%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE087+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n019.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:39:47 AM UTC 2026
% Result : Theorem 87.74s 11.52s
% Output : Proof 88.55s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE087+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.11/0.36 % Computer : n019.cluster.edu
% 0.11/0.36 % Model : x86_64 x86_64
% 0.11/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.36 % Memory : 8046.5625MB
% 0.11/0.36 % OS : Linux 6.8.0-71-generic
% 0.11/0.36 % CPULimit : 300
% 0.11/0.36 % WCLimit : 300
% 0.11/0.36 % DateTime : Sun Sep 27 13:09:48 UTC 2026
% 0.11/0.36 % CPUTime :
% 0.11/0.36 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 87.74/11.52 Command-line arguments: --stitch /export/starexec/sandbox/solver/bin/stitch --hint-skel-cost 0 --hint-skel-factor 0.5 --no-flatten-goal
% 87.74/11.52
% 87.74/11.52 % SZS status Theorem
% 87.74/11.52
% 87.74/11.60 % SZS output start Proof
% 87.74/11.60 Axiom 1 (domain4): domain(X) = antidomain(antidomain(X)).
% 87.74/11.60 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 87.74/11.60 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 87.74/11.60 Axiom 4 (additive_identity): addition(X, zero) = X.
% 87.74/11.60 Axiom 5 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 87.74/11.60 Axiom 6 (multiplicative_right_identity): multiplication(X, one) = X.
% 87.74/11.60 Axiom 7 (right_annihilation): multiplication(X, zero) = zero.
% 87.74/11.60 Axiom 8 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 87.74/11.60 Axiom 9 (multiplicative_left_identity): multiplication(one, X) = X.
% 87.74/11.60 Axiom 10 (left_annihilation): multiplication(zero, X) = zero.
% 87.74/11.60 Axiom 11 (domain1): multiplication(antidomain(X), X) = zero.
% 87.74/11.60 Axiom 12 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 87.74/11.60 Axiom 13 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 87.74/11.60 Axiom 14 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 87.74/11.60 Axiom 15 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 87.74/11.60 Axiom 16 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 87.74/11.60 Axiom 17 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 87.74/11.60 Axiom 18 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 87.74/11.60 Axiom 19 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
% 87.74/11.60 Axiom 20 (order_1): ifeq(leq(X, Y), true, addition(X, Y), Y) = Y.
% 87.74/11.60 Axiom 21 (order): ifeq2(addition(X, Y), Y, leq(X, Y), true) = true.
% 87.74/11.60 Axiom 22 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 87.74/11.60
% 87.74/11.60 Lemma 23: antidomain(one) = zero.
% 87.74/11.60 Proof:
% 87.74/11.60 antidomain(one)
% 87.74/11.60 = { by axiom 6 (multiplicative_right_identity) R->L }
% 87.74/11.60 multiplication(antidomain(one), one)
% 87.74/11.60 = { by axiom 11 (domain1) }
% 87.74/11.60 zero
% 87.74/11.60
% 87.74/11.60 Lemma 24: addition(domain(X), antidomain(X)) = one.
% 87.74/11.60 Proof:
% 87.74/11.60 addition(domain(X), antidomain(X))
% 87.74/11.60 = { by axiom 1 (domain4) }
% 87.74/11.60 addition(antidomain(antidomain(X)), antidomain(X))
% 87.74/11.60 = { by axiom 16 (domain3) }
% 87.74/11.60 one
% 87.74/11.60
% 87.74/11.60 Lemma 25: antidomain(zero) = one.
% 87.74/11.60 Proof:
% 87.74/11.60 antidomain(zero)
% 87.74/11.60 = { by lemma 23 R->L }
% 87.74/11.60 antidomain(antidomain(one))
% 87.74/11.60 = { by axiom 1 (domain4) R->L }
% 87.74/11.60 domain(one)
% 87.74/11.60 = { by axiom 4 (additive_identity) R->L }
% 87.74/11.60 addition(domain(one), zero)
% 87.74/11.60 = { by lemma 23 R->L }
% 87.74/11.60 addition(domain(one), antidomain(one))
% 87.74/11.60 = { by lemma 24 }
% 87.74/11.60 one
% 87.74/11.60
% 87.74/11.60 Lemma 26: coantidomain(one) = zero.
% 87.74/11.60 Proof:
% 87.74/11.60 coantidomain(one)
% 87.74/11.60 = { by axiom 9 (multiplicative_left_identity) R->L }
% 87.74/11.60 multiplication(one, coantidomain(one))
% 87.74/11.60 = { by axiom 8 (codomain1) }
% 87.74/11.60 zero
% 87.74/11.60
% 87.74/11.60 Lemma 27: addition(zero, X) = X.
% 87.74/11.60 Proof:
% 87.74/11.60 addition(zero, X)
% 87.74/11.60 = { by axiom 3 (additive_commutativity) R->L }
% 87.74/11.60 addition(X, zero)
% 87.74/11.60 = { by axiom 4 (additive_identity) }
% 87.74/11.60 X
% 87.74/11.60
% 87.74/11.60 Lemma 28: antidomain(domain(X)) = domain(antidomain(X)).
% 87.74/11.60 Proof:
% 87.74/11.60 antidomain(domain(X))
% 87.74/11.60 = { by axiom 1 (domain4) }
% 87.74/11.60 antidomain(antidomain(antidomain(X)))
% 87.74/11.60 = { by axiom 1 (domain4) R->L }
% 87.74/11.60 domain(antidomain(X))
% 87.74/11.60
% 87.74/11.60 Lemma 29: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 87.74/11.60 Proof:
% 87.74/11.60 multiplication(antidomain(X), addition(X, Y))
% 87.74/11.60 = { by axiom 3 (additive_commutativity) R->L }
% 87.74/11.60 multiplication(antidomain(X), addition(Y, X))
% 87.74/11.60 = { by axiom 17 (right_distributivity) }
% 87.74/11.60 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 87.74/11.60 = { by axiom 11 (domain1) }
% 87.74/11.60 addition(multiplication(antidomain(X), Y), zero)
% 87.74/11.60 = { by axiom 4 (additive_identity) }
% 87.74/11.60 multiplication(antidomain(X), Y)
% 87.74/11.60
% 87.74/11.60 Lemma 30: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
% 87.74/11.60 Proof:
% 87.74/11.60 multiplication(addition(X, antidomain(Y)), Y)
% 87.74/11.60 = { by axiom 18 (left_distributivity) }
% 87.74/11.60 addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
% 87.74/11.60 = { by axiom 11 (domain1) }
% 87.74/11.60 addition(multiplication(X, Y), zero)
% 87.74/11.60 = { by axiom 4 (additive_identity) }
% 87.74/11.60 multiplication(X, Y)
% 87.74/11.60
% 87.74/11.60 Lemma 31: multiplication(domain(X), X) = X.
% 87.74/11.60 Proof:
% 87.74/11.60 multiplication(domain(X), X)
% 87.74/11.60 = { by lemma 30 R->L }
% 87.74/11.60 multiplication(addition(domain(X), antidomain(X)), X)
% 87.74/11.60 = { by lemma 24 }
% 87.74/11.60 multiplication(one, X)
% 87.74/11.60 = { by axiom 9 (multiplicative_left_identity) }
% 87.74/11.60 X
% 87.74/11.60
% 87.74/11.60 Lemma 32: domain(antidomain(X)) = antidomain(X).
% 87.74/11.60 Proof:
% 87.74/11.60 domain(antidomain(X))
% 87.74/11.60 = { by lemma 28 R->L }
% 87.74/11.60 antidomain(domain(X))
% 87.74/11.60 = { by axiom 6 (multiplicative_right_identity) R->L }
% 87.74/11.60 multiplication(antidomain(domain(X)), one)
% 87.74/11.60 = { by lemma 24 R->L }
% 87.74/11.60 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 87.74/11.60 = { by lemma 29 }
% 87.74/11.60 multiplication(antidomain(domain(X)), antidomain(X))
% 87.74/11.60 = { by lemma 28 }
% 87.74/11.60 multiplication(domain(antidomain(X)), antidomain(X))
% 87.74/11.60 = { by lemma 31 }
% 87.74/11.60 antidomain(X)
% 87.74/11.60
% 87.74/11.60 Lemma 33: domain(domain(X)) = domain(X).
% 87.74/11.60 Proof:
% 87.74/11.60 domain(domain(X))
% 87.74/11.60 = { by axiom 1 (domain4) }
% 87.74/11.60 domain(antidomain(antidomain(X)))
% 87.74/11.60 = { by lemma 32 }
% 87.74/11.60 antidomain(antidomain(X))
% 87.74/11.60 = { by axiom 1 (domain4) R->L }
% 87.74/11.60 domain(X)
% 87.74/11.60
% 87.74/11.60 Lemma 34: addition(coantidomain(X), coantidomain(coantidomain(X))) = one.
% 87.74/11.60 Proof:
% 87.74/11.60 addition(coantidomain(X), coantidomain(coantidomain(X)))
% 87.74/11.60 = { by axiom 3 (additive_commutativity) R->L }
% 87.74/11.60 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 87.74/11.60 = { by axiom 5 (codomain3) }
% 87.74/11.60 one
% 87.74/11.60
% 87.74/11.60 Lemma 35: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
% 87.74/11.60 Proof:
% 87.74/11.60 multiplication(addition(X, Y), coantidomain(X))
% 87.74/11.60 = { by axiom 3 (additive_commutativity) R->L }
% 87.74/11.60 multiplication(addition(Y, X), coantidomain(X))
% 87.74/11.60 = { by axiom 18 (left_distributivity) }
% 87.74/11.60 addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
% 87.74/11.60 = { by axiom 8 (codomain1) }
% 87.74/11.60 addition(multiplication(Y, coantidomain(X)), zero)
% 87.74/11.60 = { by axiom 4 (additive_identity) }
% 87.74/11.60 multiplication(Y, coantidomain(X))
% 87.74/11.60
% 87.74/11.60 Lemma 36: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y).
% 87.74/11.60 Proof:
% 87.74/11.60 multiplication(X, addition(Y, coantidomain(X)))
% 87.74/11.60 = { by axiom 17 (right_distributivity) }
% 87.74/11.60 addition(multiplication(X, Y), multiplication(X, coantidomain(X)))
% 87.74/11.60 = { by axiom 8 (codomain1) }
% 87.74/11.60 addition(multiplication(X, Y), zero)
% 87.74/11.60 = { by axiom 4 (additive_identity) }
% 87.74/11.60 multiplication(X, Y)
% 87.74/11.60
% 87.74/11.60 Lemma 37: multiplication(X, addition(coantidomain(X), Y)) = multiplication(X, Y).
% 87.74/11.60 Proof:
% 88.55/11.60 multiplication(X, addition(coantidomain(X), Y))
% 88.55/11.60 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.60 multiplication(X, addition(Y, coantidomain(X)))
% 88.55/11.60 = { by lemma 36 }
% 88.55/11.60 multiplication(X, Y)
% 88.55/11.60
% 88.55/11.60 Lemma 38: multiplication(X, coantidomain(coantidomain(X))) = X.
% 88.55/11.60 Proof:
% 88.55/11.60 multiplication(X, coantidomain(coantidomain(X)))
% 88.55/11.60 = { by lemma 37 R->L }
% 88.55/11.60 multiplication(X, addition(coantidomain(X), coantidomain(coantidomain(X))))
% 88.55/11.60 = { by lemma 34 }
% 88.55/11.60 multiplication(X, one)
% 88.55/11.60 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.60 X
% 88.55/11.60
% 88.55/11.60 Lemma 39: coantidomain(coantidomain(coantidomain(X))) = coantidomain(X).
% 88.55/11.60 Proof:
% 88.55/11.60 coantidomain(coantidomain(coantidomain(X)))
% 88.55/11.60 = { by axiom 9 (multiplicative_left_identity) R->L }
% 88.55/11.60 multiplication(one, coantidomain(coantidomain(coantidomain(X))))
% 88.55/11.60 = { by lemma 34 R->L }
% 88.55/11.60 multiplication(addition(coantidomain(X), coantidomain(coantidomain(X))), coantidomain(coantidomain(coantidomain(X))))
% 88.55/11.60 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.60 multiplication(addition(coantidomain(coantidomain(X)), coantidomain(X)), coantidomain(coantidomain(coantidomain(X))))
% 88.55/11.60 = { by lemma 35 }
% 88.55/11.60 multiplication(coantidomain(X), coantidomain(coantidomain(coantidomain(X))))
% 88.55/11.60 = { by lemma 38 }
% 88.55/11.60 coantidomain(X)
% 88.55/11.60
% 88.55/11.60 Lemma 40: addition(X, addition(X, Y)) = addition(X, Y).
% 88.55/11.60 Proof:
% 88.55/11.60 addition(X, addition(X, Y))
% 88.55/11.60 = { by axiom 14 (additive_associativity) }
% 88.55/11.60 addition(addition(X, X), Y)
% 88.55/11.60 = { by axiom 2 (additive_idempotence) }
% 88.55/11.60 addition(X, Y)
% 88.55/11.60
% 88.55/11.60 Lemma 41: addition(one, antidomain(X)) = one.
% 88.55/11.60 Proof:
% 88.55/11.60 addition(one, antidomain(X))
% 88.55/11.60 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.60 addition(antidomain(X), one)
% 88.55/11.60 = { by axiom 12 (ifeq_axiom) R->L }
% 88.55/11.60 ifeq(true, true, addition(antidomain(X), one), one)
% 88.55/11.60 = { by axiom 21 (order) R->L }
% 88.55/11.60 ifeq(ifeq2(addition(antidomain(X), addition(antidomain(X), domain(X))), addition(antidomain(X), domain(X)), leq(antidomain(X), addition(antidomain(X), domain(X))), true), true, addition(antidomain(X), one), one)
% 88.55/11.60 = { by lemma 40 }
% 88.55/11.60 ifeq(ifeq2(addition(antidomain(X), domain(X)), addition(antidomain(X), domain(X)), leq(antidomain(X), addition(antidomain(X), domain(X))), true), true, addition(antidomain(X), one), one)
% 88.55/11.60 = { by axiom 13 (ifeq_axiom) }
% 88.55/11.60 ifeq(leq(antidomain(X), addition(antidomain(X), domain(X))), true, addition(antidomain(X), one), one)
% 88.55/11.60 = { by axiom 3 (additive_commutativity) }
% 88.55/11.60 ifeq(leq(antidomain(X), addition(domain(X), antidomain(X))), true, addition(antidomain(X), one), one)
% 88.55/11.60 = { by lemma 24 }
% 88.55/11.60 ifeq(leq(antidomain(X), one), true, addition(antidomain(X), one), one)
% 88.55/11.60 = { by axiom 20 (order_1) }
% 88.55/11.60 one
% 88.55/11.60
% 88.55/11.60 Lemma 42: addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))) = antidomain(multiplication(X, domain(Y))).
% 88.55/11.60 Proof:
% 88.55/11.60 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
% 88.55/11.60 = { by axiom 1 (domain4) }
% 88.55/11.60 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
% 88.55/11.60 = { by axiom 22 (domain2) }
% 88.55/11.60 antidomain(multiplication(X, antidomain(antidomain(Y))))
% 88.55/11.60 = { by axiom 1 (domain4) R->L }
% 88.55/11.60 antidomain(multiplication(X, domain(Y)))
% 88.55/11.60
% 88.55/11.60 Lemma 43: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 88.55/11.60 Proof:
% 88.55/11.60 multiplication(addition(one, Y), X)
% 88.55/11.60 = { by axiom 18 (left_distributivity) }
% 88.55/11.60 addition(multiplication(one, X), multiplication(Y, X))
% 88.55/11.60 = { by axiom 9 (multiplicative_left_identity) }
% 88.55/11.60 addition(X, multiplication(Y, X))
% 88.55/11.60
% 88.55/11.60 Lemma 44: addition(domain(X), addition(Y, antidomain(X))) = addition(Y, one).
% 88.55/11.60 Proof:
% 88.55/11.60 addition(domain(X), addition(Y, antidomain(X)))
% 88.55/11.60 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.60 addition(domain(X), addition(antidomain(X), Y))
% 88.55/11.60 = { by axiom 14 (additive_associativity) }
% 88.55/11.60 addition(addition(domain(X), antidomain(X)), Y)
% 88.55/11.60 = { by lemma 24 }
% 88.55/11.60 addition(one, Y)
% 88.55/11.60 = { by axiom 3 (additive_commutativity) }
% 88.55/11.60 addition(Y, one)
% 88.55/11.60
% 88.55/11.60 Lemma 45: addition(one, coantidomain(X)) = one.
% 88.55/11.60 Proof:
% 88.55/11.60 addition(one, coantidomain(X))
% 88.55/11.60 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.60 addition(coantidomain(X), one)
% 88.55/11.60 = { by lemma 34 R->L }
% 88.55/11.60 addition(coantidomain(X), addition(coantidomain(X), coantidomain(coantidomain(X))))
% 88.55/11.60 = { by lemma 40 }
% 88.55/11.60 addition(coantidomain(X), coantidomain(coantidomain(X)))
% 88.55/11.60 = { by lemma 34 }
% 88.55/11.60 one
% 88.55/11.60
% 88.55/11.60 Lemma 46: domain(coantidomain(X)) = coantidomain(X).
% 88.55/11.60 Proof:
% 88.55/11.60 domain(coantidomain(X))
% 88.55/11.60 = { by lemma 39 R->L }
% 88.55/11.60 domain(coantidomain(coantidomain(coantidomain(X))))
% 88.55/11.60 = { by axiom 9 (multiplicative_left_identity) R->L }
% 88.55/11.60 multiplication(one, domain(coantidomain(coantidomain(coantidomain(X)))))
% 88.55/11.60 = { by lemma 34 R->L }
% 88.55/11.60 multiplication(addition(coantidomain(coantidomain(X)), coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X)))))
% 88.55/11.60 = { by axiom 18 (left_distributivity) }
% 88.55/11.60 addition(multiplication(coantidomain(coantidomain(X)), domain(coantidomain(coantidomain(coantidomain(X))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.60 = { by lemma 38 R->L }
% 88.55/11.60 addition(multiplication(multiplication(coantidomain(coantidomain(X)), coantidomain(coantidomain(coantidomain(coantidomain(X))))), domain(coantidomain(coantidomain(coantidomain(X))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.60 = { by axiom 15 (multiplicative_associativity) R->L }
% 88.55/11.60 addition(multiplication(coantidomain(coantidomain(X)), multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X)))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.60 = { by axiom 9 (multiplicative_left_identity) R->L }
% 88.55/11.60 addition(multiplication(coantidomain(coantidomain(X)), multiplication(one, multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.60 = { by lemma 41 R->L }
% 88.55/11.60 addition(multiplication(coantidomain(coantidomain(X)), multiplication(addition(one, antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by lemma 25 R->L }
% 88.55/11.61 addition(multiplication(coantidomain(coantidomain(X)), multiplication(addition(antidomain(zero), antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by axiom 8 (codomain1) R->L }
% 88.55/11.61 addition(multiplication(coantidomain(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), coantidomain(coantidomain(coantidomain(coantidomain(coantidomain(X))))))), antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) R->L }
% 88.55/11.61 addition(multiplication(coantidomain(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), coantidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), one)))), antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by axiom 19 (codomain2) R->L }
% 88.55/11.61 addition(multiplication(coantidomain(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), addition(coantidomain(multiplication(coantidomain(coantidomain(X)), one)), coantidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), one))))), antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.61 addition(multiplication(coantidomain(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), addition(coantidomain(coantidomain(coantidomain(X))), coantidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), one))))), antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.61 addition(multiplication(coantidomain(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), addition(coantidomain(coantidomain(coantidomain(X))), coantidomain(coantidomain(coantidomain(coantidomain(coantidomain(X)))))))), antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by lemma 36 }
% 88.55/11.61 addition(multiplication(coantidomain(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), coantidomain(coantidomain(coantidomain(X))))), antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by lemma 42 }
% 88.55/11.61 addition(multiplication(coantidomain(coantidomain(X)), multiplication(antidomain(multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X)))))), multiplication(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by axiom 11 (domain1) }
% 88.55/11.61 addition(multiplication(coantidomain(coantidomain(X)), zero), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by axiom 7 (right_annihilation) }
% 88.55/11.61 addition(zero, multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by lemma 27 }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X)))))
% 88.55/11.61 = { by lemma 37 R->L }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), coantidomain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by axiom 9 (multiplicative_left_identity) R->L }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), multiplication(one, coantidomain(coantidomain(coantidomain(coantidomain(X)))))))
% 88.55/11.61 = { by lemma 41 R->L }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), multiplication(addition(one, antidomain(coantidomain(coantidomain(coantidomain(X))))), coantidomain(coantidomain(coantidomain(coantidomain(X)))))))
% 88.55/11.61 = { by lemma 43 }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), addition(coantidomain(coantidomain(coantidomain(coantidomain(X)))), multiplication(antidomain(coantidomain(coantidomain(coantidomain(X)))), coantidomain(coantidomain(coantidomain(coantidomain(X))))))))
% 88.55/11.61 = { by lemma 29 R->L }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), addition(coantidomain(coantidomain(coantidomain(coantidomain(X)))), multiplication(antidomain(coantidomain(coantidomain(coantidomain(X)))), addition(coantidomain(coantidomain(coantidomain(X))), coantidomain(coantidomain(coantidomain(coantidomain(X)))))))))
% 88.55/11.61 = { by lemma 34 }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), addition(coantidomain(coantidomain(coantidomain(coantidomain(X)))), multiplication(antidomain(coantidomain(coantidomain(coantidomain(X)))), one))))
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), addition(coantidomain(coantidomain(coantidomain(coantidomain(X)))), antidomain(coantidomain(coantidomain(coantidomain(X)))))))
% 88.55/11.61 = { by lemma 44 }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(coantidomain(coantidomain(coantidomain(coantidomain(X)))), one))
% 88.55/11.61 = { by axiom 3 (additive_commutativity) }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(one, coantidomain(coantidomain(coantidomain(coantidomain(X))))))
% 88.55/11.61 = { by lemma 45 }
% 88.55/11.61 multiplication(coantidomain(coantidomain(coantidomain(X))), one)
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.61 coantidomain(coantidomain(coantidomain(X)))
% 88.55/11.61 = { by lemma 39 }
% 88.55/11.61 coantidomain(X)
% 88.55/11.61
% 88.55/11.61 Lemma 47: multiplication(X, addition(one, Y)) = addition(X, multiplication(X, Y)).
% 88.55/11.61 Proof:
% 88.55/11.61 multiplication(X, addition(one, Y))
% 88.55/11.61 = { by axiom 17 (right_distributivity) }
% 88.55/11.61 addition(multiplication(X, one), multiplication(X, Y))
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.61 addition(X, multiplication(X, Y))
% 88.55/11.61
% 88.55/11.61 Lemma 48: addition(one, domain(X)) = one.
% 88.55/11.61 Proof:
% 88.55/11.61 addition(one, domain(X))
% 88.55/11.61 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.61 addition(domain(X), one)
% 88.55/11.61 = { by lemma 24 R->L }
% 88.55/11.61 addition(domain(X), addition(domain(X), antidomain(X)))
% 88.55/11.61 = { by lemma 40 }
% 88.55/11.61 addition(domain(X), antidomain(X))
% 88.55/11.61 = { by lemma 24 }
% 88.55/11.61 one
% 88.55/11.61
% 88.55/11.61 Lemma 49: addition(X, multiplication(X, domain(Y))) = X.
% 88.55/11.61 Proof:
% 88.55/11.61 addition(X, multiplication(X, domain(Y)))
% 88.55/11.61 = { by lemma 47 R->L }
% 88.55/11.61 multiplication(X, addition(one, domain(Y)))
% 88.55/11.61 = { by lemma 48 }
% 88.55/11.61 multiplication(X, one)
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.61 X
% 88.55/11.61
% 88.55/11.61 Lemma 50: multiplication(coantidomain(X), addition(Y, coantidomain(X))) = addition(coantidomain(X), multiplication(coantidomain(X), Y)).
% 88.55/11.61 Proof:
% 88.55/11.61 multiplication(coantidomain(X), addition(Y, coantidomain(X)))
% 88.55/11.61 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.61 multiplication(coantidomain(X), addition(coantidomain(X), Y))
% 88.55/11.61 = { by axiom 17 (right_distributivity) }
% 88.55/11.61 addition(multiplication(coantidomain(X), coantidomain(X)), multiplication(coantidomain(X), Y))
% 88.55/11.61 = { by lemma 36 R->L }
% 88.55/11.61 addition(multiplication(coantidomain(X), addition(coantidomain(X), coantidomain(coantidomain(X)))), multiplication(coantidomain(X), Y))
% 88.55/11.61 = { by lemma 34 }
% 88.55/11.61 addition(multiplication(coantidomain(X), one), multiplication(coantidomain(X), Y))
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.61 addition(coantidomain(X), multiplication(coantidomain(X), Y))
% 88.55/11.61
% 88.55/11.61 Lemma 51: addition(antidomain(X), coantidomain(domain(X))) = antidomain(X).
% 88.55/11.61 Proof:
% 88.55/11.61 addition(antidomain(X), coantidomain(domain(X)))
% 88.55/11.61 = { by axiom 9 (multiplicative_left_identity) R->L }
% 88.55/11.61 addition(antidomain(X), multiplication(one, coantidomain(domain(X))))
% 88.55/11.61 = { by lemma 24 R->L }
% 88.55/11.61 addition(antidomain(X), multiplication(addition(domain(X), antidomain(X)), coantidomain(domain(X))))
% 88.55/11.61 = { by lemma 35 }
% 88.55/11.61 addition(antidomain(X), multiplication(antidomain(X), coantidomain(domain(X))))
% 88.55/11.61 = { by lemma 47 R->L }
% 88.55/11.61 multiplication(antidomain(X), addition(one, coantidomain(domain(X))))
% 88.55/11.61 = { by lemma 45 }
% 88.55/11.61 multiplication(antidomain(X), one)
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.61 antidomain(X)
% 88.55/11.61
% 88.55/11.61 Lemma 52: multiplication(antidomain(X), multiplication(X, Y)) = zero.
% 88.55/11.61 Proof:
% 88.55/11.61 multiplication(antidomain(X), multiplication(X, Y))
% 88.55/11.61 = { by axiom 15 (multiplicative_associativity) }
% 88.55/11.61 multiplication(multiplication(antidomain(X), X), Y)
% 88.55/11.61 = { by axiom 11 (domain1) }
% 88.55/11.61 multiplication(zero, Y)
% 88.55/11.61 = { by axiom 10 (left_annihilation) }
% 88.55/11.61 zero
% 88.55/11.61
% 88.55/11.61 Lemma 53: multiplication(coantidomain(antidomain(X)), X) = X.
% 88.55/11.61 Proof:
% 88.55/11.61 multiplication(coantidomain(antidomain(X)), X)
% 88.55/11.61 = { by lemma 27 R->L }
% 88.55/11.61 addition(zero, multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by axiom 8 (codomain1) R->L }
% 88.55/11.61 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(X))), X), coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), X))), multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) R->L }
% 88.55/11.61 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(X))), X), coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), multiplication(X, one)))), multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by axiom 19 (codomain2) R->L }
% 88.55/11.61 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(X))), X), addition(coantidomain(multiplication(antidomain(X), multiplication(X, one))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), multiplication(X, one))))), multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by lemma 52 }
% 88.55/11.61 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(X))), X), addition(coantidomain(zero), coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), multiplication(X, one))))), multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by lemma 26 R->L }
% 88.55/11.61 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(X))), X), addition(coantidomain(coantidomain(one)), coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), multiplication(X, one))))), multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by lemma 27 R->L }
% 88.55/11.61 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(X))), X), addition(addition(zero, coantidomain(coantidomain(one))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), multiplication(X, one))))), multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by lemma 26 R->L }
% 88.55/11.61 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(X))), X), addition(addition(coantidomain(one), coantidomain(coantidomain(one))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), multiplication(X, one))))), multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by lemma 34 }
% 88.55/11.61 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(X))), X), addition(one, coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), multiplication(X, one))))), multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by lemma 45 }
% 88.55/11.61 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(X))), X), one), multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by axiom 15 (multiplicative_associativity) R->L }
% 88.55/11.61 addition(multiplication(coantidomain(coantidomain(antidomain(X))), multiplication(X, one)), multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.61 addition(multiplication(coantidomain(coantidomain(antidomain(X))), X), multiplication(coantidomain(antidomain(X)), X))
% 88.55/11.61 = { by axiom 18 (left_distributivity) R->L }
% 88.55/11.61 multiplication(addition(coantidomain(coantidomain(antidomain(X))), coantidomain(antidomain(X))), X)
% 88.55/11.61 = { by axiom 3 (additive_commutativity) }
% 88.55/11.61 multiplication(addition(coantidomain(antidomain(X)), coantidomain(coantidomain(antidomain(X)))), X)
% 88.55/11.61 = { by lemma 34 }
% 88.55/11.61 multiplication(one, X)
% 88.55/11.61 = { by axiom 9 (multiplicative_left_identity) }
% 88.55/11.61 X
% 88.55/11.61
% 88.55/11.61 Lemma 54: coantidomain(antidomain(X)) = domain(X).
% 88.55/11.61 Proof:
% 88.55/11.61 coantidomain(antidomain(X))
% 88.55/11.61 = { by lemma 49 R->L }
% 88.55/11.61 addition(coantidomain(antidomain(X)), multiplication(coantidomain(antidomain(X)), domain(X)))
% 88.55/11.61 = { by lemma 50 R->L }
% 88.55/11.61 multiplication(coantidomain(antidomain(X)), addition(domain(X), coantidomain(antidomain(X))))
% 88.55/11.61 = { by lemma 32 R->L }
% 88.55/11.61 multiplication(coantidomain(antidomain(X)), addition(domain(X), coantidomain(domain(antidomain(X)))))
% 88.55/11.61 = { by axiom 1 (domain4) }
% 88.55/11.61 multiplication(coantidomain(antidomain(X)), addition(antidomain(antidomain(X)), coantidomain(domain(antidomain(X)))))
% 88.55/11.61 = { by lemma 51 }
% 88.55/11.61 multiplication(coantidomain(antidomain(X)), antidomain(antidomain(X)))
% 88.55/11.61 = { by axiom 1 (domain4) R->L }
% 88.55/11.61 multiplication(coantidomain(antidomain(X)), domain(X))
% 88.55/11.61 = { by lemma 32 R->L }
% 88.55/11.61 multiplication(coantidomain(domain(antidomain(X))), domain(X))
% 88.55/11.61 = { by axiom 1 (domain4) }
% 88.55/11.61 multiplication(coantidomain(antidomain(antidomain(antidomain(X)))), domain(X))
% 88.55/11.61 = { by axiom 1 (domain4) R->L }
% 88.55/11.61 multiplication(coantidomain(antidomain(domain(X))), domain(X))
% 88.55/11.61 = { by lemma 53 }
% 88.55/11.61 domain(X)
% 88.55/11.61
% 88.55/11.61 Lemma 55: coantidomain(domain(X)) = antidomain(X).
% 88.55/11.61 Proof:
% 88.55/11.61 coantidomain(domain(X))
% 88.55/11.61 = { by lemma 49 R->L }
% 88.55/11.61 addition(coantidomain(domain(X)), multiplication(coantidomain(domain(X)), domain(antidomain(X))))
% 88.55/11.61 = { by lemma 32 }
% 88.55/11.61 addition(coantidomain(domain(X)), multiplication(coantidomain(domain(X)), antidomain(X)))
% 88.55/11.61 = { by lemma 50 R->L }
% 88.55/11.61 multiplication(coantidomain(domain(X)), addition(antidomain(X), coantidomain(domain(X))))
% 88.55/11.61 = { by lemma 51 }
% 88.55/11.61 multiplication(coantidomain(domain(X)), antidomain(X))
% 88.55/11.61 = { by axiom 1 (domain4) }
% 88.55/11.61 multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))
% 88.55/11.61 = { by lemma 53 }
% 88.55/11.61 antidomain(X)
% 88.55/11.61
% 88.55/11.61 Lemma 56: addition(Y, addition(X, Z)) = addition(X, addition(Y, Z)).
% 88.55/11.61 Proof:
% 88.55/11.61 addition(Y, addition(X, Z))
% 88.55/11.61 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.61 addition(addition(X, Z), Y)
% 88.55/11.61 = { by axiom 14 (additive_associativity) R->L }
% 88.55/11.61 addition(X, addition(Z, Y))
% 88.55/11.61 = { by axiom 3 (additive_commutativity) }
% 88.55/11.61 addition(X, addition(Y, Z))
% 88.55/11.61
% 88.55/11.61 Lemma 57: addition(multiplication(X, Y), addition(Z, multiplication(X, W))) = addition(Z, multiplication(X, addition(Y, W))).
% 88.55/11.61 Proof:
% 88.55/11.61 addition(multiplication(X, Y), addition(Z, multiplication(X, W)))
% 88.55/11.61 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.61 addition(multiplication(X, Y), addition(multiplication(X, W), Z))
% 88.55/11.61 = { by lemma 56 }
% 88.55/11.61 addition(multiplication(X, W), addition(multiplication(X, Y), Z))
% 88.55/11.61 = { by axiom 14 (additive_associativity) }
% 88.55/11.61 addition(addition(multiplication(X, W), multiplication(X, Y)), Z)
% 88.55/11.61 = { by axiom 17 (right_distributivity) R->L }
% 88.55/11.61 addition(multiplication(X, addition(W, Y)), Z)
% 88.55/11.61 = { by axiom 3 (additive_commutativity) }
% 88.55/11.61 addition(Z, multiplication(X, addition(W, Y)))
% 88.55/11.61 = { by axiom 3 (additive_commutativity) }
% 88.55/11.61 addition(Z, multiplication(X, addition(Y, W)))
% 88.55/11.61
% 88.55/11.61 Lemma 58: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y).
% 88.55/11.61 Proof:
% 88.55/11.61 multiplication(antidomain(X), addition(Y, X))
% 88.55/11.61 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.61 multiplication(antidomain(X), addition(X, Y))
% 88.55/11.61 = { by lemma 29 }
% 88.55/11.61 multiplication(antidomain(X), Y)
% 88.55/11.61
% 88.55/11.61 Lemma 59: multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y))) = zero.
% 88.55/11.61 Proof:
% 88.55/11.61 multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y)))
% 88.55/11.61 = { by lemma 30 R->L }
% 88.55/11.61 multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))), multiplication(X, domain(Y)))
% 88.55/11.61 = { by lemma 42 }
% 88.55/11.61 multiplication(antidomain(multiplication(X, domain(Y))), multiplication(X, domain(Y)))
% 88.55/11.61 = { by axiom 11 (domain1) }
% 88.55/11.61 zero
% 88.55/11.61
% 88.55/11.61 Lemma 60: domain(multiplication(X, domain(Y))) = domain(multiplication(X, Y)).
% 88.55/11.61 Proof:
% 88.55/11.61 domain(multiplication(X, domain(Y)))
% 88.55/11.61 = { by axiom 1 (domain4) }
% 88.55/11.61 antidomain(antidomain(multiplication(X, domain(Y))))
% 88.55/11.61 = { by lemma 42 R->L }
% 88.55/11.61 antidomain(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))))
% 88.55/11.61 = { by axiom 6 (multiplicative_right_identity) R->L }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), one)))
% 88.55/11.62 = { by lemma 24 R->L }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), addition(domain(multiplication(X, Y)), antidomain(multiplication(X, Y))))))
% 88.55/11.62 = { by lemma 29 R->L }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), addition(multiplication(X, domain(Y)), addition(domain(multiplication(X, Y)), antidomain(multiplication(X, Y)))))))
% 88.55/11.62 = { by lemma 56 R->L }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), addition(domain(multiplication(X, Y)), addition(multiplication(X, domain(Y)), antidomain(multiplication(X, Y)))))))
% 88.55/11.62 = { by axiom 14 (additive_associativity) }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), addition(addition(domain(multiplication(X, Y)), multiplication(X, domain(Y))), antidomain(multiplication(X, Y))))))
% 88.55/11.62 = { by lemma 57 R->L }
% 88.55/11.62 antidomain(addition(multiplication(antidomain(multiplication(X, domain(Y))), addition(domain(multiplication(X, Y)), multiplication(X, domain(Y)))), addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), antidomain(multiplication(X, Y))))))
% 88.55/11.62 = { by lemma 43 R->L }
% 88.55/11.62 antidomain(addition(multiplication(antidomain(multiplication(X, domain(Y))), addition(domain(multiplication(X, Y)), multiplication(X, domain(Y)))), multiplication(addition(one, antidomain(multiplication(X, domain(Y)))), antidomain(multiplication(X, Y)))))
% 88.55/11.62 = { by lemma 58 }
% 88.55/11.62 antidomain(addition(multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))), multiplication(addition(one, antidomain(multiplication(X, domain(Y)))), antidomain(multiplication(X, Y)))))
% 88.55/11.62 = { by lemma 41 }
% 88.55/11.62 antidomain(addition(multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))), multiplication(one, antidomain(multiplication(X, Y)))))
% 88.55/11.62 = { by axiom 9 (multiplicative_left_identity) }
% 88.55/11.62 antidomain(addition(multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))), antidomain(multiplication(X, Y))))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y)))))
% 88.55/11.62 = { by axiom 9 (multiplicative_left_identity) R->L }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(one, multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 88.55/11.62 = { by lemma 25 R->L }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(zero), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 88.55/11.62 = { by lemma 52 R->L }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(antidomain(multiplication(X, domain(Y))), multiplication(multiplication(X, domain(Y)), Y))), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 88.55/11.62 = { by axiom 15 (multiplicative_associativity) R->L }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(antidomain(multiplication(X, domain(Y))), multiplication(X, multiplication(domain(Y), Y)))), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 88.55/11.62 = { by lemma 31 }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(antidomain(multiplication(X, domain(Y))), multiplication(X, Y))), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 88.55/11.62 = { by lemma 59 }
% 88.55/11.62 antidomain(addition(antidomain(multiplication(X, Y)), zero))
% 88.55/11.62 = { by axiom 4 (additive_identity) }
% 88.55/11.62 antidomain(antidomain(multiplication(X, Y)))
% 88.55/11.62 = { by axiom 1 (domain4) R->L }
% 88.55/11.62 domain(multiplication(X, Y))
% 88.55/11.62
% 88.55/11.62 Lemma 61: addition(X, multiplication(X, coantidomain(Y))) = X.
% 88.55/11.62 Proof:
% 88.55/11.62 addition(X, multiplication(X, coantidomain(Y)))
% 88.55/11.62 = { by lemma 47 R->L }
% 88.55/11.62 multiplication(X, addition(one, coantidomain(Y)))
% 88.55/11.62 = { by lemma 45 }
% 88.55/11.62 multiplication(X, one)
% 88.55/11.62 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.62 X
% 88.55/11.62
% 88.55/11.62 Lemma 62: multiplication(domain(X), addition(Y, X)) = addition(X, multiplication(domain(X), Y)).
% 88.55/11.62 Proof:
% 88.55/11.62 multiplication(domain(X), addition(Y, X))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.62 multiplication(domain(X), addition(X, Y))
% 88.55/11.62 = { by axiom 17 (right_distributivity) }
% 88.55/11.62 addition(multiplication(domain(X), X), multiplication(domain(X), Y))
% 88.55/11.62 = { by lemma 31 }
% 88.55/11.62 addition(X, multiplication(domain(X), Y))
% 88.55/11.62
% 88.55/11.62 Lemma 63: multiplication(antidomain(X), addition(Y, domain(X))) = multiplication(antidomain(X), Y).
% 88.55/11.62 Proof:
% 88.55/11.62 multiplication(antidomain(X), addition(Y, domain(X)))
% 88.55/11.62 = { by lemma 32 R->L }
% 88.55/11.62 multiplication(domain(antidomain(X)), addition(Y, domain(X)))
% 88.55/11.62 = { by lemma 28 R->L }
% 88.55/11.62 multiplication(antidomain(domain(X)), addition(Y, domain(X)))
% 88.55/11.62 = { by lemma 58 }
% 88.55/11.62 multiplication(antidomain(domain(X)), Y)
% 88.55/11.62 = { by lemma 28 }
% 88.55/11.62 multiplication(domain(antidomain(X)), Y)
% 88.55/11.62 = { by lemma 32 }
% 88.55/11.62 multiplication(antidomain(X), Y)
% 88.55/11.62
% 88.55/11.62 Lemma 64: addition(X, multiplication(coantidomain(Y), X)) = X.
% 88.55/11.62 Proof:
% 88.55/11.62 addition(X, multiplication(coantidomain(Y), X))
% 88.55/11.62 = { by lemma 43 R->L }
% 88.55/11.62 multiplication(addition(one, coantidomain(Y)), X)
% 88.55/11.62 = { by lemma 45 }
% 88.55/11.62 multiplication(one, X)
% 88.55/11.62 = { by axiom 9 (multiplicative_left_identity) }
% 88.55/11.62 X
% 88.55/11.62
% 88.55/11.62 Lemma 65: multiplication(domain(X), antidomain(X)) = zero.
% 88.55/11.62 Proof:
% 88.55/11.62 multiplication(domain(X), antidomain(X))
% 88.55/11.62 = { by axiom 1 (domain4) }
% 88.55/11.62 multiplication(antidomain(antidomain(X)), antidomain(X))
% 88.55/11.62 = { by axiom 11 (domain1) }
% 88.55/11.62 zero
% 88.55/11.62
% 88.55/11.62 Lemma 66: multiplication(addition(X, domain(Y)), antidomain(Y)) = multiplication(X, antidomain(Y)).
% 88.55/11.62 Proof:
% 88.55/11.62 multiplication(addition(X, domain(Y)), antidomain(Y))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.62 multiplication(addition(domain(Y), X), antidomain(Y))
% 88.55/11.62 = { by axiom 18 (left_distributivity) }
% 88.55/11.62 addition(multiplication(domain(Y), antidomain(Y)), multiplication(X, antidomain(Y)))
% 88.55/11.62 = { by lemma 65 }
% 88.55/11.62 addition(zero, multiplication(X, antidomain(Y)))
% 88.55/11.62 = { by lemma 27 }
% 88.55/11.62 multiplication(X, antidomain(Y))
% 88.55/11.62
% 88.55/11.62 Lemma 67: addition(domain(X), multiplication(antidomain(X), coantidomain(Y))) = addition(coantidomain(Y), domain(X)).
% 88.55/11.62 Proof:
% 88.55/11.62 addition(domain(X), multiplication(antidomain(X), coantidomain(Y)))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.62 addition(multiplication(antidomain(X), coantidomain(Y)), domain(X))
% 88.55/11.62 = { by lemma 61 R->L }
% 88.55/11.62 addition(multiplication(antidomain(X), coantidomain(Y)), addition(domain(X), multiplication(domain(X), coantidomain(Y))))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.62 addition(multiplication(antidomain(X), coantidomain(Y)), addition(multiplication(domain(X), coantidomain(Y)), domain(X)))
% 88.55/11.62 = { by lemma 56 }
% 88.55/11.62 addition(multiplication(domain(X), coantidomain(Y)), addition(multiplication(antidomain(X), coantidomain(Y)), domain(X)))
% 88.55/11.62 = { by axiom 14 (additive_associativity) }
% 88.55/11.62 addition(addition(multiplication(domain(X), coantidomain(Y)), multiplication(antidomain(X), coantidomain(Y))), domain(X))
% 88.55/11.62 = { by axiom 18 (left_distributivity) R->L }
% 88.55/11.62 addition(multiplication(addition(domain(X), antidomain(X)), coantidomain(Y)), domain(X))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) }
% 88.55/11.62 addition(domain(X), multiplication(addition(domain(X), antidomain(X)), coantidomain(Y)))
% 88.55/11.62 = { by lemma 24 }
% 88.55/11.62 addition(domain(X), multiplication(one, coantidomain(Y)))
% 88.55/11.62 = { by axiom 9 (multiplicative_left_identity) }
% 88.55/11.62 addition(domain(X), coantidomain(Y))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) }
% 88.55/11.62 addition(coantidomain(Y), domain(X))
% 88.55/11.62
% 88.55/11.62 Lemma 68: multiplication(antidomain(X), coantidomain(Y)) = multiplication(coantidomain(Y), antidomain(X)).
% 88.55/11.62 Proof:
% 88.55/11.62 multiplication(antidomain(X), coantidomain(Y))
% 88.55/11.62 = { by lemma 63 R->L }
% 88.55/11.62 multiplication(antidomain(X), addition(coantidomain(Y), domain(X)))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.62 multiplication(antidomain(X), addition(domain(X), coantidomain(Y)))
% 88.55/11.62 = { by axiom 6 (multiplicative_right_identity) R->L }
% 88.55/11.62 multiplication(antidomain(X), addition(domain(X), multiplication(coantidomain(Y), one)))
% 88.55/11.62 = { by lemma 24 R->L }
% 88.55/11.62 multiplication(antidomain(X), addition(domain(X), multiplication(coantidomain(Y), addition(domain(X), antidomain(X)))))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.62 multiplication(antidomain(X), addition(domain(X), multiplication(coantidomain(Y), addition(antidomain(X), domain(X)))))
% 88.55/11.62 = { by lemma 57 R->L }
% 88.55/11.62 multiplication(antidomain(X), addition(multiplication(coantidomain(Y), antidomain(X)), addition(domain(X), multiplication(coantidomain(Y), domain(X)))))
% 88.55/11.62 = { by lemma 64 }
% 88.55/11.62 multiplication(antidomain(X), addition(multiplication(coantidomain(Y), antidomain(X)), domain(X)))
% 88.55/11.62 = { by lemma 63 }
% 88.55/11.62 multiplication(antidomain(X), multiplication(coantidomain(Y), antidomain(X)))
% 88.55/11.62 = { by axiom 15 (multiplicative_associativity) }
% 88.55/11.62 multiplication(multiplication(antidomain(X), coantidomain(Y)), antidomain(X))
% 88.55/11.62 = { by lemma 66 R->L }
% 88.55/11.62 multiplication(addition(multiplication(antidomain(X), coantidomain(Y)), domain(X)), antidomain(X))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) }
% 88.55/11.62 multiplication(addition(domain(X), multiplication(antidomain(X), coantidomain(Y))), antidomain(X))
% 88.55/11.62 = { by lemma 67 }
% 88.55/11.62 multiplication(addition(coantidomain(Y), domain(X)), antidomain(X))
% 88.55/11.62 = { by lemma 66 }
% 88.55/11.62 multiplication(coantidomain(Y), antidomain(X))
% 88.55/11.62
% 88.55/11.62 Lemma 69: multiplication(domain(Y), coantidomain(X)) = multiplication(coantidomain(X), domain(Y)).
% 88.55/11.62 Proof:
% 88.55/11.62 multiplication(domain(Y), coantidomain(X))
% 88.55/11.62 = { by axiom 1 (domain4) }
% 88.55/11.62 multiplication(antidomain(antidomain(Y)), coantidomain(X))
% 88.55/11.62 = { by lemma 68 }
% 88.55/11.62 multiplication(coantidomain(X), antidomain(antidomain(Y)))
% 88.55/11.62 = { by axiom 1 (domain4) R->L }
% 88.55/11.62 multiplication(coantidomain(X), domain(Y))
% 88.55/11.62
% 88.55/11.62 Lemma 70: addition(X, multiplication(domain(Y), addition(X, Z))) = addition(X, multiplication(domain(Y), Z)).
% 88.55/11.62 Proof:
% 88.55/11.62 addition(X, multiplication(domain(Y), addition(X, Z)))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.62 addition(X, multiplication(domain(Y), addition(Z, X)))
% 88.55/11.62 = { by lemma 57 R->L }
% 88.55/11.62 addition(multiplication(domain(Y), Z), addition(X, multiplication(domain(Y), X)))
% 88.55/11.62 = { by lemma 43 R->L }
% 88.55/11.62 addition(multiplication(domain(Y), Z), multiplication(addition(one, domain(Y)), X))
% 88.55/11.62 = { by lemma 48 }
% 88.55/11.62 addition(multiplication(domain(Y), Z), multiplication(one, X))
% 88.55/11.62 = { by axiom 9 (multiplicative_left_identity) }
% 88.55/11.62 addition(multiplication(domain(Y), Z), X)
% 88.55/11.62 = { by axiom 3 (additive_commutativity) }
% 88.55/11.62 addition(X, multiplication(domain(Y), Z))
% 88.55/11.62
% 88.55/11.62 Lemma 71: addition(coantidomain(X), multiplication(coantidomain(coantidomain(X)), domain(Y))) = addition(coantidomain(X), domain(Y)).
% 88.55/11.62 Proof:
% 88.55/11.62 addition(coantidomain(X), multiplication(coantidomain(coantidomain(X)), domain(Y)))
% 88.55/11.62 = { by lemma 69 R->L }
% 88.55/11.62 addition(coantidomain(X), multiplication(domain(Y), coantidomain(coantidomain(X))))
% 88.55/11.62 = { by lemma 70 R->L }
% 88.55/11.62 addition(coantidomain(X), multiplication(domain(Y), addition(coantidomain(X), coantidomain(coantidomain(X)))))
% 88.55/11.62 = { by lemma 34 }
% 88.55/11.62 addition(coantidomain(X), multiplication(domain(Y), one))
% 88.55/11.62 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.62 addition(coantidomain(X), domain(Y))
% 88.55/11.62
% 88.55/11.62 Lemma 72: addition(domain(X), multiplication(domain(Y), antidomain(X))) = addition(domain(X), domain(Y)).
% 88.55/11.62 Proof:
% 88.55/11.62 addition(domain(X), multiplication(domain(Y), antidomain(X)))
% 88.55/11.62 = { by lemma 70 R->L }
% 88.55/11.62 addition(domain(X), multiplication(domain(Y), addition(domain(X), antidomain(X))))
% 88.55/11.62 = { by lemma 24 }
% 88.55/11.62 addition(domain(X), multiplication(domain(Y), one))
% 88.55/11.62 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.62 addition(domain(X), domain(Y))
% 88.55/11.62
% 88.55/11.62 Lemma 73: domain(multiplication(coantidomain(X), antidomain(Y))) = multiplication(coantidomain(X), antidomain(Y)).
% 88.55/11.62 Proof:
% 88.55/11.62 domain(multiplication(coantidomain(X), antidomain(Y)))
% 88.55/11.62 = { by lemma 61 R->L }
% 88.55/11.62 addition(domain(multiplication(coantidomain(X), antidomain(Y))), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), coantidomain(X)))
% 88.55/11.62 = { by lemma 33 R->L }
% 88.55/11.62 addition(domain(multiplication(coantidomain(X), antidomain(Y))), multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), coantidomain(X)))
% 88.55/11.62 = { by lemma 62 R->L }
% 88.55/11.62 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), addition(coantidomain(X), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 88.55/11.62 = { by lemma 39 R->L }
% 88.55/11.62 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), addition(coantidomain(X), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), antidomain(Y)))))
% 88.55/11.62 = { by lemma 71 R->L }
% 88.55/11.62 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), addition(coantidomain(X), multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), antidomain(Y))))))
% 88.55/11.62 = { by axiom 9 (multiplicative_left_identity) R->L }
% 88.55/11.62 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), addition(coantidomain(X), multiplication(one, multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), antidomain(Y)))))))
% 88.55/11.62 = { by lemma 25 R->L }
% 88.55/11.62 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), addition(coantidomain(X), multiplication(antidomain(zero), multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), antidomain(Y)))))))
% 88.55/11.62 = { by axiom 10 (left_annihilation) R->L }
% 88.55/11.62 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), addition(coantidomain(X), multiplication(antidomain(multiplication(zero, antidomain(Y))), multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), antidomain(Y)))))))
% 88.55/11.62 = { by axiom 8 (codomain1) R->L }
% 88.55/11.62 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), addition(coantidomain(X), multiplication(antidomain(multiplication(multiplication(coantidomain(coantidomain(X)), coantidomain(coantidomain(coantidomain(X)))), antidomain(Y))), multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), antidomain(Y)))))))
% 88.55/11.62 = { by axiom 15 (multiplicative_associativity) R->L }
% 88.55/11.62 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), addition(coantidomain(X), multiplication(antidomain(multiplication(coantidomain(coantidomain(X)), multiplication(coantidomain(coantidomain(coantidomain(X))), antidomain(Y)))), multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), antidomain(Y)))))))
% 88.55/11.62 = { by lemma 59 }
% 88.55/11.62 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), addition(coantidomain(X), zero))
% 88.55/11.62 = { by axiom 4 (additive_identity) }
% 88.55/11.62 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), coantidomain(X))
% 88.55/11.62 = { by lemma 33 }
% 88.55/11.62 multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), coantidomain(X))
% 88.55/11.62 = { by lemma 69 }
% 88.55/11.62 multiplication(coantidomain(X), domain(multiplication(coantidomain(X), antidomain(Y))))
% 88.55/11.62 = { by lemma 46 R->L }
% 88.55/11.62 multiplication(domain(coantidomain(X)), domain(multiplication(coantidomain(X), antidomain(Y))))
% 88.55/11.62 = { by axiom 6 (multiplicative_right_identity) R->L }
% 88.55/11.62 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), one))
% 88.55/11.62 = { by lemma 24 R->L }
% 88.55/11.62 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), addition(domain(Y), antidomain(Y))))
% 88.55/11.62 = { by lemma 64 R->L }
% 88.55/11.62 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), addition(domain(Y), addition(antidomain(Y), multiplication(coantidomain(X), antidomain(Y))))))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.62 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), addition(domain(Y), addition(multiplication(coantidomain(X), antidomain(Y)), antidomain(Y)))))
% 88.55/11.62 = { by lemma 44 }
% 88.55/11.62 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), addition(multiplication(coantidomain(X), antidomain(Y)), one)))
% 88.55/11.62 = { by axiom 3 (additive_commutativity) }
% 88.55/11.62 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), addition(one, multiplication(coantidomain(X), antidomain(Y)))))
% 88.55/11.62 = { by lemma 62 }
% 88.55/11.62 multiplication(domain(coantidomain(X)), addition(multiplication(coantidomain(X), antidomain(Y)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), one)))
% 88.55/11.62 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.63 multiplication(domain(coantidomain(X)), addition(multiplication(coantidomain(X), antidomain(Y)), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 88.55/11.63 = { by axiom 17 (right_distributivity) }
% 88.55/11.63 addition(multiplication(domain(coantidomain(X)), multiplication(coantidomain(X), antidomain(Y))), multiplication(domain(coantidomain(X)), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 88.55/11.63 = { by axiom 15 (multiplicative_associativity) }
% 88.55/11.63 addition(multiplication(multiplication(domain(coantidomain(X)), coantidomain(X)), antidomain(Y)), multiplication(domain(coantidomain(X)), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 88.55/11.63 = { by lemma 31 }
% 88.55/11.63 addition(multiplication(coantidomain(X), antidomain(Y)), multiplication(domain(coantidomain(X)), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 88.55/11.63 = { by lemma 46 }
% 88.55/11.63 addition(multiplication(coantidomain(X), antidomain(Y)), multiplication(coantidomain(X), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 88.55/11.63 = { by axiom 17 (right_distributivity) R->L }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 88.55/11.63 = { by lemma 32 R->L }
% 88.55/11.63 multiplication(coantidomain(X), addition(domain(antidomain(Y)), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 88.55/11.63 = { by lemma 72 R->L }
% 88.55/11.63 multiplication(coantidomain(X), addition(domain(antidomain(Y)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), antidomain(antidomain(Y)))))
% 88.55/11.63 = { by axiom 1 (domain4) R->L }
% 88.55/11.63 multiplication(coantidomain(X), addition(domain(antidomain(Y)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), domain(Y))))
% 88.55/11.63 = { by lemma 32 }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), domain(Y))))
% 88.55/11.63 = { by lemma 54 R->L }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), coantidomain(antidomain(Y)))))
% 88.55/11.63 = { by lemma 69 }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(coantidomain(antidomain(Y)), domain(multiplication(coantidomain(X), antidomain(Y))))))
% 88.55/11.63 = { by lemma 54 }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y))))))
% 88.55/11.63 = { by axiom 9 (multiplicative_left_identity) R->L }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(one, multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))))
% 88.55/11.63 = { by lemma 25 R->L }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(antidomain(zero), multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))))
% 88.55/11.63 = { by lemma 65 R->L }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(antidomain(multiplication(domain(Y), antidomain(Y))), multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))))
% 88.55/11.63 = { by lemma 64 R->L }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(antidomain(multiplication(domain(Y), addition(antidomain(Y), multiplication(coantidomain(X), antidomain(Y))))), multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))))
% 88.55/11.63 = { by axiom 17 (right_distributivity) }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(antidomain(addition(multiplication(domain(Y), antidomain(Y)), multiplication(domain(Y), multiplication(coantidomain(X), antidomain(Y))))), multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))))
% 88.55/11.63 = { by lemma 65 }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(antidomain(addition(zero, multiplication(domain(Y), multiplication(coantidomain(X), antidomain(Y))))), multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))))
% 88.55/11.63 = { by lemma 27 }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(antidomain(multiplication(domain(Y), multiplication(coantidomain(X), antidomain(Y)))), multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))))
% 88.55/11.63 = { by lemma 59 }
% 88.55/11.63 multiplication(coantidomain(X), addition(antidomain(Y), zero))
% 88.55/11.63 = { by axiom 4 (additive_identity) }
% 88.55/11.63 multiplication(coantidomain(X), antidomain(Y))
% 88.55/11.63
% 88.55/11.63 Lemma 74: addition(coantidomain(X), domain(multiplication(coantidomain(coantidomain(X)), Y))) = addition(coantidomain(X), domain(Y)).
% 88.55/11.63 Proof:
% 88.55/11.63 addition(coantidomain(X), domain(multiplication(coantidomain(coantidomain(X)), Y)))
% 88.55/11.63 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.63 addition(domain(multiplication(coantidomain(coantidomain(X)), Y)), coantidomain(X))
% 88.55/11.63 = { by lemma 60 R->L }
% 88.55/11.63 addition(domain(multiplication(coantidomain(coantidomain(X)), domain(Y))), coantidomain(X))
% 88.55/11.63 = { by axiom 1 (domain4) }
% 88.55/11.63 addition(domain(multiplication(coantidomain(coantidomain(X)), antidomain(antidomain(Y)))), coantidomain(X))
% 88.55/11.63 = { by lemma 73 }
% 88.55/11.63 addition(multiplication(coantidomain(coantidomain(X)), antidomain(antidomain(Y))), coantidomain(X))
% 88.55/11.63 = { by axiom 1 (domain4) R->L }
% 88.55/11.63 addition(multiplication(coantidomain(coantidomain(X)), domain(Y)), coantidomain(X))
% 88.55/11.63 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.63 addition(coantidomain(X), multiplication(coantidomain(coantidomain(X)), domain(Y)))
% 88.55/11.63 = { by lemma 71 }
% 88.55/11.63 addition(coantidomain(X), domain(Y))
% 88.55/11.63
% 88.55/11.63 Lemma 75: multiplication(antidomain(addition(X, Y)), X) = zero.
% 88.55/11.63 Proof:
% 88.55/11.63 multiplication(antidomain(addition(X, Y)), X)
% 88.55/11.63 = { by lemma 58 R->L }
% 88.55/11.63 multiplication(antidomain(addition(X, Y)), addition(X, addition(X, Y)))
% 88.55/11.63 = { by lemma 40 }
% 88.55/11.63 multiplication(antidomain(addition(X, Y)), addition(X, Y))
% 88.55/11.63 = { by axiom 11 (domain1) }
% 88.55/11.63 zero
% 88.55/11.63
% 88.55/11.63 Lemma 76: addition(domain(X), domain(Y)) = domain(addition(X, domain(Y))).
% 88.55/11.63 Proof:
% 88.55/11.63 addition(domain(X), domain(Y))
% 88.55/11.63 = { by lemma 54 R->L }
% 88.55/11.63 addition(domain(X), coantidomain(antidomain(Y)))
% 88.55/11.63 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.63 addition(coantidomain(antidomain(Y)), domain(X))
% 88.55/11.63 = { by lemma 74 R->L }
% 88.55/11.63 addition(coantidomain(antidomain(Y)), domain(multiplication(coantidomain(coantidomain(antidomain(Y))), X)))
% 88.55/11.63 = { by lemma 36 R->L }
% 88.55/11.63 addition(coantidomain(antidomain(Y)), domain(multiplication(coantidomain(coantidomain(antidomain(Y))), addition(X, coantidomain(coantidomain(coantidomain(antidomain(Y))))))))
% 88.55/11.63 = { by lemma 74 }
% 88.55/11.63 addition(coantidomain(antidomain(Y)), domain(addition(X, coantidomain(coantidomain(coantidomain(antidomain(Y)))))))
% 88.55/11.63 = { by lemma 39 }
% 88.55/11.63 addition(coantidomain(antidomain(Y)), domain(addition(X, coantidomain(antidomain(Y)))))
% 88.55/11.63 = { by lemma 67 R->L }
% 88.55/11.63 addition(domain(addition(X, coantidomain(antidomain(Y)))), multiplication(antidomain(addition(X, coantidomain(antidomain(Y)))), coantidomain(antidomain(Y))))
% 88.55/11.63 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.63 addition(domain(addition(X, coantidomain(antidomain(Y)))), multiplication(antidomain(addition(coantidomain(antidomain(Y)), X)), coantidomain(antidomain(Y))))
% 88.55/11.63 = { by lemma 75 }
% 88.55/11.63 addition(domain(addition(X, coantidomain(antidomain(Y)))), zero)
% 88.55/11.63 = { by axiom 4 (additive_identity) }
% 88.55/11.63 domain(addition(X, coantidomain(antidomain(Y))))
% 88.55/11.63 = { by lemma 54 }
% 88.55/11.63 domain(addition(X, domain(Y)))
% 88.55/11.63
% 88.55/11.63 Lemma 77: domain(addition(X, domain(multiplication(antidomain(X), Y)))) = domain(addition(X, domain(Y))).
% 88.55/11.63 Proof:
% 88.55/11.63 domain(addition(X, domain(multiplication(antidomain(X), Y))))
% 88.55/11.63 = { by lemma 76 R->L }
% 88.55/11.63 addition(domain(X), domain(multiplication(antidomain(X), Y)))
% 88.55/11.63 = { by lemma 55 R->L }
% 88.55/11.63 addition(domain(X), domain(multiplication(coantidomain(domain(X)), Y)))
% 88.55/11.63 = { by lemma 60 R->L }
% 88.55/11.63 addition(domain(X), domain(multiplication(coantidomain(domain(X)), domain(Y))))
% 88.55/11.63 = { by lemma 69 R->L }
% 88.55/11.63 addition(domain(X), domain(multiplication(domain(Y), coantidomain(domain(X)))))
% 88.55/11.63 = { by lemma 55 }
% 88.55/11.63 addition(domain(X), domain(multiplication(domain(Y), antidomain(X))))
% 88.55/11.63 = { by lemma 54 R->L }
% 88.55/11.63 addition(domain(X), domain(multiplication(coantidomain(antidomain(Y)), antidomain(X))))
% 88.55/11.63 = { by lemma 73 }
% 88.55/11.63 addition(domain(X), multiplication(coantidomain(antidomain(Y)), antidomain(X)))
% 88.55/11.63 = { by lemma 54 }
% 88.55/11.63 addition(domain(X), multiplication(domain(Y), antidomain(X)))
% 88.55/11.63 = { by lemma 70 R->L }
% 88.55/11.63 addition(domain(X), multiplication(domain(Y), addition(domain(X), antidomain(X))))
% 88.55/11.63 = { by lemma 24 }
% 88.55/11.63 addition(domain(X), multiplication(domain(Y), one))
% 88.55/11.63 = { by axiom 6 (multiplicative_right_identity) }
% 88.55/11.63 addition(domain(X), domain(Y))
% 88.55/11.63 = { by lemma 76 }
% 88.55/11.63 domain(addition(X, domain(Y)))
% 88.55/11.63
% 88.55/11.63 Goal 1 (goals): domain(addition(x0, x1)) = addition(domain(x0), domain(x1)).
% 88.55/11.63 Proof:
% 88.55/11.63 domain(addition(x0, x1))
% 88.55/11.63 = { by axiom 4 (additive_identity) R->L }
% 88.55/11.63 addition(domain(addition(x0, x1)), zero)
% 88.55/11.63 = { by lemma 59 R->L }
% 88.55/11.63 addition(domain(addition(x0, x1)), multiplication(antidomain(multiplication(antidomain(addition(x0, x1)), x0)), multiplication(antidomain(addition(x0, x1)), domain(x0))))
% 88.55/11.63 = { by lemma 75 }
% 88.55/11.63 addition(domain(addition(x0, x1)), multiplication(antidomain(zero), multiplication(antidomain(addition(x0, x1)), domain(x0))))
% 88.55/11.63 = { by lemma 25 }
% 88.55/11.63 addition(domain(addition(x0, x1)), multiplication(one, multiplication(antidomain(addition(x0, x1)), domain(x0))))
% 88.55/11.63 = { by axiom 9 (multiplicative_left_identity) }
% 88.55/11.63 addition(domain(addition(x0, x1)), multiplication(antidomain(addition(x0, x1)), domain(x0)))
% 88.55/11.63 = { by axiom 3 (additive_commutativity) }
% 88.55/11.63 addition(domain(addition(x0, x1)), multiplication(antidomain(addition(x1, x0)), domain(x0)))
% 88.55/11.63 = { by lemma 54 R->L }
% 88.55/11.63 addition(domain(addition(x0, x1)), multiplication(antidomain(addition(x1, x0)), coantidomain(antidomain(x0))))
% 88.55/11.63 = { by lemma 68 }
% 88.55/11.63 addition(domain(addition(x0, x1)), multiplication(coantidomain(antidomain(x0)), antidomain(addition(x1, x0))))
% 88.55/11.63 = { by lemma 54 }
% 88.55/11.63 addition(domain(addition(x0, x1)), multiplication(domain(x0), antidomain(addition(x1, x0))))
% 88.55/11.63 = { by axiom 3 (additive_commutativity) }
% 88.55/11.63 addition(domain(addition(x0, x1)), multiplication(domain(x0), antidomain(addition(x0, x1))))
% 88.55/11.63 = { by lemma 72 }
% 88.55/11.63 addition(domain(addition(x0, x1)), domain(x0))
% 88.55/11.63 = { by axiom 3 (additive_commutativity) }
% 88.55/11.63 addition(domain(x0), domain(addition(x0, x1)))
% 88.55/11.63 = { by lemma 76 }
% 88.55/11.63 domain(addition(x0, domain(addition(x0, x1))))
% 88.55/11.63 = { by axiom 3 (additive_commutativity) R->L }
% 88.55/11.63 domain(addition(x0, domain(addition(x1, x0))))
% 88.55/11.63 = { by lemma 77 R->L }
% 88.55/11.63 domain(addition(x0, domain(multiplication(antidomain(x0), addition(x1, x0)))))
% 88.55/11.63 = { by lemma 58 }
% 88.55/11.63 domain(addition(x0, domain(multiplication(antidomain(x0), x1))))
% 88.55/11.63 = { by lemma 77 }
% 88.55/11.63 domain(addition(x0, domain(x1)))
% 88.55/11.63 = { by lemma 76 R->L }
% 88.55/11.63 addition(domain(x0), domain(x1))
% 88.55/11.63 % SZS output end Proof
% 88.55/11.63
% 88.55/11.63 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------