TSTP Solution File: KLE100+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE100+1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n026.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 05:35:53 EDT 2023
% Result : Theorem 8.23s 1.58s
% Output : Proof 9.96s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.14 % Problem : KLE100+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.36 % Computer : n026.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Tue Aug 29 11:22:35 EDT 2023
% 0.14/0.36 % CPUTime :
% 8.23/1.58 Command-line arguments: --flatten
% 8.23/1.58
% 8.23/1.58 % SZS status Theorem
% 8.23/1.58
% 8.23/1.64 % SZS output start Proof
% 8.23/1.64 Take the following subset of the input axioms:
% 9.82/1.64 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 9.82/1.64 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 9.82/1.64 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 9.82/1.64 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 9.82/1.64 fof(codomain1, axiom, ![X0]: multiplication(X0, coantidomain(X0))=zero).
% 9.82/1.64 fof(codomain3, axiom, ![X0_2]: addition(coantidomain(coantidomain(X0_2)), coantidomain(X0_2))=one).
% 9.82/1.64 fof(codomain4, axiom, ![X0_2]: codomain(X0_2)=coantidomain(coantidomain(X0_2))).
% 9.82/1.64 fof(complement, axiom, ![X0_2]: c(X0_2)=antidomain(domain(X0_2))).
% 9.82/1.64 fof(domain1, axiom, ![X0_2]: multiplication(antidomain(X0_2), X0_2)=zero).
% 9.82/1.64 fof(domain2, axiom, ![X1, X0_2]: addition(antidomain(multiplication(X0_2, X1)), antidomain(multiplication(X0_2, antidomain(antidomain(X1)))))=antidomain(multiplication(X0_2, antidomain(antidomain(X1))))).
% 9.82/1.64 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 9.82/1.64 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 9.82/1.64 fof(domain_difference, axiom, ![X0_2, X1_2]: domain_difference(X0_2, X1_2)=multiplication(domain(X0_2), antidomain(X1_2))).
% 9.82/1.64 fof(forward_box, axiom, ![X0_2, X1_2]: forward_box(X0_2, X1_2)=c(forward_diamond(X0_2, c(X1_2)))).
% 9.82/1.64 fof(forward_diamond, axiom, ![X0_2, X1_2]: forward_diamond(X0_2, X1_2)=domain(multiplication(X0_2, domain(X1_2)))).
% 9.82/1.64 fof(goals, conjecture, ![X2, X0_2, X1_2]: (addition(forward_diamond(X0_2, domain(X1_2)), domain(X2))=domain(X2) <= multiplication(antidomain(X2), multiplication(X0_2, domain(X1_2)))=zero)).
% 9.82/1.64 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 9.82/1.64 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 9.82/1.64 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 9.82/1.64 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 9.82/1.64 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 9.82/1.64 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 9.82/1.64 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 9.82/1.64
% 9.82/1.64 Now clausify the problem and encode Horn clauses using encoding 3 of
% 9.82/1.64 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 9.82/1.64 We repeatedly replace C & s=t => u=v by the two clauses:
% 9.82/1.64 fresh(y, y, x1...xn) = u
% 9.82/1.64 C => fresh(s, t, x1...xn) = v
% 9.82/1.64 where fresh is a fresh function symbol and x1..xn are the free
% 9.82/1.64 variables of u and v.
% 9.82/1.64 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 9.82/1.64 input problem has no model of domain size 1).
% 9.82/1.64
% 9.82/1.64 The encoding turns the above axioms into the following unit equations and goals:
% 9.82/1.64
% 9.82/1.64 Axiom 1 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 9.82/1.64 Axiom 2 (complement): c(X) = antidomain(domain(X)).
% 9.82/1.64 Axiom 3 (domain4): domain(X) = antidomain(antidomain(X)).
% 9.82/1.64 Axiom 4 (additive_idempotence): addition(X, X) = X.
% 9.82/1.64 Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 9.82/1.64 Axiom 6 (additive_identity): addition(X, zero) = X.
% 9.82/1.64 Axiom 7 (multiplicative_right_identity): multiplication(X, one) = X.
% 9.82/1.64 Axiom 8 (multiplicative_left_identity): multiplication(one, X) = X.
% 9.82/1.64 Axiom 9 (left_annihilation): multiplication(zero, X) = zero.
% 9.82/1.64 Axiom 10 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 9.82/1.64 Axiom 11 (domain1): multiplication(antidomain(X), X) = zero.
% 9.82/1.64 Axiom 12 (order): fresh(X, X, Y, Z) = true.
% 9.82/1.64 Axiom 13 (order_1): fresh2(X, X, Y, Z) = Z.
% 9.82/1.64 Axiom 14 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
% 9.82/1.64 Axiom 15 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 9.82/1.64 Axiom 16 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 9.82/1.64 Axiom 17 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 9.82/1.64 Axiom 18 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 9.82/1.64 Axiom 19 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 9.82/1.64 Axiom 20 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 9.82/1.64 Axiom 21 (order): fresh(addition(X, Y), Y, X, Y) = leq(X, Y).
% 9.82/1.64 Axiom 22 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 9.82/1.64 Axiom 23 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 9.82/1.64 Axiom 24 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 9.82/1.64 Axiom 25 (goals): multiplication(antidomain(x2), multiplication(x0, domain(x1))) = zero.
% 9.82/1.64 Axiom 26 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 9.82/1.64
% 9.82/1.64 Lemma 27: addition(domain(X), antidomain(X)) = one.
% 9.82/1.64 Proof:
% 9.82/1.64 addition(domain(X), antidomain(X))
% 9.82/1.64 = { by axiom 3 (domain4) }
% 9.82/1.64 addition(antidomain(antidomain(X)), antidomain(X))
% 9.82/1.64 = { by axiom 20 (domain3) }
% 9.82/1.64 one
% 9.82/1.64
% 9.82/1.64 Lemma 28: multiplication(domain(X), X) = X.
% 9.82/1.64 Proof:
% 9.82/1.64 multiplication(domain(X), X)
% 9.82/1.64 = { by axiom 6 (additive_identity) R->L }
% 9.82/1.64 addition(multiplication(domain(X), X), zero)
% 9.82/1.64 = { by axiom 11 (domain1) R->L }
% 9.82/1.64 addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
% 9.82/1.64 = { by axiom 24 (left_distributivity) R->L }
% 9.82/1.64 multiplication(addition(domain(X), antidomain(X)), X)
% 9.82/1.64 = { by lemma 27 }
% 9.82/1.64 multiplication(one, X)
% 9.82/1.64 = { by axiom 8 (multiplicative_left_identity) }
% 9.82/1.64 X
% 9.82/1.64
% 9.82/1.64 Lemma 29: domain_difference(antidomain(X), X) = antidomain(X).
% 9.82/1.64 Proof:
% 9.82/1.64 domain_difference(antidomain(X), X)
% 9.82/1.64 = { by axiom 17 (domain_difference) }
% 9.82/1.64 multiplication(domain(antidomain(X)), antidomain(X))
% 9.82/1.64 = { by lemma 28 }
% 9.82/1.64 antidomain(X)
% 9.82/1.64
% 9.82/1.64 Lemma 30: domain(antidomain(X)) = c(X).
% 9.82/1.64 Proof:
% 9.82/1.64 domain(antidomain(X))
% 9.82/1.64 = { by axiom 3 (domain4) }
% 9.82/1.65 antidomain(antidomain(antidomain(X)))
% 9.82/1.65 = { by axiom 3 (domain4) R->L }
% 9.82/1.65 antidomain(domain(X))
% 9.82/1.65 = { by axiom 2 (complement) R->L }
% 9.82/1.65 c(X)
% 9.82/1.65
% 9.82/1.65 Lemma 31: multiplication(c(X), antidomain(Y)) = domain_difference(antidomain(X), Y).
% 9.82/1.65 Proof:
% 9.82/1.65 multiplication(c(X), antidomain(Y))
% 9.82/1.65 = { by lemma 30 R->L }
% 9.82/1.65 multiplication(domain(antidomain(X)), antidomain(Y))
% 9.82/1.65 = { by axiom 17 (domain_difference) R->L }
% 9.82/1.65 domain_difference(antidomain(X), Y)
% 9.82/1.65
% 9.82/1.65 Lemma 32: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 9.82/1.65 Proof:
% 9.82/1.65 multiplication(antidomain(X), addition(X, Y))
% 9.82/1.65 = { by axiom 5 (additive_commutativity) R->L }
% 9.82/1.65 multiplication(antidomain(X), addition(Y, X))
% 9.82/1.65 = { by axiom 23 (right_distributivity) }
% 9.82/1.65 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 9.82/1.65 = { by axiom 11 (domain1) }
% 9.82/1.65 addition(multiplication(antidomain(X), Y), zero)
% 9.82/1.65 = { by axiom 6 (additive_identity) }
% 9.82/1.65 multiplication(antidomain(X), Y)
% 9.82/1.65
% 9.82/1.65 Lemma 33: antidomain(X) = c(X).
% 9.82/1.65 Proof:
% 9.82/1.65 antidomain(X)
% 9.82/1.65 = { by lemma 29 R->L }
% 9.82/1.65 domain_difference(antidomain(X), X)
% 9.82/1.65 = { by lemma 31 R->L }
% 9.82/1.65 multiplication(c(X), antidomain(X))
% 9.82/1.65 = { by axiom 2 (complement) }
% 9.82/1.65 multiplication(antidomain(domain(X)), antidomain(X))
% 9.82/1.65 = { by lemma 32 R->L }
% 9.82/1.65 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 9.82/1.65 = { by lemma 27 }
% 9.82/1.65 multiplication(antidomain(domain(X)), one)
% 9.82/1.65 = { by axiom 7 (multiplicative_right_identity) }
% 9.82/1.65 antidomain(domain(X))
% 9.82/1.65 = { by axiom 2 (complement) R->L }
% 9.82/1.65 c(X)
% 9.82/1.65
% 9.82/1.65 Lemma 34: addition(X, multiplication(antidomain(x2), multiplication(x0, domain(x1)))) = X.
% 9.82/1.65 Proof:
% 9.82/1.65 addition(X, multiplication(antidomain(x2), multiplication(x0, domain(x1))))
% 9.82/1.65 = { by axiom 5 (additive_commutativity) R->L }
% 9.82/1.65 addition(multiplication(antidomain(x2), multiplication(x0, domain(x1))), X)
% 9.82/1.65 = { by axiom 25 (goals) }
% 9.82/1.65 addition(zero, X)
% 9.82/1.65 = { by axiom 5 (additive_commutativity) R->L }
% 9.82/1.65 addition(X, zero)
% 9.82/1.65 = { by axiom 6 (additive_identity) }
% 9.82/1.65 X
% 9.82/1.65
% 9.82/1.65 Lemma 35: multiplication(antidomain(x2), multiplication(x0, domain(x1))) = antidomain(one).
% 9.82/1.65 Proof:
% 9.82/1.65 multiplication(antidomain(x2), multiplication(x0, domain(x1)))
% 9.82/1.65 = { by axiom 25 (goals) }
% 9.82/1.65 zero
% 9.82/1.65 = { by axiom 11 (domain1) R->L }
% 9.82/1.65 multiplication(antidomain(one), one)
% 9.82/1.65 = { by axiom 7 (multiplicative_right_identity) }
% 9.82/1.65 antidomain(one)
% 9.82/1.65
% 9.82/1.65 Lemma 36: domain(one) = one.
% 9.82/1.65 Proof:
% 9.82/1.65 domain(one)
% 9.82/1.65 = { by lemma 34 R->L }
% 9.82/1.65 addition(domain(one), multiplication(antidomain(x2), multiplication(x0, domain(x1))))
% 9.82/1.65 = { by lemma 35 }
% 9.82/1.65 addition(domain(one), antidomain(one))
% 9.82/1.65 = { by lemma 27 }
% 9.82/1.65 one
% 9.82/1.65
% 9.82/1.65 Lemma 37: antidomain(c(X)) = c(antidomain(X)).
% 9.82/1.65 Proof:
% 9.82/1.65 antidomain(c(X))
% 9.82/1.65 = { by lemma 30 R->L }
% 9.82/1.65 antidomain(domain(antidomain(X)))
% 9.82/1.65 = { by axiom 2 (complement) R->L }
% 9.82/1.65 c(antidomain(X))
% 9.82/1.65
% 9.82/1.65 Lemma 38: multiplication(c(X), domain(Y)) = domain_difference(antidomain(X), antidomain(Y)).
% 9.82/1.65 Proof:
% 9.82/1.65 multiplication(c(X), domain(Y))
% 9.82/1.65 = { by lemma 30 R->L }
% 9.82/1.65 multiplication(domain(antidomain(X)), domain(Y))
% 9.82/1.65 = { by axiom 3 (domain4) }
% 9.82/1.65 multiplication(domain(antidomain(X)), antidomain(antidomain(Y)))
% 9.82/1.65 = { by axiom 17 (domain_difference) R->L }
% 9.82/1.65 domain_difference(antidomain(X), antidomain(Y))
% 9.82/1.65
% 9.82/1.65 Lemma 39: c(c(X)) = domain(X).
% 9.82/1.65 Proof:
% 9.82/1.65 c(c(X))
% 9.82/1.65 = { by lemma 33 R->L }
% 9.82/1.65 c(antidomain(X))
% 9.82/1.65 = { by lemma 37 R->L }
% 9.82/1.65 antidomain(c(X))
% 9.82/1.65 = { by axiom 7 (multiplicative_right_identity) R->L }
% 9.82/1.65 multiplication(antidomain(c(X)), one)
% 9.82/1.65 = { by lemma 27 R->L }
% 9.82/1.65 multiplication(antidomain(c(X)), addition(domain(antidomain(X)), antidomain(antidomain(X))))
% 9.82/1.65 = { by lemma 30 }
% 9.82/1.65 multiplication(antidomain(c(X)), addition(c(X), antidomain(antidomain(X))))
% 9.82/1.65 = { by axiom 3 (domain4) R->L }
% 9.82/1.65 multiplication(antidomain(c(X)), addition(c(X), domain(X)))
% 9.82/1.65 = { by lemma 32 }
% 9.82/1.65 multiplication(antidomain(c(X)), domain(X))
% 9.82/1.65 = { by lemma 37 }
% 9.82/1.65 multiplication(c(antidomain(X)), domain(X))
% 9.82/1.65 = { by lemma 38 }
% 9.82/1.65 domain_difference(antidomain(antidomain(X)), antidomain(X))
% 9.82/1.65 = { by lemma 29 }
% 9.82/1.65 antidomain(antidomain(X))
% 9.82/1.65 = { by axiom 3 (domain4) R->L }
% 9.82/1.65 domain(X)
% 9.82/1.65
% 9.82/1.65 Lemma 40: addition(X, addition(X, Y)) = addition(X, Y).
% 9.82/1.65 Proof:
% 9.82/1.65 addition(X, addition(X, Y))
% 9.82/1.65 = { by axiom 16 (additive_associativity) }
% 9.82/1.65 addition(addition(X, X), Y)
% 9.82/1.65 = { by axiom 4 (additive_idempotence) }
% 9.82/1.65 addition(X, Y)
% 9.82/1.65
% 9.82/1.65 Lemma 41: addition(codomain(X), coantidomain(X)) = one.
% 9.82/1.65 Proof:
% 9.82/1.65 addition(codomain(X), coantidomain(X))
% 9.82/1.65 = { by axiom 1 (codomain4) }
% 9.82/1.65 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 9.82/1.65 = { by axiom 19 (codomain3) }
% 9.82/1.65 one
% 9.82/1.65
% 9.82/1.65 Lemma 42: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
% 9.82/1.65 Proof:
% 9.82/1.65 multiplication(X, addition(Y, one))
% 9.82/1.65 = { by axiom 5 (additive_commutativity) R->L }
% 9.82/1.65 multiplication(X, addition(one, Y))
% 9.82/1.65 = { by axiom 23 (right_distributivity) }
% 9.82/1.65 addition(multiplication(X, one), multiplication(X, Y))
% 9.82/1.65 = { by axiom 7 (multiplicative_right_identity) }
% 9.82/1.65 addition(X, multiplication(X, Y))
% 9.82/1.65
% 9.82/1.65 Lemma 43: leq(X, addition(Y, X)) = true.
% 9.82/1.65 Proof:
% 9.82/1.65 leq(X, addition(Y, X))
% 9.82/1.65 = { by axiom 5 (additive_commutativity) R->L }
% 9.82/1.65 leq(X, addition(X, Y))
% 9.82/1.65 = { by axiom 21 (order) R->L }
% 9.82/1.65 fresh(addition(X, addition(X, Y)), addition(X, Y), X, addition(X, Y))
% 9.82/1.65 = { by lemma 40 }
% 9.82/1.65 fresh(addition(X, Y), addition(X, Y), X, addition(X, Y))
% 9.82/1.65 = { by axiom 12 (order) }
% 9.82/1.65 true
% 9.82/1.65
% 9.82/1.65 Lemma 44: c(multiplication(X, c(Y))) = forward_box(X, Y).
% 9.82/1.65 Proof:
% 9.82/1.65 c(multiplication(X, c(Y)))
% 9.82/1.65 = { by lemma 30 R->L }
% 9.82/1.65 c(multiplication(X, domain(antidomain(Y))))
% 9.82/1.65 = { by axiom 2 (complement) }
% 9.82/1.65 antidomain(domain(multiplication(X, domain(antidomain(Y)))))
% 9.82/1.65 = { by axiom 15 (forward_diamond) R->L }
% 9.82/1.65 antidomain(forward_diamond(X, antidomain(Y)))
% 9.82/1.65 = { by lemma 33 }
% 9.82/1.65 c(forward_diamond(X, antidomain(Y)))
% 9.82/1.65 = { by lemma 33 }
% 9.82/1.65 c(forward_diamond(X, c(Y)))
% 9.82/1.65 = { by axiom 14 (forward_box) R->L }
% 9.82/1.65 forward_box(X, Y)
% 9.82/1.65
% 9.82/1.65 Lemma 45: multiplication(antidomain(x2), multiplication(x0, domain(x1))) = coantidomain(one).
% 9.82/1.65 Proof:
% 9.82/1.65 multiplication(antidomain(x2), multiplication(x0, domain(x1)))
% 9.82/1.65 = { by axiom 25 (goals) }
% 9.82/1.65 zero
% 9.82/1.65 = { by axiom 10 (codomain1) R->L }
% 9.82/1.65 multiplication(one, coantidomain(one))
% 9.82/1.65 = { by axiom 8 (multiplicative_left_identity) }
% 9.82/1.65 coantidomain(one)
% 9.82/1.65
% 9.82/1.65 Lemma 46: multiplication(antidomain(x2), multiplication(x0, domain(x1))) = multiplication(antidomain(X), multiplication(X, Y)).
% 9.82/1.65 Proof:
% 9.82/1.65 multiplication(antidomain(x2), multiplication(x0, domain(x1)))
% 9.82/1.65 = { by axiom 25 (goals) }
% 9.82/1.65 zero
% 9.82/1.65 = { by axiom 9 (left_annihilation) R->L }
% 9.82/1.65 multiplication(zero, Y)
% 9.82/1.65 = { by axiom 11 (domain1) R->L }
% 9.82/1.65 multiplication(multiplication(antidomain(X), X), Y)
% 9.82/1.65 = { by axiom 18 (multiplicative_associativity) R->L }
% 9.82/1.65 multiplication(antidomain(X), multiplication(X, Y))
% 9.82/1.65
% 9.82/1.65 Lemma 47: addition(multiplication(antidomain(x2), multiplication(x0, domain(x1))), X) = X.
% 9.82/1.65 Proof:
% 9.82/1.65 addition(multiplication(antidomain(x2), multiplication(x0, domain(x1))), X)
% 9.82/1.65 = { by axiom 5 (additive_commutativity) R->L }
% 9.82/1.65 addition(X, multiplication(antidomain(x2), multiplication(x0, domain(x1))))
% 9.82/1.65 = { by lemma 34 }
% 9.82/1.65 X
% 9.82/1.65
% 9.82/1.65 Goal 1 (goals_1): addition(forward_diamond(x0, domain(x1)), domain(x2)) = domain(x2).
% 9.82/1.65 Proof:
% 9.82/1.65 addition(forward_diamond(x0, domain(x1)), domain(x2))
% 9.82/1.65 = { by axiom 5 (additive_commutativity) }
% 9.82/1.65 addition(domain(x2), forward_diamond(x0, domain(x1)))
% 9.82/1.65 = { by axiom 8 (multiplicative_left_identity) R->L }
% 9.82/1.65 addition(domain(x2), multiplication(one, forward_diamond(x0, domain(x1))))
% 9.82/1.65 = { by lemma 27 R->L }
% 9.82/1.65 addition(domain(x2), multiplication(addition(domain(x2), antidomain(x2)), forward_diamond(x0, domain(x1))))
% 9.82/1.65 = { by axiom 5 (additive_commutativity) R->L }
% 9.82/1.65 addition(domain(x2), multiplication(addition(antidomain(x2), domain(x2)), forward_diamond(x0, domain(x1))))
% 9.82/1.65 = { by axiom 24 (left_distributivity) }
% 9.82/1.65 addition(domain(x2), addition(multiplication(antidomain(x2), forward_diamond(x0, domain(x1))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.65 = { by lemma 33 }
% 9.82/1.65 addition(domain(x2), addition(multiplication(c(x2), forward_diamond(x0, domain(x1))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.65 = { by axiom 2 (complement) }
% 9.82/1.65 addition(domain(x2), addition(multiplication(antidomain(domain(x2)), forward_diamond(x0, domain(x1))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.65 = { by lemma 32 R->L }
% 9.82/1.65 addition(domain(x2), addition(multiplication(antidomain(domain(x2)), addition(domain(x2), forward_diamond(x0, domain(x1)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.65 = { by axiom 5 (additive_commutativity) R->L }
% 9.82/1.65 addition(domain(x2), addition(multiplication(antidomain(domain(x2)), addition(forward_diamond(x0, domain(x1)), domain(x2))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.65 = { by axiom 2 (complement) R->L }
% 9.82/1.65 addition(domain(x2), addition(multiplication(c(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.65 = { by lemma 33 R->L }
% 9.82/1.65 addition(domain(x2), addition(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 8 (multiplicative_left_identity) R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(one, multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 41 R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(addition(codomain(one), coantidomain(one)), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 45 R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(addition(codomain(one), multiplication(antidomain(x2), multiplication(x0, domain(x1)))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 34 }
% 9.82/1.66 addition(domain(x2), addition(multiplication(codomain(one), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 1 (codomain4) }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(coantidomain(one)), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 45 R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(multiplication(antidomain(x2), multiplication(x0, domain(x1)))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 35 }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(antidomain(one)), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 36 R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(antidomain(domain(one))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 2 (complement) R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(one)), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 13 (order_1) R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(fresh2(true, true, antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), one))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 43 R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(fresh2(leq(antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), addition(domain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1)))))))), true, antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), one))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 27 }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(fresh2(leq(antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), one), true, antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), one))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 22 (order_1) }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(addition(antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1)))))), one))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 5 (additive_commutativity) }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(addition(one, antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 36 R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(addition(domain(one), antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 3 (domain4) }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(addition(antidomain(antidomain(one)), antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 35 R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(addition(antidomain(multiplication(antidomain(x2), multiplication(x0, domain(x1)))), antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 46 }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(addition(antidomain(multiplication(antidomain(domain(x2)), multiplication(domain(x2), multiplication(x0, domain(x1))))), antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 2 (complement) R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(addition(antidomain(multiplication(c(x2), multiplication(domain(x2), multiplication(x0, domain(x1))))), antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 33 R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(addition(antidomain(multiplication(antidomain(x2), multiplication(domain(x2), multiplication(x0, domain(x1))))), antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 3 (domain4) }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(addition(antidomain(multiplication(antidomain(x2), multiplication(domain(x2), multiplication(x0, domain(x1))))), antidomain(multiplication(antidomain(x2), antidomain(antidomain(multiplication(domain(x2), multiplication(x0, domain(x1)))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 26 (domain2) }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(antidomain(multiplication(antidomain(x2), antidomain(antidomain(multiplication(domain(x2), multiplication(x0, domain(x1))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 3 (domain4) R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(antidomain(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1)))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 33 }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(c(multiplication(antidomain(x2), domain(multiplication(domain(x2), multiplication(x0, domain(x1)))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 39 R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(c(multiplication(antidomain(x2), c(c(multiplication(domain(x2), multiplication(x0, domain(x1))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 44 }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(forward_box(antidomain(x2), c(multiplication(domain(x2), multiplication(x0, domain(x1))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 47 R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(forward_box(antidomain(x2), c(addition(multiplication(antidomain(x2), multiplication(x0, domain(x1))), multiplication(domain(x2), multiplication(x0, domain(x1)))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 24 (left_distributivity) R->L }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(forward_box(antidomain(x2), c(multiplication(addition(antidomain(x2), domain(x2)), multiplication(x0, domain(x1))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by axiom 5 (additive_commutativity) }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(forward_box(antidomain(x2), c(multiplication(addition(domain(x2), antidomain(x2)), multiplication(x0, domain(x1))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.82/1.66 = { by lemma 27 }
% 9.82/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(forward_box(antidomain(x2), c(multiplication(one, multiplication(x0, domain(x1))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by axiom 8 (multiplicative_left_identity) }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(forward_box(antidomain(x2), c(multiplication(x0, domain(x1)))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 33 R->L }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(antidomain(forward_box(antidomain(x2), c(multiplication(x0, domain(x1)))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 44 R->L }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(antidomain(c(multiplication(antidomain(x2), c(c(multiplication(x0, domain(x1)))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by axiom 2 (complement) }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(antidomain(antidomain(domain(multiplication(antidomain(x2), c(c(multiplication(x0, domain(x1))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by axiom 3 (domain4) R->L }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(domain(domain(multiplication(antidomain(x2), c(c(multiplication(x0, domain(x1)))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by axiom 3 (domain4) }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(domain(antidomain(antidomain(multiplication(antidomain(x2), c(c(multiplication(x0, domain(x1))))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 30 }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(antidomain(multiplication(antidomain(x2), c(c(multiplication(x0, domain(x1)))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 33 }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(c(c(multiplication(antidomain(x2), c(c(multiplication(x0, domain(x1)))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 39 }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(antidomain(x2), c(c(multiplication(x0, domain(x1))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 33 R->L }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(antidomain(x2), antidomain(c(multiplication(x0, domain(x1))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 33 }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(c(x2), antidomain(c(multiplication(x0, domain(x1))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 31 }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(domain(domain_difference(antidomain(x2), c(multiplication(x0, domain(x1)))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 33 R->L }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(domain(domain_difference(antidomain(x2), antidomain(multiplication(x0, domain(x1)))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 38 R->L }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(c(x2), domain(multiplication(x0, domain(x1)))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 33 R->L }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(antidomain(x2), domain(multiplication(x0, domain(x1)))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by axiom 15 (forward_diamond) R->L }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(forward_diamond(antidomain(x2), multiplication(x0, domain(x1)))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by lemma 39 R->L }
% 9.96/1.66 addition(domain(x2), addition(multiplication(coantidomain(forward_diamond(antidomain(x2), multiplication(x0, c(c(x1))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.66 = { by axiom 2 (complement) }
% 9.96/1.67 addition(domain(x2), addition(multiplication(coantidomain(forward_diamond(antidomain(x2), multiplication(x0, c(antidomain(domain(x1)))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 33 }
% 9.96/1.67 addition(domain(x2), addition(multiplication(coantidomain(forward_diamond(antidomain(x2), multiplication(x0, c(c(domain(x1)))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 39 }
% 9.96/1.67 addition(domain(x2), addition(multiplication(coantidomain(forward_diamond(antidomain(x2), multiplication(x0, domain(domain(x1))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 15 (forward_diamond) }
% 9.96/1.67 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(antidomain(x2), domain(multiplication(x0, domain(domain(x1))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 15 (forward_diamond) R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(antidomain(x2), forward_diamond(x0, domain(x1))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 33 }
% 9.96/1.67 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(c(x2), forward_diamond(x0, domain(x1))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 2 (complement) }
% 9.96/1.67 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(antidomain(domain(x2)), forward_diamond(x0, domain(x1))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 32 R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(antidomain(domain(x2)), addition(domain(x2), forward_diamond(x0, domain(x1)))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 5 (additive_commutativity) R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(antidomain(domain(x2)), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 2 (complement) R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(c(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 33 R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 47 R->L }
% 9.96/1.67 addition(domain(x2), addition(addition(multiplication(antidomain(x2), multiplication(x0, domain(x1))), multiplication(coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 25 (goals) }
% 9.96/1.67 addition(domain(x2), addition(addition(zero, multiplication(coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 11 (domain1) R->L }
% 9.96/1.67 addition(domain(x2), addition(addition(multiplication(antidomain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 24 (left_distributivity) R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(addition(antidomain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 33 }
% 9.96/1.67 addition(domain(x2), addition(multiplication(addition(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 8 (multiplicative_left_identity) R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(addition(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(one, coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 27 R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(addition(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(addition(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), antidomain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 5 (additive_commutativity) R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(addition(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(addition(antidomain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 24 (left_distributivity) }
% 9.96/1.67 addition(domain(x2), addition(multiplication(addition(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), addition(multiplication(antidomain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))))), multiplication(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 10 (codomain1) }
% 9.96/1.67 addition(domain(x2), addition(multiplication(addition(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), addition(multiplication(antidomain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))))), zero)), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 6 (additive_identity) }
% 9.96/1.67 addition(domain(x2), addition(multiplication(addition(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(antidomain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 33 }
% 9.96/1.67 addition(domain(x2), addition(multiplication(addition(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 42 R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(multiplication(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), addition(coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), one)), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 22 (order_1) R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(multiplication(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), fresh2(leq(coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), one), true, coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), one)), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 41 R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(multiplication(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), fresh2(leq(coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), addition(codomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))))), true, coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), one)), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 43 }
% 9.96/1.67 addition(domain(x2), addition(multiplication(multiplication(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), fresh2(true, true, coantidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), one)), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 13 (order_1) }
% 9.96/1.67 addition(domain(x2), addition(multiplication(multiplication(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), one), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 7 (multiplicative_right_identity) }
% 9.96/1.67 addition(domain(x2), addition(multiplication(c(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by axiom 2 (complement) }
% 9.96/1.67 addition(domain(x2), addition(multiplication(antidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 28 R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(antidomain(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(domain(multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2)))), multiplication(antidomain(x2), addition(forward_diamond(x0, domain(x1)), domain(x2))))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 46 R->L }
% 9.96/1.67 addition(domain(x2), addition(multiplication(antidomain(x2), multiplication(x0, domain(x1))), multiplication(domain(x2), forward_diamond(x0, domain(x1)))))
% 9.96/1.67 = { by lemma 47 }
% 9.96/1.67 addition(domain(x2), multiplication(domain(x2), forward_diamond(x0, domain(x1))))
% 9.96/1.67 = { by lemma 42 R->L }
% 9.96/1.67 multiplication(domain(x2), addition(forward_diamond(x0, domain(x1)), one))
% 9.96/1.67 = { by axiom 5 (additive_commutativity) R->L }
% 9.96/1.67 multiplication(domain(x2), addition(one, forward_diamond(x0, domain(x1))))
% 9.96/1.67 = { by axiom 15 (forward_diamond) }
% 9.96/1.67 multiplication(domain(x2), addition(one, domain(multiplication(x0, domain(domain(x1))))))
% 9.96/1.67 = { by axiom 5 (additive_commutativity) R->L }
% 9.96/1.67 multiplication(domain(x2), addition(domain(multiplication(x0, domain(domain(x1)))), one))
% 9.96/1.67 = { by lemma 27 R->L }
% 9.96/1.67 multiplication(domain(x2), addition(domain(multiplication(x0, domain(domain(x1)))), addition(domain(multiplication(x0, domain(domain(x1)))), antidomain(multiplication(x0, domain(domain(x1)))))))
% 9.96/1.67 = { by lemma 40 }
% 9.96/1.67 multiplication(domain(x2), addition(domain(multiplication(x0, domain(domain(x1)))), antidomain(multiplication(x0, domain(domain(x1))))))
% 9.96/1.67 = { by lemma 27 }
% 9.96/1.67 multiplication(domain(x2), one)
% 9.96/1.67 = { by axiom 7 (multiplicative_right_identity) }
% 9.96/1.67 domain(x2)
% 9.96/1.67 % SZS output end Proof
% 9.96/1.67
% 9.96/1.67 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------