TSTP Solution File: KLE107+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE107+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 : n006.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 05:35:54 EDT 2023
% Result : Theorem 116.21s 15.08s
% Output : Proof 117.01s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : KLE107+1 : TPTP v8.1.2. Released v4.0.0.
% 0.06/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n006.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Aug 29 11:14:51 EDT 2023
% 0.12/0.33 % CPUTime :
% 116.21/15.08 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 116.21/15.08
% 116.21/15.08 % SZS status Theorem
% 116.21/15.08
% 116.79/15.16 % SZS output start Proof
% 116.79/15.16 Take the following subset of the input axioms:
% 116.79/15.16 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 116.79/15.16 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 116.79/15.16 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 116.79/15.16 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 116.79/15.16 fof(backward_diamond, axiom, ![X0, X1]: backward_diamond(X0, X1)=codomain(multiplication(codomain(X1), X0))).
% 116.79/15.16 fof(codomain1, axiom, ![X0_2]: multiplication(X0_2, coantidomain(X0_2))=zero).
% 116.79/15.16 fof(codomain2, axiom, ![X0_2, X1_2]: addition(coantidomain(multiplication(X0_2, X1_2)), coantidomain(multiplication(coantidomain(coantidomain(X0_2)), X1_2)))=coantidomain(multiplication(coantidomain(coantidomain(X0_2)), X1_2))).
% 116.79/15.16 fof(codomain3, axiom, ![X0_2]: addition(coantidomain(coantidomain(X0_2)), coantidomain(X0_2))=one).
% 116.79/15.16 fof(codomain4, axiom, ![X0_2]: codomain(X0_2)=coantidomain(coantidomain(X0_2))).
% 116.79/15.16 fof(complement, axiom, ![X0_2]: c(X0_2)=antidomain(domain(X0_2))).
% 116.79/15.16 fof(domain1, axiom, ![X0_2]: multiplication(antidomain(X0_2), X0_2)=zero).
% 116.79/15.16 fof(domain2, axiom, ![X0_2, X1_2]: addition(antidomain(multiplication(X0_2, X1_2)), antidomain(multiplication(X0_2, antidomain(antidomain(X1_2)))))=antidomain(multiplication(X0_2, antidomain(antidomain(X1_2))))).
% 116.79/15.16 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 116.79/15.16 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 116.79/15.16 fof(domain_difference, axiom, ![X0_2, X1_2]: domain_difference(X0_2, X1_2)=multiplication(domain(X0_2), antidomain(X1_2))).
% 116.79/15.16 fof(forward_box, axiom, ![X0_2, X1_2]: forward_box(X0_2, X1_2)=c(forward_diamond(X0_2, c(X1_2)))).
% 116.79/15.16 fof(forward_diamond, axiom, ![X0_2, X1_2]: forward_diamond(X0_2, X1_2)=domain(multiplication(X0_2, domain(X1_2)))).
% 116.79/15.16 fof(goals, conjecture, ![X2, X0_2, X1_2]: (addition(backward_diamond(X0_2, domain(X1_2)), domain(X2))=domain(X2) => addition(domain(X1_2), forward_box(X0_2, domain(X2)))=forward_box(X0_2, domain(X2)))).
% 116.79/15.16 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 116.79/15.16 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 116.79/15.16 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 116.79/15.16 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 116.79/15.16 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 116.79/15.16 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 116.79/15.16 fof(right_annihilation, axiom, ![A3]: multiplication(A3, zero)=zero).
% 116.79/15.16 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 116.79/15.16
% 116.79/15.16 Now clausify the problem and encode Horn clauses using encoding 3 of
% 116.79/15.16 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 116.79/15.16 We repeatedly replace C & s=t => u=v by the two clauses:
% 116.79/15.16 fresh(y, y, x1...xn) = u
% 116.79/15.16 C => fresh(s, t, x1...xn) = v
% 116.79/15.16 where fresh is a fresh function symbol and x1..xn are the free
% 116.79/15.16 variables of u and v.
% 116.79/15.16 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 116.79/15.16 input problem has no model of domain size 1).
% 116.79/15.16
% 116.79/15.16 The encoding turns the above axioms into the following unit equations and goals:
% 116.79/15.16
% 116.79/15.16 Axiom 1 (right_annihilation): multiplication(X, zero) = zero.
% 116.79/15.16 Axiom 2 (multiplicative_right_identity): multiplication(X, one) = X.
% 116.79/15.16 Axiom 3 (left_annihilation): multiplication(zero, X) = zero.
% 116.79/15.16 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 116.79/15.16 Axiom 5 (additive_idempotence): addition(X, X) = X.
% 116.79/15.16 Axiom 6 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 116.79/15.16 Axiom 7 (additive_identity): addition(X, zero) = X.
% 116.79/15.16 Axiom 8 (complement): c(X) = antidomain(domain(X)).
% 116.79/15.16 Axiom 9 (domain4): domain(X) = antidomain(antidomain(X)).
% 116.79/15.16 Axiom 10 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 116.79/15.16 Axiom 11 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 116.79/15.16 Axiom 12 (domain1): multiplication(antidomain(X), X) = zero.
% 116.79/15.17 Axiom 13 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 116.79/15.17 Axiom 14 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 116.79/15.17 Axiom 15 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 116.79/15.17 Axiom 16 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 116.79/15.17 Axiom 17 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
% 116.79/15.17 Axiom 18 (backward_diamond): backward_diamond(X, Y) = codomain(multiplication(codomain(Y), X)).
% 116.79/15.17 Axiom 19 (order): fresh(X, X, Y, Z) = true.
% 116.79/15.17 Axiom 20 (order_1): fresh2(X, X, Y, Z) = Z.
% 116.79/15.17 Axiom 21 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 116.79/15.17 Axiom 22 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 116.79/15.17 Axiom 23 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 116.79/15.17 Axiom 24 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 116.79/15.17 Axiom 25 (goals): addition(backward_diamond(x0, domain(x1)), domain(x2)) = domain(x2).
% 116.79/15.17 Axiom 26 (order): fresh(addition(X, Y), Y, X, Y) = leq(X, Y).
% 116.79/15.17 Axiom 27 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 116.79/15.17 Axiom 28 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 116.79/15.17 Axiom 29 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
% 116.79/15.17
% 116.79/15.17 Lemma 30: antidomain(one) = zero.
% 116.79/15.17 Proof:
% 116.79/15.17 antidomain(one)
% 116.79/15.17 = { by axiom 2 (multiplicative_right_identity) R->L }
% 116.79/15.17 multiplication(antidomain(one), one)
% 116.79/15.17 = { by axiom 12 (domain1) }
% 116.79/15.17 zero
% 116.79/15.17
% 116.79/15.17 Lemma 31: coantidomain(one) = zero.
% 116.79/15.17 Proof:
% 116.79/15.17 coantidomain(one)
% 116.79/15.17 = { by axiom 4 (multiplicative_left_identity) R->L }
% 116.79/15.17 multiplication(one, coantidomain(one))
% 116.79/15.17 = { by axiom 11 (codomain1) }
% 116.79/15.17 zero
% 116.79/15.17
% 116.79/15.17 Lemma 32: addition(zero, X) = X.
% 116.79/15.17 Proof:
% 116.79/15.17 addition(zero, X)
% 116.79/15.17 = { by axiom 6 (additive_commutativity) R->L }
% 116.79/15.17 addition(X, zero)
% 116.79/15.17 = { by axiom 7 (additive_identity) }
% 116.79/15.17 X
% 116.79/15.17
% 116.79/15.17 Lemma 33: addition(coantidomain(X), codomain(X)) = one.
% 116.79/15.17 Proof:
% 116.79/15.17 addition(coantidomain(X), codomain(X))
% 116.79/15.17 = { by axiom 6 (additive_commutativity) R->L }
% 116.79/15.17 addition(codomain(X), coantidomain(X))
% 116.79/15.17 = { by axiom 10 (codomain4) }
% 116.79/15.17 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 116.79/15.17 = { by axiom 22 (codomain3) }
% 116.79/15.17 one
% 116.79/15.17
% 116.79/15.17 Lemma 34: coantidomain(zero) = one.
% 116.79/15.17 Proof:
% 116.79/15.17 coantidomain(zero)
% 116.79/15.17 = { by lemma 31 R->L }
% 116.79/15.17 coantidomain(coantidomain(one))
% 116.79/15.17 = { by axiom 10 (codomain4) R->L }
% 116.79/15.17 codomain(one)
% 116.79/15.17 = { by lemma 32 R->L }
% 116.79/15.17 addition(zero, codomain(one))
% 116.79/15.17 = { by lemma 31 R->L }
% 116.79/15.17 addition(coantidomain(one), codomain(one))
% 116.79/15.17 = { by lemma 33 }
% 116.79/15.17 one
% 116.79/15.17
% 116.79/15.17 Lemma 35: addition(domain(X), antidomain(X)) = one.
% 116.79/15.17 Proof:
% 116.79/15.17 addition(domain(X), antidomain(X))
% 116.79/15.17 = { by axiom 9 (domain4) }
% 116.79/15.17 addition(antidomain(antidomain(X)), antidomain(X))
% 116.79/15.17 = { by axiom 21 (domain3) }
% 116.79/15.17 one
% 116.79/15.17
% 116.79/15.17 Lemma 36: domain(antidomain(X)) = c(X).
% 116.79/15.17 Proof:
% 116.79/15.17 domain(antidomain(X))
% 116.79/15.17 = { by axiom 9 (domain4) }
% 116.79/15.17 antidomain(antidomain(antidomain(X)))
% 116.79/15.17 = { by axiom 9 (domain4) R->L }
% 116.79/15.17 antidomain(domain(X))
% 116.79/15.17 = { by axiom 8 (complement) R->L }
% 116.79/15.17 c(X)
% 116.79/15.17
% 116.79/15.17 Lemma 37: multiplication(domain(X), X) = X.
% 116.79/15.17 Proof:
% 116.79/15.17 multiplication(domain(X), X)
% 116.79/15.17 = { by axiom 7 (additive_identity) R->L }
% 116.79/15.17 addition(multiplication(domain(X), X), zero)
% 116.79/15.17 = { by axiom 12 (domain1) R->L }
% 116.79/15.17 addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
% 116.79/15.17 = { by axiom 24 (left_distributivity) R->L }
% 116.79/15.17 multiplication(addition(domain(X), antidomain(X)), X)
% 116.79/15.17 = { by lemma 35 }
% 116.79/15.17 multiplication(one, X)
% 116.79/15.17 = { by axiom 4 (multiplicative_left_identity) }
% 116.79/15.17 X
% 116.79/15.17
% 116.79/15.17 Lemma 38: c(X) = antidomain(X).
% 116.79/15.17 Proof:
% 116.79/15.17 c(X)
% 116.79/15.17 = { by axiom 8 (complement) }
% 116.79/15.17 antidomain(domain(X))
% 116.79/15.17 = { by axiom 2 (multiplicative_right_identity) R->L }
% 116.79/15.17 multiplication(antidomain(domain(X)), one)
% 116.79/15.17 = { by lemma 35 R->L }
% 116.79/15.17 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 116.79/15.17 = { by axiom 6 (additive_commutativity) R->L }
% 116.79/15.17 multiplication(antidomain(domain(X)), addition(antidomain(X), domain(X)))
% 116.79/15.17 = { by axiom 23 (right_distributivity) }
% 116.79/15.17 addition(multiplication(antidomain(domain(X)), antidomain(X)), multiplication(antidomain(domain(X)), domain(X)))
% 116.79/15.17 = { by axiom 12 (domain1) }
% 116.79/15.17 addition(multiplication(antidomain(domain(X)), antidomain(X)), zero)
% 116.79/15.17 = { by axiom 7 (additive_identity) }
% 116.79/15.17 multiplication(antidomain(domain(X)), antidomain(X))
% 116.79/15.17 = { by axiom 8 (complement) R->L }
% 116.79/15.17 multiplication(c(X), antidomain(X))
% 116.79/15.17 = { by lemma 36 R->L }
% 116.79/15.17 multiplication(domain(antidomain(X)), antidomain(X))
% 116.79/15.17 = { by lemma 37 }
% 116.79/15.17 antidomain(X)
% 116.79/15.17
% 116.79/15.17 Lemma 39: multiplication(X, codomain(X)) = X.
% 116.79/15.17 Proof:
% 116.79/15.17 multiplication(X, codomain(X))
% 116.79/15.17 = { by axiom 7 (additive_identity) R->L }
% 116.79/15.17 addition(multiplication(X, codomain(X)), zero)
% 116.79/15.17 = { by axiom 11 (codomain1) R->L }
% 116.79/15.17 addition(multiplication(X, codomain(X)), multiplication(X, coantidomain(X)))
% 116.79/15.17 = { by axiom 23 (right_distributivity) R->L }
% 116.79/15.17 multiplication(X, addition(codomain(X), coantidomain(X)))
% 116.79/15.17 = { by axiom 6 (additive_commutativity) }
% 116.79/15.17 multiplication(X, addition(coantidomain(X), codomain(X)))
% 116.79/15.17 = { by lemma 33 }
% 116.79/15.17 multiplication(X, one)
% 116.79/15.17 = { by axiom 2 (multiplicative_right_identity) }
% 116.79/15.17 X
% 116.79/15.17
% 116.79/15.17 Lemma 40: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 116.79/15.17 Proof:
% 116.79/15.17 addition(X, multiplication(Y, X))
% 116.79/15.17 = { by axiom 4 (multiplicative_left_identity) R->L }
% 116.79/15.17 addition(multiplication(one, X), multiplication(Y, X))
% 116.79/15.17 = { by axiom 24 (left_distributivity) R->L }
% 116.79/15.17 multiplication(addition(one, Y), X)
% 116.79/15.17 = { by axiom 6 (additive_commutativity) }
% 116.79/15.17 multiplication(addition(Y, one), X)
% 116.79/15.17
% 116.79/15.17 Lemma 41: addition(X, addition(X, Y)) = addition(X, Y).
% 116.79/15.17 Proof:
% 116.79/15.17 addition(X, addition(X, Y))
% 116.79/15.17 = { by axiom 15 (additive_associativity) }
% 116.79/15.17 addition(addition(X, X), Y)
% 116.79/15.17 = { by axiom 5 (additive_idempotence) }
% 116.79/15.17 addition(X, Y)
% 116.79/15.17
% 116.79/15.17 Lemma 42: addition(one, domain(X)) = one.
% 116.79/15.17 Proof:
% 116.79/15.17 addition(one, domain(X))
% 116.79/15.17 = { by axiom 6 (additive_commutativity) R->L }
% 116.79/15.17 addition(domain(X), one)
% 116.79/15.17 = { by lemma 35 R->L }
% 116.79/15.17 addition(domain(X), addition(domain(X), antidomain(X)))
% 116.79/15.17 = { by lemma 41 }
% 116.79/15.17 addition(domain(X), antidomain(X))
% 116.79/15.17 = { by lemma 35 }
% 116.79/15.17 one
% 116.79/15.17
% 116.79/15.17 Lemma 43: addition(one, c(X)) = one.
% 116.79/15.17 Proof:
% 116.79/15.17 addition(one, c(X))
% 116.79/15.17 = { by lemma 36 R->L }
% 116.79/15.17 addition(one, domain(antidomain(X)))
% 116.79/15.17 = { by lemma 42 }
% 116.79/15.17 one
% 116.79/15.17
% 116.79/15.17 Lemma 44: addition(antidomain(X), codomain(antidomain(X))) = codomain(antidomain(X)).
% 116.79/15.17 Proof:
% 116.79/15.17 addition(antidomain(X), codomain(antidomain(X)))
% 116.79/15.17 = { by lemma 38 R->L }
% 116.79/15.17 addition(c(X), codomain(antidomain(X)))
% 116.79/15.17 = { by lemma 38 R->L }
% 116.79/15.17 addition(c(X), codomain(c(X)))
% 116.79/15.17 = { by axiom 6 (additive_commutativity) R->L }
% 116.79/15.17 addition(codomain(c(X)), c(X))
% 116.79/15.17 = { by lemma 39 R->L }
% 116.79/15.17 addition(codomain(c(X)), multiplication(c(X), codomain(c(X))))
% 116.79/15.17 = { by lemma 40 }
% 116.79/15.17 multiplication(addition(c(X), one), codomain(c(X)))
% 116.79/15.17 = { by axiom 6 (additive_commutativity) }
% 116.79/15.17 multiplication(addition(one, c(X)), codomain(c(X)))
% 116.79/15.17 = { by lemma 43 }
% 116.79/15.17 multiplication(one, codomain(c(X)))
% 116.79/15.17 = { by axiom 4 (multiplicative_left_identity) }
% 116.79/15.17 codomain(c(X))
% 116.79/15.17 = { by lemma 38 }
% 116.79/15.17 codomain(antidomain(X))
% 116.79/15.17
% 116.79/15.17 Lemma 45: addition(domain(X), addition(Y, antidomain(X))) = addition(Y, one).
% 116.79/15.17 Proof:
% 116.79/15.17 addition(domain(X), addition(Y, antidomain(X)))
% 116.79/15.17 = { by axiom 6 (additive_commutativity) R->L }
% 116.79/15.17 addition(domain(X), addition(antidomain(X), Y))
% 116.79/15.17 = { by axiom 15 (additive_associativity) }
% 116.79/15.17 addition(addition(domain(X), antidomain(X)), Y)
% 116.79/15.17 = { by lemma 35 }
% 116.79/15.17 addition(one, Y)
% 116.79/15.17 = { by axiom 6 (additive_commutativity) }
% 116.79/15.17 addition(Y, one)
% 116.79/15.17
% 116.79/15.17 Lemma 46: addition(one, coantidomain(X)) = one.
% 116.79/15.17 Proof:
% 116.79/15.17 addition(one, coantidomain(X))
% 116.79/15.17 = { by axiom 6 (additive_commutativity) R->L }
% 116.79/15.17 addition(coantidomain(X), one)
% 116.79/15.17 = { by lemma 33 R->L }
% 116.79/15.17 addition(coantidomain(X), addition(coantidomain(X), codomain(X)))
% 116.79/15.17 = { by lemma 41 }
% 116.79/15.17 addition(coantidomain(X), codomain(X))
% 116.79/15.17 = { by lemma 33 }
% 116.79/15.17 one
% 116.79/15.17
% 116.79/15.17 Lemma 47: addition(one, codomain(X)) = one.
% 116.79/15.17 Proof:
% 116.79/15.17 addition(one, codomain(X))
% 116.79/15.17 = { by axiom 10 (codomain4) }
% 116.79/15.17 addition(one, coantidomain(coantidomain(X)))
% 116.79/15.17 = { by lemma 46 }
% 116.79/15.17 one
% 116.79/15.17
% 116.79/15.17 Lemma 48: addition(domain(X), codomain(antidomain(X))) = one.
% 116.79/15.17 Proof:
% 116.79/15.17 addition(domain(X), codomain(antidomain(X)))
% 116.79/15.17 = { by lemma 44 R->L }
% 116.79/15.17 addition(domain(X), addition(antidomain(X), codomain(antidomain(X))))
% 116.79/15.17 = { by axiom 6 (additive_commutativity) R->L }
% 116.79/15.17 addition(domain(X), addition(codomain(antidomain(X)), antidomain(X)))
% 116.79/15.17 = { by lemma 45 }
% 116.79/15.17 addition(codomain(antidomain(X)), one)
% 116.79/15.17 = { by axiom 6 (additive_commutativity) }
% 116.79/15.17 addition(one, codomain(antidomain(X)))
% 116.79/15.17 = { by lemma 47 }
% 116.79/15.17 one
% 116.79/15.17
% 116.79/15.17 Lemma 49: multiplication(coantidomain(X), addition(Y, codomain(X))) = multiplication(coantidomain(X), Y).
% 116.79/15.17 Proof:
% 116.79/15.17 multiplication(coantidomain(X), addition(Y, codomain(X)))
% 116.79/15.17 = { by axiom 6 (additive_commutativity) R->L }
% 116.79/15.17 multiplication(coantidomain(X), addition(codomain(X), Y))
% 116.79/15.17 = { by axiom 23 (right_distributivity) }
% 116.79/15.17 addition(multiplication(coantidomain(X), codomain(X)), multiplication(coantidomain(X), Y))
% 116.79/15.17 = { by axiom 10 (codomain4) }
% 116.79/15.17 addition(multiplication(coantidomain(X), coantidomain(coantidomain(X))), multiplication(coantidomain(X), Y))
% 116.79/15.17 = { by axiom 11 (codomain1) }
% 116.79/15.17 addition(zero, multiplication(coantidomain(X), Y))
% 116.79/15.17 = { by lemma 32 }
% 116.79/15.17 multiplication(coantidomain(X), Y)
% 116.79/15.17
% 116.79/15.17 Lemma 50: multiplication(coantidomain(antidomain(X)), X) = X.
% 116.79/15.17 Proof:
% 116.79/15.17 multiplication(coantidomain(antidomain(X)), X)
% 116.79/15.17 = { by lemma 32 R->L }
% 116.79/15.17 addition(zero, multiplication(coantidomain(antidomain(X)), X))
% 116.79/15.17 = { by axiom 11 (codomain1) R->L }
% 116.79/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), coantidomain(multiplication(codomain(antidomain(X)), X))), multiplication(coantidomain(antidomain(X)), X))
% 116.79/15.17 = { by lemma 38 R->L }
% 116.79/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), coantidomain(multiplication(codomain(c(X)), X))), multiplication(coantidomain(antidomain(X)), X))
% 116.79/15.17 = { by axiom 10 (codomain4) }
% 116.79/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), coantidomain(multiplication(coantidomain(coantidomain(c(X))), X))), multiplication(coantidomain(antidomain(X)), X))
% 116.79/15.17 = { by axiom 29 (codomain2) R->L }
% 116.79/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), addition(coantidomain(multiplication(c(X), X)), coantidomain(multiplication(coantidomain(coantidomain(c(X))), X)))), multiplication(coantidomain(antidomain(X)), X))
% 116.79/15.17 = { by axiom 10 (codomain4) R->L }
% 117.01/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), addition(coantidomain(multiplication(c(X), X)), coantidomain(multiplication(codomain(c(X)), X)))), multiplication(coantidomain(antidomain(X)), X))
% 117.01/15.17 = { by axiom 8 (complement) }
% 117.01/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), addition(coantidomain(multiplication(antidomain(domain(X)), X)), coantidomain(multiplication(codomain(c(X)), X)))), multiplication(coantidomain(antidomain(X)), X))
% 117.01/15.17 = { by lemma 37 R->L }
% 117.01/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), addition(coantidomain(multiplication(antidomain(domain(X)), multiplication(domain(X), X))), coantidomain(multiplication(codomain(c(X)), X)))), multiplication(coantidomain(antidomain(X)), X))
% 117.01/15.17 = { by axiom 14 (multiplicative_associativity) }
% 117.01/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), addition(coantidomain(multiplication(multiplication(antidomain(domain(X)), domain(X)), X)), coantidomain(multiplication(codomain(c(X)), X)))), multiplication(coantidomain(antidomain(X)), X))
% 117.01/15.17 = { by axiom 12 (domain1) }
% 117.01/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), addition(coantidomain(multiplication(zero, X)), coantidomain(multiplication(codomain(c(X)), X)))), multiplication(coantidomain(antidomain(X)), X))
% 117.01/15.17 = { by axiom 3 (left_annihilation) }
% 117.01/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), addition(coantidomain(zero), coantidomain(multiplication(codomain(c(X)), X)))), multiplication(coantidomain(antidomain(X)), X))
% 117.01/15.17 = { by lemma 34 }
% 117.01/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), addition(one, coantidomain(multiplication(codomain(c(X)), X)))), multiplication(coantidomain(antidomain(X)), X))
% 117.01/15.17 = { by lemma 46 }
% 117.01/15.17 addition(multiplication(multiplication(codomain(antidomain(X)), X), one), multiplication(coantidomain(antidomain(X)), X))
% 117.01/15.17 = { by axiom 14 (multiplicative_associativity) R->L }
% 117.01/15.17 addition(multiplication(codomain(antidomain(X)), multiplication(X, one)), multiplication(coantidomain(antidomain(X)), X))
% 117.01/15.17 = { by axiom 2 (multiplicative_right_identity) }
% 117.01/15.17 addition(multiplication(codomain(antidomain(X)), X), multiplication(coantidomain(antidomain(X)), X))
% 117.01/15.17 = { by axiom 24 (left_distributivity) R->L }
% 117.01/15.17 multiplication(addition(codomain(antidomain(X)), coantidomain(antidomain(X))), X)
% 117.01/15.17 = { by axiom 6 (additive_commutativity) }
% 117.01/15.17 multiplication(addition(coantidomain(antidomain(X)), codomain(antidomain(X))), X)
% 117.01/15.17 = { by lemma 33 }
% 117.01/15.17 multiplication(one, X)
% 117.01/15.17 = { by axiom 4 (multiplicative_left_identity) }
% 117.01/15.17 X
% 117.01/15.17
% 117.01/15.17 Lemma 51: coantidomain(antidomain(X)) = domain(X).
% 117.01/15.17 Proof:
% 117.01/15.17 coantidomain(antidomain(X))
% 117.01/15.17 = { by axiom 2 (multiplicative_right_identity) R->L }
% 117.01/15.17 multiplication(coantidomain(antidomain(X)), one)
% 117.01/15.17 = { by lemma 48 R->L }
% 117.01/15.17 multiplication(coantidomain(antidomain(X)), addition(domain(X), codomain(antidomain(X))))
% 117.01/15.17 = { by lemma 49 }
% 117.01/15.17 multiplication(coantidomain(antidomain(X)), domain(X))
% 117.01/15.17 = { by lemma 38 R->L }
% 117.01/15.17 multiplication(coantidomain(c(X)), domain(X))
% 117.01/15.17 = { by axiom 8 (complement) }
% 117.01/15.17 multiplication(coantidomain(antidomain(domain(X))), domain(X))
% 117.01/15.17 = { by lemma 50 }
% 117.01/15.17 domain(X)
% 117.01/15.17
% 117.01/15.17 Lemma 52: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
% 117.01/15.17 Proof:
% 117.01/15.17 multiplication(addition(X, Y), coantidomain(X))
% 117.01/15.17 = { by axiom 6 (additive_commutativity) R->L }
% 117.01/15.17 multiplication(addition(Y, X), coantidomain(X))
% 117.01/15.17 = { by axiom 24 (left_distributivity) }
% 117.01/15.17 addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
% 117.01/15.17 = { by axiom 11 (codomain1) }
% 117.01/15.17 addition(multiplication(Y, coantidomain(X)), zero)
% 117.01/15.17 = { by axiom 7 (additive_identity) }
% 117.01/15.17 multiplication(Y, coantidomain(X))
% 117.01/15.17
% 117.01/15.17 Lemma 53: multiplication(addition(X, Y), coantidomain(Y)) = multiplication(X, coantidomain(Y)).
% 117.01/15.17 Proof:
% 117.01/15.17 multiplication(addition(X, Y), coantidomain(Y))
% 117.01/15.17 = { by axiom 6 (additive_commutativity) R->L }
% 117.01/15.17 multiplication(addition(Y, X), coantidomain(Y))
% 117.01/15.17 = { by lemma 52 }
% 117.01/15.17 multiplication(X, coantidomain(Y))
% 117.01/15.17
% 117.01/15.17 Lemma 54: codomain(coantidomain(X)) = coantidomain(codomain(X)).
% 117.01/15.17 Proof:
% 117.01/15.17 codomain(coantidomain(X))
% 117.01/15.17 = { by axiom 10 (codomain4) }
% 117.01/15.17 coantidomain(coantidomain(coantidomain(X)))
% 117.01/15.17 = { by axiom 10 (codomain4) R->L }
% 117.01/15.18 coantidomain(codomain(X))
% 117.01/15.18
% 117.01/15.18 Lemma 55: coantidomain(codomain(X)) = coantidomain(X).
% 117.01/15.18 Proof:
% 117.01/15.18 coantidomain(codomain(X))
% 117.01/15.18 = { by axiom 4 (multiplicative_left_identity) R->L }
% 117.01/15.18 multiplication(one, coantidomain(codomain(X)))
% 117.01/15.18 = { by lemma 33 R->L }
% 117.01/15.18 multiplication(addition(coantidomain(X), codomain(X)), coantidomain(codomain(X)))
% 117.01/15.18 = { by lemma 53 }
% 117.01/15.18 multiplication(coantidomain(X), coantidomain(codomain(X)))
% 117.01/15.18 = { by lemma 54 R->L }
% 117.01/15.18 multiplication(coantidomain(X), codomain(coantidomain(X)))
% 117.01/15.18 = { by lemma 39 }
% 117.01/15.18 coantidomain(X)
% 117.01/15.18
% 117.01/15.18 Lemma 56: antidomain(forward_diamond(X, antidomain(Y))) = forward_box(X, Y).
% 117.01/15.18 Proof:
% 117.01/15.18 antidomain(forward_diamond(X, antidomain(Y)))
% 117.01/15.18 = { by lemma 38 R->L }
% 117.01/15.18 antidomain(forward_diamond(X, c(Y)))
% 117.01/15.18 = { by lemma 38 R->L }
% 117.01/15.18 c(forward_diamond(X, c(Y)))
% 117.01/15.18 = { by axiom 17 (forward_box) R->L }
% 117.01/15.18 forward_box(X, Y)
% 117.01/15.18
% 117.01/15.18 Lemma 57: domain(forward_box(X, Y)) = forward_box(X, Y).
% 117.01/15.18 Proof:
% 117.01/15.18 domain(forward_box(X, Y))
% 117.01/15.18 = { by lemma 56 R->L }
% 117.01/15.18 domain(antidomain(forward_diamond(X, antidomain(Y))))
% 117.01/15.18 = { by lemma 36 }
% 117.01/15.18 c(forward_diamond(X, antidomain(Y)))
% 117.01/15.18 = { by lemma 38 }
% 117.01/15.18 antidomain(forward_diamond(X, antidomain(Y)))
% 117.01/15.18 = { by lemma 56 }
% 117.01/15.18 forward_box(X, Y)
% 117.01/15.18
% 117.01/15.18 Lemma 58: forward_diamond(X, antidomain(Y)) = antidomain(forward_box(X, Y)).
% 117.01/15.18 Proof:
% 117.01/15.18 forward_diamond(X, antidomain(Y))
% 117.01/15.18 = { by axiom 16 (forward_diamond) }
% 117.01/15.18 domain(multiplication(X, domain(antidomain(Y))))
% 117.01/15.18 = { by axiom 9 (domain4) }
% 117.01/15.18 antidomain(antidomain(multiplication(X, domain(antidomain(Y)))))
% 117.01/15.18 = { by lemma 38 R->L }
% 117.01/15.18 antidomain(c(multiplication(X, domain(antidomain(Y)))))
% 117.01/15.18 = { by axiom 8 (complement) }
% 117.01/15.18 antidomain(antidomain(domain(multiplication(X, domain(antidomain(Y))))))
% 117.01/15.18 = { by axiom 9 (domain4) R->L }
% 117.01/15.18 domain(domain(multiplication(X, domain(antidomain(Y)))))
% 117.01/15.18 = { by axiom 16 (forward_diamond) R->L }
% 117.01/15.18 domain(forward_diamond(X, antidomain(Y)))
% 117.01/15.18 = { by axiom 9 (domain4) }
% 117.01/15.18 antidomain(antidomain(forward_diamond(X, antidomain(Y))))
% 117.01/15.18 = { by lemma 56 }
% 117.01/15.18 antidomain(forward_box(X, Y))
% 117.01/15.18
% 117.01/15.18 Lemma 59: multiplication(domain(X), c(Y)) = domain_difference(X, domain(Y)).
% 117.01/15.18 Proof:
% 117.01/15.18 multiplication(domain(X), c(Y))
% 117.01/15.18 = { by axiom 8 (complement) }
% 117.01/15.18 multiplication(domain(X), antidomain(domain(Y)))
% 117.01/15.18 = { by axiom 13 (domain_difference) R->L }
% 117.01/15.18 domain_difference(X, domain(Y))
% 117.01/15.18
% 117.01/15.18 Lemma 60: antidomain(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)).
% 117.01/15.18 Proof:
% 117.01/15.18 antidomain(multiplication(X, domain(Y)))
% 117.01/15.18 = { by lemma 38 R->L }
% 117.01/15.18 c(multiplication(X, domain(Y)))
% 117.01/15.18 = { by axiom 8 (complement) }
% 117.01/15.18 antidomain(domain(multiplication(X, domain(Y))))
% 117.01/15.18 = { by axiom 16 (forward_diamond) R->L }
% 117.01/15.18 antidomain(forward_diamond(X, Y))
% 117.01/15.18
% 117.01/15.18 Lemma 61: domain(multiplication(X, c(Y))) = forward_diamond(X, antidomain(Y)).
% 117.01/15.18 Proof:
% 117.01/15.18 domain(multiplication(X, c(Y)))
% 117.01/15.18 = { by lemma 36 R->L }
% 117.01/15.18 domain(multiplication(X, domain(antidomain(Y))))
% 117.01/15.18 = { by axiom 16 (forward_diamond) R->L }
% 117.01/15.18 forward_diamond(X, antidomain(Y))
% 117.01/15.18
% 117.01/15.18 Lemma 62: multiplication(X, multiplication(codomain(X), Y)) = multiplication(X, Y).
% 117.01/15.18 Proof:
% 117.01/15.18 multiplication(X, multiplication(codomain(X), Y))
% 117.01/15.18 = { by axiom 14 (multiplicative_associativity) }
% 117.01/15.18 multiplication(multiplication(X, codomain(X)), Y)
% 117.01/15.18 = { by lemma 39 }
% 117.01/15.18 multiplication(X, Y)
% 117.01/15.18
% 117.01/15.18 Lemma 63: codomain(multiplication(backward_diamond(X, Y), Z)) = backward_diamond(Z, backward_diamond(X, Y)).
% 117.01/15.18 Proof:
% 117.01/15.18 codomain(multiplication(backward_diamond(X, Y), Z))
% 117.01/15.18 = { by axiom 18 (backward_diamond) }
% 117.01/15.18 codomain(multiplication(codomain(multiplication(codomain(Y), X)), Z))
% 117.01/15.18 = { by lemma 39 R->L }
% 117.01/15.18 codomain(multiplication(multiplication(codomain(multiplication(codomain(Y), X)), codomain(codomain(multiplication(codomain(Y), X)))), Z))
% 117.01/15.18 = { by axiom 10 (codomain4) }
% 117.01/15.18 codomain(multiplication(multiplication(codomain(multiplication(codomain(Y), X)), coantidomain(coantidomain(codomain(multiplication(codomain(Y), X))))), Z))
% 117.01/15.18 = { by lemma 53 R->L }
% 117.01/15.18 codomain(multiplication(multiplication(addition(codomain(multiplication(codomain(Y), X)), coantidomain(codomain(multiplication(codomain(Y), X)))), coantidomain(coantidomain(codomain(multiplication(codomain(Y), X))))), Z))
% 117.01/15.18 = { by axiom 10 (codomain4) }
% 117.01/15.18 codomain(multiplication(multiplication(addition(coantidomain(coantidomain(multiplication(codomain(Y), X))), coantidomain(codomain(multiplication(codomain(Y), X)))), coantidomain(coantidomain(codomain(multiplication(codomain(Y), X))))), Z))
% 117.01/15.18 = { by lemma 54 R->L }
% 117.01/15.18 codomain(multiplication(multiplication(addition(coantidomain(coantidomain(multiplication(codomain(Y), X))), codomain(coantidomain(multiplication(codomain(Y), X)))), coantidomain(coantidomain(codomain(multiplication(codomain(Y), X))))), Z))
% 117.01/15.18 = { by lemma 33 }
% 117.01/15.18 codomain(multiplication(multiplication(one, coantidomain(coantidomain(codomain(multiplication(codomain(Y), X))))), Z))
% 117.01/15.18 = { by axiom 4 (multiplicative_left_identity) }
% 117.01/15.18 codomain(multiplication(coantidomain(coantidomain(codomain(multiplication(codomain(Y), X)))), Z))
% 117.01/15.18 = { by axiom 10 (codomain4) R->L }
% 117.01/15.18 codomain(multiplication(codomain(codomain(multiplication(codomain(Y), X))), Z))
% 117.01/15.18 = { by axiom 18 (backward_diamond) R->L }
% 117.01/15.18 codomain(multiplication(codomain(backward_diamond(X, Y)), Z))
% 117.01/15.18 = { by axiom 18 (backward_diamond) R->L }
% 117.01/15.18 backward_diamond(Z, backward_diamond(X, Y))
% 117.01/15.18
% 117.01/15.18 Lemma 64: backward_diamond(X, multiplication(codomain(Y), Z)) = backward_diamond(X, backward_diamond(Z, Y)).
% 117.01/15.18 Proof:
% 117.01/15.18 backward_diamond(X, multiplication(codomain(Y), Z))
% 117.01/15.18 = { by axiom 18 (backward_diamond) }
% 117.01/15.18 codomain(multiplication(codomain(multiplication(codomain(Y), Z)), X))
% 117.01/15.18 = { by axiom 18 (backward_diamond) R->L }
% 117.01/15.18 codomain(multiplication(backward_diamond(Z, Y), X))
% 117.01/15.18 = { by lemma 63 }
% 117.01/15.18 backward_diamond(X, backward_diamond(Z, Y))
% 117.01/15.18
% 117.01/15.18 Lemma 65: backward_diamond(addition(domain(X), forward_box(Y, Z)), domain(X)) = domain(X).
% 117.01/15.18 Proof:
% 117.01/15.18 backward_diamond(addition(domain(X), forward_box(Y, Z)), domain(X))
% 117.01/15.18 = { by lemma 51 R->L }
% 117.01/15.18 backward_diamond(addition(domain(X), forward_box(Y, Z)), coantidomain(antidomain(X)))
% 117.01/15.18 = { by axiom 18 (backward_diamond) }
% 117.01/15.18 codomain(multiplication(codomain(coantidomain(antidomain(X))), addition(domain(X), forward_box(Y, Z))))
% 117.01/15.18 = { by lemma 54 }
% 117.01/15.18 codomain(multiplication(coantidomain(codomain(antidomain(X))), addition(domain(X), forward_box(Y, Z))))
% 117.01/15.18 = { by lemma 55 }
% 117.01/15.18 codomain(multiplication(coantidomain(antidomain(X)), addition(domain(X), forward_box(Y, Z))))
% 117.01/15.18 = { by lemma 49 R->L }
% 117.01/15.18 codomain(multiplication(coantidomain(antidomain(X)), addition(addition(domain(X), forward_box(Y, Z)), codomain(antidomain(X)))))
% 117.01/15.18 = { by lemma 44 R->L }
% 117.01/15.18 codomain(multiplication(coantidomain(antidomain(X)), addition(addition(domain(X), forward_box(Y, Z)), addition(antidomain(X), codomain(antidomain(X))))))
% 117.01/15.18 = { by axiom 15 (additive_associativity) }
% 117.01/15.18 codomain(multiplication(coantidomain(antidomain(X)), addition(addition(addition(domain(X), forward_box(Y, Z)), antidomain(X)), codomain(antidomain(X)))))
% 117.01/15.18 = { by axiom 15 (additive_associativity) R->L }
% 117.01/15.18 codomain(multiplication(coantidomain(antidomain(X)), addition(addition(domain(X), addition(forward_box(Y, Z), antidomain(X))), codomain(antidomain(X)))))
% 117.01/15.18 = { by lemma 45 }
% 117.01/15.18 codomain(multiplication(coantidomain(antidomain(X)), addition(addition(forward_box(Y, Z), one), codomain(antidomain(X)))))
% 117.01/15.18 = { by axiom 6 (additive_commutativity) }
% 117.01/15.18 codomain(multiplication(coantidomain(antidomain(X)), addition(addition(one, forward_box(Y, Z)), codomain(antidomain(X)))))
% 117.01/15.18 = { by axiom 17 (forward_box) }
% 117.01/15.18 codomain(multiplication(coantidomain(antidomain(X)), addition(addition(one, c(forward_diamond(Y, c(Z)))), codomain(antidomain(X)))))
% 117.01/15.18 = { by lemma 43 }
% 117.01/15.18 codomain(multiplication(coantidomain(antidomain(X)), addition(one, codomain(antidomain(X)))))
% 117.01/15.18 = { by lemma 47 }
% 117.01/15.18 codomain(multiplication(coantidomain(antidomain(X)), one))
% 117.01/15.18 = { by axiom 2 (multiplicative_right_identity) }
% 117.01/15.18 codomain(coantidomain(antidomain(X)))
% 117.01/15.18 = { by axiom 10 (codomain4) }
% 117.01/15.18 coantidomain(coantidomain(coantidomain(antidomain(X))))
% 117.01/15.18 = { by axiom 10 (codomain4) R->L }
% 117.01/15.18 coantidomain(codomain(antidomain(X)))
% 117.01/15.18 = { by lemma 55 }
% 117.01/15.18 coantidomain(antidomain(X))
% 117.01/15.18 = { by lemma 51 }
% 117.01/15.18 domain(X)
% 117.01/15.18
% 117.01/15.18 Goal 1 (goals_1): addition(domain(x1), forward_box(x0, domain(x2))) = forward_box(x0, domain(x2)).
% 117.01/15.18 Proof:
% 117.01/15.18 addition(domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.18 = { by axiom 27 (order_1) R->L }
% 117.01/15.18 fresh2(leq(domain(x1), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.18 = { by axiom 2 (multiplicative_right_identity) R->L }
% 117.01/15.18 fresh2(leq(multiplication(domain(x1), one), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.18 = { by lemma 35 R->L }
% 117.01/15.18 fresh2(leq(multiplication(domain(x1), addition(domain(forward_box(x0, domain(x2))), antidomain(forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.18 = { by axiom 6 (additive_commutativity) R->L }
% 117.01/15.18 fresh2(leq(multiplication(domain(x1), addition(antidomain(forward_box(x0, domain(x2))), domain(forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.18 = { by axiom 23 (right_distributivity) }
% 117.01/15.18 fresh2(leq(addition(multiplication(domain(x1), antidomain(forward_box(x0, domain(x2)))), multiplication(domain(x1), domain(forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.18 = { by axiom 13 (domain_difference) R->L }
% 117.01/15.18 fresh2(leq(addition(domain_difference(x1, forward_box(x0, domain(x2))), multiplication(domain(x1), domain(forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.18 = { by axiom 6 (additive_commutativity) }
% 117.01/15.18 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), domain_difference(x1, forward_box(x0, domain(x2)))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.18 = { by axiom 4 (multiplicative_left_identity) R->L }
% 117.01/15.18 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(one, domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.18 = { by axiom 20 (order_1) R->L }
% 117.01/15.18 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(fresh2(true, true, antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), one), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.18 = { by axiom 19 (order) R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(fresh2(fresh(addition(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), domain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), addition(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), domain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), addition(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), domain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))))), true, antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), one), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 41 R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(fresh2(fresh(addition(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), addition(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), domain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))))), addition(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), domain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), addition(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), domain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))))), true, antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), one), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 26 (order) }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(fresh2(leq(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), addition(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), domain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))))), true, antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), one), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 6 (additive_commutativity) }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(fresh2(leq(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), addition(domain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))))), true, antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), one), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 35 }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(fresh2(leq(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), one), true, antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), one), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 27 (order_1) }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), one), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 6 (additive_commutativity) }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(one, antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 35 R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(addition(domain(one), antidomain(one)), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 30 }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(addition(domain(one), zero), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 7 (additive_identity) }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(domain(one), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 9 (domain4) }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(antidomain(one)), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 30 }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(zero), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 1 (right_annihilation) R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(domain(x1), zero)), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 1 (right_annihilation) R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(domain(x1), multiplication(x0, zero))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 1 (right_annihilation) R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(domain(x1), multiplication(x0, multiplication(antidomain(x2), zero)))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 14 (multiplicative_associativity) }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), zero))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 31 R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), coantidomain(one)))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 34 R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), coantidomain(coantidomain(zero))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 10 (codomain4) R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), codomain(zero)))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 11 (codomain1) R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), codomain(multiplication(domain(x2), coantidomain(domain(x2))))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 25 (goals) R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), codomain(multiplication(addition(backward_diamond(x0, domain(x1)), domain(x2)), coantidomain(domain(x2))))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 6 (additive_commutativity) }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), codomain(multiplication(addition(domain(x2), backward_diamond(x0, domain(x1))), coantidomain(domain(x2))))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 52 }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), codomain(multiplication(backward_diamond(x0, domain(x1)), coantidomain(domain(x2))))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 63 }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(coantidomain(domain(x2)), backward_diamond(x0, domain(x1)))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 65 R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(coantidomain(domain(x2)), backward_diamond(x0, backward_diamond(addition(domain(x1), forward_box(X, Y)), domain(x1))))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 64 R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(coantidomain(domain(x2)), backward_diamond(x0, multiplication(codomain(domain(x1)), addition(domain(x1), forward_box(X, Y)))))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by lemma 64 R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(coantidomain(domain(x2)), multiplication(codomain(multiplication(codomain(domain(x1)), addition(domain(x1), forward_box(X, Y)))), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.19 = { by axiom 18 (backward_diamond) R->L }
% 117.01/15.19 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(coantidomain(domain(x2)), multiplication(backward_diamond(addition(domain(x1), forward_box(X, Y)), domain(x1)), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 65 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(coantidomain(domain(x2)), multiplication(domain(x1), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 2 (multiplicative_right_identity) R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(multiplication(coantidomain(domain(x2)), one), multiplication(domain(x1), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 48 R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(multiplication(coantidomain(domain(x2)), addition(domain(antidomain(x2)), codomain(antidomain(antidomain(x2))))), multiplication(domain(x1), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 9 (domain4) R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(multiplication(coantidomain(domain(x2)), addition(domain(antidomain(x2)), codomain(domain(x2)))), multiplication(domain(x1), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 9 (domain4) }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(multiplication(coantidomain(domain(x2)), addition(antidomain(antidomain(antidomain(x2))), codomain(domain(x2)))), multiplication(domain(x1), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 9 (domain4) R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(multiplication(coantidomain(domain(x2)), addition(antidomain(domain(x2)), codomain(domain(x2)))), multiplication(domain(x1), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 8 (complement) R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(multiplication(coantidomain(domain(x2)), addition(c(x2), codomain(domain(x2)))), multiplication(domain(x1), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 38 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(multiplication(coantidomain(domain(x2)), addition(antidomain(x2), codomain(domain(x2)))), multiplication(domain(x1), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 49 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(multiplication(coantidomain(domain(x2)), antidomain(x2)), multiplication(domain(x1), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 9 (domain4) }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(multiplication(coantidomain(antidomain(antidomain(x2))), antidomain(x2)), multiplication(domain(x1), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 50 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), backward_diamond(antidomain(x2), multiplication(domain(x1), x0))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 18 (backward_diamond) }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(antidomain(x2), codomain(multiplication(codomain(multiplication(domain(x1), x0)), antidomain(x2)))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 62 R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(codomain(multiplication(domain(x1), x0)), multiplication(antidomain(x2), codomain(multiplication(codomain(multiplication(domain(x1), x0)), antidomain(x2))))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 14 (multiplicative_associativity) }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(multiplication(codomain(multiplication(domain(x1), x0)), antidomain(x2)), codomain(multiplication(codomain(multiplication(domain(x1), x0)), antidomain(x2)))))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 39 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), multiplication(codomain(multiplication(domain(x1), x0)), antidomain(x2)))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 62 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(multiplication(domain(x1), x0), antidomain(x2))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 14 (multiplicative_associativity) R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(domain(x1), multiplication(x0, antidomain(x2)))), antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 60 R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(domain(x1), multiplication(x0, antidomain(x2)))), antidomain(multiplication(domain(x1), domain(multiplication(x0, antidomain(x2)))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 9 (domain4) }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(addition(antidomain(multiplication(domain(x1), multiplication(x0, antidomain(x2)))), antidomain(multiplication(domain(x1), antidomain(antidomain(multiplication(x0, antidomain(x2))))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 28 (domain2) }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(multiplication(domain(x1), antidomain(antidomain(multiplication(x0, antidomain(x2)))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 9 (domain4) R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(multiplication(domain(x1), domain(multiplication(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 60 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(forward_diamond(domain(x1), multiplication(x0, antidomain(x2)))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 38 R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(forward_diamond(domain(x1), multiplication(x0, c(x2)))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 36 R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(forward_diamond(domain(x1), multiplication(x0, domain(antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 16 (forward_diamond) }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(domain(multiplication(domain(x1), domain(multiplication(x0, domain(antidomain(x2))))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 16 (forward_diamond) R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(domain(multiplication(domain(x1), forward_diamond(x0, antidomain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 58 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(domain(multiplication(domain(x1), antidomain(forward_box(x0, x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 38 R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(domain(multiplication(domain(x1), c(forward_box(x0, x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 61 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(forward_diamond(domain(x1), antidomain(forward_box(x0, x2)))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 58 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(antidomain(forward_box(domain(x1), forward_box(x0, x2)))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 9 (domain4) R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(domain(forward_box(domain(x1), forward_box(x0, x2))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 57 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(forward_box(domain(x1), forward_box(x0, x2)), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 56 R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(forward_box(domain(x1), antidomain(forward_diamond(x0, antidomain(x2)))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 38 R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(forward_box(domain(x1), antidomain(forward_diamond(x0, c(x2)))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 8 (complement) }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(forward_box(domain(x1), antidomain(forward_diamond(x0, antidomain(domain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 56 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(forward_box(domain(x1), forward_box(x0, domain(x2))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 56 R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(forward_diamond(domain(x1), antidomain(forward_box(x0, domain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 61 R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(domain(multiplication(domain(x1), c(forward_box(x0, domain(x2)))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 59 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(domain(domain_difference(x1, domain(forward_box(x0, domain(x2)))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by axiom 8 (complement) R->L }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(c(domain_difference(x1, domain(forward_box(x0, domain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 38 }
% 117.01/15.20 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(domain_difference(x1, domain(forward_box(x0, domain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.20 = { by lemma 59 R->L }
% 117.01/15.21 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(multiplication(domain(x1), c(forward_box(x0, domain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by lemma 38 }
% 117.01/15.21 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(multiplication(domain(x1), antidomain(forward_box(x0, domain(x2))))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 13 (domain_difference) R->L }
% 117.01/15.21 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), multiplication(antidomain(domain_difference(x1, forward_box(x0, domain(x2)))), domain_difference(x1, forward_box(x0, domain(x2))))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 12 (domain1) }
% 117.01/15.21 fresh2(leq(addition(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), zero), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 7 (additive_identity) }
% 117.01/15.21 fresh2(leq(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by lemma 57 }
% 117.01/15.21 fresh2(leq(multiplication(domain(x1), forward_box(x0, domain(x2))), forward_box(x0, domain(x2))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by lemma 57 R->L }
% 117.01/15.21 fresh2(leq(multiplication(domain(x1), forward_box(x0, domain(x2))), domain(forward_box(x0, domain(x2)))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 9 (domain4) }
% 117.01/15.21 fresh2(leq(multiplication(domain(x1), forward_box(x0, domain(x2))), antidomain(antidomain(forward_box(x0, domain(x2))))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by lemma 57 R->L }
% 117.01/15.21 fresh2(leq(multiplication(domain(x1), domain(forward_box(x0, domain(x2)))), antidomain(antidomain(forward_box(x0, domain(x2))))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 9 (domain4) }
% 117.01/15.21 fresh2(leq(multiplication(domain(x1), antidomain(antidomain(forward_box(x0, domain(x2))))), antidomain(antidomain(forward_box(x0, domain(x2))))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 13 (domain_difference) R->L }
% 117.01/15.21 fresh2(leq(domain_difference(x1, antidomain(forward_box(x0, domain(x2)))), antidomain(antidomain(forward_box(x0, domain(x2))))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 26 (order) R->L }
% 117.01/15.21 fresh2(fresh(addition(domain_difference(x1, antidomain(forward_box(x0, domain(x2)))), antidomain(antidomain(forward_box(x0, domain(x2))))), antidomain(antidomain(forward_box(x0, domain(x2)))), domain_difference(x1, antidomain(forward_box(x0, domain(x2)))), antidomain(antidomain(forward_box(x0, domain(x2))))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 6 (additive_commutativity) }
% 117.01/15.21 fresh2(fresh(addition(antidomain(antidomain(forward_box(x0, domain(x2)))), domain_difference(x1, antidomain(forward_box(x0, domain(x2))))), antidomain(antidomain(forward_box(x0, domain(x2)))), domain_difference(x1, antidomain(forward_box(x0, domain(x2)))), antidomain(antidomain(forward_box(x0, domain(x2))))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 13 (domain_difference) }
% 117.01/15.21 fresh2(fresh(addition(antidomain(antidomain(forward_box(x0, domain(x2)))), multiplication(domain(x1), antidomain(antidomain(forward_box(x0, domain(x2)))))), antidomain(antidomain(forward_box(x0, domain(x2)))), domain_difference(x1, antidomain(forward_box(x0, domain(x2)))), antidomain(antidomain(forward_box(x0, domain(x2))))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by lemma 40 }
% 117.01/15.21 fresh2(fresh(multiplication(addition(domain(x1), one), antidomain(antidomain(forward_box(x0, domain(x2))))), antidomain(antidomain(forward_box(x0, domain(x2)))), domain_difference(x1, antidomain(forward_box(x0, domain(x2)))), antidomain(antidomain(forward_box(x0, domain(x2))))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 6 (additive_commutativity) }
% 117.01/15.21 fresh2(fresh(multiplication(addition(one, domain(x1)), antidomain(antidomain(forward_box(x0, domain(x2))))), antidomain(antidomain(forward_box(x0, domain(x2)))), domain_difference(x1, antidomain(forward_box(x0, domain(x2)))), antidomain(antidomain(forward_box(x0, domain(x2))))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by lemma 42 }
% 117.01/15.21 fresh2(fresh(multiplication(one, antidomain(antidomain(forward_box(x0, domain(x2))))), antidomain(antidomain(forward_box(x0, domain(x2)))), domain_difference(x1, antidomain(forward_box(x0, domain(x2)))), antidomain(antidomain(forward_box(x0, domain(x2))))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 4 (multiplicative_left_identity) }
% 117.01/15.21 fresh2(fresh(antidomain(antidomain(forward_box(x0, domain(x2)))), antidomain(antidomain(forward_box(x0, domain(x2)))), domain_difference(x1, antidomain(forward_box(x0, domain(x2)))), antidomain(antidomain(forward_box(x0, domain(x2))))), true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 19 (order) }
% 117.01/15.21 fresh2(true, true, domain(x1), forward_box(x0, domain(x2)))
% 117.01/15.21 = { by axiom 20 (order_1) }
% 117.01/15.21 forward_box(x0, domain(x2))
% 117.01/15.21 % SZS output end Proof
% 117.01/15.21
% 117.01/15.21 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------