3,
mult(x2, i(x1))))

i(opl(i(x1), x2, x3)
)
= by Lemma 517 RL with {x3 ← x3, x2 ← x2, x1 ← x1}
i(

opl(i(x1), i(x2), i(x3))

i(opl(
x1 , x2, x3))
Lemma 520: opr(x1, x2, x2) = rd(opt(opr(mult(x2, x1), x2, x2), x2), opl(x2, x2, x1))
opr(x1, x2, x2)

)
rd(opt(mult(x2, opr(x1, x2, x2))
rd(opt(
Lemma 521: opl(x1, i(x1), i(x2)) = opl(x1, opt(x2, i(x1)), i(x1))
opl( x1
, i(x1), i(x2))
opl(
opl(i(i(x1)), i(x1),


opl(i(i(x1))
opl(
opl(x1, x1, x2
opl(x1, x1,


opl(x1, rd(i(x2), x1), i(x2))

, x2), x2, i(opl(x1, rd(i(x2), x1), i(x2))))
= by Lemma 57