TSTP Solution File: KLE132+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE132+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 : n029.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 05:35:59 EDT 2023
% Result : Theorem 3.40s 0.77s
% Output : Proof 4.01s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.07 % Problem : KLE132+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.07 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.06/0.26 % Computer : n029.cluster.edu
% 0.06/0.26 % Model : x86_64 x86_64
% 0.06/0.26 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.26 % Memory : 8042.1875MB
% 0.06/0.26 % OS : Linux 3.10.0-693.el7.x86_64
% 0.06/0.26 % CPULimit : 300
% 0.06/0.26 % WCLimit : 300
% 0.06/0.26 % DateTime : Tue Aug 29 12:15:41 EDT 2023
% 0.06/0.26 % CPUTime :
% 3.40/0.77 Command-line arguments: --flatten
% 3.40/0.77
% 3.40/0.77 % SZS status Theorem
% 3.40/0.77
% 4.01/0.82 % SZS output start Proof
% 4.01/0.82 Take the following subset of the input axioms:
% 4.01/0.82 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 4.01/0.82 fof(additive_commutativity, axiom, ![A2, B2]: addition(A2, B2)=addition(B2, A2)).
% 4.01/0.82 fof(additive_idempotence, axiom, ![A2]: addition(A2, A2)=A2).
% 4.01/0.82 fof(additive_identity, axiom, ![A2]: addition(A2, zero)=A2).
% 4.01/0.82 fof(codomain1, axiom, ![X0]: multiplication(X0, coantidomain(X0))=zero).
% 4.01/0.82 fof(codomain3, axiom, ![X0_2]: addition(coantidomain(coantidomain(X0_2)), coantidomain(X0_2))=one).
% 4.01/0.82 fof(codomain4, axiom, ![X0_2]: codomain(X0_2)=coantidomain(coantidomain(X0_2))).
% 4.01/0.82 fof(complement, axiom, ![X0_2]: c(X0_2)=antidomain(domain(X0_2))).
% 4.01/0.82 fof(divergence1, axiom, ![X0_2]: forward_diamond(X0_2, divergence(X0_2))=divergence(X0_2)).
% 4.01/0.82 fof(domain1, axiom, ![X0_2]: multiplication(antidomain(X0_2), X0_2)=zero).
% 4.01/0.82 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 4.01/0.82 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 4.01/0.82 fof(domain_difference, axiom, ![X1, X0_2]: domain_difference(X0_2, X1)=multiplication(domain(X0_2), antidomain(X1))).
% 4.01/0.82 fof(forward_box, axiom, ![X1_2, X0_2]: forward_box(X0_2, X1_2)=c(forward_diamond(X0_2, c(X1_2)))).
% 4.01/0.82 fof(forward_diamond, axiom, ![X1_2, X0_2]: forward_diamond(X0_2, X1_2)=domain(multiplication(X0_2, domain(X1_2)))).
% 4.01/0.82 fof(goals, conjecture, ![X0_2]: (![X1_2]: addition(forward_diamond(X0_2, domain(X1_2)), forward_diamond(star(X0_2), domain_difference(domain(X1_2), forward_diamond(X0_2, domain(X1_2)))))=forward_diamond(star(X0_2), domain_difference(domain(X1_2), forward_diamond(X0_2, domain(X1_2)))) => ![X2]: (addition(domain(X2), forward_diamond(X0_2, domain(X2)))=forward_diamond(X0_2, domain(X2)) => domain(X2)=zero))).
% 4.01/0.82 fof(left_annihilation, axiom, ![A2]: multiplication(zero, A2)=zero).
% 4.01/0.82 fof(left_distributivity, axiom, ![A2, B2, C2]: multiplication(addition(A2, B2), C2)=addition(multiplication(A2, C2), multiplication(B2, C2))).
% 4.01/0.82 fof(multiplicative_associativity, axiom, ![A2, B2, C2]: multiplication(A2, multiplication(B2, C2))=multiplication(multiplication(A2, B2), C2)).
% 4.01/0.82 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 4.01/0.82 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 4.01/0.82 fof(right_annihilation, axiom, ![A2]: multiplication(A2, zero)=zero).
% 4.01/0.82 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 4.01/0.82
% 4.01/0.82 Now clausify the problem and encode Horn clauses using encoding 3 of
% 4.01/0.82 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 4.01/0.82 We repeatedly replace C & s=t => u=v by the two clauses:
% 4.01/0.82 fresh(y, y, x1...xn) = u
% 4.01/0.82 C => fresh(s, t, x1...xn) = v
% 4.01/0.82 where fresh is a fresh function symbol and x1..xn are the free
% 4.01/0.82 variables of u and v.
% 4.01/0.82 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 4.01/0.82 input problem has no model of domain size 1).
% 4.01/0.82
% 4.01/0.82 The encoding turns the above axioms into the following unit equations and goals:
% 4.01/0.82
% 4.01/0.82 Axiom 1 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 4.01/0.82 Axiom 2 (domain4): domain(X) = antidomain(antidomain(X)).
% 4.01/0.82 Axiom 3 (complement): c(X) = antidomain(domain(X)).
% 4.01/0.82 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 4.01/0.82 Axiom 5 (right_annihilation): multiplication(X, zero) = zero.
% 4.01/0.82 Axiom 6 (multiplicative_left_identity): multiplication(one, X) = X.
% 4.01/0.82 Axiom 7 (left_annihilation): multiplication(zero, X) = zero.
% 4.01/0.82 Axiom 8 (additive_idempotence): addition(X, X) = X.
% 4.01/0.82 Axiom 9 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 4.01/0.82 Axiom 10 (additive_identity): addition(X, zero) = X.
% 4.01/0.82 Axiom 11 (divergence1): forward_diamond(X, divergence(X)) = divergence(X).
% 4.01/0.82 Axiom 12 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 4.01/0.82 Axiom 13 (domain1): multiplication(antidomain(X), X) = zero.
% 4.01/0.82 Axiom 14 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
% 4.01/0.82 Axiom 15 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 4.01/0.82 Axiom 16 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 4.01/0.82 Axiom 17 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 4.01/0.82 Axiom 18 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 4.01/0.82 Axiom 19 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 4.01/0.82 Axiom 20 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 4.01/0.82 Axiom 21 (goals): addition(domain(x2), forward_diamond(x0, domain(x2))) = forward_diamond(x0, domain(x2)).
% 4.01/0.82 Axiom 22 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 4.01/0.82 Axiom 23 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 4.01/0.82 Axiom 24 (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)))).
% 4.01/0.82
% 4.01/0.82 Lemma 25: domain(zero) = divergence(zero).
% 4.01/0.82 Proof:
% 4.01/0.82 domain(zero)
% 4.01/0.82 = { by axiom 7 (left_annihilation) R->L }
% 4.01/0.82 domain(multiplication(zero, domain(divergence(zero))))
% 4.01/0.82 = { by axiom 17 (forward_diamond) R->L }
% 4.01/0.82 forward_diamond(zero, divergence(zero))
% 4.01/0.82 = { by axiom 11 (divergence1) }
% 4.01/0.82 divergence(zero)
% 4.01/0.82
% 4.01/0.82 Lemma 26: antidomain(one) = zero.
% 4.01/0.82 Proof:
% 4.01/0.82 antidomain(one)
% 4.01/0.82 = { by axiom 4 (multiplicative_right_identity) R->L }
% 4.01/0.82 multiplication(antidomain(one), one)
% 4.01/0.82 = { by axiom 13 (domain1) }
% 4.01/0.82 zero
% 4.01/0.82
% 4.01/0.82 Lemma 27: addition(antidomain(X), domain(X)) = one.
% 4.01/0.82 Proof:
% 4.01/0.82 addition(antidomain(X), domain(X))
% 4.01/0.82 = { by axiom 9 (additive_commutativity) R->L }
% 4.01/0.82 addition(domain(X), antidomain(X))
% 4.01/0.82 = { by axiom 2 (domain4) }
% 4.01/0.82 addition(antidomain(antidomain(X)), antidomain(X))
% 4.01/0.82 = { by axiom 20 (domain3) }
% 4.01/0.82 one
% 4.01/0.82
% 4.01/0.82 Lemma 28: antidomain(zero) = one.
% 4.01/0.82 Proof:
% 4.01/0.82 antidomain(zero)
% 4.01/0.82 = { by axiom 10 (additive_identity) R->L }
% 4.01/0.82 addition(antidomain(zero), zero)
% 4.01/0.82 = { by axiom 9 (additive_commutativity) }
% 4.01/0.82 addition(zero, antidomain(zero))
% 4.01/0.82 = { by lemma 26 R->L }
% 4.01/0.82 addition(antidomain(one), antidomain(zero))
% 4.01/0.82 = { by lemma 26 R->L }
% 4.01/0.82 addition(antidomain(one), antidomain(antidomain(one)))
% 4.01/0.82 = { by axiom 2 (domain4) R->L }
% 4.01/0.82 addition(antidomain(one), domain(one))
% 4.01/0.82 = { by lemma 27 }
% 4.01/0.82 one
% 4.01/0.82
% 4.01/0.82 Lemma 29: divergence(zero) = zero.
% 4.01/0.82 Proof:
% 4.01/0.82 divergence(zero)
% 4.01/0.82 = { by lemma 25 R->L }
% 4.01/0.82 domain(zero)
% 4.01/0.82 = { by axiom 2 (domain4) }
% 4.01/0.82 antidomain(antidomain(zero))
% 4.01/0.82 = { by lemma 28 }
% 4.01/0.82 antidomain(one)
% 4.01/0.82 = { by lemma 26 }
% 4.01/0.82 zero
% 4.01/0.82
% 4.01/0.82 Lemma 30: multiplication(domain(X), X) = X.
% 4.01/0.82 Proof:
% 4.01/0.82 multiplication(domain(X), X)
% 4.01/0.82 = { by axiom 10 (additive_identity) R->L }
% 4.01/0.82 addition(multiplication(domain(X), X), zero)
% 4.01/0.82 = { by axiom 13 (domain1) R->L }
% 4.01/0.82 addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
% 4.01/0.82 = { by axiom 23 (left_distributivity) R->L }
% 4.01/0.82 multiplication(addition(domain(X), antidomain(X)), X)
% 4.01/0.82 = { by axiom 9 (additive_commutativity) }
% 4.01/0.82 multiplication(addition(antidomain(X), domain(X)), X)
% 4.01/0.82 = { by lemma 27 }
% 4.01/0.82 multiplication(one, X)
% 4.01/0.82 = { by axiom 6 (multiplicative_left_identity) }
% 4.01/0.82 X
% 4.01/0.82
% 4.01/0.82 Lemma 31: domain(antidomain(X)) = c(X).
% 4.01/0.82 Proof:
% 4.01/0.82 domain(antidomain(X))
% 4.01/0.82 = { by axiom 2 (domain4) }
% 4.01/0.82 antidomain(antidomain(antidomain(X)))
% 4.01/0.82 = { by axiom 2 (domain4) R->L }
% 4.01/0.82 antidomain(domain(X))
% 4.01/0.82 = { by axiom 3 (complement) R->L }
% 4.01/0.83 c(X)
% 4.01/0.83
% 4.01/0.83 Lemma 32: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 4.01/0.83 Proof:
% 4.01/0.83 multiplication(antidomain(X), addition(X, Y))
% 4.01/0.83 = { by axiom 9 (additive_commutativity) R->L }
% 4.01/0.83 multiplication(antidomain(X), addition(Y, X))
% 4.01/0.83 = { by axiom 22 (right_distributivity) }
% 4.01/0.83 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 4.01/0.83 = { by axiom 13 (domain1) }
% 4.01/0.83 addition(multiplication(antidomain(X), Y), zero)
% 4.01/0.83 = { by axiom 10 (additive_identity) }
% 4.01/0.83 multiplication(antidomain(X), Y)
% 4.01/0.83
% 4.01/0.83 Lemma 33: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y).
% 4.01/0.83 Proof:
% 4.01/0.83 multiplication(antidomain(X), addition(Y, X))
% 4.01/0.83 = { by axiom 9 (additive_commutativity) R->L }
% 4.01/0.83 multiplication(antidomain(X), addition(X, Y))
% 4.01/0.83 = { by lemma 32 }
% 4.01/0.83 multiplication(antidomain(X), Y)
% 4.01/0.83
% 4.01/0.83 Lemma 34: antidomain(X) = c(X).
% 4.01/0.83 Proof:
% 4.01/0.83 antidomain(X)
% 4.01/0.83 = { by lemma 30 R->L }
% 4.01/0.83 multiplication(domain(antidomain(X)), antidomain(X))
% 4.01/0.83 = { by lemma 31 }
% 4.01/0.83 multiplication(c(X), antidomain(X))
% 4.01/0.83 = { by axiom 3 (complement) }
% 4.01/0.83 multiplication(antidomain(domain(X)), antidomain(X))
% 4.01/0.83 = { by lemma 33 R->L }
% 4.01/0.83 multiplication(antidomain(domain(X)), addition(antidomain(X), domain(X)))
% 4.01/0.83 = { by lemma 27 }
% 4.01/0.83 multiplication(antidomain(domain(X)), one)
% 4.01/0.83 = { by axiom 4 (multiplicative_right_identity) }
% 4.01/0.83 antidomain(domain(X))
% 4.01/0.83 = { by axiom 3 (complement) R->L }
% 4.01/0.83 c(X)
% 4.01/0.83
% 4.01/0.83 Lemma 35: coantidomain(one) = zero.
% 4.01/0.83 Proof:
% 4.01/0.83 coantidomain(one)
% 4.01/0.83 = { by axiom 6 (multiplicative_left_identity) R->L }
% 4.01/0.83 multiplication(one, coantidomain(one))
% 4.01/0.83 = { by axiom 12 (codomain1) }
% 4.01/0.83 zero
% 4.01/0.83
% 4.01/0.83 Lemma 36: antidomain(c(X)) = c(antidomain(X)).
% 4.01/0.83 Proof:
% 4.01/0.83 antidomain(c(X))
% 4.01/0.83 = { by lemma 31 R->L }
% 4.01/0.83 antidomain(domain(antidomain(X)))
% 4.01/0.83 = { by axiom 3 (complement) R->L }
% 4.01/0.83 c(antidomain(X))
% 4.01/0.83
% 4.01/0.83 Lemma 37: c(zero) = one.
% 4.01/0.83 Proof:
% 4.01/0.83 c(zero)
% 4.01/0.83 = { by lemma 26 R->L }
% 4.01/0.83 c(antidomain(one))
% 4.01/0.83 = { by lemma 36 R->L }
% 4.01/0.83 antidomain(c(one))
% 4.01/0.83 = { by lemma 31 R->L }
% 4.01/0.83 antidomain(domain(antidomain(one)))
% 4.01/0.83 = { by lemma 26 }
% 4.01/0.83 antidomain(domain(zero))
% 4.01/0.83 = { by lemma 25 }
% 4.01/0.83 antidomain(divergence(zero))
% 4.01/0.83 = { by lemma 29 }
% 4.01/0.83 antidomain(zero)
% 4.01/0.83 = { by lemma 28 }
% 4.01/0.83 one
% 4.01/0.83
% 4.01/0.83 Lemma 38: multiplication(domain(X), domain(Y)) = domain_difference(X, antidomain(Y)).
% 4.01/0.83 Proof:
% 4.01/0.83 multiplication(domain(X), domain(Y))
% 4.01/0.83 = { by axiom 2 (domain4) }
% 4.01/0.83 multiplication(domain(X), antidomain(antidomain(Y)))
% 4.01/0.83 = { by axiom 15 (domain_difference) R->L }
% 4.01/0.83 domain_difference(X, antidomain(Y))
% 4.01/0.83
% 4.01/0.83 Lemma 39: multiplication(c(X), domain(Y)) = domain_difference(antidomain(X), antidomain(Y)).
% 4.01/0.83 Proof:
% 4.01/0.83 multiplication(c(X), domain(Y))
% 4.01/0.83 = { by lemma 31 R->L }
% 4.01/0.83 multiplication(domain(antidomain(X)), domain(Y))
% 4.01/0.83 = { by lemma 38 }
% 4.01/0.83 domain_difference(antidomain(X), antidomain(Y))
% 4.01/0.83
% 4.01/0.83 Lemma 40: domain_difference(domain(X), antidomain(X)) = c(antidomain(X)).
% 4.01/0.83 Proof:
% 4.01/0.83 domain_difference(domain(X), antidomain(X))
% 4.01/0.83 = { by axiom 2 (domain4) }
% 4.01/0.83 domain_difference(antidomain(antidomain(X)), antidomain(X))
% 4.01/0.83 = { by lemma 39 R->L }
% 4.01/0.83 multiplication(c(antidomain(X)), domain(X))
% 4.01/0.83 = { by lemma 36 R->L }
% 4.01/0.83 multiplication(antidomain(c(X)), domain(X))
% 4.01/0.83 = { by lemma 32 R->L }
% 4.01/0.83 multiplication(antidomain(c(X)), addition(c(X), domain(X)))
% 4.01/0.83 = { by axiom 9 (additive_commutativity) R->L }
% 4.01/0.83 multiplication(antidomain(c(X)), addition(domain(X), c(X)))
% 4.01/0.83 = { by axiom 2 (domain4) }
% 4.01/0.83 multiplication(antidomain(c(X)), addition(antidomain(antidomain(X)), c(X)))
% 4.01/0.83 = { by lemma 31 R->L }
% 4.01/0.83 multiplication(antidomain(c(X)), addition(antidomain(antidomain(X)), domain(antidomain(X))))
% 4.01/0.83 = { by lemma 27 }
% 4.01/0.83 multiplication(antidomain(c(X)), one)
% 4.01/0.83 = { by axiom 4 (multiplicative_right_identity) }
% 4.01/0.83 antidomain(c(X))
% 4.01/0.83 = { by lemma 36 }
% 4.01/0.83 c(antidomain(X))
% 4.01/0.83
% 4.01/0.83 Lemma 41: domain(domain(X)) = domain(X).
% 4.01/0.83 Proof:
% 4.01/0.83 domain(domain(X))
% 4.01/0.83 = { by axiom 2 (domain4) }
% 4.01/0.83 domain(antidomain(antidomain(X)))
% 4.01/0.83 = { by lemma 31 }
% 4.01/0.83 c(antidomain(X))
% 4.01/0.83 = { by lemma 40 R->L }
% 4.01/0.83 domain_difference(domain(X), antidomain(X))
% 4.01/0.83 = { by lemma 38 R->L }
% 4.01/0.83 multiplication(domain(domain(X)), domain(X))
% 4.01/0.83 = { by lemma 30 }
% 4.01/0.83 domain(X)
% 4.01/0.83
% 4.01/0.83 Lemma 42: domain(domain(X)) = antidomain(c(X)).
% 4.01/0.83 Proof:
% 4.01/0.83 domain(domain(X))
% 4.01/0.83 = { by axiom 2 (domain4) }
% 4.01/0.83 antidomain(antidomain(domain(X)))
% 4.01/0.83 = { by axiom 3 (complement) R->L }
% 4.01/0.83 antidomain(c(X))
% 4.01/0.83
% 4.01/0.83 Lemma 43: c(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)).
% 4.01/0.83 Proof:
% 4.01/0.83 c(multiplication(X, domain(Y)))
% 4.01/0.83 = { by axiom 3 (complement) }
% 4.01/0.83 antidomain(domain(multiplication(X, domain(Y))))
% 4.01/0.83 = { by axiom 17 (forward_diamond) R->L }
% 4.01/0.83 antidomain(forward_diamond(X, Y))
% 4.01/0.83
% 4.01/0.83 Lemma 44: forward_diamond(X, zero) = zero.
% 4.01/0.83 Proof:
% 4.01/0.83 forward_diamond(X, zero)
% 4.01/0.83 = { by axiom 6 (multiplicative_left_identity) R->L }
% 4.01/0.83 multiplication(one, forward_diamond(X, zero))
% 4.01/0.83 = { by lemma 37 R->L }
% 4.01/0.83 multiplication(c(zero), forward_diamond(X, zero))
% 4.01/0.83 = { by axiom 5 (right_annihilation) R->L }
% 4.01/0.83 multiplication(c(multiplication(X, zero)), forward_diamond(X, zero))
% 4.01/0.83 = { by lemma 29 R->L }
% 4.01/0.83 multiplication(c(multiplication(X, divergence(zero))), forward_diamond(X, zero))
% 4.01/0.83 = { by lemma 25 R->L }
% 4.01/0.83 multiplication(c(multiplication(X, domain(zero))), forward_diamond(X, zero))
% 4.01/0.83 = { by lemma 43 }
% 4.01/0.83 multiplication(antidomain(forward_diamond(X, zero)), forward_diamond(X, zero))
% 4.01/0.83 = { by axiom 13 (domain1) }
% 4.01/0.83 zero
% 4.01/0.83
% 4.01/0.83 Lemma 45: c(antidomain(X)) = domain(X).
% 4.01/0.83 Proof:
% 4.01/0.83 c(antidomain(X))
% 4.01/0.83 = { by lemma 40 R->L }
% 4.01/0.83 domain_difference(domain(X), antidomain(X))
% 4.01/0.83 = { by axiom 2 (domain4) }
% 4.01/0.83 domain_difference(antidomain(antidomain(X)), antidomain(X))
% 4.01/0.83 = { by lemma 39 R->L }
% 4.01/0.83 multiplication(c(antidomain(X)), domain(X))
% 4.01/0.83 = { by lemma 36 R->L }
% 4.01/0.83 multiplication(antidomain(c(X)), domain(X))
% 4.01/0.83 = { by lemma 42 R->L }
% 4.01/0.83 multiplication(domain(domain(X)), domain(X))
% 4.01/0.83 = { by lemma 30 }
% 4.01/0.83 domain(X)
% 4.01/0.83
% 4.01/0.83 Lemma 46: domain(multiplication(X, c(Y))) = forward_diamond(X, antidomain(Y)).
% 4.01/0.83 Proof:
% 4.01/0.83 domain(multiplication(X, c(Y)))
% 4.01/0.83 = { by lemma 31 R->L }
% 4.01/0.83 domain(multiplication(X, domain(antidomain(Y))))
% 4.01/0.83 = { by axiom 17 (forward_diamond) R->L }
% 4.01/0.83 forward_diamond(X, antidomain(Y))
% 4.01/0.83
% 4.01/0.83 Lemma 47: c(domain(X)) = forward_box(X, zero).
% 4.01/0.83 Proof:
% 4.01/0.83 c(domain(X))
% 4.01/0.83 = { by axiom 4 (multiplicative_right_identity) R->L }
% 4.01/0.83 c(domain(multiplication(X, one)))
% 4.01/0.83 = { by lemma 37 R->L }
% 4.01/0.83 c(domain(multiplication(X, c(zero))))
% 4.01/0.83 = { by lemma 46 }
% 4.01/0.83 c(forward_diamond(X, antidomain(zero)))
% 4.01/0.83 = { by lemma 28 }
% 4.01/0.83 c(forward_diamond(X, one))
% 4.01/0.83 = { by lemma 37 R->L }
% 4.01/0.83 c(forward_diamond(X, c(zero)))
% 4.01/0.83 = { by axiom 14 (forward_box) R->L }
% 4.01/0.83 forward_box(X, zero)
% 4.01/0.83
% 4.01/0.83 Lemma 48: forward_box(X, zero) = c(X).
% 4.01/0.83 Proof:
% 4.01/0.83 forward_box(X, zero)
% 4.01/0.83 = { by lemma 47 R->L }
% 4.01/0.83 c(domain(X))
% 4.01/0.83 = { by axiom 2 (domain4) }
% 4.01/0.83 c(antidomain(antidomain(X)))
% 4.01/0.83 = { by lemma 45 }
% 4.01/0.83 domain(antidomain(X))
% 4.01/0.83 = { by lemma 31 }
% 4.01/0.83 c(X)
% 4.01/0.83
% 4.01/0.83 Lemma 49: c(c(X)) = domain(X).
% 4.01/0.83 Proof:
% 4.01/0.83 c(c(X))
% 4.01/0.83 = { by axiom 6 (multiplicative_left_identity) R->L }
% 4.01/0.83 c(multiplication(one, c(X)))
% 4.01/0.83 = { by lemma 31 R->L }
% 4.01/0.83 c(multiplication(one, domain(antidomain(X))))
% 4.01/0.83 = { by lemma 43 }
% 4.01/0.83 antidomain(forward_diamond(one, antidomain(X)))
% 4.01/0.83 = { by axiom 17 (forward_diamond) }
% 4.01/0.83 antidomain(domain(multiplication(one, domain(antidomain(X)))))
% 4.01/0.83 = { by axiom 6 (multiplicative_left_identity) }
% 4.01/0.83 antidomain(domain(domain(antidomain(X))))
% 4.01/0.83 = { by lemma 42 }
% 4.01/0.83 antidomain(antidomain(c(antidomain(X))))
% 4.01/0.83 = { by axiom 2 (domain4) R->L }
% 4.01/0.83 domain(c(antidomain(X)))
% 4.01/0.83 = { by axiom 3 (complement) }
% 4.01/0.83 domain(antidomain(domain(antidomain(X))))
% 4.01/0.83 = { by lemma 31 }
% 4.01/0.83 c(domain(antidomain(X)))
% 4.01/0.83 = { by lemma 47 }
% 4.01/0.83 forward_box(antidomain(X), zero)
% 4.01/0.83 = { by lemma 48 }
% 4.01/0.83 c(antidomain(X))
% 4.01/0.83 = { by lemma 45 }
% 4.01/0.83 domain(X)
% 4.01/0.83
% 4.01/0.83 Lemma 50: antidomain(c(X)) = domain(X).
% 4.01/0.83 Proof:
% 4.01/0.83 antidomain(c(X))
% 4.01/0.83 = { by lemma 42 R->L }
% 4.01/0.83 domain(domain(X))
% 4.01/0.83 = { by lemma 45 R->L }
% 4.01/0.83 c(antidomain(domain(X)))
% 4.01/0.83 = { by axiom 3 (complement) R->L }
% 4.01/0.83 c(c(X))
% 4.01/0.83 = { by lemma 49 }
% 4.01/0.83 domain(X)
% 4.01/0.83
% 4.01/0.83 Lemma 51: addition(domain(X), antidomain(X)) = one.
% 4.01/0.83 Proof:
% 4.01/0.83 addition(domain(X), antidomain(X))
% 4.01/0.83 = { by axiom 9 (additive_commutativity) R->L }
% 4.01/0.83 addition(antidomain(X), domain(X))
% 4.01/0.83 = { by lemma 27 }
% 4.01/0.83 one
% 4.01/0.83
% 4.01/0.83 Lemma 52: multiplication(domain(X), c(Y)) = domain_difference(X, domain(Y)).
% 4.01/0.83 Proof:
% 4.01/0.83 multiplication(domain(X), c(Y))
% 4.01/0.83 = { by axiom 3 (complement) }
% 4.01/0.83 multiplication(domain(X), antidomain(domain(Y)))
% 4.01/0.83 = { by axiom 15 (domain_difference) R->L }
% 4.01/0.83 domain_difference(X, domain(Y))
% 4.01/0.83
% 4.01/0.83 Lemma 53: multiplication(domain(X), domain(Y)) = domain_difference(domain(X), c(Y)).
% 4.01/0.83 Proof:
% 4.01/0.83 multiplication(domain(X), domain(Y))
% 4.01/0.83 = { by lemma 41 R->L }
% 4.01/0.83 multiplication(domain(domain(X)), domain(Y))
% 4.01/0.83 = { by lemma 45 R->L }
% 4.01/0.83 multiplication(domain(domain(X)), c(antidomain(Y)))
% 4.01/0.83 = { by lemma 52 }
% 4.01/0.83 domain_difference(domain(X), domain(antidomain(Y)))
% 4.01/0.83 = { by lemma 31 }
% 4.01/0.83 domain_difference(domain(X), c(Y))
% 4.01/0.83
% 4.01/0.83 Lemma 54: multiplication(c(X), antidomain(Y)) = domain_difference(antidomain(X), Y).
% 4.01/0.83 Proof:
% 4.01/0.83 multiplication(c(X), antidomain(Y))
% 4.01/0.83 = { by lemma 31 R->L }
% 4.01/0.83 multiplication(domain(antidomain(X)), antidomain(Y))
% 4.01/0.83 = { by axiom 15 (domain_difference) R->L }
% 4.01/0.83 domain_difference(antidomain(X), Y)
% 4.01/0.83
% 4.01/0.83 Lemma 55: domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))) = zero.
% 4.01/0.83 Proof:
% 4.01/0.83 domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2))))
% 4.01/0.83 = { by axiom 15 (domain_difference) }
% 4.01/0.83 multiplication(domain(domain(x2)), antidomain(addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 4.01/0.83 = { by lemma 34 }
% 4.01/0.83 multiplication(domain(domain(x2)), c(addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 4.01/0.83 = { by lemma 52 }
% 4.01/0.83 domain_difference(domain(x2), domain(addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 4.01/0.83 = { by lemma 49 R->L }
% 4.01/0.83 domain_difference(domain(x2), c(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by lemma 53 R->L }
% 4.01/0.83 multiplication(domain(x2), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by lemma 41 R->L }
% 4.01/0.83 multiplication(domain(domain(x2)), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by lemma 50 R->L }
% 4.01/0.83 multiplication(antidomain(c(domain(x2))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by axiom 4 (multiplicative_right_identity) R->L }
% 4.01/0.83 multiplication(multiplication(antidomain(c(domain(x2))), one), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by lemma 27 R->L }
% 4.01/0.83 multiplication(multiplication(antidomain(c(domain(x2))), addition(antidomain(antidomain(multiplication(x0, domain(domain(x2))))), domain(antidomain(multiplication(x0, domain(domain(x2))))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by axiom 8 (additive_idempotence) R->L }
% 4.01/0.83 multiplication(multiplication(antidomain(c(domain(x2))), addition(addition(antidomain(antidomain(multiplication(x0, domain(domain(x2))))), antidomain(antidomain(multiplication(x0, domain(domain(x2)))))), domain(antidomain(multiplication(x0, domain(domain(x2))))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by axiom 18 (additive_associativity) R->L }
% 4.01/0.83 multiplication(multiplication(antidomain(c(domain(x2))), addition(antidomain(antidomain(multiplication(x0, domain(domain(x2))))), addition(antidomain(antidomain(multiplication(x0, domain(domain(x2))))), domain(antidomain(multiplication(x0, domain(domain(x2)))))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by lemma 27 }
% 4.01/0.83 multiplication(multiplication(antidomain(c(domain(x2))), addition(antidomain(antidomain(multiplication(x0, domain(domain(x2))))), one)), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by axiom 9 (additive_commutativity) }
% 4.01/0.83 multiplication(multiplication(antidomain(c(domain(x2))), addition(one, antidomain(antidomain(multiplication(x0, domain(domain(x2))))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by axiom 2 (domain4) R->L }
% 4.01/0.83 multiplication(multiplication(antidomain(c(domain(x2))), addition(one, domain(multiplication(x0, domain(domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by axiom 17 (forward_diamond) R->L }
% 4.01/0.83 multiplication(multiplication(antidomain(c(domain(x2))), addition(one, forward_diamond(x0, domain(x2)))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by axiom 21 (goals) R->L }
% 4.01/0.83 multiplication(multiplication(antidomain(c(domain(x2))), addition(one, addition(domain(x2), forward_diamond(x0, domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by lemma 51 R->L }
% 4.01/0.83 multiplication(multiplication(antidomain(c(domain(x2))), addition(addition(domain(x2), antidomain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.83 = { by axiom 18 (additive_associativity) R->L }
% 4.01/0.83 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(antidomain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 30 R->L }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(multiplication(domain(antidomain(x2)), antidomain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 15 (domain_difference) R->L }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(domain_difference(antidomain(x2), x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 54 R->L }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(multiplication(c(x2), antidomain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 3 (complement) }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(multiplication(antidomain(domain(x2)), antidomain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 32 R->L }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(multiplication(antidomain(domain(x2)), addition(domain(x2), antidomain(x2))), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 51 }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(multiplication(antidomain(domain(x2)), one), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 4 (multiplicative_right_identity) }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(antidomain(domain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 3 (complement) R->L }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(c(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 48 R->L }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(forward_box(x2, zero), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 47 R->L }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(c(domain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 9 (additive_commutativity) R->L }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(addition(domain(x2), forward_diamond(x0, domain(x2))), c(domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 21 (goals) }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), addition(forward_diamond(x0, domain(x2)), c(domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 18 (additive_associativity) }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(addition(domain(x2), forward_diamond(x0, domain(x2))), c(domain(x2)))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 33 }
% 4.01/0.84 multiplication(multiplication(antidomain(c(domain(x2))), addition(domain(x2), forward_diamond(x0, domain(x2)))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 50 }
% 4.01/0.84 multiplication(multiplication(domain(domain(x2)), addition(domain(x2), forward_diamond(x0, domain(x2)))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 41 }
% 4.01/0.84 multiplication(multiplication(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 21 (goals) }
% 4.01/0.84 multiplication(multiplication(domain(x2), forward_diamond(x0, domain(x2))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 17 (forward_diamond) }
% 4.01/0.84 multiplication(multiplication(domain(x2), domain(multiplication(x0, domain(domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 41 }
% 4.01/0.84 multiplication(multiplication(domain(x2), domain(multiplication(x0, domain(x2)))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 45 R->L }
% 4.01/0.84 multiplication(multiplication(domain(x2), c(antidomain(multiplication(x0, domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 36 R->L }
% 4.01/0.84 multiplication(multiplication(domain(x2), antidomain(c(multiplication(x0, domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 42 R->L }
% 4.01/0.84 multiplication(multiplication(domain(x2), domain(domain(multiplication(x0, domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 45 R->L }
% 4.01/0.84 multiplication(multiplication(domain(x2), domain(domain(multiplication(x0, c(antidomain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 46 }
% 4.01/0.84 multiplication(multiplication(domain(x2), domain(forward_diamond(x0, antidomain(antidomain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 2 (domain4) R->L }
% 4.01/0.84 multiplication(multiplication(domain(x2), domain(forward_diamond(x0, domain(x2)))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 21 (goals) R->L }
% 4.01/0.84 multiplication(multiplication(domain(x2), domain(addition(domain(x2), forward_diamond(x0, domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 53 }
% 4.01/0.84 multiplication(domain_difference(domain(x2), c(addition(domain(x2), forward_diamond(x0, domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 2 (domain4) }
% 4.01/0.84 multiplication(domain_difference(antidomain(antidomain(x2)), c(addition(domain(x2), forward_diamond(x0, domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 54 R->L }
% 4.01/0.84 multiplication(multiplication(c(antidomain(x2)), antidomain(c(addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by lemma 45 }
% 4.01/0.84 multiplication(multiplication(domain(x2), antidomain(c(addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 16 (multiplicative_associativity) R->L }
% 4.01/0.84 multiplication(domain(x2), multiplication(antidomain(c(addition(domain(x2), forward_diamond(x0, domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2)))))))
% 4.01/0.84 = { by lemma 34 }
% 4.01/0.84 multiplication(domain(x2), multiplication(c(c(addition(domain(x2), forward_diamond(x0, domain(x2))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2)))))))
% 4.01/0.84 = { by axiom 3 (complement) }
% 4.01/0.84 multiplication(domain(x2), multiplication(antidomain(domain(c(addition(domain(x2), forward_diamond(x0, domain(x2)))))), domain(c(addition(domain(x2), forward_diamond(x0, domain(x2)))))))
% 4.01/0.84 = { by axiom 13 (domain1) }
% 4.01/0.84 multiplication(domain(x2), zero)
% 4.01/0.84 = { by axiom 5 (right_annihilation) }
% 4.01/0.84 zero
% 4.01/0.84
% 4.01/0.84 Goal 1 (goals_2): domain(x2) = zero.
% 4.01/0.84 Proof:
% 4.01/0.84 domain(x2)
% 4.01/0.84 = { by axiom 4 (multiplicative_right_identity) R->L }
% 4.01/0.84 multiplication(domain(x2), one)
% 4.01/0.84 = { by axiom 19 (codomain3) R->L }
% 4.01/0.84 multiplication(domain(x2), addition(coantidomain(coantidomain(one)), coantidomain(one)))
% 4.01/0.84 = { by axiom 1 (codomain4) R->L }
% 4.01/0.84 multiplication(domain(x2), addition(codomain(one), coantidomain(one)))
% 4.01/0.84 = { by lemma 35 }
% 4.01/0.84 multiplication(domain(x2), addition(codomain(one), zero))
% 4.01/0.84 = { by axiom 10 (additive_identity) }
% 4.01/0.84 multiplication(domain(x2), codomain(one))
% 4.01/0.84 = { by axiom 1 (codomain4) }
% 4.01/0.84 multiplication(domain(x2), coantidomain(coantidomain(one)))
% 4.01/0.84 = { by lemma 35 }
% 4.01/0.84 multiplication(domain(x2), coantidomain(zero))
% 4.01/0.84 = { by lemma 44 R->L }
% 4.01/0.84 multiplication(domain(x2), coantidomain(forward_diamond(star(x0), zero)))
% 4.01/0.84 = { by lemma 55 R->L }
% 4.01/0.84 multiplication(domain(x2), coantidomain(forward_diamond(star(x0), domain_difference(domain(x2), addition(domain(x2), forward_diamond(x0, domain(x2)))))))
% 4.01/0.84 = { by axiom 21 (goals) }
% 4.01/0.84 multiplication(domain(x2), coantidomain(forward_diamond(star(x0), domain_difference(domain(x2), forward_diamond(x0, domain(x2))))))
% 4.01/0.84 = { by axiom 24 (goals_1) R->L }
% 4.01/0.84 multiplication(domain(x2), coantidomain(addition(forward_diamond(x0, domain(x2)), forward_diamond(star(x0), domain_difference(domain(x2), forward_diamond(x0, domain(x2)))))))
% 4.01/0.84 = { by axiom 21 (goals) R->L }
% 4.01/0.84 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)))))))
% 4.01/0.84 = { by axiom 21 (goals) R->L }
% 4.01/0.84 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))))))))
% 4.01/0.84 = { by lemma 55 }
% 4.01/0.84 multiplication(domain(x2), coantidomain(addition(addition(domain(x2), forward_diamond(x0, domain(x2))), forward_diamond(star(x0), zero))))
% 4.01/0.84 = { by lemma 44 }
% 4.01/0.84 multiplication(domain(x2), coantidomain(addition(addition(domain(x2), forward_diamond(x0, domain(x2))), zero)))
% 4.01/0.84 = { by axiom 10 (additive_identity) }
% 4.01/0.84 multiplication(domain(x2), coantidomain(addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 4.01/0.84 = { by axiom 21 (goals) }
% 4.01/0.84 multiplication(domain(x2), coantidomain(forward_diamond(x0, domain(x2))))
% 4.01/0.84 = { by axiom 10 (additive_identity) R->L }
% 4.01/0.84 addition(multiplication(domain(x2), coantidomain(forward_diamond(x0, domain(x2)))), zero)
% 4.01/0.84 = { by axiom 12 (codomain1) R->L }
% 4.01/0.84 addition(multiplication(domain(x2), coantidomain(forward_diamond(x0, domain(x2)))), multiplication(forward_diamond(x0, domain(x2)), coantidomain(forward_diamond(x0, domain(x2)))))
% 4.01/0.84 = { by axiom 23 (left_distributivity) R->L }
% 4.01/0.84 multiplication(addition(domain(x2), forward_diamond(x0, domain(x2))), coantidomain(forward_diamond(x0, domain(x2))))
% 4.01/0.84 = { by axiom 21 (goals) R->L }
% 4.01/0.84 multiplication(addition(domain(x2), forward_diamond(x0, domain(x2))), coantidomain(addition(domain(x2), forward_diamond(x0, domain(x2)))))
% 4.01/0.84 = { by axiom 12 (codomain1) }
% 4.01/0.84 zero
% 4.01/0.84 % SZS output end Proof
% 4.01/0.84
% 4.01/0.84 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------