%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE120+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n004.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 11:39:51 AM UTC 2026
% Result : Theorem 16.54s 2.56s
% Output : Proof 17.29s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : KLE120+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.36 % Computer : n004.cluster.edu
% 0.10/0.36 % Model : x86_64 x86_64
% 0.10/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.36 % Memory : 8046.5625MB
% 0.10/0.36 % OS : Linux 6.8.0-71-generic
% 0.10/0.36 % CPULimit : 300
% 0.10/0.36 % WCLimit : 300
% 0.10/0.36 % DateTime : Sun Sep 27 13:11:21 UTC 2026
% 0.10/0.36 % CPUTime :
% 0.10/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 16.54/2.56 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 16.54/2.56
% 16.54/2.56 % SZS status Theorem
% 16.54/2.56
% 17.29/2.62 % SZS output start Proof
% 17.29/2.62 Axiom 1 (domain4): domain(X) = antidomain(antidomain(X)).
% 17.29/2.62 Axiom 2 (multiplicative_right_identity): multiplication(X, one) = X.
% 17.29/2.62 Axiom 3 (multiplicative_left_identity): multiplication(one, X) = X.
% 17.29/2.62 Axiom 4 (domain1): multiplication(antidomain(X), X) = zero.
% 17.29/2.62 Axiom 5 (additive_idempotence): addition(X, X) = X.
% 17.29/2.62 Axiom 6 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 17.29/2.62 Axiom 7 (additive_identity): addition(X, zero) = X.
% 17.29/2.62 Axiom 8 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 17.29/2.62 Axiom 9 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 17.29/2.62 Axiom 10 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 17.29/2.62 Axiom 11 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 17.29/2.62 Axiom 12 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 17.29/2.62 Axiom 13 (backward_diamond): backward_diamond(X, Y) = codomain(multiplication(codomain(Y), X)).
% 17.29/2.62 Axiom 14 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 17.29/2.62 Axiom 15 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 17.29/2.62 Axiom 16 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 17.29/2.62 Axiom 17 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
% 17.29/2.62
% 17.29/2.62 Lemma 18: coantidomain(one) = zero.
% 17.29/2.62 Proof:
% 17.29/2.62 coantidomain(one)
% 17.29/2.62 = { by axiom 3 (multiplicative_left_identity) R->L }
% 17.29/2.62 multiplication(one, coantidomain(one))
% 17.29/2.62 = { by axiom 10 (codomain1) }
% 17.29/2.62 zero
% 17.29/2.62
% 17.29/2.62 Lemma 19: addition(zero, X) = X.
% 17.29/2.62 Proof:
% 17.29/2.62 addition(zero, X)
% 17.29/2.62 = { by axiom 6 (additive_commutativity) R->L }
% 17.29/2.62 addition(X, zero)
% 17.29/2.63 = { by axiom 7 (additive_identity) }
% 17.29/2.63 X
% 17.29/2.63
% 17.29/2.63 Lemma 20: addition(coantidomain(X), codomain(X)) = one.
% 17.29/2.63 Proof:
% 17.29/2.63 addition(coantidomain(X), codomain(X))
% 17.29/2.63 = { by axiom 6 (additive_commutativity) R->L }
% 17.29/2.63 addition(codomain(X), coantidomain(X))
% 17.29/2.63 = { by axiom 9 (codomain4) }
% 17.29/2.63 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 17.29/2.63 = { by axiom 14 (codomain3) }
% 17.29/2.63 one
% 17.29/2.63
% 17.29/2.63 Lemma 21: multiplication(addition(X, Y), coantidomain(Y)) = multiplication(X, coantidomain(Y)).
% 17.29/2.63 Proof:
% 17.29/2.63 multiplication(addition(X, Y), coantidomain(Y))
% 17.29/2.63 = { by axiom 16 (left_distributivity) }
% 17.29/2.63 addition(multiplication(X, coantidomain(Y)), multiplication(Y, coantidomain(Y)))
% 17.29/2.63 = { by axiom 10 (codomain1) }
% 17.29/2.63 addition(multiplication(X, coantidomain(Y)), zero)
% 17.29/2.63 = { by axiom 7 (additive_identity) }
% 17.29/2.63 multiplication(X, coantidomain(Y))
% 17.29/2.63
% 17.29/2.63 Lemma 22: codomain(coantidomain(X)) = coantidomain(codomain(X)).
% 17.29/2.63 Proof:
% 17.29/2.63 codomain(coantidomain(X))
% 17.29/2.63 = { by axiom 9 (codomain4) }
% 17.29/2.63 coantidomain(coantidomain(coantidomain(X)))
% 17.29/2.63 = { by axiom 9 (codomain4) R->L }
% 17.29/2.63 coantidomain(codomain(X))
% 17.29/2.63
% 17.29/2.63 Lemma 23: multiplication(X, codomain(X)) = X.
% 17.29/2.63 Proof:
% 17.29/2.63 multiplication(X, codomain(X))
% 17.29/2.63 = { by axiom 7 (additive_identity) R->L }
% 17.29/2.63 addition(multiplication(X, codomain(X)), zero)
% 17.29/2.63 = { by axiom 10 (codomain1) R->L }
% 17.29/2.63 addition(multiplication(X, codomain(X)), multiplication(X, coantidomain(X)))
% 17.29/2.63 = { by axiom 15 (right_distributivity) R->L }
% 17.29/2.63 multiplication(X, addition(codomain(X), coantidomain(X)))
% 17.29/2.63 = { by axiom 6 (additive_commutativity) }
% 17.29/2.63 multiplication(X, addition(coantidomain(X), codomain(X)))
% 17.29/2.63 = { by lemma 20 }
% 17.29/2.63 multiplication(X, one)
% 17.29/2.63 = { by axiom 2 (multiplicative_right_identity) }
% 17.29/2.63 X
% 17.29/2.63
% 17.29/2.63 Lemma 24: coantidomain(codomain(X)) = coantidomain(X).
% 17.29/2.63 Proof:
% 17.29/2.63 coantidomain(codomain(X))
% 17.29/2.63 = { by axiom 3 (multiplicative_left_identity) R->L }
% 17.29/2.63 multiplication(one, coantidomain(codomain(X)))
% 17.29/2.63 = { by lemma 20 R->L }
% 17.29/2.63 multiplication(addition(coantidomain(X), codomain(X)), coantidomain(codomain(X)))
% 17.29/2.63 = { by lemma 21 }
% 17.29/2.63 multiplication(coantidomain(X), coantidomain(codomain(X)))
% 17.29/2.63 = { by lemma 22 R->L }
% 17.29/2.63 multiplication(coantidomain(X), codomain(coantidomain(X)))
% 17.29/2.63 = { by lemma 23 }
% 17.29/2.63 coantidomain(X)
% 17.29/2.63
% 17.29/2.63 Lemma 25: backward_diamond(one, X) = codomain(X).
% 17.29/2.63 Proof:
% 17.29/2.63 backward_diamond(one, X)
% 17.29/2.63 = { by axiom 13 (backward_diamond) }
% 17.29/2.63 codomain(multiplication(codomain(X), one))
% 17.29/2.63 = { by axiom 2 (multiplicative_right_identity) }
% 17.29/2.63 codomain(codomain(X))
% 17.29/2.63 = { by axiom 9 (codomain4) }
% 17.29/2.63 coantidomain(coantidomain(codomain(X)))
% 17.29/2.63 = { by axiom 3 (multiplicative_left_identity) R->L }
% 17.29/2.63 multiplication(one, coantidomain(coantidomain(codomain(X))))
% 17.29/2.63 = { by lemma 20 R->L }
% 17.29/2.63 multiplication(addition(coantidomain(coantidomain(X)), codomain(coantidomain(X))), coantidomain(coantidomain(codomain(X))))
% 17.29/2.63 = { by lemma 22 }
% 17.29/2.63 multiplication(addition(coantidomain(coantidomain(X)), coantidomain(codomain(X))), coantidomain(coantidomain(codomain(X))))
% 17.29/2.63 = { by axiom 9 (codomain4) R->L }
% 17.29/2.63 multiplication(addition(codomain(X), coantidomain(codomain(X))), coantidomain(coantidomain(codomain(X))))
% 17.29/2.63 = { by lemma 21 }
% 17.29/2.63 multiplication(codomain(X), coantidomain(coantidomain(codomain(X))))
% 17.29/2.63 = { by axiom 9 (codomain4) R->L }
% 17.29/2.63 multiplication(codomain(X), codomain(codomain(X)))
% 17.29/2.63 = { by lemma 23 }
% 17.29/2.63 codomain(X)
% 17.29/2.63
% 17.29/2.63 Lemma 26: addition(X, addition(X, Y)) = addition(X, Y).
% 17.29/2.63 Proof:
% 17.29/2.63 addition(X, addition(X, Y))
% 17.29/2.63 = { by axiom 12 (additive_associativity) }
% 17.29/2.63 addition(addition(X, X), Y)
% 17.29/2.63 = { by axiom 5 (additive_idempotence) }
% 17.29/2.63 addition(X, Y)
% 17.29/2.63
% 17.29/2.63 Lemma 27: addition(one, coantidomain(X)) = one.
% 17.29/2.63 Proof:
% 17.29/2.63 addition(one, coantidomain(X))
% 17.29/2.63 = { by axiom 6 (additive_commutativity) R->L }
% 17.29/2.63 addition(coantidomain(X), one)
% 17.29/2.63 = { by lemma 20 R->L }
% 17.29/2.63 addition(coantidomain(X), addition(coantidomain(X), codomain(X)))
% 17.29/2.63 = { by lemma 26 }
% 17.29/2.63 addition(coantidomain(X), codomain(X))
% 17.29/2.63 = { by lemma 20 }
% 17.29/2.63 one
% 17.29/2.63
% 17.29/2.63 Lemma 28: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 17.29/2.63 Proof:
% 17.29/2.63 addition(X, multiplication(Y, X))
% 17.29/2.63 = { by axiom 3 (multiplicative_left_identity) R->L }
% 17.29/2.63 addition(multiplication(one, X), multiplication(Y, X))
% 17.29/2.63 = { by axiom 16 (left_distributivity) R->L }
% 17.29/2.63 multiplication(addition(one, Y), X)
% 17.29/2.63 = { by axiom 6 (additive_commutativity) }
% 17.29/2.63 multiplication(addition(Y, one), X)
% 17.29/2.63
% 17.29/2.63 Lemma 29: coantidomain(multiplication(codomain(X), Y)) = coantidomain(backward_diamond(Y, X)).
% 17.29/2.63 Proof:
% 17.29/2.63 coantidomain(multiplication(codomain(X), Y))
% 17.29/2.63 = { by lemma 24 R->L }
% 17.29/2.63 coantidomain(codomain(multiplication(codomain(X), Y)))
% 17.29/2.63 = { by axiom 13 (backward_diamond) R->L }
% 17.29/2.63 coantidomain(backward_diamond(Y, X))
% 17.29/2.63
% 17.29/2.63 Lemma 30: addition(antidomain(X), antidomain(antidomain(X))) = one.
% 17.29/2.63 Proof:
% 17.29/2.63 addition(antidomain(X), antidomain(antidomain(X)))
% 17.29/2.63 = { by axiom 6 (additive_commutativity) R->L }
% 17.29/2.63 addition(antidomain(antidomain(X)), antidomain(X))
% 17.29/2.63 = { by axiom 8 (domain3) }
% 17.29/2.63 one
% 17.29/2.63
% 17.29/2.63 Lemma 31: addition(domain(X), antidomain(domain(X))) = one.
% 17.29/2.63 Proof:
% 17.29/2.63 addition(domain(X), antidomain(domain(X)))
% 17.29/2.63 = { by axiom 1 (domain4) }
% 17.29/2.63 addition(domain(X), antidomain(antidomain(antidomain(X))))
% 17.29/2.63 = { by axiom 1 (domain4) }
% 17.29/2.63 addition(antidomain(antidomain(X)), antidomain(antidomain(antidomain(X))))
% 17.29/2.63 = { by lemma 30 }
% 17.29/2.64 one
% 17.29/2.64
% 17.29/2.64 Goal 1 (goals): addition(backward_diamond(one, domain(x0)), domain(x0)) = domain(x0).
% 17.29/2.64 Proof:
% 17.29/2.64 addition(backward_diamond(one, domain(x0)), domain(x0))
% 17.29/2.64 = { by axiom 2 (multiplicative_right_identity) R->L }
% 17.29/2.64 multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), one)
% 17.29/2.64 = { by lemma 30 R->L }
% 17.29/2.64 multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), addition(antidomain(x0), antidomain(antidomain(x0))))
% 17.29/2.64 = { by axiom 1 (domain4) R->L }
% 17.29/2.64 multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), addition(antidomain(x0), domain(x0)))
% 17.29/2.64 = { by axiom 15 (right_distributivity) }
% 17.29/2.64 addition(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by axiom 2 (multiplicative_right_identity) R->L }
% 17.29/2.64 addition(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), multiplication(antidomain(x0), one)), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by axiom 11 (multiplicative_associativity) }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), one), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by lemma 27 R->L }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), addition(one, coantidomain(backward_diamond(antidomain(x0), domain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by lemma 20 R->L }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), addition(addition(coantidomain(one), codomain(one)), coantidomain(backward_diamond(antidomain(x0), domain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by lemma 18 }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), addition(addition(zero, codomain(one)), coantidomain(backward_diamond(antidomain(x0), domain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by lemma 19 }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), addition(codomain(one), coantidomain(backward_diamond(antidomain(x0), domain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by axiom 9 (codomain4) }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), addition(coantidomain(coantidomain(one)), coantidomain(backward_diamond(antidomain(x0), domain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by lemma 18 }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), addition(coantidomain(zero), coantidomain(backward_diamond(antidomain(x0), domain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by axiom 4 (domain1) R->L }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), addition(coantidomain(multiplication(antidomain(antidomain(x0)), antidomain(x0))), coantidomain(backward_diamond(antidomain(x0), domain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by axiom 1 (domain4) R->L }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), addition(coantidomain(multiplication(domain(x0), antidomain(x0))), coantidomain(backward_diamond(antidomain(x0), domain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by lemma 29 R->L }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), addition(coantidomain(multiplication(domain(x0), antidomain(x0))), coantidomain(multiplication(codomain(domain(x0)), antidomain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by axiom 9 (codomain4) }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), addition(coantidomain(multiplication(domain(x0), antidomain(x0))), coantidomain(multiplication(coantidomain(coantidomain(domain(x0))), antidomain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by axiom 17 (codomain2) }
% 17.29/2.64 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(multiplication(coantidomain(coantidomain(domain(x0))), antidomain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.64 = { by axiom 9 (codomain4) R->L }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(multiplication(codomain(domain(x0)), antidomain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 29 }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(backward_diamond(antidomain(x0), domain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by axiom 13 (backward_diamond) }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(codomain(multiplication(codomain(domain(x0)), antidomain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 25 R->L }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(codomain(multiplication(backward_diamond(one, domain(x0)), antidomain(x0))))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 24 }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(multiplication(backward_diamond(one, domain(x0)), antidomain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by axiom 3 (multiplicative_left_identity) R->L }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(multiplication(multiplication(one, backward_diamond(one, domain(x0))), antidomain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 31 R->L }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(multiplication(multiplication(addition(domain(x0), antidomain(domain(x0))), backward_diamond(one, domain(x0))), antidomain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 26 R->L }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(multiplication(multiplication(addition(domain(x0), addition(domain(x0), antidomain(domain(x0)))), backward_diamond(one, domain(x0))), antidomain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 31 }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(multiplication(multiplication(addition(domain(x0), one), backward_diamond(one, domain(x0))), antidomain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 28 R->L }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(multiplication(addition(backward_diamond(one, domain(x0)), multiplication(domain(x0), backward_diamond(one, domain(x0)))), antidomain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 25 }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(multiplication(addition(backward_diamond(one, domain(x0)), multiplication(domain(x0), codomain(domain(x0)))), antidomain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 23 }
% 17.29/2.65 addition(multiplication(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)), coantidomain(multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), antidomain(x0)))), multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by axiom 10 (codomain1) }
% 17.29/2.65 addition(zero, multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 19 }
% 17.29/2.65 multiplication(addition(backward_diamond(one, domain(x0)), domain(x0)), domain(x0))
% 17.29/2.65 = { by axiom 6 (additive_commutativity) R->L }
% 17.29/2.65 multiplication(addition(domain(x0), backward_diamond(one, domain(x0))), domain(x0))
% 17.29/2.65 = { by axiom 16 (left_distributivity) }
% 17.29/2.65 addition(multiplication(domain(x0), domain(x0)), multiplication(backward_diamond(one, domain(x0)), domain(x0)))
% 17.29/2.65 = { by axiom 7 (additive_identity) R->L }
% 17.29/2.65 addition(addition(multiplication(domain(x0), domain(x0)), zero), multiplication(backward_diamond(one, domain(x0)), domain(x0)))
% 17.29/2.65 = { by axiom 4 (domain1) R->L }
% 17.29/2.65 addition(addition(multiplication(domain(x0), domain(x0)), multiplication(antidomain(domain(x0)), domain(x0))), multiplication(backward_diamond(one, domain(x0)), domain(x0)))
% 17.29/2.65 = { by axiom 16 (left_distributivity) R->L }
% 17.29/2.65 addition(multiplication(addition(domain(x0), antidomain(domain(x0))), domain(x0)), multiplication(backward_diamond(one, domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 31 }
% 17.29/2.65 addition(multiplication(one, domain(x0)), multiplication(backward_diamond(one, domain(x0)), domain(x0)))
% 17.29/2.65 = { by axiom 3 (multiplicative_left_identity) }
% 17.29/2.65 addition(domain(x0), multiplication(backward_diamond(one, domain(x0)), domain(x0)))
% 17.29/2.65 = { by lemma 28 }
% 17.29/2.65 multiplication(addition(backward_diamond(one, domain(x0)), one), domain(x0))
% 17.29/2.65 = { by axiom 6 (additive_commutativity) }
% 17.29/2.65 multiplication(addition(one, backward_diamond(one, domain(x0))), domain(x0))
% 17.29/2.65 = { by axiom 13 (backward_diamond) }
% 17.29/2.65 multiplication(addition(one, codomain(multiplication(codomain(domain(x0)), one))), domain(x0))
% 17.29/2.65 = { by axiom 9 (codomain4) }
% 17.29/2.65 multiplication(addition(one, coantidomain(coantidomain(multiplication(codomain(domain(x0)), one)))), domain(x0))
% 17.29/2.65 = { by lemma 27 }
% 17.29/2.65 multiplication(one, domain(x0))
% 17.29/2.65 = { by axiom 3 (multiplicative_left_identity) }
% 17.29/2.65 domain(x0)
% 17.29/2.65 % SZS output end Proof
% 17.29/2.65
% 17.29/2.65 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------