%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE132+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n009.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 11:39:52 AM UTC 2026
% Result : Theorem 5.96s 1.29s
% Output : Proof 7.44s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE132+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.35 % Computer : n009.cluster.edu
% 0.09/0.35 % Model : x86_64 x86_64
% 0.09/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.35 % Memory : 8046.5625MB
% 0.09/0.35 % OS : Linux 6.8.0-71-generic
% 0.09/0.35 % CPULimit : 300
% 0.09/0.35 % WCLimit : 300
% 0.09/0.35 % DateTime : Sun Sep 27 13:12:29 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 5.96/1.29 Command-line arguments: --flatten --complete-subsets
% 5.96/1.29
% 5.96/1.29 % SZS status Theorem
% 5.96/1.29
% 6.77/1.39 % SZS output start Proof
% 6.77/1.39 Axiom 1 (complement): c(X) = antidomain(domain(X)).
% 6.77/1.39 Axiom 2 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 6.77/1.39 Axiom 3 (domain4): domain(X) = antidomain(antidomain(X)).
% 6.77/1.39 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 6.77/1.39 Axiom 5 (right_annihilation): multiplication(X, zero) = zero.
% 6.77/1.39 Axiom 6 (multiplicative_left_identity): multiplication(one, X) = X.
% 6.77/1.39 Axiom 7 (left_annihilation): multiplication(zero, X) = zero.
% 6.77/1.39 Axiom 8 (additive_idempotence): addition(X, X) = X.
% 6.77/1.39 Axiom 9 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 6.77/1.40 Axiom 10 (additive_identity): addition(X, zero) = X.
% 6.77/1.40 Axiom 11 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 6.77/1.40 Axiom 12 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 6.77/1.40 Axiom 13 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 6.77/1.40 Axiom 14 (domain1): multiplication(antidomain(X), X) = zero.
% 6.77/1.40 Axiom 15 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
% 6.77/1.40 Axiom 16 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 6.77/1.40 Axiom 17 (goals): addition(domain(x2), forward_diamond(x0, domain(x2))) = forward_diamond(x0, domain(x2)).
% 6.77/1.40 Axiom 18 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 6.77/1.40 Axiom 19 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 6.77/1.40 Axiom 20 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 6.77/1.40 Axiom 21 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 6.77/1.40 Axiom 22 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 6.77/1.40 Axiom 23 (goals_1): addition(forward_diamond(x0, domain(X)), forward_diamond(star(x0), domain_difference(domain(X), forward_diamond(x0, domain(X))))) = forward_diamond(star(x0), domain_difference(domain(X), forward_diamond(x0, domain(X)))).
% 6.77/1.40
% 6.77/1.40 Lemma 24: addition(domain(X), antidomain(X)) = one.
% 6.77/1.40 Proof:
% 6.77/1.40 addition(domain(X), antidomain(X))
% 6.77/1.40 = { by axiom 3 (domain4) }
% 6.77/1.40 addition(antidomain(antidomain(X)), antidomain(X))
% 6.77/1.40 = { by axiom 20 (domain3) }
% 6.77/1.40 one
% 6.77/1.40
% 6.77/1.40 Lemma 25: multiplication(domain(X), X) = X.
% 6.77/1.40 Proof:
% 6.77/1.40 multiplication(domain(X), X)
% 6.77/1.40 = { by axiom 10 (additive_identity) R->L }
% 6.77/1.40 addition(multiplication(domain(X), X), zero)
% 6.77/1.40 = { by axiom 14 (domain1) R->L }
% 6.77/1.40 addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
% 6.77/1.40 = { by axiom 22 (left_distributivity) R->L }
% 6.77/1.40 multiplication(addition(domain(X), antidomain(X)), X)
% 6.77/1.40 = { by lemma 24 }
% 6.77/1.40 multiplication(one, X)
% 6.77/1.40 = { by axiom 6 (multiplicative_left_identity) }
% 6.77/1.40 X
% 6.77/1.40
% 6.77/1.40 Lemma 26: domain(antidomain(X)) = c(X).
% 6.77/1.40 Proof:
% 6.77/1.40 domain(antidomain(X))
% 6.77/1.40 = { by axiom 3 (domain4) }
% 6.77/1.40 antidomain(antidomain(antidomain(X)))
% 6.77/1.40 = { by axiom 3 (domain4) R->L }
% 6.77/1.40 antidomain(domain(X))
% 6.77/1.40 = { by axiom 1 (complement) R->L }
% 6.77/1.40 c(X)
% 6.77/1.40
% 6.77/1.40 Lemma 27: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 6.77/1.40 Proof:
% 6.77/1.40 multiplication(antidomain(X), addition(X, Y))
% 6.77/1.40 = { by axiom 9 (additive_commutativity) R->L }
% 6.77/1.40 multiplication(antidomain(X), addition(Y, X))
% 6.77/1.40 = { by axiom 21 (right_distributivity) }
% 6.77/1.40 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 6.77/1.40 = { by axiom 14 (domain1) }
% 6.77/1.40 addition(multiplication(antidomain(X), Y), zero)
% 6.77/1.40 = { by axiom 10 (additive_identity) }
% 6.77/1.40 multiplication(antidomain(X), Y)
% 6.77/1.40
% 6.77/1.40 Lemma 28: antidomain(X) = c(X).
% 6.77/1.40 Proof:
% 6.77/1.40 antidomain(X)
% 6.77/1.40 = { by lemma 25 R->L }
% 6.77/1.40 multiplication(domain(antidomain(X)), antidomain(X))
% 6.77/1.40 = { by lemma 26 }
% 6.77/1.40 multiplication(c(X), antidomain(X))
% 6.77/1.40 = { by axiom 1 (complement) }
% 6.77/1.40 multiplication(antidomain(domain(X)), antidomain(X))
% 6.77/1.40 = { by lemma 27 R->L }
% 6.77/1.40 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 6.77/1.40 = { by lemma 24 }
% 6.77/1.40 multiplication(antidomain(domain(X)), one)
% 6.77/1.40 = { by axiom 4 (multiplicative_right_identity) }
% 6.77/1.40 antidomain(domain(X))
% 6.77/1.40 = { by axiom 1 (complement) R->L }
% 6.77/1.40 c(X)
% 6.77/1.40
% 6.77/1.40 Lemma 29: antidomain(one) = zero.
% 6.77/1.40 Proof:
% 6.77/1.40 antidomain(one)
% 6.77/1.40 = { by axiom 4 (multiplicative_right_identity) R->L }
% 6.77/1.40 multiplication(antidomain(one), one)
% 6.77/1.40 = { by axiom 14 (domain1) }
% 6.77/1.40 zero
% 6.77/1.40
% 6.77/1.40 Lemma 30: domain(one) = one.
% 6.77/1.40 Proof:
% 6.77/1.40 domain(one)
% 6.77/1.40 = { by axiom 10 (additive_identity) R->L }
% 6.77/1.40 addition(domain(one), zero)
% 6.77/1.40 = { by lemma 29 R->L }
% 6.77/1.40 addition(domain(one), antidomain(one))
% 6.77/1.40 = { by lemma 24 }
% 6.77/1.40 one
% 6.77/1.40
% 6.77/1.40 Lemma 31: antidomain(zero) = one.
% 6.77/1.40 Proof:
% 6.77/1.40 antidomain(zero)
% 6.77/1.40 = { by lemma 29 R->L }
% 6.77/1.40 antidomain(antidomain(one))
% 6.77/1.40 = { by axiom 3 (domain4) R->L }
% 6.77/1.40 domain(one)
% 6.77/1.40 = { by lemma 30 }
% 6.77/1.40 one
% 6.77/1.40
% 6.77/1.40 Lemma 32: domain(zero) = zero.
% 6.77/1.40 Proof:
% 6.77/1.40 domain(zero)
% 6.77/1.40 = { by axiom 3 (domain4) }
% 6.77/1.40 antidomain(antidomain(zero))
% 6.77/1.40 = { by lemma 31 }
% 6.77/1.40 antidomain(one)
% 6.77/1.40 = { by lemma 29 }
% 6.77/1.40 zero
% 6.77/1.40
% 6.77/1.40 Lemma 33: coantidomain(one) = zero.
% 6.77/1.40 Proof:
% 6.77/1.40 coantidomain(one)
% 6.77/1.40 = { by axiom 6 (multiplicative_left_identity) R->L }
% 6.77/1.40 multiplication(one, coantidomain(one))
% 6.77/1.40 = { by axiom 12 (codomain1) }
% 6.77/1.40 zero
% 6.77/1.40
% 6.77/1.40 Lemma 34: antidomain(c(X)) = domain(domain(X)).
% 6.77/1.40 Proof:
% 6.77/1.40 antidomain(c(X))
% 6.77/1.40 = { by axiom 1 (complement) }
% 6.77/1.40 antidomain(antidomain(domain(X)))
% 6.77/1.40 = { by axiom 3 (domain4) R->L }
% 6.77/1.40 domain(domain(X))
% 6.77/1.40
% 6.77/1.40 Lemma 35: multiplication(domain(X), c(Y)) = domain_difference(X, domain(Y)).
% 6.77/1.40 Proof:
% 6.77/1.40 multiplication(domain(X), c(Y))
% 6.77/1.40 = { by axiom 1 (complement) }
% 6.77/1.40 multiplication(domain(X), antidomain(domain(Y)))
% 6.77/1.40 = { by axiom 13 (domain_difference) R->L }
% 6.77/1.40 domain_difference(X, domain(Y))
% 6.77/1.40
% 6.77/1.40 Lemma 36: c(domain(X)) = c(X).
% 6.77/1.40 Proof:
% 6.77/1.40 c(domain(X))
% 6.77/1.40 = { by axiom 1 (complement) }
% 7.44/1.40 antidomain(domain(domain(X)))
% 7.44/1.40 = { by axiom 4 (multiplicative_right_identity) R->L }
% 7.44/1.40 multiplication(antidomain(domain(domain(X))), one)
% 7.44/1.40 = { by lemma 24 R->L }
% 7.44/1.40 multiplication(antidomain(domain(domain(X))), addition(domain(domain(X)), antidomain(domain(X))))
% 7.44/1.40 = { by axiom 1 (complement) R->L }
% 7.44/1.40 multiplication(antidomain(domain(domain(X))), addition(domain(domain(X)), c(X)))
% 7.44/1.40 = { by lemma 27 }
% 7.44/1.40 multiplication(antidomain(domain(domain(X))), c(X))
% 7.44/1.40 = { by axiom 1 (complement) R->L }
% 7.44/1.40 multiplication(c(domain(X)), c(X))
% 7.44/1.40 = { by lemma 26 R->L }
% 7.44/1.40 multiplication(domain(antidomain(domain(X))), c(X))
% 7.44/1.40 = { by lemma 35 }
% 7.44/1.40 domain_difference(antidomain(domain(X)), domain(X))
% 7.44/1.40 = { by axiom 1 (complement) R->L }
% 7.44/1.40 domain_difference(c(X), domain(X))
% 7.44/1.40 = { by lemma 35 R->L }
% 7.44/1.40 multiplication(domain(c(X)), c(X))
% 7.44/1.40 = { by lemma 25 }
% 7.44/1.40 c(X)
% 7.44/1.40
% 7.44/1.40 Lemma 37: c(c(X)) = domain(domain(domain(X))).
% 7.44/1.40 Proof:
% 7.44/1.40 c(c(X))
% 7.44/1.40 = { by lemma 26 R->L }
% 7.44/1.40 domain(antidomain(c(X)))
% 7.44/1.40 = { by lemma 34 }
% 7.44/1.40 domain(domain(domain(X)))
% 7.44/1.40
% 7.44/1.40 Lemma 38: domain(domain(domain(X))) = domain(X).
% 7.44/1.40 Proof:
% 7.44/1.40 domain(domain(domain(X)))
% 7.44/1.40 = { by lemma 37 R->L }
% 7.44/1.40 c(c(X))
% 7.44/1.40 = { by lemma 28 R->L }
% 7.44/1.40 c(antidomain(X))
% 7.44/1.40 = { by lemma 28 R->L }
% 7.44/1.40 antidomain(antidomain(X))
% 7.44/1.40 = { by axiom 3 (domain4) R->L }
% 7.44/1.40 domain(X)
% 7.44/1.40
% 7.44/1.40 Lemma 39: domain(domain(X)) = domain(X).
% 7.44/1.40 Proof:
% 7.44/1.40 domain(domain(X))
% 7.44/1.40 = { by lemma 34 R->L }
% 7.44/1.40 antidomain(c(X))
% 7.44/1.40 = { by lemma 36 R->L }
% 7.44/1.40 antidomain(c(domain(X)))
% 7.44/1.40 = { by lemma 34 }
% 7.44/1.40 domain(domain(domain(X)))
% 7.44/1.40 = { by lemma 38 }
% 7.44/1.40 domain(X)
% 7.44/1.40
% 7.44/1.40 Lemma 40: forward_diamond(X, zero) = zero.
% 7.44/1.40 Proof:
% 7.44/1.40 forward_diamond(X, zero)
% 7.44/1.40 = { by axiom 11 (forward_diamond) }
% 7.44/1.40 domain(multiplication(X, domain(zero)))
% 7.44/1.40 = { by lemma 32 }
% 7.44/1.40 domain(multiplication(X, zero))
% 7.44/1.40 = { by axiom 5 (right_annihilation) }
% 7.44/1.40 domain(zero)
% 7.44/1.40 = { by lemma 32 }
% 7.44/1.40 zero
% 7.44/1.40
% 7.44/1.40 Lemma 41: addition(X, addition(X, Y)) = addition(X, Y).
% 7.44/1.40 Proof:
% 7.44/1.40 addition(X, addition(X, Y))
% 7.44/1.40 = { by axiom 18 (additive_associativity) }
% 7.44/1.40 addition(addition(X, X), Y)
% 7.44/1.40 = { by axiom 8 (additive_idempotence) }
% 7.44/1.40 addition(X, Y)
% 7.44/1.40
% 7.44/1.40 Lemma 42: multiplication(domain(X), c(Y)) = domain_difference(X, Y).
% 7.44/1.40 Proof:
% 7.44/1.40 multiplication(domain(X), c(Y))
% 7.44/1.40 = { by lemma 28 R->L }
% 7.44/1.40 multiplication(domain(X), antidomain(Y))
% 7.44/1.40 = { by axiom 13 (domain_difference) R->L }
% 7.44/1.40 domain_difference(X, Y)
% 7.44/1.40
% 7.44/1.40 Lemma 43: c(multiplication(X, c(Y))) = forward_box(X, Y).
% 7.44/1.40 Proof:
% 7.44/1.40 c(multiplication(X, c(Y)))
% 7.44/1.40 = { by lemma 36 R->L }
% 7.44/1.40 c(domain(multiplication(X, c(Y))))
% 7.44/1.40 = { by lemma 26 R->L }
% 7.44/1.40 c(domain(multiplication(X, domain(antidomain(Y)))))
% 7.44/1.40 = { by axiom 11 (forward_diamond) R->L }
% 7.44/1.40 c(forward_diamond(X, antidomain(Y)))
% 7.44/1.40 = { by lemma 28 }
% 7.44/1.40 c(forward_diamond(X, c(Y)))
% 7.44/1.40 = { by axiom 15 (forward_box) R->L }
% 7.44/1.40 forward_box(X, Y)
% 7.44/1.40
% 7.44/1.40 Lemma 44: domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))) = zero.
% 7.44/1.40 Proof:
% 7.44/1.40 domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2))))
% 7.44/1.40 = { by lemma 25 R->L }
% 7.44/1.40 multiplication(domain(domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 39 R->L }
% 7.44/1.41 multiplication(domain(domain(domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 34 R->L }
% 7.44/1.41 multiplication(antidomain(c(domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 42 R->L }
% 7.44/1.41 multiplication(antidomain(c(multiplication(domain(domain(x2)), c(addition(domain(x2), forward_diamond(x0, domain(x2))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 35 }
% 7.44/1.41 multiplication(antidomain(c(domain_difference(domain(x2), domain(addition(domain(x2), forward_diamond(x0, domain(x2))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by axiom 13 (domain_difference) }
% 7.44/1.41 multiplication(antidomain(c(multiplication(domain(domain(x2)), antidomain(domain(addition(domain(x2), forward_diamond(x0, domain(x2)))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 39 }
% 7.44/1.41 multiplication(antidomain(c(multiplication(domain(x2), antidomain(domain(addition(domain(x2), forward_diamond(x0, domain(x2)))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 28 }
% 7.44/1.41 multiplication(antidomain(c(multiplication(domain(x2), c(domain(addition(domain(x2), forward_diamond(x0, domain(x2)))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 42 }
% 7.44/1.41 multiplication(antidomain(c(domain_difference(x2, domain(addition(domain(x2), forward_diamond(x0, domain(x2))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 35 R->L }
% 7.44/1.41 multiplication(antidomain(c(multiplication(domain(x2), c(addition(domain(x2), forward_diamond(x0, domain(x2))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 43 }
% 7.44/1.41 multiplication(antidomain(forward_box(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 28 }
% 7.44/1.41 multiplication(c(forward_box(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 39 R->L }
% 7.44/1.41 multiplication(c(forward_box(domain(domain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 38 R->L }
% 7.44/1.41 multiplication(c(forward_box(domain(domain(domain(domain(x2)))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 37 R->L }
% 7.44/1.41 multiplication(c(forward_box(c(c(domain(x2))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by axiom 4 (multiplicative_right_identity) R->L }
% 7.44/1.41 multiplication(c(forward_box(multiplication(c(c(domain(x2))), one), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 24 R->L }
% 7.44/1.41 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(domain(multiplication(x0, domain(domain(x2)))), antidomain(multiplication(x0, domain(domain(x2)))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 41 R->L }
% 7.44/1.41 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(domain(multiplication(x0, domain(domain(x2)))), addition(domain(multiplication(x0, domain(domain(x2)))), antidomain(multiplication(x0, domain(domain(x2))))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by lemma 24 }
% 7.44/1.41 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(domain(multiplication(x0, domain(domain(x2)))), one)), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by axiom 9 (additive_commutativity) }
% 7.44/1.41 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(one, domain(multiplication(x0, domain(domain(x2)))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.41 = { by axiom 11 (forward_diamond) R->L }
% 7.44/1.41 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(one, forward_diamond(x0, domain(x2)))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 17 (goals) R->L }
% 7.44/1.42 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(one, addition(domain(x2), forward_diamond(x0, domain(x2))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 24 R->L }
% 7.44/1.42 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(addition(domain(x2), antidomain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 18 (additive_associativity) R->L }
% 7.44/1.42 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(domain(x2), addition(antidomain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 28 }
% 7.44/1.42 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(domain(x2), addition(c(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 36 R->L }
% 7.44/1.42 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(domain(x2), addition(c(domain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2)))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 9 (additive_commutativity) R->L }
% 7.44/1.42 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(domain(x2), addition(addition(domain(x2), forward_diamond(x0, domain(x2))), c(domain(x2))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 17 (goals) }
% 7.44/1.42 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(domain(x2), addition(forward_diamond(x0, domain(x2)), c(domain(x2))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 18 (additive_associativity) }
% 7.44/1.42 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(addition(domain(x2), forward_diamond(x0, domain(x2))), c(domain(x2)))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 28 R->L }
% 7.44/1.42 multiplication(c(forward_box(multiplication(antidomain(c(domain(x2))), addition(addition(domain(x2), forward_diamond(x0, domain(x2))), c(domain(x2)))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 9 (additive_commutativity) R->L }
% 7.44/1.42 multiplication(c(forward_box(multiplication(antidomain(c(domain(x2))), addition(c(domain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 27 }
% 7.44/1.42 multiplication(c(forward_box(multiplication(antidomain(c(domain(x2))), addition(domain(x2), forward_diamond(x0, domain(x2)))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 28 }
% 7.44/1.42 multiplication(c(forward_box(multiplication(c(c(domain(x2))), addition(domain(x2), forward_diamond(x0, domain(x2)))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 37 }
% 7.44/1.42 multiplication(c(forward_box(multiplication(domain(domain(domain(domain(x2)))), addition(domain(x2), forward_diamond(x0, domain(x2)))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 38 }
% 7.44/1.42 multiplication(c(forward_box(multiplication(domain(domain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2)))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 39 }
% 7.44/1.42 multiplication(c(forward_box(multiplication(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 17 (goals) }
% 7.44/1.42 multiplication(c(forward_box(multiplication(domain(x2), forward_diamond(x0, domain(x2))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 11 (forward_diamond) }
% 7.44/1.42 multiplication(c(forward_box(multiplication(domain(x2), domain(multiplication(x0, domain(domain(x2))))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 39 }
% 7.44/1.42 multiplication(c(forward_box(multiplication(domain(x2), domain(multiplication(x0, domain(x2)))), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 43 R->L }
% 7.44/1.42 multiplication(c(c(multiplication(multiplication(domain(x2), domain(multiplication(x0, domain(x2)))), c(addition(domain(x2), forward_diamond(x0, domain(x2))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 28 R->L }
% 7.44/1.42 multiplication(c(c(multiplication(multiplication(domain(x2), domain(multiplication(x0, domain(x2)))), antidomain(addition(domain(x2), forward_diamond(x0, domain(x2))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 17 (goals) }
% 7.44/1.42 multiplication(c(c(multiplication(multiplication(domain(x2), domain(multiplication(x0, domain(x2)))), antidomain(forward_diamond(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 11 (forward_diamond) }
% 7.44/1.42 multiplication(c(c(multiplication(multiplication(domain(x2), domain(multiplication(x0, domain(x2)))), antidomain(domain(multiplication(x0, domain(domain(x2)))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 1 (complement) R->L }
% 7.44/1.42 multiplication(c(c(multiplication(multiplication(domain(x2), domain(multiplication(x0, domain(x2)))), c(multiplication(x0, domain(domain(x2))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 43 }
% 7.44/1.42 multiplication(c(forward_box(multiplication(domain(x2), domain(multiplication(x0, domain(x2)))), multiplication(x0, domain(domain(x2))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 39 }
% 7.44/1.42 multiplication(c(forward_box(multiplication(domain(x2), domain(multiplication(x0, domain(x2)))), multiplication(x0, domain(x2)))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 39 R->L }
% 7.44/1.42 multiplication(c(forward_box(multiplication(domain(domain(x2)), domain(multiplication(x0, domain(x2)))), multiplication(x0, domain(x2)))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 3 (domain4) }
% 7.44/1.42 multiplication(c(forward_box(multiplication(domain(domain(x2)), antidomain(antidomain(multiplication(x0, domain(x2))))), multiplication(x0, domain(x2)))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 13 (domain_difference) R->L }
% 7.44/1.42 multiplication(c(forward_box(domain_difference(domain(x2), antidomain(multiplication(x0, domain(x2)))), multiplication(x0, domain(x2)))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 28 }
% 7.44/1.42 multiplication(c(forward_box(domain_difference(domain(x2), c(multiplication(x0, domain(x2)))), multiplication(x0, domain(x2)))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by lemma 43 R->L }
% 7.44/1.42 multiplication(c(c(multiplication(domain_difference(domain(x2), c(multiplication(x0, domain(x2)))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 10 (additive_identity) R->L }
% 7.44/1.42 multiplication(c(c(addition(multiplication(domain_difference(domain(x2), c(multiplication(x0, domain(x2)))), c(multiplication(x0, domain(x2)))), zero))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 9 (additive_commutativity) }
% 7.44/1.42 multiplication(c(c(addition(zero, multiplication(domain_difference(domain(x2), c(multiplication(x0, domain(x2)))), c(multiplication(x0, domain(x2))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.42 = { by axiom 14 (domain1) R->L }
% 7.44/1.42 multiplication(c(c(addition(multiplication(antidomain(c(multiplication(x0, domain(x2)))), c(multiplication(x0, domain(x2)))), multiplication(domain_difference(domain(x2), c(multiplication(x0, domain(x2)))), c(multiplication(x0, domain(x2))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 22 (left_distributivity) R->L }
% 7.44/1.43 multiplication(c(c(multiplication(addition(antidomain(c(multiplication(x0, domain(x2)))), domain_difference(domain(x2), c(multiplication(x0, domain(x2))))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 13 (domain_difference) }
% 7.44/1.43 multiplication(c(c(multiplication(addition(antidomain(c(multiplication(x0, domain(x2)))), multiplication(domain(domain(x2)), antidomain(c(multiplication(x0, domain(x2)))))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 6 (multiplicative_left_identity) R->L }
% 7.44/1.43 multiplication(c(c(multiplication(addition(multiplication(one, antidomain(c(multiplication(x0, domain(x2))))), multiplication(domain(domain(x2)), antidomain(c(multiplication(x0, domain(x2)))))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 22 (left_distributivity) R->L }
% 7.44/1.43 multiplication(c(c(multiplication(multiplication(addition(one, domain(domain(x2))), antidomain(c(multiplication(x0, domain(x2))))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 9 (additive_commutativity) }
% 7.44/1.43 multiplication(c(c(multiplication(multiplication(addition(domain(domain(x2)), one), antidomain(c(multiplication(x0, domain(x2))))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by lemma 24 R->L }
% 7.44/1.43 multiplication(c(c(multiplication(multiplication(addition(domain(domain(x2)), addition(domain(domain(x2)), antidomain(domain(x2)))), antidomain(c(multiplication(x0, domain(x2))))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by lemma 41 }
% 7.44/1.43 multiplication(c(c(multiplication(multiplication(addition(domain(domain(x2)), antidomain(domain(x2))), antidomain(c(multiplication(x0, domain(x2))))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by lemma 24 }
% 7.44/1.43 multiplication(c(c(multiplication(multiplication(one, antidomain(c(multiplication(x0, domain(x2))))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 6 (multiplicative_left_identity) }
% 7.44/1.43 multiplication(c(c(multiplication(antidomain(c(multiplication(x0, domain(x2)))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by lemma 28 }
% 7.44/1.43 multiplication(c(c(multiplication(c(c(multiplication(x0, domain(x2)))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 1 (complement) }
% 7.44/1.43 multiplication(c(c(multiplication(antidomain(domain(c(multiplication(x0, domain(x2))))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by lemma 25 R->L }
% 7.44/1.43 multiplication(c(c(multiplication(antidomain(domain(c(multiplication(x0, domain(x2))))), multiplication(domain(c(multiplication(x0, domain(x2)))), c(multiplication(x0, domain(x2))))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 16 (multiplicative_associativity) }
% 7.44/1.43 multiplication(c(c(multiplication(multiplication(antidomain(domain(c(multiplication(x0, domain(x2))))), domain(c(multiplication(x0, domain(x2))))), c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 14 (domain1) }
% 7.44/1.43 multiplication(c(c(multiplication(zero, c(multiplication(x0, domain(x2)))))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 7 (left_annihilation) }
% 7.44/1.43 multiplication(c(c(zero)), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by lemma 26 R->L }
% 7.44/1.43 multiplication(c(domain(antidomain(zero))), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by lemma 31 }
% 7.44/1.43 multiplication(c(domain(one)), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by lemma 30 }
% 7.44/1.43 multiplication(c(one), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 1 (complement) }
% 7.44/1.43 multiplication(antidomain(domain(one)), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by lemma 30 }
% 7.44/1.43 multiplication(antidomain(one), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by lemma 29 }
% 7.44/1.43 multiplication(zero, domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 7 (left_annihilation) }
% 7.44/1.43 zero
% 7.44/1.43
% 7.44/1.43 Goal 1 (goals_2): domain(x2) = zero.
% 7.44/1.43 Proof:
% 7.44/1.43 domain(x2)
% 7.44/1.43 = { by axiom 4 (multiplicative_right_identity) R->L }
% 7.44/1.43 multiplication(domain(x2), one)
% 7.44/1.43 = { by axiom 19 (codomain3) R->L }
% 7.44/1.43 multiplication(domain(x2), addition(coantidomain(coantidomain(one)), coantidomain(one)))
% 7.44/1.43 = { by axiom 2 (codomain4) R->L }
% 7.44/1.43 multiplication(domain(x2), addition(codomain(one), coantidomain(one)))
% 7.44/1.43 = { by lemma 33 }
% 7.44/1.43 multiplication(domain(x2), addition(codomain(one), zero))
% 7.44/1.43 = { by axiom 10 (additive_identity) }
% 7.44/1.43 multiplication(domain(x2), codomain(one))
% 7.44/1.43 = { by axiom 2 (codomain4) }
% 7.44/1.43 multiplication(domain(x2), coantidomain(coantidomain(one)))
% 7.44/1.43 = { by lemma 33 }
% 7.44/1.43 multiplication(domain(x2), coantidomain(zero))
% 7.44/1.43 = { by lemma 40 R->L }
% 7.44/1.43 multiplication(domain(x2), coantidomain(forward_diamond(star(x0), zero)))
% 7.44/1.43 = { by lemma 44 R->L }
% 7.44/1.43 multiplication(domain(x2), coantidomain(forward_diamond(star(x0), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))))
% 7.44/1.43 = { by axiom 17 (goals) }
% 7.44/1.43 multiplication(domain(x2), coantidomain(forward_diamond(star(x0), domain_difference(domain(x2), forward_diamond(x0, domain(x2))))))
% 7.44/1.43 = { by axiom 23 (goals_1) R->L }
% 7.44/1.43 multiplication(domain(x2), coantidomain(addition(forward_diamond(x0, domain(x2)), forward_diamond(star(x0), domain_difference(domain(x2), forward_diamond(x0, domain(x2)))))))
% 7.44/1.43 = { by axiom 17 (goals) R->L }
% 7.44/1.43 multiplication(domain(x2), coantidomain(addition(addition(domain(x2), forward_diamond(x0, domain(x2))), forward_diamond(star(x0), domain_difference(domain(x2), forward_diamond(x0, domain(x2)))))))
% 7.44/1.43 = { by axiom 17 (goals) R->L }
% 7.44/1.43 multiplication(domain(x2), coantidomain(addition(addition(domain(x2), forward_diamond(x0, domain(x2))), forward_diamond(star(x0), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2))))))))
% 7.44/1.43 = { by lemma 44 }
% 7.44/1.43 multiplication(domain(x2), coantidomain(addition(addition(domain(x2), forward_diamond(x0, domain(x2))), forward_diamond(star(x0), zero))))
% 7.44/1.43 = { by lemma 40 }
% 7.44/1.43 multiplication(domain(x2), coantidomain(addition(addition(domain(x2), forward_diamond(x0, domain(x2))), zero)))
% 7.44/1.43 = { by axiom 10 (additive_identity) }
% 7.44/1.43 multiplication(domain(x2), coantidomain(addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 17 (goals) }
% 7.44/1.43 multiplication(domain(x2), coantidomain(forward_diamond(x0, domain(x2))))
% 7.44/1.43 = { by axiom 10 (additive_identity) R->L }
% 7.44/1.43 addition(multiplication(domain(x2), coantidomain(forward_diamond(x0, domain(x2)))), zero)
% 7.44/1.43 = { by axiom 12 (codomain1) R->L }
% 7.44/1.43 addition(multiplication(domain(x2), coantidomain(forward_diamond(x0, domain(x2)))), multiplication(forward_diamond(x0, domain(x2)), coantidomain(forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 22 (left_distributivity) R->L }
% 7.44/1.43 multiplication(addition(domain(x2), forward_diamond(x0, domain(x2))), coantidomain(forward_diamond(x0, domain(x2))))
% 7.44/1.43 = { by axiom 17 (goals) R->L }
% 7.44/1.43 multiplication(addition(domain(x2), forward_diamond(x0, domain(x2))), coantidomain(addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 7.44/1.43 = { by axiom 12 (codomain1) }
% 7.44/1.43 zero
% 7.44/1.43 % SZS output end Proof
% 7.44/1.43
% 7.44/1.43 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------