%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE073+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n002.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 11:39:46 AM UTC 2026
% Result : Theorem 106.75s 13.92s
% Output : Proof 108.92s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE073+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.39 % Computer : n002.cluster.edu
% 0.10/0.39 % Model : x86_64 x86_64
% 0.10/0.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.39 % Memory : 8046.5625MB
% 0.10/0.39 % OS : Linux 6.8.0-71-generic
% 0.10/0.39 % CPULimit : 300
% 0.10/0.39 % WCLimit : 300
% 0.10/0.39 % DateTime : Sun Sep 27 13:11:38 UTC 2026
% 0.10/0.39 % CPUTime :
% 0.10/0.39 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 106.75/13.92 Command-line arguments: --no-flatten-goal
% 106.75/13.92
% 106.75/13.92 % SZS status Theorem
% 106.75/13.92
% 108.08/14.14 % SZS output start Proof
% 108.08/14.14 Axiom 1 (domain4): domain(X) = antidomain(antidomain(X)).
% 108.08/14.14 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 108.08/14.14 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 108.08/14.14 Axiom 4 (additive_identity): addition(X, zero) = X.
% 108.08/14.14 Axiom 5 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 108.08/14.14 Axiom 6 (multiplicative_right_identity): multiplication(X, one) = X.
% 108.08/14.14 Axiom 7 (right_annihilation): multiplication(X, zero) = zero.
% 108.08/14.14 Axiom 8 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 108.08/14.14 Axiom 9 (multiplicative_left_identity): multiplication(one, X) = X.
% 108.08/14.14 Axiom 10 (left_annihilation): multiplication(zero, X) = zero.
% 108.08/14.14 Axiom 11 (domain1): multiplication(antidomain(X), X) = zero.
% 108.08/14.14 Axiom 12 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 108.08/14.14 Axiom 13 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 108.08/14.14 Axiom 14 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 108.08/14.14 Axiom 15 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 108.08/14.14 Axiom 16 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 108.08/14.14 Axiom 17 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 108.08/14.14 Axiom 18 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 108.08/14.14 Axiom 19 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
% 108.08/14.14 Axiom 20 (order_1): ifeq(leq(X, Y), true, addition(X, Y), Y) = Y.
% 108.08/14.14 Axiom 21 (order): ifeq2(addition(X, Y), Y, leq(X, Y), true) = true.
% 108.08/14.14 Axiom 22 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 108.08/14.14
% 108.08/14.14 Lemma 23: antidomain(one) = zero.
% 108.08/14.14 Proof:
% 108.08/14.14 antidomain(one)
% 108.08/14.14 = { by axiom 6 (multiplicative_right_identity) R->L }
% 108.08/14.14 multiplication(antidomain(one), one)
% 108.08/14.14 = { by axiom 11 (domain1) }
% 108.08/14.14 zero
% 108.08/14.14
% 108.08/14.14 Lemma 24: addition(domain(X), antidomain(X)) = one.
% 108.08/14.14 Proof:
% 108.08/14.14 addition(domain(X), antidomain(X))
% 108.08/14.14 = { by axiom 1 (domain4) }
% 108.08/14.14 addition(antidomain(antidomain(X)), antidomain(X))
% 108.08/14.14 = { by axiom 16 (domain3) }
% 108.08/14.14 one
% 108.08/14.14
% 108.08/14.14 Lemma 25: antidomain(zero) = one.
% 108.08/14.14 Proof:
% 108.08/14.14 antidomain(zero)
% 108.08/14.14 = { by lemma 23 R->L }
% 108.08/14.14 antidomain(antidomain(one))
% 108.08/14.14 = { by axiom 1 (domain4) R->L }
% 108.08/14.14 domain(one)
% 108.08/14.14 = { by axiom 4 (additive_identity) R->L }
% 108.08/14.14 addition(domain(one), zero)
% 108.08/14.14 = { by lemma 23 R->L }
% 108.08/14.14 addition(domain(one), antidomain(one))
% 108.08/14.14 = { by lemma 24 }
% 108.08/14.14 one
% 108.08/14.14
% 108.08/14.14 Lemma 26: coantidomain(one) = zero.
% 108.08/14.14 Proof:
% 108.08/14.14 coantidomain(one)
% 108.08/14.14 = { by axiom 9 (multiplicative_left_identity) R->L }
% 108.08/14.14 multiplication(one, coantidomain(one))
% 108.08/14.14 = { by axiom 8 (codomain1) }
% 108.08/14.14 zero
% 108.08/14.14
% 108.08/14.14 Lemma 27: addition(zero, X) = X.
% 108.08/14.14 Proof:
% 108.08/14.14 addition(zero, X)
% 108.08/14.14 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.14 addition(X, zero)
% 108.08/14.14 = { by axiom 4 (additive_identity) }
% 108.08/14.14 X
% 108.08/14.14
% 108.08/14.14 Lemma 28: antidomain(domain(X)) = domain(antidomain(X)).
% 108.08/14.14 Proof:
% 108.08/14.14 antidomain(domain(X))
% 108.08/14.14 = { by axiom 1 (domain4) }
% 108.08/14.14 antidomain(antidomain(antidomain(X)))
% 108.08/14.14 = { by axiom 1 (domain4) R->L }
% 108.08/14.14 domain(antidomain(X))
% 108.08/14.14
% 108.08/14.14 Lemma 29: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 108.08/14.14 Proof:
% 108.08/14.14 multiplication(antidomain(X), addition(X, Y))
% 108.08/14.14 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.14 multiplication(antidomain(X), addition(Y, X))
% 108.08/14.14 = { by axiom 17 (right_distributivity) }
% 108.08/14.14 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 108.08/14.14 = { by axiom 11 (domain1) }
% 108.08/14.14 addition(multiplication(antidomain(X), Y), zero)
% 108.08/14.14 = { by axiom 4 (additive_identity) }
% 108.08/14.14 multiplication(antidomain(X), Y)
% 108.08/14.14
% 108.08/14.14 Lemma 30: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
% 108.08/14.14 Proof:
% 108.08/14.14 multiplication(addition(X, antidomain(Y)), Y)
% 108.08/14.14 = { by axiom 18 (left_distributivity) }
% 108.08/14.14 addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
% 108.08/14.14 = { by axiom 11 (domain1) }
% 108.08/14.14 addition(multiplication(X, Y), zero)
% 108.08/14.14 = { by axiom 4 (additive_identity) }
% 108.08/14.14 multiplication(X, Y)
% 108.08/14.14
% 108.08/14.14 Lemma 31: multiplication(domain(X), X) = X.
% 108.08/14.14 Proof:
% 108.08/14.14 multiplication(domain(X), X)
% 108.08/14.14 = { by lemma 30 R->L }
% 108.08/14.14 multiplication(addition(domain(X), antidomain(X)), X)
% 108.08/14.14 = { by lemma 24 }
% 108.08/14.14 multiplication(one, X)
% 108.08/14.14 = { by axiom 9 (multiplicative_left_identity) }
% 108.08/14.14 X
% 108.08/14.14
% 108.08/14.14 Lemma 32: domain(antidomain(X)) = antidomain(X).
% 108.08/14.14 Proof:
% 108.08/14.14 domain(antidomain(X))
% 108.08/14.14 = { by lemma 28 R->L }
% 108.08/14.14 antidomain(domain(X))
% 108.08/14.14 = { by axiom 6 (multiplicative_right_identity) R->L }
% 108.08/14.14 multiplication(antidomain(domain(X)), one)
% 108.08/14.14 = { by lemma 24 R->L }
% 108.08/14.14 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 108.08/14.14 = { by lemma 29 }
% 108.08/14.14 multiplication(antidomain(domain(X)), antidomain(X))
% 108.08/14.14 = { by lemma 28 }
% 108.08/14.14 multiplication(domain(antidomain(X)), antidomain(X))
% 108.08/14.14 = { by lemma 31 }
% 108.08/14.14 antidomain(X)
% 108.08/14.14
% 108.08/14.14 Lemma 33: domain(domain(X)) = domain(X).
% 108.08/14.14 Proof:
% 108.08/14.14 domain(domain(X))
% 108.08/14.14 = { by axiom 1 (domain4) }
% 108.08/14.14 domain(antidomain(antidomain(X)))
% 108.08/14.14 = { by lemma 32 }
% 108.08/14.14 antidomain(antidomain(X))
% 108.08/14.14 = { by axiom 1 (domain4) R->L }
% 108.08/14.14 domain(X)
% 108.08/14.14
% 108.08/14.14 Lemma 34: addition(coantidomain(X), coantidomain(coantidomain(X))) = one.
% 108.08/14.14 Proof:
% 108.08/14.14 addition(coantidomain(X), coantidomain(coantidomain(X)))
% 108.08/14.14 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.14 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 108.08/14.14 = { by axiom 5 (codomain3) }
% 108.08/14.15 one
% 108.08/14.15
% 108.08/14.15 Lemma 35: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
% 108.08/14.15 Proof:
% 108.08/14.15 multiplication(addition(X, Y), coantidomain(X))
% 108.08/14.15 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.15 multiplication(addition(Y, X), coantidomain(X))
% 108.08/14.15 = { by axiom 18 (left_distributivity) }
% 108.08/14.15 addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
% 108.08/14.15 = { by axiom 8 (codomain1) }
% 108.08/14.15 addition(multiplication(Y, coantidomain(X)), zero)
% 108.08/14.15 = { by axiom 4 (additive_identity) }
% 108.08/14.15 multiplication(Y, coantidomain(X))
% 108.08/14.15
% 108.08/14.15 Lemma 36: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y).
% 108.08/14.15 Proof:
% 108.08/14.15 multiplication(X, addition(Y, coantidomain(X)))
% 108.08/14.15 = { by axiom 17 (right_distributivity) }
% 108.08/14.15 addition(multiplication(X, Y), multiplication(X, coantidomain(X)))
% 108.08/14.15 = { by axiom 8 (codomain1) }
% 108.08/14.15 addition(multiplication(X, Y), zero)
% 108.08/14.15 = { by axiom 4 (additive_identity) }
% 108.08/14.15 multiplication(X, Y)
% 108.08/14.15
% 108.08/14.15 Lemma 37: multiplication(X, addition(coantidomain(X), Y)) = multiplication(X, Y).
% 108.08/14.15 Proof:
% 108.08/14.15 multiplication(X, addition(coantidomain(X), Y))
% 108.08/14.15 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.15 multiplication(X, addition(Y, coantidomain(X)))
% 108.08/14.15 = { by lemma 36 }
% 108.08/14.15 multiplication(X, Y)
% 108.08/14.15
% 108.08/14.15 Lemma 38: multiplication(X, coantidomain(coantidomain(X))) = X.
% 108.08/14.15 Proof:
% 108.08/14.15 multiplication(X, coantidomain(coantidomain(X)))
% 108.08/14.15 = { by lemma 37 R->L }
% 108.08/14.15 multiplication(X, addition(coantidomain(X), coantidomain(coantidomain(X))))
% 108.08/14.15 = { by lemma 34 }
% 108.08/14.15 multiplication(X, one)
% 108.08/14.15 = { by axiom 6 (multiplicative_right_identity) }
% 108.08/14.15 X
% 108.08/14.15
% 108.08/14.15 Lemma 39: coantidomain(coantidomain(coantidomain(X))) = coantidomain(X).
% 108.08/14.15 Proof:
% 108.08/14.15 coantidomain(coantidomain(coantidomain(X)))
% 108.08/14.15 = { by axiom 9 (multiplicative_left_identity) R->L }
% 108.08/14.15 multiplication(one, coantidomain(coantidomain(coantidomain(X))))
% 108.08/14.15 = { by lemma 34 R->L }
% 108.08/14.15 multiplication(addition(coantidomain(X), coantidomain(coantidomain(X))), coantidomain(coantidomain(coantidomain(X))))
% 108.08/14.15 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.15 multiplication(addition(coantidomain(coantidomain(X)), coantidomain(X)), coantidomain(coantidomain(coantidomain(X))))
% 108.08/14.15 = { by lemma 35 }
% 108.08/14.15 multiplication(coantidomain(X), coantidomain(coantidomain(coantidomain(X))))
% 108.08/14.15 = { by lemma 38 }
% 108.08/14.15 coantidomain(X)
% 108.08/14.15
% 108.08/14.15 Lemma 40: addition(X, addition(X, Y)) = addition(X, Y).
% 108.08/14.15 Proof:
% 108.08/14.15 addition(X, addition(X, Y))
% 108.08/14.15 = { by axiom 14 (additive_associativity) }
% 108.08/14.15 addition(addition(X, X), Y)
% 108.08/14.15 = { by axiom 2 (additive_idempotence) }
% 108.08/14.15 addition(X, Y)
% 108.08/14.15
% 108.08/14.15 Lemma 41: addition(one, antidomain(X)) = one.
% 108.08/14.15 Proof:
% 108.08/14.15 addition(one, antidomain(X))
% 108.08/14.15 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.15 addition(antidomain(X), one)
% 108.08/14.15 = { by axiom 12 (ifeq_axiom) R->L }
% 108.08/14.15 ifeq(true, true, addition(antidomain(X), one), one)
% 108.08/14.15 = { by axiom 21 (order) R->L }
% 108.08/14.15 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)
% 108.08/14.15 = { by lemma 40 }
% 108.08/14.15 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)
% 108.08/14.15 = { by axiom 13 (ifeq_axiom) }
% 108.08/14.15 ifeq(leq(antidomain(X), addition(antidomain(X), domain(X))), true, addition(antidomain(X), one), one)
% 108.08/14.15 = { by axiom 3 (additive_commutativity) }
% 108.08/14.15 ifeq(leq(antidomain(X), addition(domain(X), antidomain(X))), true, addition(antidomain(X), one), one)
% 108.08/14.15 = { by lemma 24 }
% 108.08/14.15 ifeq(leq(antidomain(X), one), true, addition(antidomain(X), one), one)
% 108.08/14.15 = { by axiom 20 (order_1) }
% 108.08/14.15 one
% 108.08/14.15
% 108.08/14.15 Lemma 42: addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))) = antidomain(multiplication(X, domain(Y))).
% 108.08/14.15 Proof:
% 108.08/14.15 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
% 108.08/14.15 = { by axiom 1 (domain4) }
% 108.08/14.15 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
% 108.08/14.15 = { by axiom 22 (domain2) }
% 108.08/14.15 antidomain(multiplication(X, antidomain(antidomain(Y))))
% 108.08/14.15 = { by axiom 1 (domain4) R->L }
% 108.08/14.15 antidomain(multiplication(X, domain(Y)))
% 108.08/14.15
% 108.08/14.15 Lemma 43: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 108.08/14.15 Proof:
% 108.08/14.15 multiplication(addition(one, Y), X)
% 108.08/14.15 = { by axiom 18 (left_distributivity) }
% 108.08/14.15 addition(multiplication(one, X), multiplication(Y, X))
% 108.08/14.15 = { by axiom 9 (multiplicative_left_identity) }
% 108.08/14.15 addition(X, multiplication(Y, X))
% 108.08/14.15
% 108.08/14.15 Lemma 44: addition(domain(X), addition(Y, antidomain(X))) = addition(Y, one).
% 108.08/14.15 Proof:
% 108.08/14.15 addition(domain(X), addition(Y, antidomain(X)))
% 108.08/14.15 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.15 addition(domain(X), addition(antidomain(X), Y))
% 108.08/14.15 = { by axiom 14 (additive_associativity) }
% 108.08/14.15 addition(addition(domain(X), antidomain(X)), Y)
% 108.08/14.15 = { by lemma 24 }
% 108.08/14.15 addition(one, Y)
% 108.08/14.15 = { by axiom 3 (additive_commutativity) }
% 108.08/14.15 addition(Y, one)
% 108.08/14.15
% 108.08/14.15 Lemma 45: addition(one, coantidomain(X)) = one.
% 108.08/14.15 Proof:
% 108.08/14.15 addition(one, coantidomain(X))
% 108.08/14.15 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.15 addition(coantidomain(X), one)
% 108.08/14.15 = { by lemma 34 R->L }
% 108.08/14.15 addition(coantidomain(X), addition(coantidomain(X), coantidomain(coantidomain(X))))
% 108.08/14.15 = { by lemma 40 }
% 108.08/14.15 addition(coantidomain(X), coantidomain(coantidomain(X)))
% 108.08/14.15 = { by lemma 34 }
% 108.08/14.15 one
% 108.08/14.15
% 108.08/14.15 Lemma 46: domain(coantidomain(X)) = coantidomain(X).
% 108.08/14.15 Proof:
% 108.08/14.15 domain(coantidomain(X))
% 108.08/14.15 = { by lemma 39 R->L }
% 108.08/14.15 domain(coantidomain(coantidomain(coantidomain(X))))
% 108.08/14.15 = { by axiom 9 (multiplicative_left_identity) R->L }
% 108.08/14.15 multiplication(one, domain(coantidomain(coantidomain(coantidomain(X)))))
% 108.08/14.15 = { by lemma 34 R->L }
% 108.08/14.15 multiplication(addition(coantidomain(coantidomain(X)), coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X)))))
% 108.08/14.15 = { by axiom 18 (left_distributivity) }
% 108.08/14.15 addition(multiplication(coantidomain(coantidomain(X)), domain(coantidomain(coantidomain(coantidomain(X))))), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 108.08/14.15 = { by lemma 38 R->L }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by axiom 15 (multiplicative_associativity) R->L }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by axiom 9 (multiplicative_left_identity) R->L }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by lemma 41 R->L }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by lemma 25 R->L }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by axiom 8 (codomain1) R->L }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by axiom 6 (multiplicative_right_identity) R->L }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by axiom 19 (codomain2) R->L }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by axiom 6 (multiplicative_right_identity) }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by axiom 6 (multiplicative_right_identity) }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by lemma 36 }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by lemma 42 }
% 108.08/14.15 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))))))
% 108.08/14.15 = { by axiom 11 (domain1) }
% 108.08/14.15 addition(multiplication(coantidomain(coantidomain(X)), zero), multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 108.08/14.15 = { by axiom 7 (right_annihilation) }
% 108.08/14.15 addition(zero, multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 108.08/14.15 = { by lemma 27 }
% 108.08/14.15 multiplication(coantidomain(coantidomain(coantidomain(X))), domain(coantidomain(coantidomain(coantidomain(X)))))
% 108.08/14.15 = { by lemma 37 R->L }
% 108.08/14.15 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(coantidomain(coantidomain(coantidomain(coantidomain(X)))), domain(coantidomain(coantidomain(coantidomain(X))))))
% 108.08/14.15 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.15 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), coantidomain(coantidomain(coantidomain(coantidomain(X))))))
% 108.08/14.15 = { by axiom 9 (multiplicative_left_identity) R->L }
% 108.08/14.15 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), multiplication(one, coantidomain(coantidomain(coantidomain(coantidomain(X)))))))
% 108.08/14.15 = { by lemma 41 R->L }
% 108.08/14.15 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), multiplication(addition(one, antidomain(coantidomain(coantidomain(coantidomain(X))))), coantidomain(coantidomain(coantidomain(coantidomain(X)))))))
% 108.08/14.15 = { by lemma 43 }
% 108.08/14.15 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))))))))
% 108.08/14.15 = { by lemma 29 R->L }
% 108.08/14.15 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)))))))))
% 108.08/14.15 = { by lemma 34 }
% 108.08/14.15 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), addition(coantidomain(coantidomain(coantidomain(coantidomain(X)))), multiplication(antidomain(coantidomain(coantidomain(coantidomain(X)))), one))))
% 108.08/14.15 = { by axiom 6 (multiplicative_right_identity) }
% 108.08/14.15 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(domain(coantidomain(coantidomain(coantidomain(X)))), addition(coantidomain(coantidomain(coantidomain(coantidomain(X)))), antidomain(coantidomain(coantidomain(coantidomain(X)))))))
% 108.08/14.16 = { by lemma 44 }
% 108.08/14.16 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(coantidomain(coantidomain(coantidomain(coantidomain(X)))), one))
% 108.08/14.16 = { by axiom 3 (additive_commutativity) }
% 108.08/14.16 multiplication(coantidomain(coantidomain(coantidomain(X))), addition(one, coantidomain(coantidomain(coantidomain(coantidomain(X))))))
% 108.08/14.16 = { by lemma 45 }
% 108.08/14.16 multiplication(coantidomain(coantidomain(coantidomain(X))), one)
% 108.08/14.16 = { by axiom 6 (multiplicative_right_identity) }
% 108.08/14.16 coantidomain(coantidomain(coantidomain(X)))
% 108.08/14.16 = { by lemma 39 }
% 108.08/14.16 coantidomain(X)
% 108.08/14.16
% 108.08/14.16 Lemma 47: addition(one, domain(X)) = one.
% 108.08/14.16 Proof:
% 108.08/14.16 addition(one, domain(X))
% 108.08/14.16 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.16 addition(domain(X), one)
% 108.08/14.16 = { by lemma 24 R->L }
% 108.08/14.16 addition(domain(X), addition(domain(X), antidomain(X)))
% 108.08/14.16 = { by lemma 40 }
% 108.08/14.16 addition(domain(X), antidomain(X))
% 108.08/14.16 = { by lemma 24 }
% 108.08/14.16 one
% 108.08/14.16
% 108.08/14.16 Lemma 48: multiplication(X, addition(one, Y)) = addition(X, multiplication(X, Y)).
% 108.08/14.16 Proof:
% 108.08/14.16 multiplication(X, addition(one, Y))
% 108.08/14.16 = { by axiom 17 (right_distributivity) }
% 108.08/14.16 addition(multiplication(X, one), multiplication(X, Y))
% 108.08/14.16 = { by axiom 6 (multiplicative_right_identity) }
% 108.08/14.16 addition(X, multiplication(X, Y))
% 108.08/14.16
% 108.08/14.16 Lemma 49: addition(antidomain(X), coantidomain(domain(X))) = antidomain(X).
% 108.08/14.16 Proof:
% 108.08/14.16 addition(antidomain(X), coantidomain(domain(X)))
% 108.08/14.16 = { by axiom 9 (multiplicative_left_identity) R->L }
% 108.08/14.16 addition(antidomain(X), multiplication(one, coantidomain(domain(X))))
% 108.08/14.16 = { by lemma 24 R->L }
% 108.08/14.16 addition(antidomain(X), multiplication(addition(domain(X), antidomain(X)), coantidomain(domain(X))))
% 108.08/14.16 = { by lemma 35 }
% 108.08/14.16 addition(antidomain(X), multiplication(antidomain(X), coantidomain(domain(X))))
% 108.08/14.16 = { by lemma 48 R->L }
% 108.08/14.16 multiplication(antidomain(X), addition(one, coantidomain(domain(X))))
% 108.08/14.16 = { by lemma 45 }
% 108.08/14.16 multiplication(antidomain(X), one)
% 108.08/14.16 = { by axiom 6 (multiplicative_right_identity) }
% 108.08/14.16 antidomain(X)
% 108.08/14.16
% 108.08/14.16 Lemma 50: multiplication(antidomain(X), multiplication(X, Y)) = zero.
% 108.08/14.16 Proof:
% 108.08/14.16 multiplication(antidomain(X), multiplication(X, Y))
% 108.08/14.16 = { by axiom 15 (multiplicative_associativity) }
% 108.08/14.16 multiplication(multiplication(antidomain(X), X), Y)
% 108.08/14.16 = { by axiom 11 (domain1) }
% 108.08/14.16 multiplication(zero, Y)
% 108.08/14.16 = { by axiom 10 (left_annihilation) }
% 108.08/14.16 zero
% 108.08/14.16
% 108.08/14.16 Lemma 51: coantidomain(domain(X)) = antidomain(X).
% 108.08/14.16 Proof:
% 108.08/14.16 coantidomain(domain(X))
% 108.08/14.16 = { by axiom 6 (multiplicative_right_identity) R->L }
% 108.08/14.16 multiplication(coantidomain(domain(X)), one)
% 108.08/14.16 = { by lemma 47 R->L }
% 108.08/14.16 multiplication(coantidomain(domain(X)), addition(one, domain(antidomain(X))))
% 108.08/14.16 = { by lemma 48 }
% 108.08/14.16 addition(coantidomain(domain(X)), multiplication(coantidomain(domain(X)), domain(antidomain(X))))
% 108.08/14.16 = { by lemma 32 }
% 108.08/14.16 addition(coantidomain(domain(X)), multiplication(coantidomain(domain(X)), antidomain(X)))
% 108.08/14.16 = { by axiom 6 (multiplicative_right_identity) R->L }
% 108.08/14.16 addition(multiplication(coantidomain(domain(X)), one), multiplication(coantidomain(domain(X)), antidomain(X)))
% 108.08/14.16 = { by lemma 34 R->L }
% 108.08/14.16 addition(multiplication(coantidomain(domain(X)), addition(coantidomain(domain(X)), coantidomain(coantidomain(domain(X))))), multiplication(coantidomain(domain(X)), antidomain(X)))
% 108.08/14.16 = { by lemma 36 }
% 108.08/14.16 addition(multiplication(coantidomain(domain(X)), coantidomain(domain(X))), multiplication(coantidomain(domain(X)), antidomain(X)))
% 108.08/14.16 = { by axiom 17 (right_distributivity) R->L }
% 108.08/14.16 multiplication(coantidomain(domain(X)), addition(coantidomain(domain(X)), antidomain(X)))
% 108.08/14.16 = { by axiom 3 (additive_commutativity) }
% 108.08/14.16 multiplication(coantidomain(domain(X)), addition(antidomain(X), coantidomain(domain(X))))
% 108.08/14.16 = { by lemma 49 }
% 108.08/14.16 multiplication(coantidomain(domain(X)), antidomain(X))
% 108.08/14.16 = { by axiom 1 (domain4) }
% 108.08/14.16 multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))
% 108.08/14.16 = { by lemma 27 R->L }
% 108.08/14.16 addition(zero, multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by axiom 8 (codomain1) R->L }
% 108.08/14.16 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)), coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by axiom 6 (multiplicative_right_identity) R->L }
% 108.08/14.16 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)), coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one)))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by axiom 19 (codomain2) R->L }
% 108.08/14.16 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)), addition(coantidomain(multiplication(antidomain(antidomain(X)), multiplication(antidomain(X), one))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one))))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by lemma 50 }
% 108.08/14.16 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)), addition(coantidomain(zero), coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one))))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by lemma 26 R->L }
% 108.08/14.16 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)), addition(coantidomain(coantidomain(one)), coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one))))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by lemma 27 R->L }
% 108.08/14.16 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)), addition(addition(zero, coantidomain(coantidomain(one))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one))))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by lemma 26 R->L }
% 108.08/14.16 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)), addition(addition(coantidomain(one), coantidomain(coantidomain(one))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one))))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by lemma 34 }
% 108.08/14.16 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)), addition(one, coantidomain(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one))))), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by lemma 45 }
% 108.08/14.16 addition(multiplication(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)), one), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by axiom 15 (multiplicative_associativity) R->L }
% 108.08/14.16 addition(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), multiplication(antidomain(X), one)), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by axiom 6 (multiplicative_right_identity) }
% 108.08/14.16 addition(multiplication(coantidomain(coantidomain(antidomain(antidomain(X)))), antidomain(X)), multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X)))
% 108.08/14.16 = { by axiom 18 (left_distributivity) R->L }
% 108.08/14.16 multiplication(addition(coantidomain(coantidomain(antidomain(antidomain(X)))), coantidomain(antidomain(antidomain(X)))), antidomain(X))
% 108.08/14.16 = { by axiom 3 (additive_commutativity) }
% 108.08/14.16 multiplication(addition(coantidomain(antidomain(antidomain(X))), coantidomain(coantidomain(antidomain(antidomain(X))))), antidomain(X))
% 108.08/14.16 = { by lemma 34 }
% 108.08/14.16 multiplication(one, antidomain(X))
% 108.08/14.16 = { by axiom 9 (multiplicative_left_identity) }
% 108.08/14.16 antidomain(X)
% 108.08/14.16
% 108.08/14.16 Lemma 52: addition(X, multiplication(domain(Y), X)) = X.
% 108.08/14.16 Proof:
% 108.08/14.16 addition(X, multiplication(domain(Y), X))
% 108.08/14.16 = { by lemma 43 R->L }
% 108.08/14.16 multiplication(addition(one, domain(Y)), X)
% 108.08/14.16 = { by lemma 47 }
% 108.08/14.16 multiplication(one, X)
% 108.08/14.16 = { by axiom 9 (multiplicative_left_identity) }
% 108.08/14.16 X
% 108.08/14.16
% 108.08/14.16 Lemma 53: coantidomain(antidomain(X)) = domain(X).
% 108.08/14.16 Proof:
% 108.08/14.16 coantidomain(antidomain(X))
% 108.08/14.16 = { by lemma 51 R->L }
% 108.08/14.16 coantidomain(coantidomain(domain(X)))
% 108.08/14.16 = { by lemma 52 R->L }
% 108.08/14.16 addition(coantidomain(coantidomain(domain(X))), multiplication(domain(X), coantidomain(coantidomain(domain(X)))))
% 108.08/14.16 = { by lemma 38 }
% 108.08/14.16 addition(coantidomain(coantidomain(domain(X))), domain(X))
% 108.08/14.16 = { by axiom 3 (additive_commutativity) }
% 108.08/14.16 addition(domain(X), coantidomain(coantidomain(domain(X))))
% 108.08/14.16 = { by lemma 51 }
% 108.08/14.16 addition(domain(X), coantidomain(antidomain(X)))
% 108.08/14.16 = { by lemma 32 R->L }
% 108.08/14.16 addition(domain(X), coantidomain(domain(antidomain(X))))
% 108.08/14.16 = { by axiom 1 (domain4) }
% 108.08/14.16 addition(antidomain(antidomain(X)), coantidomain(domain(antidomain(X))))
% 108.08/14.16 = { by lemma 49 }
% 108.08/14.16 antidomain(antidomain(X))
% 108.08/14.16 = { by axiom 1 (domain4) R->L }
% 108.08/14.16 domain(X)
% 108.08/14.16
% 108.08/14.16 Lemma 54: addition(Y, addition(X, Z)) = addition(X, addition(Y, Z)).
% 108.08/14.16 Proof:
% 108.08/14.16 addition(Y, addition(X, Z))
% 108.08/14.16 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.16 addition(addition(X, Z), Y)
% 108.08/14.16 = { by axiom 14 (additive_associativity) R->L }
% 108.08/14.16 addition(X, addition(Z, Y))
% 108.08/14.16 = { by axiom 3 (additive_commutativity) }
% 108.08/14.16 addition(X, addition(Y, Z))
% 108.08/14.16
% 108.08/14.16 Lemma 55: addition(multiplication(X, Y), addition(Z, multiplication(X, W))) = addition(Z, multiplication(X, addition(Y, W))).
% 108.08/14.16 Proof:
% 108.08/14.16 addition(multiplication(X, Y), addition(Z, multiplication(X, W)))
% 108.08/14.16 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.16 addition(multiplication(X, Y), addition(multiplication(X, W), Z))
% 108.08/14.16 = { by lemma 54 }
% 108.08/14.16 addition(multiplication(X, W), addition(multiplication(X, Y), Z))
% 108.08/14.16 = { by axiom 14 (additive_associativity) }
% 108.08/14.16 addition(addition(multiplication(X, W), multiplication(X, Y)), Z)
% 108.08/14.16 = { by axiom 17 (right_distributivity) R->L }
% 108.08/14.16 addition(multiplication(X, addition(W, Y)), Z)
% 108.08/14.16 = { by axiom 3 (additive_commutativity) }
% 108.08/14.16 addition(Z, multiplication(X, addition(W, Y)))
% 108.08/14.16 = { by axiom 3 (additive_commutativity) }
% 108.08/14.16 addition(Z, multiplication(X, addition(Y, W)))
% 108.08/14.16
% 108.08/14.16 Lemma 56: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y).
% 108.08/14.16 Proof:
% 108.08/14.16 multiplication(antidomain(X), addition(Y, X))
% 108.08/14.16 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.16 multiplication(antidomain(X), addition(X, Y))
% 108.08/14.16 = { by lemma 29 }
% 108.08/14.16 multiplication(antidomain(X), Y)
% 108.08/14.16
% 108.08/14.16 Lemma 57: multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y))) = zero.
% 108.08/14.16 Proof:
% 108.08/14.16 multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y)))
% 108.08/14.16 = { by lemma 30 R->L }
% 108.08/14.16 multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))), multiplication(X, domain(Y)))
% 108.08/14.16 = { by lemma 42 }
% 108.08/14.16 multiplication(antidomain(multiplication(X, domain(Y))), multiplication(X, domain(Y)))
% 108.08/14.16 = { by axiom 11 (domain1) }
% 108.08/14.16 zero
% 108.08/14.16
% 108.08/14.16 Lemma 58: domain(multiplication(X, domain(Y))) = domain(multiplication(X, Y)).
% 108.08/14.16 Proof:
% 108.08/14.16 domain(multiplication(X, domain(Y)))
% 108.08/14.16 = { by axiom 1 (domain4) }
% 108.08/14.16 antidomain(antidomain(multiplication(X, domain(Y))))
% 108.08/14.16 = { by lemma 42 R->L }
% 108.08/14.16 antidomain(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))))
% 108.08/14.16 = { by axiom 6 (multiplicative_right_identity) R->L }
% 108.08/14.16 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), one)))
% 108.08/14.16 = { by lemma 24 R->L }
% 108.08/14.16 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), addition(domain(multiplication(X, Y)), antidomain(multiplication(X, Y))))))
% 108.08/14.16 = { by lemma 29 R->L }
% 108.08/14.16 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)))))))
% 108.08/14.16 = { by lemma 54 R->L }
% 108.08/14.16 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)))))))
% 108.08/14.16 = { by axiom 14 (additive_associativity) }
% 108.08/14.16 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))))))
% 108.08/14.16 = { by lemma 55 R->L }
% 108.08/14.16 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))))))
% 108.08/14.16 = { by lemma 43 R->L }
% 108.08/14.16 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)))))
% 108.08/14.16 = { by lemma 56 }
% 108.08/14.16 antidomain(addition(multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))), multiplication(addition(one, antidomain(multiplication(X, domain(Y)))), antidomain(multiplication(X, Y)))))
% 108.08/14.16 = { by lemma 41 }
% 108.08/14.17 antidomain(addition(multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))), multiplication(one, antidomain(multiplication(X, Y)))))
% 108.08/14.17 = { by axiom 9 (multiplicative_left_identity) }
% 108.08/14.17 antidomain(addition(multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))), antidomain(multiplication(X, Y))))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) }
% 108.08/14.17 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y)))))
% 108.08/14.17 = { by axiom 9 (multiplicative_left_identity) R->L }
% 108.08/14.17 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(one, multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 108.08/14.17 = { by lemma 25 R->L }
% 108.08/14.17 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(zero), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 108.08/14.17 = { by lemma 50 R->L }
% 108.08/14.17 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))))))
% 108.08/14.17 = { by axiom 15 (multiplicative_associativity) R->L }
% 108.08/14.17 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))))))
% 108.08/14.17 = { by lemma 31 }
% 108.08/14.17 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))))))
% 108.08/14.17 = { by lemma 57 }
% 108.08/14.17 antidomain(addition(antidomain(multiplication(X, Y)), zero))
% 108.08/14.17 = { by axiom 4 (additive_identity) }
% 108.08/14.17 antidomain(antidomain(multiplication(X, Y)))
% 108.08/14.17 = { by axiom 1 (domain4) R->L }
% 108.08/14.17 domain(multiplication(X, Y))
% 108.08/14.17
% 108.08/14.17 Lemma 59: addition(X, multiplication(X, coantidomain(Y))) = X.
% 108.08/14.17 Proof:
% 108.08/14.17 addition(X, multiplication(X, coantidomain(Y)))
% 108.08/14.17 = { by lemma 48 R->L }
% 108.08/14.17 multiplication(X, addition(one, coantidomain(Y)))
% 108.08/14.17 = { by lemma 45 }
% 108.08/14.17 multiplication(X, one)
% 108.08/14.17 = { by axiom 6 (multiplicative_right_identity) }
% 108.08/14.17 X
% 108.08/14.17
% 108.08/14.17 Lemma 60: multiplication(domain(X), addition(Y, X)) = addition(X, multiplication(domain(X), Y)).
% 108.08/14.17 Proof:
% 108.08/14.17 multiplication(domain(X), addition(Y, X))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.17 multiplication(domain(X), addition(X, Y))
% 108.08/14.17 = { by axiom 17 (right_distributivity) }
% 108.08/14.17 addition(multiplication(domain(X), X), multiplication(domain(X), Y))
% 108.08/14.17 = { by lemma 31 }
% 108.08/14.17 addition(X, multiplication(domain(X), Y))
% 108.08/14.17
% 108.08/14.17 Lemma 61: addition(X, multiplication(domain(Y), addition(X, Z))) = addition(X, multiplication(domain(Y), Z)).
% 108.08/14.17 Proof:
% 108.08/14.17 addition(X, multiplication(domain(Y), addition(X, Z)))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.17 addition(X, multiplication(domain(Y), addition(Z, X)))
% 108.08/14.17 = { by lemma 55 R->L }
% 108.08/14.17 addition(multiplication(domain(Y), Z), addition(X, multiplication(domain(Y), X)))
% 108.08/14.17 = { by lemma 52 }
% 108.08/14.17 addition(multiplication(domain(Y), Z), X)
% 108.08/14.17 = { by axiom 3 (additive_commutativity) }
% 108.08/14.17 addition(X, multiplication(domain(Y), Z))
% 108.08/14.17
% 108.08/14.17 Lemma 62: multiplication(antidomain(X), addition(Y, domain(X))) = multiplication(antidomain(X), Y).
% 108.08/14.17 Proof:
% 108.08/14.17 multiplication(antidomain(X), addition(Y, domain(X)))
% 108.08/14.17 = { by lemma 32 R->L }
% 108.08/14.17 multiplication(domain(antidomain(X)), addition(Y, domain(X)))
% 108.08/14.17 = { by lemma 28 R->L }
% 108.08/14.17 multiplication(antidomain(domain(X)), addition(Y, domain(X)))
% 108.08/14.17 = { by lemma 56 }
% 108.08/14.17 multiplication(antidomain(domain(X)), Y)
% 108.08/14.17 = { by lemma 28 }
% 108.08/14.17 multiplication(domain(antidomain(X)), Y)
% 108.08/14.17 = { by lemma 32 }
% 108.08/14.17 multiplication(antidomain(X), Y)
% 108.08/14.17
% 108.08/14.17 Lemma 63: addition(X, multiplication(coantidomain(Y), X)) = X.
% 108.08/14.17 Proof:
% 108.08/14.17 addition(X, multiplication(coantidomain(Y), X))
% 108.08/14.17 = { by lemma 43 R->L }
% 108.08/14.17 multiplication(addition(one, coantidomain(Y)), X)
% 108.08/14.17 = { by lemma 45 }
% 108.08/14.17 multiplication(one, X)
% 108.08/14.17 = { by axiom 9 (multiplicative_left_identity) }
% 108.08/14.17 X
% 108.08/14.17
% 108.08/14.17 Lemma 64: addition(X, multiplication(coantidomain(Y), addition(X, Z))) = addition(X, multiplication(coantidomain(Y), Z)).
% 108.08/14.17 Proof:
% 108.08/14.17 addition(X, multiplication(coantidomain(Y), addition(X, Z)))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.17 addition(X, multiplication(coantidomain(Y), addition(Z, X)))
% 108.08/14.17 = { by lemma 55 R->L }
% 108.08/14.17 addition(multiplication(coantidomain(Y), Z), addition(X, multiplication(coantidomain(Y), X)))
% 108.08/14.17 = { by lemma 63 }
% 108.08/14.17 addition(multiplication(coantidomain(Y), Z), X)
% 108.08/14.17 = { by axiom 3 (additive_commutativity) }
% 108.08/14.17 addition(X, multiplication(coantidomain(Y), Z))
% 108.08/14.17
% 108.08/14.17 Lemma 65: multiplication(domain(X), antidomain(X)) = zero.
% 108.08/14.17 Proof:
% 108.08/14.17 multiplication(domain(X), antidomain(X))
% 108.08/14.17 = { by axiom 1 (domain4) }
% 108.08/14.17 multiplication(antidomain(antidomain(X)), antidomain(X))
% 108.08/14.17 = { by axiom 11 (domain1) }
% 108.08/14.17 zero
% 108.08/14.17
% 108.08/14.17 Lemma 66: multiplication(addition(X, domain(Y)), antidomain(Y)) = multiplication(X, antidomain(Y)).
% 108.08/14.17 Proof:
% 108.08/14.17 multiplication(addition(X, domain(Y)), antidomain(Y))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.17 multiplication(addition(domain(Y), X), antidomain(Y))
% 108.08/14.17 = { by axiom 18 (left_distributivity) }
% 108.08/14.17 addition(multiplication(domain(Y), antidomain(Y)), multiplication(X, antidomain(Y)))
% 108.08/14.17 = { by lemma 65 }
% 108.08/14.17 addition(zero, multiplication(X, antidomain(Y)))
% 108.08/14.17 = { by lemma 27 }
% 108.08/14.17 multiplication(X, antidomain(Y))
% 108.08/14.17
% 108.08/14.17 Lemma 67: addition(domain(X), multiplication(antidomain(X), coantidomain(Y))) = addition(coantidomain(Y), domain(X)).
% 108.08/14.17 Proof:
% 108.08/14.17 addition(domain(X), multiplication(antidomain(X), coantidomain(Y)))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.17 addition(multiplication(antidomain(X), coantidomain(Y)), domain(X))
% 108.08/14.17 = { by lemma 59 R->L }
% 108.08/14.17 addition(multiplication(antidomain(X), coantidomain(Y)), addition(domain(X), multiplication(domain(X), coantidomain(Y))))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.17 addition(multiplication(antidomain(X), coantidomain(Y)), addition(multiplication(domain(X), coantidomain(Y)), domain(X)))
% 108.08/14.17 = { by lemma 54 }
% 108.08/14.17 addition(multiplication(domain(X), coantidomain(Y)), addition(multiplication(antidomain(X), coantidomain(Y)), domain(X)))
% 108.08/14.17 = { by axiom 14 (additive_associativity) }
% 108.08/14.17 addition(addition(multiplication(domain(X), coantidomain(Y)), multiplication(antidomain(X), coantidomain(Y))), domain(X))
% 108.08/14.17 = { by axiom 18 (left_distributivity) R->L }
% 108.08/14.17 addition(multiplication(addition(domain(X), antidomain(X)), coantidomain(Y)), domain(X))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) }
% 108.08/14.17 addition(domain(X), multiplication(addition(domain(X), antidomain(X)), coantidomain(Y)))
% 108.08/14.17 = { by lemma 24 }
% 108.08/14.17 addition(domain(X), multiplication(one, coantidomain(Y)))
% 108.08/14.17 = { by axiom 9 (multiplicative_left_identity) }
% 108.08/14.17 addition(domain(X), coantidomain(Y))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) }
% 108.08/14.17 addition(coantidomain(Y), domain(X))
% 108.08/14.17
% 108.08/14.17 Lemma 68: multiplication(antidomain(X), coantidomain(Y)) = multiplication(coantidomain(Y), antidomain(X)).
% 108.08/14.17 Proof:
% 108.08/14.17 multiplication(antidomain(X), coantidomain(Y))
% 108.08/14.17 = { by lemma 62 R->L }
% 108.08/14.17 multiplication(antidomain(X), addition(coantidomain(Y), domain(X)))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.17 multiplication(antidomain(X), addition(domain(X), coantidomain(Y)))
% 108.08/14.17 = { by axiom 6 (multiplicative_right_identity) R->L }
% 108.08/14.17 multiplication(antidomain(X), addition(domain(X), multiplication(coantidomain(Y), one)))
% 108.08/14.17 = { by lemma 24 R->L }
% 108.08/14.17 multiplication(antidomain(X), addition(domain(X), multiplication(coantidomain(Y), addition(domain(X), antidomain(X)))))
% 108.08/14.17 = { by lemma 64 }
% 108.08/14.17 multiplication(antidomain(X), addition(domain(X), multiplication(coantidomain(Y), antidomain(X))))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.17 multiplication(antidomain(X), addition(multiplication(coantidomain(Y), antidomain(X)), domain(X)))
% 108.08/14.17 = { by lemma 62 }
% 108.08/14.17 multiplication(antidomain(X), multiplication(coantidomain(Y), antidomain(X)))
% 108.08/14.17 = { by axiom 15 (multiplicative_associativity) }
% 108.08/14.17 multiplication(multiplication(antidomain(X), coantidomain(Y)), antidomain(X))
% 108.08/14.17 = { by lemma 66 R->L }
% 108.08/14.17 multiplication(addition(multiplication(antidomain(X), coantidomain(Y)), domain(X)), antidomain(X))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) }
% 108.08/14.17 multiplication(addition(domain(X), multiplication(antidomain(X), coantidomain(Y))), antidomain(X))
% 108.08/14.17 = { by lemma 67 }
% 108.08/14.17 multiplication(addition(coantidomain(Y), domain(X)), antidomain(X))
% 108.08/14.17 = { by lemma 66 }
% 108.08/14.17 multiplication(coantidomain(Y), antidomain(X))
% 108.08/14.17
% 108.08/14.17 Lemma 69: multiplication(domain(Y), coantidomain(X)) = multiplication(coantidomain(X), domain(Y)).
% 108.08/14.17 Proof:
% 108.08/14.17 multiplication(domain(Y), coantidomain(X))
% 108.08/14.17 = { by axiom 1 (domain4) }
% 108.08/14.17 multiplication(antidomain(antidomain(Y)), coantidomain(X))
% 108.08/14.17 = { by lemma 68 }
% 108.08/14.17 multiplication(coantidomain(X), antidomain(antidomain(Y)))
% 108.08/14.17 = { by axiom 1 (domain4) R->L }
% 108.08/14.17 multiplication(coantidomain(X), domain(Y))
% 108.08/14.17
% 108.08/14.17 Lemma 70: addition(coantidomain(X), domain(multiplication(coantidomain(X), Y))) = coantidomain(X).
% 108.08/14.17 Proof:
% 108.08/14.17 addition(coantidomain(X), domain(multiplication(coantidomain(X), Y)))
% 108.08/14.17 = { by lemma 39 R->L }
% 108.08/14.17 addition(coantidomain(X), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), Y)))
% 108.08/14.17 = { by axiom 6 (multiplicative_right_identity) R->L }
% 108.08/14.17 addition(coantidomain(X), multiplication(domain(multiplication(coantidomain(coantidomain(coantidomain(X))), Y)), one))
% 108.08/14.17 = { by lemma 34 R->L }
% 108.08/14.17 addition(coantidomain(X), multiplication(domain(multiplication(coantidomain(coantidomain(coantidomain(X))), Y)), addition(coantidomain(X), coantidomain(coantidomain(X)))))
% 108.08/14.17 = { by lemma 61 }
% 108.08/14.17 addition(coantidomain(X), multiplication(domain(multiplication(coantidomain(coantidomain(coantidomain(X))), Y)), coantidomain(coantidomain(X))))
% 108.08/14.17 = { by lemma 69 }
% 108.08/14.17 addition(coantidomain(X), multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), Y))))
% 108.08/14.17 = { by axiom 9 (multiplicative_left_identity) R->L }
% 108.08/14.17 addition(coantidomain(X), multiplication(one, multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), Y)))))
% 108.08/14.17 = { by lemma 25 R->L }
% 108.08/14.17 addition(coantidomain(X), multiplication(antidomain(zero), multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), Y)))))
% 108.08/14.17 = { by axiom 10 (left_annihilation) R->L }
% 108.08/14.17 addition(coantidomain(X), multiplication(antidomain(multiplication(zero, Y)), multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), Y)))))
% 108.08/14.17 = { by axiom 8 (codomain1) R->L }
% 108.08/14.17 addition(coantidomain(X), multiplication(antidomain(multiplication(multiplication(coantidomain(coantidomain(X)), coantidomain(coantidomain(coantidomain(X)))), Y)), multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), Y)))))
% 108.08/14.17 = { by axiom 15 (multiplicative_associativity) R->L }
% 108.08/14.17 addition(coantidomain(X), multiplication(antidomain(multiplication(coantidomain(coantidomain(X)), multiplication(coantidomain(coantidomain(coantidomain(X))), Y))), multiplication(coantidomain(coantidomain(X)), domain(multiplication(coantidomain(coantidomain(coantidomain(X))), Y)))))
% 108.08/14.17 = { by lemma 57 }
% 108.08/14.17 addition(coantidomain(X), zero)
% 108.08/14.17 = { by axiom 4 (additive_identity) }
% 108.08/14.17 coantidomain(X)
% 108.08/14.17
% 108.08/14.17 Lemma 71: domain(multiplication(coantidomain(X), antidomain(Y))) = multiplication(coantidomain(X), antidomain(Y)).
% 108.08/14.17 Proof:
% 108.08/14.17 domain(multiplication(coantidomain(X), antidomain(Y)))
% 108.08/14.17 = { by lemma 59 R->L }
% 108.08/14.17 addition(domain(multiplication(coantidomain(X), antidomain(Y))), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), coantidomain(X)))
% 108.08/14.17 = { by lemma 33 R->L }
% 108.08/14.17 addition(domain(multiplication(coantidomain(X), antidomain(Y))), multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), coantidomain(X)))
% 108.08/14.17 = { by lemma 60 R->L }
% 108.08/14.17 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), addition(coantidomain(X), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 108.08/14.17 = { by lemma 70 }
% 108.08/14.17 multiplication(domain(domain(multiplication(coantidomain(X), antidomain(Y)))), coantidomain(X))
% 108.08/14.17 = { by lemma 33 }
% 108.08/14.17 multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), coantidomain(X))
% 108.08/14.17 = { by lemma 69 }
% 108.08/14.17 multiplication(coantidomain(X), domain(multiplication(coantidomain(X), antidomain(Y))))
% 108.08/14.17 = { by lemma 46 R->L }
% 108.08/14.17 multiplication(domain(coantidomain(X)), domain(multiplication(coantidomain(X), antidomain(Y))))
% 108.08/14.17 = { by axiom 6 (multiplicative_right_identity) R->L }
% 108.08/14.17 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), one))
% 108.08/14.17 = { by lemma 24 R->L }
% 108.08/14.17 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), addition(domain(Y), antidomain(Y))))
% 108.08/14.17 = { by lemma 63 R->L }
% 108.08/14.17 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), addition(domain(Y), addition(antidomain(Y), multiplication(coantidomain(X), antidomain(Y))))))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) R->L }
% 108.08/14.17 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), addition(domain(Y), addition(multiplication(coantidomain(X), antidomain(Y)), antidomain(Y)))))
% 108.08/14.17 = { by lemma 44 }
% 108.08/14.17 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), addition(multiplication(coantidomain(X), antidomain(Y)), one)))
% 108.08/14.17 = { by axiom 3 (additive_commutativity) }
% 108.08/14.17 multiplication(domain(coantidomain(X)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), addition(one, multiplication(coantidomain(X), antidomain(Y)))))
% 108.08/14.17 = { by lemma 60 }
% 108.08/14.17 multiplication(domain(coantidomain(X)), addition(multiplication(coantidomain(X), antidomain(Y)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), one)))
% 108.08/14.17 = { by axiom 6 (multiplicative_right_identity) }
% 108.08/14.17 multiplication(domain(coantidomain(X)), addition(multiplication(coantidomain(X), antidomain(Y)), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 108.08/14.17 = { by axiom 17 (right_distributivity) }
% 108.08/14.17 addition(multiplication(domain(coantidomain(X)), multiplication(coantidomain(X), antidomain(Y))), multiplication(domain(coantidomain(X)), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 108.08/14.17 = { by axiom 15 (multiplicative_associativity) }
% 108.08/14.17 addition(multiplication(multiplication(domain(coantidomain(X)), coantidomain(X)), antidomain(Y)), multiplication(domain(coantidomain(X)), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 108.08/14.17 = { by lemma 31 }
% 108.08/14.17 addition(multiplication(coantidomain(X), antidomain(Y)), multiplication(domain(coantidomain(X)), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 108.08/14.17 = { by lemma 46 }
% 108.08/14.17 addition(multiplication(coantidomain(X), antidomain(Y)), multiplication(coantidomain(X), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 108.08/14.17 = { by axiom 17 (right_distributivity) R->L }
% 108.08/14.17 multiplication(coantidomain(X), addition(antidomain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 108.08/14.17 = { by lemma 32 R->L }
% 108.08/14.17 multiplication(coantidomain(X), addition(domain(antidomain(Y)), domain(multiplication(coantidomain(X), antidomain(Y)))))
% 108.08/14.17 = { by axiom 6 (multiplicative_right_identity) R->L }
% 108.08/14.17 multiplication(coantidomain(X), addition(domain(antidomain(Y)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), one)))
% 108.08/14.17 = { by lemma 24 R->L }
% 108.08/14.17 multiplication(coantidomain(X), addition(domain(antidomain(Y)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), addition(domain(antidomain(Y)), antidomain(antidomain(Y))))))
% 108.08/14.17 = { by lemma 61 }
% 108.08/14.18 multiplication(coantidomain(X), addition(domain(antidomain(Y)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), antidomain(antidomain(Y)))))
% 108.08/14.18 = { by axiom 1 (domain4) R->L }
% 108.08/14.18 multiplication(coantidomain(X), addition(domain(antidomain(Y)), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), domain(Y))))
% 108.08/14.18 = { by lemma 32 }
% 108.08/14.18 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), domain(Y))))
% 108.08/14.18 = { by lemma 53 R->L }
% 108.08/14.18 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(domain(multiplication(coantidomain(X), antidomain(Y))), coantidomain(antidomain(Y)))))
% 108.08/14.18 = { by lemma 69 }
% 108.08/14.18 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(coantidomain(antidomain(Y)), domain(multiplication(coantidomain(X), antidomain(Y))))))
% 108.08/14.18 = { by lemma 53 }
% 108.08/14.18 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y))))))
% 108.08/14.18 = { by axiom 9 (multiplicative_left_identity) R->L }
% 108.08/14.18 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(one, multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))))
% 108.08/14.18 = { by lemma 25 R->L }
% 108.08/14.18 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(antidomain(zero), multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))))
% 108.08/14.18 = { by lemma 65 R->L }
% 108.08/14.18 multiplication(coantidomain(X), addition(antidomain(Y), multiplication(antidomain(multiplication(domain(Y), antidomain(Y))), multiplication(domain(Y), domain(multiplication(coantidomain(X), antidomain(Y)))))))
% 108.08/14.18 = { by lemma 63 R->L }
% 108.08/14.18 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)))))))
% 108.92/14.18 = { by axiom 17 (right_distributivity) }
% 108.92/14.18 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)))))))
% 108.92/14.18 = { by lemma 65 }
% 108.92/14.18 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)))))))
% 108.92/14.18 = { by lemma 27 }
% 108.92/14.18 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)))))))
% 108.92/14.18 = { by lemma 57 }
% 108.92/14.18 multiplication(coantidomain(X), addition(antidomain(Y), zero))
% 108.92/14.18 = { by axiom 4 (additive_identity) }
% 108.92/14.18 multiplication(coantidomain(X), antidomain(Y))
% 108.92/14.18
% 108.92/14.18 Lemma 72: addition(coantidomain(X), domain(multiplication(coantidomain(coantidomain(X)), Y))) = addition(coantidomain(X), domain(Y)).
% 108.92/14.18 Proof:
% 108.92/14.18 addition(coantidomain(X), domain(multiplication(coantidomain(coantidomain(X)), Y)))
% 108.92/14.18 = { by lemma 58 R->L }
% 108.92/14.18 addition(coantidomain(X), domain(multiplication(coantidomain(coantidomain(X)), domain(Y))))
% 108.92/14.18 = { by axiom 1 (domain4) }
% 108.92/14.18 addition(coantidomain(X), domain(multiplication(coantidomain(coantidomain(X)), antidomain(antidomain(Y)))))
% 108.92/14.18 = { by lemma 71 }
% 108.92/14.18 addition(coantidomain(X), multiplication(coantidomain(coantidomain(X)), antidomain(antidomain(Y))))
% 108.92/14.18 = { by axiom 1 (domain4) R->L }
% 108.92/14.18 addition(coantidomain(X), multiplication(coantidomain(coantidomain(X)), domain(Y)))
% 108.92/14.18 = { by lemma 69 R->L }
% 108.92/14.18 addition(coantidomain(X), multiplication(domain(Y), coantidomain(coantidomain(X))))
% 108.92/14.18 = { by lemma 53 R->L }
% 108.92/14.18 addition(coantidomain(X), multiplication(coantidomain(antidomain(Y)), coantidomain(coantidomain(X))))
% 108.92/14.18 = { by lemma 64 R->L }
% 108.92/14.18 addition(coantidomain(X), multiplication(coantidomain(antidomain(Y)), addition(coantidomain(X), coantidomain(coantidomain(X)))))
% 108.92/14.18 = { by lemma 34 }
% 108.92/14.18 addition(coantidomain(X), multiplication(coantidomain(antidomain(Y)), one))
% 108.92/14.18 = { by axiom 6 (multiplicative_right_identity) }
% 108.92/14.18 addition(coantidomain(X), coantidomain(antidomain(Y)))
% 108.92/14.18 = { by lemma 53 }
% 108.92/14.18 addition(coantidomain(X), domain(Y))
% 108.92/14.18
% 108.92/14.18 Lemma 73: multiplication(antidomain(addition(X, Y)), X) = zero.
% 108.92/14.18 Proof:
% 108.92/14.18 multiplication(antidomain(addition(X, Y)), X)
% 108.92/14.18 = { by lemma 56 R->L }
% 108.92/14.18 multiplication(antidomain(addition(X, Y)), addition(X, addition(X, Y)))
% 108.92/14.18 = { by lemma 40 }
% 108.92/14.18 multiplication(antidomain(addition(X, Y)), addition(X, Y))
% 108.92/14.18 = { by axiom 11 (domain1) }
% 108.92/14.18 zero
% 108.92/14.18
% 108.92/14.18 Lemma 74: addition(domain(X), domain(Y)) = domain(addition(X, domain(Y))).
% 108.92/14.18 Proof:
% 108.92/14.18 addition(domain(X), domain(Y))
% 108.92/14.18 = { by lemma 53 R->L }
% 108.92/14.18 addition(domain(X), coantidomain(antidomain(Y)))
% 108.92/14.18 = { by axiom 3 (additive_commutativity) R->L }
% 108.92/14.18 addition(coantidomain(antidomain(Y)), domain(X))
% 108.92/14.18 = { by lemma 72 R->L }
% 108.92/14.18 addition(coantidomain(antidomain(Y)), domain(multiplication(coantidomain(coantidomain(antidomain(Y))), X)))
% 108.92/14.18 = { by lemma 36 R->L }
% 108.92/14.18 addition(coantidomain(antidomain(Y)), domain(multiplication(coantidomain(coantidomain(antidomain(Y))), addition(X, coantidomain(coantidomain(coantidomain(antidomain(Y))))))))
% 108.92/14.18 = { by lemma 72 }
% 108.92/14.18 addition(coantidomain(antidomain(Y)), domain(addition(X, coantidomain(coantidomain(coantidomain(antidomain(Y)))))))
% 108.92/14.18 = { by lemma 39 }
% 108.92/14.18 addition(coantidomain(antidomain(Y)), domain(addition(X, coantidomain(antidomain(Y)))))
% 108.92/14.18 = { by lemma 67 R->L }
% 108.92/14.18 addition(domain(addition(X, coantidomain(antidomain(Y)))), multiplication(antidomain(addition(X, coantidomain(antidomain(Y)))), coantidomain(antidomain(Y))))
% 108.92/14.18 = { by axiom 3 (additive_commutativity) R->L }
% 108.92/14.18 addition(domain(addition(X, coantidomain(antidomain(Y)))), multiplication(antidomain(addition(coantidomain(antidomain(Y)), X)), coantidomain(antidomain(Y))))
% 108.92/14.18 = { by lemma 73 }
% 108.92/14.18 addition(domain(addition(X, coantidomain(antidomain(Y)))), zero)
% 108.92/14.18 = { by axiom 4 (additive_identity) }
% 108.92/14.18 domain(addition(X, coantidomain(antidomain(Y))))
% 108.92/14.18 = { by lemma 53 }
% 108.92/14.18 domain(addition(X, domain(Y)))
% 108.92/14.18
% 108.92/14.18 Lemma 75: domain(addition(X, domain(multiplication(antidomain(X), Y)))) = domain(addition(X, domain(Y))).
% 108.92/14.18 Proof:
% 108.92/14.18 domain(addition(X, domain(multiplication(antidomain(X), Y))))
% 108.92/14.18 = { by lemma 74 R->L }
% 108.92/14.18 addition(domain(X), domain(multiplication(antidomain(X), Y)))
% 108.92/14.18 = { by lemma 58 R->L }
% 108.92/14.18 addition(domain(X), domain(multiplication(antidomain(X), domain(Y))))
% 108.92/14.18 = { by lemma 53 R->L }
% 108.92/14.18 addition(domain(X), domain(multiplication(antidomain(X), coantidomain(antidomain(Y)))))
% 108.92/14.18 = { by lemma 68 }
% 108.92/14.18 addition(domain(X), domain(multiplication(coantidomain(antidomain(Y)), antidomain(X))))
% 108.92/14.18 = { by lemma 71 }
% 108.92/14.18 addition(domain(X), multiplication(coantidomain(antidomain(Y)), antidomain(X)))
% 108.92/14.18 = { by lemma 53 }
% 108.92/14.18 addition(domain(X), multiplication(domain(Y), antidomain(X)))
% 108.92/14.18 = { by lemma 61 R->L }
% 108.92/14.18 addition(domain(X), multiplication(domain(Y), addition(domain(X), antidomain(X))))
% 108.92/14.18 = { by lemma 24 }
% 108.92/14.18 addition(domain(X), multiplication(domain(Y), one))
% 108.92/14.18 = { by axiom 6 (multiplicative_right_identity) }
% 108.92/14.18 addition(domain(X), domain(Y))
% 108.92/14.18 = { by lemma 74 }
% 108.92/14.18 domain(addition(X, domain(Y)))
% 108.92/14.18
% 108.92/14.18 Lemma 76: domain(addition(X, domain(Y))) = domain(addition(X, Y)).
% 108.92/14.18 Proof:
% 108.92/14.18 domain(addition(X, domain(Y)))
% 108.92/14.18 = { by lemma 75 R->L }
% 108.92/14.18 domain(addition(X, domain(multiplication(antidomain(X), Y))))
% 108.92/14.18 = { by lemma 56 R->L }
% 108.92/14.18 domain(addition(X, domain(multiplication(antidomain(X), addition(Y, X)))))
% 108.92/14.18 = { by lemma 75 }
% 108.92/14.18 domain(addition(X, domain(addition(Y, X))))
% 108.92/14.18 = { by axiom 3 (additive_commutativity) }
% 108.92/14.18 domain(addition(X, domain(addition(X, Y))))
% 108.92/14.18 = { by lemma 74 R->L }
% 108.92/14.18 addition(domain(X), domain(addition(X, Y)))
% 108.92/14.18 = { by axiom 3 (additive_commutativity) R->L }
% 108.92/14.18 addition(domain(X), domain(addition(Y, X)))
% 108.92/14.18 = { by axiom 3 (additive_commutativity) R->L }
% 108.92/14.18 addition(domain(addition(Y, X)), domain(X))
% 108.92/14.18 = { by axiom 9 (multiplicative_left_identity) R->L }
% 108.92/14.18 addition(domain(addition(Y, X)), domain(multiplication(one, X)))
% 108.92/14.18 = { by lemma 24 R->L }
% 108.92/14.18 addition(domain(addition(Y, X)), domain(multiplication(addition(domain(addition(X, Y)), antidomain(addition(X, Y))), X)))
% 108.92/14.18 = { by axiom 3 (additive_commutativity) R->L }
% 108.92/14.18 addition(domain(addition(Y, X)), domain(multiplication(addition(antidomain(addition(X, Y)), domain(addition(X, Y))), X)))
% 108.92/14.18 = { by axiom 18 (left_distributivity) }
% 108.92/14.18 addition(domain(addition(Y, X)), domain(addition(multiplication(antidomain(addition(X, Y)), X), multiplication(domain(addition(X, Y)), X))))
% 108.92/14.18 = { by lemma 73 }
% 108.92/14.18 addition(domain(addition(Y, X)), domain(addition(zero, multiplication(domain(addition(X, Y)), X))))
% 108.92/14.18 = { by lemma 27 }
% 108.92/14.18 addition(domain(addition(Y, X)), domain(multiplication(domain(addition(X, Y)), X)))
% 108.92/14.18 = { by axiom 3 (additive_commutativity) }
% 108.92/14.18 addition(domain(addition(Y, X)), domain(multiplication(domain(addition(Y, X)), X)))
% 108.92/14.18 = { by lemma 53 R->L }
% 108.92/14.18 addition(domain(addition(Y, X)), domain(multiplication(coantidomain(antidomain(addition(Y, X))), X)))
% 108.92/14.18 = { by lemma 53 R->L }
% 108.92/14.18 addition(coantidomain(antidomain(addition(Y, X))), domain(multiplication(coantidomain(antidomain(addition(Y, X))), X)))
% 108.92/14.18 = { by lemma 70 }
% 108.92/14.18 coantidomain(antidomain(addition(Y, X)))
% 108.92/14.18 = { by lemma 53 }
% 108.92/14.18 domain(addition(Y, X))
% 108.92/14.18 = { by axiom 3 (additive_commutativity) }
% 108.92/14.18 domain(addition(X, Y))
% 108.92/14.18
% 108.92/14.18 Goal 1 (goals): domain(multiplication(x0, addition(domain(x1), domain(x2)))) = addition(domain(multiplication(x0, domain(x1))), domain(multiplication(x0, domain(x2)))).
% 108.92/14.18 Proof:
% 108.92/14.18 domain(multiplication(x0, addition(domain(x1), domain(x2))))
% 108.92/14.18 = { by lemma 74 }
% 108.92/14.18 domain(multiplication(x0, domain(addition(x1, domain(x2)))))
% 108.92/14.18 = { by lemma 76 }
% 108.92/14.18 domain(multiplication(x0, domain(addition(x1, x2))))
% 108.92/14.18 = { by lemma 58 }
% 108.92/14.18 domain(multiplication(x0, addition(x1, x2)))
% 108.92/14.18 = { by axiom 17 (right_distributivity) }
% 108.92/14.18 domain(addition(multiplication(x0, x1), multiplication(x0, x2)))
% 108.92/14.19 = { by lemma 76 R->L }
% 108.92/14.19 domain(addition(multiplication(x0, x1), domain(multiplication(x0, x2))))
% 108.92/14.19 = { by lemma 74 R->L }
% 108.92/14.19 addition(domain(multiplication(x0, x1)), domain(multiplication(x0, x2)))
% 108.92/14.19 = { by lemma 58 R->L }
% 108.92/14.19 addition(domain(multiplication(x0, x1)), domain(multiplication(x0, domain(x2))))
% 108.92/14.19 = { by lemma 58 R->L }
% 108.92/14.19 addition(domain(multiplication(x0, domain(x1))), domain(multiplication(x0, domain(x2))))
% 108.92/14.19 % SZS output end Proof
% 108.92/14.19
% 108.92/14.19 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------