%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : KLE097+1 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/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 11:39:49 AM UTC 2026
% Result : Theorem 31.34s 4.40s
% Output : Proof 31.34s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : KLE097+1 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.08/0.37 % Computer : n004.cluster.edu
% 0.08/0.37 % Model : x86_64 x86_64
% 0.08/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.37 % Memory : 8046.5625MB
% 0.08/0.37 % OS : Linux 6.8.0-71-generic
% 0.08/0.37 % CPULimit : 300
% 0.08/0.37 % WCLimit : 300
% 0.08/0.37 % DateTime : Sun Sep 27 13:09:52 UTC 2026
% 0.08/0.37 % CPUTime :
% 0.08/0.37 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 31.34/4.40 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-regeneralise
% 31.34/4.40
% 31.34/4.40 % SZS status Theorem
% 31.34/4.40
% 31.34/4.46 % SZS output start Proof
% 31.34/4.46 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 31.34/4.46 Axiom 2 (left_annihilation): multiplication(zero, X) = zero.
% 31.34/4.46 Axiom 3 (multiplicative_left_identity): multiplication(one, X) = X.
% 31.34/4.46 Axiom 4 (additive_idempotence): addition(X, X) = X.
% 31.34/4.46 Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 31.34/4.46 Axiom 6 (additive_identity): addition(X, zero) = X.
% 31.34/4.46 Axiom 7 (domain4): domain(X) = antidomain(antidomain(X)).
% 31.34/4.46 Axiom 8 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 31.34/4.46 Axiom 9 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 31.34/4.46 Axiom 10 (domain1): multiplication(antidomain(X), X) = zero.
% 31.34/4.46 Axiom 11 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 31.34/4.46 Axiom 12 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 31.34/4.46 Axiom 13 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 31.34/4.46 Axiom 14 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 31.34/4.46 Axiom 15 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 31.34/4.46 Axiom 16 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 31.34/4.46 Axiom 17 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 31.34/4.46 Axiom 18 (goals): addition(multiplication(x0, domain(x1)), multiplication(domain(x2), x0)) = multiplication(domain(x2), x0).
% 31.34/4.46 Axiom 19 (order_1): ifeq(leq(X, Y), true, addition(X, Y), Y) = Y.
% 31.34/4.46 Axiom 20 (order): ifeq2(addition(X, Y), Y, leq(X, Y), true) = true.
% 31.34/4.46 Axiom 21 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 31.34/4.46
% 31.34/4.46 Lemma 22: antidomain(one) = zero.
% 31.34/4.46 Proof:
% 31.34/4.46 antidomain(one)
% 31.34/4.46 = { by axiom 1 (multiplicative_right_identity) R->L }
% 31.34/4.46 multiplication(antidomain(one), one)
% 31.34/4.46 = { by axiom 10 (domain1) }
% 31.34/4.46 zero
% 31.34/4.46
% 31.34/4.46 Lemma 23: addition(zero, X) = X.
% 31.34/4.46 Proof:
% 31.34/4.46 addition(zero, X)
% 31.34/4.46 = { by axiom 5 (additive_commutativity) R->L }
% 31.34/4.46 addition(X, zero)
% 31.34/4.46 = { by axiom 6 (additive_identity) }
% 31.34/4.46 X
% 31.34/4.46
% 31.34/4.46 Lemma 24: domain(antidomain(X)) = antidomain(domain(X)).
% 31.34/4.46 Proof:
% 31.34/4.46 domain(antidomain(X))
% 31.34/4.46 = { by axiom 7 (domain4) }
% 31.34/4.46 antidomain(antidomain(antidomain(X)))
% 31.34/4.46 = { by axiom 7 (domain4) R->L }
% 31.34/4.46 antidomain(domain(X))
% 31.34/4.46
% 31.34/4.46 Lemma 25: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 31.34/4.46 Proof:
% 31.34/4.46 multiplication(antidomain(X), addition(X, Y))
% 31.34/4.46 = { by axiom 5 (additive_commutativity) R->L }
% 31.34/4.46 multiplication(antidomain(X), addition(Y, X))
% 31.34/4.46 = { by axiom 16 (right_distributivity) }
% 31.34/4.46 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 31.34/4.46 = { by axiom 10 (domain1) }
% 31.34/4.46 addition(multiplication(antidomain(X), Y), zero)
% 31.34/4.46 = { by axiom 6 (additive_identity) }
% 31.34/4.46 multiplication(antidomain(X), Y)
% 31.34/4.46
% 31.34/4.46 Lemma 26: addition(domain(X), antidomain(X)) = one.
% 31.34/4.46 Proof:
% 31.34/4.46 addition(domain(X), antidomain(X))
% 31.34/4.46 = { by axiom 7 (domain4) }
% 31.34/4.46 addition(antidomain(antidomain(X)), antidomain(X))
% 31.34/4.46 = { by axiom 15 (domain3) }
% 31.34/4.46 one
% 31.34/4.46
% 31.34/4.46 Lemma 27: multiplication(antidomain(domain(X)), antidomain(X)) = antidomain(domain(X)).
% 31.34/4.46 Proof:
% 31.34/4.46 multiplication(antidomain(domain(X)), antidomain(X))
% 31.34/4.46 = { by lemma 25 R->L }
% 31.34/4.46 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 31.34/4.46 = { by lemma 26 }
% 31.34/4.46 multiplication(antidomain(domain(X)), one)
% 31.34/4.46 = { by axiom 1 (multiplicative_right_identity) }
% 31.34/4.46 antidomain(domain(X))
% 31.34/4.46
% 31.34/4.46 Lemma 28: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
% 31.34/4.46 Proof:
% 31.34/4.46 multiplication(addition(X, antidomain(Y)), Y)
% 31.34/4.46 = { by axiom 17 (left_distributivity) }
% 31.34/4.46 addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
% 31.34/4.46 = { by axiom 10 (domain1) }
% 31.34/4.46 addition(multiplication(X, Y), zero)
% 31.34/4.46 = { by axiom 6 (additive_identity) }
% 31.34/4.46 multiplication(X, Y)
% 31.34/4.46
% 31.34/4.46 Lemma 29: multiplication(domain(X), X) = X.
% 31.34/4.46 Proof:
% 31.34/4.46 multiplication(domain(X), X)
% 31.34/4.46 = { by lemma 28 R->L }
% 31.34/4.46 multiplication(addition(domain(X), antidomain(X)), X)
% 31.34/4.46 = { by lemma 26 }
% 31.34/4.46 multiplication(one, X)
% 31.34/4.46 = { by axiom 3 (multiplicative_left_identity) }
% 31.34/4.46 X
% 31.34/4.46
% 31.34/4.46 Lemma 30: domain(domain(X)) = domain(X).
% 31.34/4.46 Proof:
% 31.34/4.46 domain(domain(X))
% 31.34/4.46 = { by axiom 7 (domain4) }
% 31.34/4.46 antidomain(antidomain(domain(X)))
% 31.34/4.46 = { by lemma 24 R->L }
% 31.34/4.46 antidomain(domain(antidomain(X)))
% 31.34/4.46 = { by lemma 27 R->L }
% 31.34/4.46 multiplication(antidomain(domain(antidomain(X))), antidomain(antidomain(X)))
% 31.34/4.46 = { by axiom 7 (domain4) R->L }
% 31.34/4.46 multiplication(antidomain(domain(antidomain(X))), domain(X))
% 31.34/4.46 = { by lemma 24 }
% 31.34/4.46 multiplication(antidomain(antidomain(domain(X))), domain(X))
% 31.34/4.46 = { by axiom 7 (domain4) R->L }
% 31.34/4.46 multiplication(domain(domain(X)), domain(X))
% 31.34/4.46 = { by lemma 29 }
% 31.34/4.46 domain(X)
% 31.34/4.46
% 31.34/4.46 Lemma 31: domain(forward_diamond(X, Y)) = forward_diamond(X, Y).
% 31.34/4.46 Proof:
% 31.34/4.46 domain(forward_diamond(X, Y))
% 31.34/4.46 = { by axiom 7 (domain4) }
% 31.34/4.46 antidomain(antidomain(forward_diamond(X, Y)))
% 31.34/4.46 = { by axiom 1 (multiplicative_right_identity) R->L }
% 31.34/4.46 multiplication(antidomain(antidomain(forward_diamond(X, Y))), one)
% 31.34/4.46 = { by lemma 26 R->L }
% 31.34/4.46 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(domain(antidomain(multiplication(X, domain(Y)))), antidomain(antidomain(multiplication(X, domain(Y))))))
% 31.34/4.46 = { by axiom 7 (domain4) R->L }
% 31.34/4.47 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(domain(antidomain(multiplication(X, domain(Y)))), domain(multiplication(X, domain(Y)))))
% 31.34/4.47 = { by lemma 24 }
% 31.34/4.47 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(antidomain(domain(multiplication(X, domain(Y)))), domain(multiplication(X, domain(Y)))))
% 31.34/4.47 = { by axiom 5 (additive_commutativity) }
% 31.34/4.47 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(domain(multiplication(X, domain(Y))), antidomain(domain(multiplication(X, domain(Y))))))
% 31.34/4.47 = { by axiom 8 (forward_diamond) R->L }
% 31.34/4.47 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(forward_diamond(X, Y), antidomain(domain(multiplication(X, domain(Y))))))
% 31.34/4.47 = { by axiom 8 (forward_diamond) R->L }
% 31.34/4.47 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))))
% 31.34/4.47 = { by axiom 5 (additive_commutativity) }
% 31.34/4.47 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(antidomain(forward_diamond(X, Y)), forward_diamond(X, Y)))
% 31.34/4.47 = { by lemma 25 }
% 31.34/4.47 multiplication(antidomain(antidomain(forward_diamond(X, Y))), forward_diamond(X, Y))
% 31.34/4.47 = { by axiom 7 (domain4) R->L }
% 31.34/4.47 multiplication(domain(forward_diamond(X, Y)), forward_diamond(X, Y))
% 31.34/4.47 = { by lemma 29 }
% 31.34/4.47 forward_diamond(X, Y)
% 31.34/4.47
% 31.34/4.47 Lemma 32: multiplication(antidomain(X), multiplication(X, Y)) = zero.
% 31.34/4.47 Proof:
% 31.34/4.47 multiplication(antidomain(X), multiplication(X, Y))
% 31.34/4.47 = { by axiom 11 (multiplicative_associativity) }
% 31.34/4.47 multiplication(multiplication(antidomain(X), X), Y)
% 31.34/4.47 = { by axiom 10 (domain1) }
% 31.34/4.47 multiplication(zero, Y)
% 31.34/4.47 = { by axiom 2 (left_annihilation) }
% 31.34/4.47 zero
% 31.34/4.47
% 31.34/4.47 Goal 1 (goals_1): addition(forward_diamond(x0, domain(x1)), domain(x2)) = domain(x2).
% 31.34/4.47 Proof:
% 31.34/4.47 addition(forward_diamond(x0, domain(x1)), domain(x2))
% 31.34/4.47 = { by axiom 5 (additive_commutativity) }
% 31.34/4.47 addition(domain(x2), forward_diamond(x0, domain(x1)))
% 31.34/4.47 = { by axiom 13 (ifeq_axiom) R->L }
% 31.34/4.47 ifeq(true, true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 20 (order) R->L }
% 31.34/4.47 ifeq(ifeq2(addition(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 5 (additive_commutativity) R->L }
% 31.34/4.47 ifeq(ifeq2(addition(domain(x2), domain_difference(x2, antidomain(forward_diamond(x0, domain(x1))))), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 9 (domain_difference) }
% 31.34/4.47 ifeq(ifeq2(addition(domain(x2), multiplication(domain(x2), antidomain(antidomain(forward_diamond(x0, domain(x1)))))), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 1 (multiplicative_right_identity) R->L }
% 31.34/4.47 ifeq(ifeq2(addition(multiplication(domain(x2), one), multiplication(domain(x2), antidomain(antidomain(forward_diamond(x0, domain(x1)))))), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 16 (right_distributivity) R->L }
% 31.34/4.47 ifeq(ifeq2(multiplication(domain(x2), addition(one, antidomain(antidomain(forward_diamond(x0, domain(x1)))))), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 5 (additive_commutativity) R->L }
% 31.34/4.47 ifeq(ifeq2(multiplication(domain(x2), addition(antidomain(antidomain(forward_diamond(x0, domain(x1)))), one)), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by lemma 26 R->L }
% 31.34/4.47 ifeq(ifeq2(multiplication(domain(x2), addition(antidomain(antidomain(forward_diamond(x0, domain(x1)))), addition(domain(antidomain(forward_diamond(x0, domain(x1)))), antidomain(antidomain(forward_diamond(x0, domain(x1))))))), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 5 (additive_commutativity) }
% 31.34/4.47 ifeq(ifeq2(multiplication(domain(x2), addition(antidomain(antidomain(forward_diamond(x0, domain(x1)))), addition(antidomain(antidomain(forward_diamond(x0, domain(x1)))), domain(antidomain(forward_diamond(x0, domain(x1))))))), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 12 (additive_associativity) }
% 31.34/4.47 ifeq(ifeq2(multiplication(domain(x2), addition(addition(antidomain(antidomain(forward_diamond(x0, domain(x1)))), antidomain(antidomain(forward_diamond(x0, domain(x1))))), domain(antidomain(forward_diamond(x0, domain(x1)))))), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 4 (additive_idempotence) }
% 31.34/4.47 ifeq(ifeq2(multiplication(domain(x2), addition(antidomain(antidomain(forward_diamond(x0, domain(x1)))), domain(antidomain(forward_diamond(x0, domain(x1)))))), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 5 (additive_commutativity) R->L }
% 31.34/4.47 ifeq(ifeq2(multiplication(domain(x2), addition(domain(antidomain(forward_diamond(x0, domain(x1)))), antidomain(antidomain(forward_diamond(x0, domain(x1)))))), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by lemma 26 }
% 31.34/4.47 ifeq(ifeq2(multiplication(domain(x2), one), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 1 (multiplicative_right_identity) }
% 31.34/4.47 ifeq(ifeq2(domain(x2), domain(x2), leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 14 (ifeq_axiom) }
% 31.34/4.47 ifeq(leq(domain_difference(x2, antidomain(forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 9 (domain_difference) }
% 31.34/4.47 ifeq(leq(multiplication(domain(x2), antidomain(antidomain(forward_diamond(x0, domain(x1))))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 7 (domain4) R->L }
% 31.34/4.47 ifeq(leq(multiplication(domain(x2), domain(forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by lemma 31 }
% 31.34/4.47 ifeq(leq(multiplication(domain(x2), forward_diamond(x0, domain(x1))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by lemma 23 R->L }
% 31.34/4.47 ifeq(leq(addition(zero, multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 10 (domain1) R->L }
% 31.34/4.47 ifeq(leq(addition(multiplication(antidomain(multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 7 (domain4) }
% 31.34/4.47 ifeq(leq(addition(multiplication(antidomain(multiplication(antidomain(x2), antidomain(antidomain(multiplication(x0, domain(x1)))))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 21 (domain2) R->L }
% 31.34/4.47 ifeq(leq(addition(multiplication(addition(antidomain(multiplication(antidomain(x2), multiplication(x0, domain(x1)))), antidomain(multiplication(antidomain(x2), antidomain(antidomain(multiplication(x0, domain(x1))))))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by axiom 7 (domain4) R->L }
% 31.34/4.47 ifeq(leq(addition(multiplication(addition(antidomain(multiplication(antidomain(x2), multiplication(x0, domain(x1)))), antidomain(multiplication(antidomain(x2), domain(multiplication(x0, domain(x1)))))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by lemma 28 }
% 31.34/4.47 ifeq(leq(addition(multiplication(antidomain(multiplication(antidomain(x2), multiplication(x0, domain(x1)))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by lemma 29 R->L }
% 31.34/4.47 ifeq(leq(addition(multiplication(antidomain(multiplication(multiplication(domain(antidomain(x2)), antidomain(x2)), multiplication(x0, domain(x1)))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by lemma 24 }
% 31.34/4.47 ifeq(leq(addition(multiplication(antidomain(multiplication(multiplication(antidomain(domain(x2)), antidomain(x2)), multiplication(x0, domain(x1)))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.47 = { by lemma 27 }
% 31.34/4.47 ifeq(leq(addition(multiplication(antidomain(multiplication(antidomain(domain(x2)), multiplication(x0, domain(x1)))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by lemma 23 R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(addition(zero, multiplication(antidomain(domain(x2)), multiplication(x0, domain(x1))))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by lemma 32 R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(addition(multiplication(antidomain(domain(x2)), multiplication(domain(x2), x0)), multiplication(antidomain(domain(x2)), multiplication(x0, domain(x1))))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 16 (right_distributivity) R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(multiplication(antidomain(domain(x2)), addition(multiplication(domain(x2), x0), multiplication(x0, domain(x1))))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 5 (additive_commutativity) }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(multiplication(antidomain(domain(x2)), addition(multiplication(x0, domain(x1)), multiplication(domain(x2), x0)))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 18 (goals) }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(multiplication(antidomain(domain(x2)), multiplication(domain(x2), x0))), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by lemma 32 }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(zero), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by lemma 22 R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(antidomain(one)), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 7 (domain4) R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(domain(one), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 6 (additive_identity) R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(addition(domain(one), zero), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by lemma 22 R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(addition(domain(one), antidomain(one)), multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by lemma 26 }
% 31.34/4.48 ifeq(leq(addition(multiplication(one, multiplication(antidomain(x2), domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 3 (multiplicative_left_identity) }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(x2), domain(multiplication(x0, domain(x1)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by lemma 30 R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(x2), domain(domain(multiplication(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 3 (multiplicative_left_identity) R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(x2), domain(multiplication(one, domain(multiplication(x0, domain(x1)))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 8 (forward_diamond) R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(x2), forward_diamond(one, multiplication(x0, domain(x1)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by lemma 30 R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(x2), forward_diamond(one, multiplication(x0, domain(domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 8 (forward_diamond) }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(x2), domain(multiplication(one, domain(multiplication(x0, domain(domain(x1))))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 8 (forward_diamond) R->L }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(x2), domain(multiplication(one, forward_diamond(x0, domain(x1))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 3 (multiplicative_left_identity) }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(x2), domain(forward_diamond(x0, domain(x1)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by lemma 31 }
% 31.34/4.48 ifeq(leq(addition(multiplication(antidomain(x2), forward_diamond(x0, domain(x1))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 17 (left_distributivity) R->L }
% 31.34/4.48 ifeq(leq(multiplication(addition(antidomain(x2), domain(x2)), forward_diamond(x0, domain(x1))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 5 (additive_commutativity) }
% 31.34/4.48 ifeq(leq(multiplication(addition(domain(x2), antidomain(x2)), forward_diamond(x0, domain(x1))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by lemma 26 }
% 31.34/4.48 ifeq(leq(multiplication(one, forward_diamond(x0, domain(x1))), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 3 (multiplicative_left_identity) }
% 31.34/4.48 ifeq(leq(forward_diamond(x0, domain(x1)), domain(x2)), true, addition(domain(x2), forward_diamond(x0, domain(x1))), domain(x2))
% 31.34/4.48 = { by axiom 5 (additive_commutativity) }
% 31.34/4.48 ifeq(leq(forward_diamond(x0, domain(x1)), domain(x2)), true, addition(forward_diamond(x0, domain(x1)), domain(x2)), domain(x2))
% 31.34/4.48 = { by axiom 19 (order_1) }
% 31.34/4.48 domain(x2)
% 31.34/4.48 % SZS output end Proof
% 31.34/4.48
% 31.34/4.48 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------