TSTP Solution File: KLE087+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE087+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 : n021.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 05:35:50 EDT 2023
% Result : Theorem 89.13s 11.84s
% Output : Proof 90.15s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.14 % Problem : KLE087+1 : TPTP v8.1.2. Released v4.0.0.
% 0.08/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.36 % Computer : n021.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Tue Aug 29 12:15:14 EDT 2023
% 0.14/0.36 % CPUTime :
% 89.13/11.84 Command-line arguments: --no-flatten-goal
% 89.13/11.84
% 89.13/11.84 % SZS status Theorem
% 89.13/11.84
% 90.05/11.90 % SZS output start Proof
% 90.05/11.90 Take the following subset of the input axioms:
% 90.05/11.91 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 90.05/11.91 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 90.05/11.91 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 90.05/11.91 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 90.05/11.91 fof(codomain1, axiom, ![X0]: multiplication(X0, coantidomain(X0))=zero).
% 90.05/11.91 fof(codomain2, axiom, ![X1, X0_2]: addition(coantidomain(multiplication(X0_2, X1)), coantidomain(multiplication(coantidomain(coantidomain(X0_2)), X1)))=coantidomain(multiplication(coantidomain(coantidomain(X0_2)), X1))).
% 90.05/11.91 fof(codomain3, axiom, ![X0_2]: addition(coantidomain(coantidomain(X0_2)), coantidomain(X0_2))=one).
% 90.05/11.91 fof(codomain4, axiom, ![X0_2]: codomain(X0_2)=coantidomain(coantidomain(X0_2))).
% 90.05/11.91 fof(domain1, axiom, ![X0_2]: multiplication(antidomain(X0_2), X0_2)=zero).
% 90.05/11.91 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))))).
% 90.05/11.91 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 90.05/11.91 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 90.05/11.91 fof(goals, conjecture, ![X0_2, X1_2]: domain(addition(X0_2, X1_2))=addition(domain(X0_2), domain(X1_2))).
% 90.05/11.91 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 90.05/11.91 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 90.05/11.91 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 90.05/11.91 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 90.05/11.91 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 90.05/11.91 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 90.05/11.91 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 90.05/11.91
% 90.05/11.91 Now clausify the problem and encode Horn clauses using encoding 3 of
% 90.05/11.91 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 90.05/11.91 We repeatedly replace C & s=t => u=v by the two clauses:
% 90.05/11.91 fresh(y, y, x1...xn) = u
% 90.05/11.91 C => fresh(s, t, x1...xn) = v
% 90.05/11.91 where fresh is a fresh function symbol and x1..xn are the free
% 90.05/11.91 variables of u and v.
% 90.05/11.91 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 90.05/11.91 input problem has no model of domain size 1).
% 90.05/11.91
% 90.05/11.91 The encoding turns the above axioms into the following unit equations and goals:
% 90.05/11.91
% 90.05/11.91 Axiom 1 (domain4): domain(X) = antidomain(antidomain(X)).
% 90.05/11.91 Axiom 2 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 90.05/11.91 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 90.05/11.91 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 90.05/11.91 Axiom 5 (additive_identity): addition(X, zero) = X.
% 90.05/11.91 Axiom 6 (multiplicative_right_identity): multiplication(X, one) = X.
% 90.05/11.91 Axiom 7 (multiplicative_left_identity): multiplication(one, X) = X.
% 90.05/11.91 Axiom 8 (left_annihilation): multiplication(zero, X) = zero.
% 90.05/11.91 Axiom 9 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 90.05/11.91 Axiom 10 (domain1): multiplication(antidomain(X), X) = zero.
% 90.05/11.91 Axiom 11 (order): fresh(X, X, Y, Z) = true.
% 90.05/11.91 Axiom 12 (order_1): fresh2(X, X, Y, Z) = Z.
% 90.05/11.91 Axiom 13 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 90.05/11.91 Axiom 14 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 90.05/11.91 Axiom 15 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 90.05/11.91 Axiom 16 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 90.05/11.91 Axiom 17 (order): fresh(addition(X, Y), Y, X, Y) = leq(X, Y).
% 90.05/11.91 Axiom 18 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 90.05/11.91 Axiom 19 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 90.05/11.91 Axiom 20 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 90.05/11.91 Axiom 21 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 90.05/11.91 Axiom 22 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
% 90.05/11.91
% 90.05/11.91 Lemma 23: antidomain(one) = zero.
% 90.05/11.91 Proof:
% 90.05/11.91 antidomain(one)
% 90.05/11.91 = { by axiom 6 (multiplicative_right_identity) R->L }
% 90.05/11.91 multiplication(antidomain(one), one)
% 90.05/11.91 = { by axiom 10 (domain1) }
% 90.05/11.91 zero
% 90.05/11.91
% 90.05/11.91 Lemma 24: addition(domain(X), antidomain(X)) = one.
% 90.05/11.91 Proof:
% 90.05/11.91 addition(domain(X), antidomain(X))
% 90.05/11.91 = { by axiom 1 (domain4) }
% 90.05/11.91 addition(antidomain(antidomain(X)), antidomain(X))
% 90.05/11.91 = { by axiom 15 (domain3) }
% 90.05/11.91 one
% 90.05/11.91
% 90.05/11.91 Lemma 25: antidomain(zero) = one.
% 90.05/11.91 Proof:
% 90.05/11.91 antidomain(zero)
% 90.05/11.91 = { by lemma 23 R->L }
% 90.05/11.91 antidomain(antidomain(one))
% 90.05/11.91 = { by axiom 1 (domain4) R->L }
% 90.05/11.91 domain(one)
% 90.05/11.91 = { by axiom 5 (additive_identity) R->L }
% 90.05/11.91 addition(domain(one), zero)
% 90.05/11.91 = { by lemma 23 R->L }
% 90.05/11.91 addition(domain(one), antidomain(one))
% 90.05/11.91 = { by lemma 24 }
% 90.05/11.91 one
% 90.05/11.91
% 90.05/11.91 Lemma 26: coantidomain(one) = zero.
% 90.05/11.91 Proof:
% 90.05/11.91 coantidomain(one)
% 90.05/11.91 = { by axiom 7 (multiplicative_left_identity) R->L }
% 90.05/11.91 multiplication(one, coantidomain(one))
% 90.05/11.91 = { by axiom 9 (codomain1) }
% 90.05/11.91 zero
% 90.05/11.91
% 90.05/11.91 Lemma 27: addition(zero, X) = X.
% 90.05/11.91 Proof:
% 90.05/11.91 addition(zero, X)
% 90.05/11.91 = { by axiom 4 (additive_commutativity) R->L }
% 90.05/11.91 addition(X, zero)
% 90.05/11.91 = { by axiom 5 (additive_identity) }
% 90.05/11.91 X
% 90.05/11.91
% 90.05/11.91 Lemma 28: antidomain(domain(X)) = domain(antidomain(X)).
% 90.05/11.91 Proof:
% 90.05/11.91 antidomain(domain(X))
% 90.05/11.91 = { by axiom 1 (domain4) }
% 90.05/11.91 antidomain(antidomain(antidomain(X)))
% 90.05/11.91 = { by axiom 1 (domain4) R->L }
% 90.05/11.91 domain(antidomain(X))
% 90.05/11.91
% 90.05/11.91 Lemma 29: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 90.05/11.91 Proof:
% 90.05/11.91 multiplication(antidomain(X), addition(X, Y))
% 90.05/11.91 = { by axiom 4 (additive_commutativity) R->L }
% 90.05/11.91 multiplication(antidomain(X), addition(Y, X))
% 90.05/11.91 = { by axiom 19 (right_distributivity) }
% 90.05/11.91 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 90.05/11.91 = { by axiom 10 (domain1) }
% 90.05/11.91 addition(multiplication(antidomain(X), Y), zero)
% 90.05/11.91 = { by axiom 5 (additive_identity) }
% 90.05/11.91 multiplication(antidomain(X), Y)
% 90.05/11.91
% 90.05/11.91 Lemma 30: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
% 90.05/11.91 Proof:
% 90.05/11.91 multiplication(addition(X, antidomain(Y)), Y)
% 90.05/11.91 = { by axiom 20 (left_distributivity) }
% 90.05/11.91 addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
% 90.05/11.91 = { by axiom 10 (domain1) }
% 90.05/11.91 addition(multiplication(X, Y), zero)
% 90.05/11.91 = { by axiom 5 (additive_identity) }
% 90.05/11.91 multiplication(X, Y)
% 90.05/11.91
% 90.05/11.91 Lemma 31: multiplication(domain(X), X) = X.
% 90.05/11.91 Proof:
% 90.05/11.91 multiplication(domain(X), X)
% 90.05/11.91 = { by lemma 30 R->L }
% 90.05/11.91 multiplication(addition(domain(X), antidomain(X)), X)
% 90.05/11.91 = { by lemma 24 }
% 90.05/11.91 multiplication(one, X)
% 90.05/11.91 = { by axiom 7 (multiplicative_left_identity) }
% 90.05/11.91 X
% 90.05/11.91
% 90.05/11.91 Lemma 32: domain(antidomain(X)) = antidomain(X).
% 90.05/11.91 Proof:
% 90.05/11.91 domain(antidomain(X))
% 90.05/11.91 = { by lemma 28 R->L }
% 90.05/11.91 antidomain(domain(X))
% 90.05/11.91 = { by axiom 6 (multiplicative_right_identity) R->L }
% 90.05/11.91 multiplication(antidomain(domain(X)), one)
% 90.05/11.91 = { by lemma 24 R->L }
% 90.05/11.91 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 90.05/11.91 = { by lemma 29 }
% 90.05/11.91 multiplication(antidomain(domain(X)), antidomain(X))
% 90.05/11.91 = { by lemma 28 }
% 90.05/11.91 multiplication(domain(antidomain(X)), antidomain(X))
% 90.05/11.91 = { by lemma 31 }
% 90.05/11.91 antidomain(X)
% 90.05/11.91
% 90.05/11.91 Lemma 33: domain(domain(X)) = domain(X).
% 90.05/11.91 Proof:
% 90.05/11.91 domain(domain(X))
% 90.05/11.91 = { by axiom 1 (domain4) }
% 90.05/11.91 domain(antidomain(antidomain(X)))
% 90.05/11.91 = { by lemma 32 }
% 90.05/11.91 antidomain(antidomain(X))
% 90.05/11.91 = { by axiom 1 (domain4) R->L }
% 90.05/11.91 domain(X)
% 90.05/11.91
% 90.05/11.91 Lemma 34: addition(X, addition(X, Y)) = addition(X, Y).
% 90.05/11.91 Proof:
% 90.05/11.91 addition(X, addition(X, Y))
% 90.05/11.91 = { by axiom 13 (additive_associativity) }
% 90.05/11.91 addition(addition(X, X), Y)
% 90.05/11.91 = { by axiom 3 (additive_idempotence) }
% 90.05/11.91 addition(X, Y)
% 90.05/11.91
% 90.05/11.91 Lemma 35: leq(X, addition(Y, X)) = true.
% 90.05/11.91 Proof:
% 90.05/11.91 leq(X, addition(Y, X))
% 90.05/11.91 = { by axiom 4 (additive_commutativity) R->L }
% 90.05/11.91 leq(X, addition(X, Y))
% 90.05/11.91 = { by axiom 17 (order) R->L }
% 90.05/11.91 fresh(addition(X, addition(X, Y)), addition(X, Y), X, addition(X, Y))
% 90.05/11.91 = { by lemma 34 }
% 90.05/11.91 fresh(addition(X, Y), addition(X, Y), X, addition(X, Y))
% 90.05/11.91 = { by axiom 11 (order) }
% 90.05/11.91 true
% 90.05/11.91
% 90.05/11.91 Lemma 36: addition(one, antidomain(X)) = one.
% 90.05/11.91 Proof:
% 90.05/11.91 addition(one, antidomain(X))
% 90.05/11.91 = { by axiom 4 (additive_commutativity) R->L }
% 90.05/11.91 addition(antidomain(X), one)
% 90.05/11.91 = { by axiom 18 (order_1) R->L }
% 90.05/11.91 fresh2(leq(antidomain(X), one), true, antidomain(X), one)
% 90.05/11.91 = { by lemma 24 R->L }
% 90.05/11.91 fresh2(leq(antidomain(X), addition(domain(X), antidomain(X))), true, antidomain(X), one)
% 90.05/11.91 = { by lemma 35 }
% 90.05/11.91 fresh2(true, true, antidomain(X), one)
% 90.05/11.91 = { by axiom 12 (order_1) }
% 90.05/11.91 one
% 90.05/11.91
% 90.05/11.91 Lemma 37: addition(codomain(X), coantidomain(X)) = one.
% 90.05/11.91 Proof:
% 90.05/11.91 addition(codomain(X), coantidomain(X))
% 90.05/11.91 = { by axiom 2 (codomain4) }
% 90.05/11.91 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 90.05/11.91 = { by axiom 16 (codomain3) }
% 90.05/11.91 one
% 90.05/11.91
% 90.05/11.91 Lemma 38: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
% 90.05/11.91 Proof:
% 90.05/11.91 multiplication(addition(X, Y), coantidomain(X))
% 90.05/11.91 = { by axiom 4 (additive_commutativity) R->L }
% 90.05/11.91 multiplication(addition(Y, X), coantidomain(X))
% 90.05/11.91 = { by axiom 20 (left_distributivity) }
% 90.05/11.91 addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
% 90.05/11.91 = { by axiom 9 (codomain1) }
% 90.05/11.91 addition(multiplication(Y, coantidomain(X)), zero)
% 90.05/11.91 = { by axiom 5 (additive_identity) }
% 90.05/11.91 multiplication(Y, coantidomain(X))
% 90.05/11.91
% 90.05/11.91 Lemma 39: multiplication(X, addition(one, Y)) = addition(X, multiplication(X, Y)).
% 90.05/11.91 Proof:
% 90.05/11.91 multiplication(X, addition(one, Y))
% 90.05/11.91 = { by axiom 19 (right_distributivity) }
% 90.05/11.91 addition(multiplication(X, one), multiplication(X, Y))
% 90.05/11.91 = { by axiom 6 (multiplicative_right_identity) }
% 90.05/11.91 addition(X, multiplication(X, Y))
% 90.05/11.91
% 90.05/11.91 Lemma 40: addition(one, coantidomain(X)) = one.
% 90.05/11.91 Proof:
% 90.05/11.91 addition(one, coantidomain(X))
% 90.05/11.91 = { by axiom 4 (additive_commutativity) R->L }
% 90.05/11.91 addition(coantidomain(X), one)
% 90.05/11.91 = { by axiom 18 (order_1) R->L }
% 90.05/11.91 fresh2(leq(coantidomain(X), one), true, coantidomain(X), one)
% 90.05/11.91 = { by lemma 37 R->L }
% 90.05/11.91 fresh2(leq(coantidomain(X), addition(codomain(X), coantidomain(X))), true, coantidomain(X), one)
% 90.05/11.91 = { by lemma 35 }
% 90.05/11.91 fresh2(true, true, coantidomain(X), one)
% 90.05/11.91 = { by axiom 12 (order_1) }
% 90.05/11.91 one
% 90.05/11.91
% 90.05/11.91 Lemma 41: multiplication(domain(X), antidomain(X)) = zero.
% 90.05/11.91 Proof:
% 90.05/11.91 multiplication(domain(X), antidomain(X))
% 90.05/11.91 = { by axiom 1 (domain4) }
% 90.05/11.91 multiplication(antidomain(antidomain(X)), antidomain(X))
% 90.05/11.91 = { by axiom 10 (domain1) }
% 90.05/11.91 zero
% 90.05/11.91
% 90.05/11.91 Lemma 42: coantidomain(antidomain(X)) = domain(X).
% 90.05/11.91 Proof:
% 90.05/11.91 coantidomain(antidomain(X))
% 90.05/11.91 = { by axiom 6 (multiplicative_right_identity) R->L }
% 90.05/11.91 multiplication(coantidomain(antidomain(X)), one)
% 90.05/11.91 = { by lemma 36 R->L }
% 90.05/11.91 multiplication(coantidomain(antidomain(X)), addition(one, antidomain(antidomain(X))))
% 90.05/11.91 = { by lemma 37 R->L }
% 90.05/11.91 multiplication(coantidomain(antidomain(X)), addition(addition(codomain(domain(antidomain(X))), coantidomain(domain(antidomain(X)))), antidomain(antidomain(X))))
% 90.05/11.91 = { by axiom 13 (additive_associativity) R->L }
% 90.05/11.91 multiplication(coantidomain(antidomain(X)), addition(codomain(domain(antidomain(X))), addition(coantidomain(domain(antidomain(X))), antidomain(antidomain(X)))))
% 90.05/11.91 = { by axiom 4 (additive_commutativity) }
% 90.05/11.91 multiplication(coantidomain(antidomain(X)), addition(codomain(domain(antidomain(X))), addition(antidomain(antidomain(X)), coantidomain(domain(antidomain(X))))))
% 90.05/11.91 = { by axiom 7 (multiplicative_left_identity) R->L }
% 90.05/11.91 multiplication(coantidomain(antidomain(X)), addition(codomain(domain(antidomain(X))), addition(antidomain(antidomain(X)), multiplication(one, coantidomain(domain(antidomain(X)))))))
% 90.05/11.91 = { by lemma 24 R->L }
% 90.05/11.91 multiplication(coantidomain(antidomain(X)), addition(codomain(domain(antidomain(X))), addition(antidomain(antidomain(X)), multiplication(addition(domain(antidomain(X)), antidomain(antidomain(X))), coantidomain(domain(antidomain(X)))))))
% 90.05/11.91 = { by lemma 38 }
% 90.15/11.91 multiplication(coantidomain(antidomain(X)), addition(codomain(domain(antidomain(X))), addition(antidomain(antidomain(X)), multiplication(antidomain(antidomain(X)), coantidomain(domain(antidomain(X)))))))
% 90.15/11.91 = { by lemma 39 R->L }
% 90.15/11.91 multiplication(coantidomain(antidomain(X)), addition(codomain(domain(antidomain(X))), multiplication(antidomain(antidomain(X)), addition(one, coantidomain(domain(antidomain(X)))))))
% 90.15/11.91 = { by lemma 40 }
% 90.15/11.91 multiplication(coantidomain(antidomain(X)), addition(codomain(domain(antidomain(X))), multiplication(antidomain(antidomain(X)), one)))
% 90.15/11.91 = { by axiom 6 (multiplicative_right_identity) }
% 90.15/11.91 multiplication(coantidomain(antidomain(X)), addition(codomain(domain(antidomain(X))), antidomain(antidomain(X))))
% 90.15/11.91 = { by axiom 4 (additive_commutativity) }
% 90.15/11.91 multiplication(coantidomain(antidomain(X)), addition(antidomain(antidomain(X)), codomain(domain(antidomain(X)))))
% 90.15/11.91 = { by axiom 1 (domain4) R->L }
% 90.15/11.91 multiplication(coantidomain(antidomain(X)), addition(domain(X), codomain(domain(antidomain(X)))))
% 90.15/11.91 = { by lemma 32 }
% 90.15/11.91 multiplication(coantidomain(antidomain(X)), addition(domain(X), codomain(antidomain(X))))
% 90.15/11.91 = { by axiom 4 (additive_commutativity) R->L }
% 90.15/11.91 multiplication(coantidomain(antidomain(X)), addition(codomain(antidomain(X)), domain(X)))
% 90.15/11.91 = { by axiom 19 (right_distributivity) }
% 90.15/11.92 addition(multiplication(coantidomain(antidomain(X)), codomain(antidomain(X))), multiplication(coantidomain(antidomain(X)), domain(X)))
% 90.15/11.92 = { by axiom 2 (codomain4) }
% 90.15/11.92 addition(multiplication(coantidomain(antidomain(X)), coantidomain(coantidomain(antidomain(X)))), multiplication(coantidomain(antidomain(X)), domain(X)))
% 90.15/11.92 = { by axiom 9 (codomain1) }
% 90.15/11.92 addition(zero, multiplication(coantidomain(antidomain(X)), domain(X)))
% 90.15/11.92 = { by lemma 27 }
% 90.15/11.92 multiplication(coantidomain(antidomain(X)), domain(X))
% 90.15/11.92 = { by lemma 32 R->L }
% 90.15/11.92 multiplication(coantidomain(domain(antidomain(X))), domain(X))
% 90.15/11.92 = { by axiom 1 (domain4) }
% 90.15/11.92 multiplication(coantidomain(antidomain(antidomain(antidomain(X)))), domain(X))
% 90.15/11.92 = { by axiom 1 (domain4) R->L }
% 90.15/11.92 multiplication(coantidomain(antidomain(domain(X))), domain(X))
% 90.15/11.92 = { by axiom 5 (additive_identity) R->L }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), zero), domain(X))
% 90.15/11.92 = { by axiom 9 (codomain1) R->L }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(domain(X)))))), domain(X))
% 90.15/11.92 = { by axiom 2 (codomain4) }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(domain(X)))), domain(domain(X)))))), domain(X))
% 90.15/11.92 = { by axiom 22 (codomain2) R->L }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), addition(coantidomain(multiplication(antidomain(domain(X)), domain(domain(X)))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(domain(X)))), domain(domain(X))))))), domain(X))
% 90.15/11.92 = { by axiom 2 (codomain4) R->L }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), addition(coantidomain(multiplication(antidomain(domain(X)), domain(domain(X)))), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(domain(X))))))), domain(X))
% 90.15/11.92 = { by axiom 1 (domain4) }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), addition(coantidomain(multiplication(antidomain(domain(X)), antidomain(antidomain(domain(X))))), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(domain(X))))))), domain(X))
% 90.15/11.92 = { by lemma 32 R->L }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), addition(coantidomain(multiplication(domain(antidomain(domain(X))), antidomain(antidomain(domain(X))))), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(domain(X))))))), domain(X))
% 90.15/11.92 = { by lemma 41 }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), addition(coantidomain(zero), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(domain(X))))))), domain(X))
% 90.15/11.92 = { by lemma 26 R->L }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), addition(coantidomain(coantidomain(one)), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(domain(X))))))), domain(X))
% 90.15/11.92 = { by axiom 2 (codomain4) R->L }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), addition(codomain(one), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(domain(X))))))), domain(X))
% 90.15/11.92 = { by axiom 5 (additive_identity) R->L }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), addition(addition(codomain(one), zero), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(domain(X))))))), domain(X))
% 90.15/11.92 = { by lemma 26 R->L }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), addition(addition(codomain(one), coantidomain(one)), coantidomain(multiplication(codomain(antidomain(domain(X))), domain(domain(X))))))), domain(X))
% 90.15/11.92 = { by lemma 37 }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), addition(one, coantidomain(multiplication(codomain(antidomain(domain(X))), domain(domain(X))))))), domain(X))
% 90.15/11.92 = { by lemma 40 }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), one)), domain(X))
% 90.15/11.92 = { by axiom 14 (multiplicative_associativity) R->L }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(codomain(antidomain(domain(X))), multiplication(domain(domain(X)), one))), domain(X))
% 90.15/11.92 = { by axiom 6 (multiplicative_right_identity) }
% 90.15/11.92 multiplication(addition(coantidomain(antidomain(domain(X))), multiplication(codomain(antidomain(domain(X))), domain(domain(X)))), domain(X))
% 90.15/11.92 = { by axiom 4 (additive_commutativity) R->L }
% 90.15/11.92 multiplication(addition(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), coantidomain(antidomain(domain(X)))), domain(X))
% 90.15/11.92 = { by axiom 20 (left_distributivity) }
% 90.15/11.92 addition(multiplication(multiplication(codomain(antidomain(domain(X))), domain(domain(X))), domain(X)), multiplication(coantidomain(antidomain(domain(X))), domain(X)))
% 90.15/11.92 = { by axiom 14 (multiplicative_associativity) R->L }
% 90.15/11.92 addition(multiplication(codomain(antidomain(domain(X))), multiplication(domain(domain(X)), domain(X))), multiplication(coantidomain(antidomain(domain(X))), domain(X)))
% 90.15/11.92 = { by lemma 31 }
% 90.15/11.92 addition(multiplication(codomain(antidomain(domain(X))), domain(X)), multiplication(coantidomain(antidomain(domain(X))), domain(X)))
% 90.15/11.92 = { by axiom 20 (left_distributivity) R->L }
% 90.15/11.92 multiplication(addition(codomain(antidomain(domain(X))), coantidomain(antidomain(domain(X)))), domain(X))
% 90.15/11.92 = { by lemma 37 }
% 90.15/11.92 multiplication(one, domain(X))
% 90.15/11.92 = { by axiom 7 (multiplicative_left_identity) }
% 90.15/11.92 domain(X)
% 90.15/11.92
% 90.15/11.92 Lemma 43: coantidomain(codomain(X)) = codomain(coantidomain(X)).
% 90.15/11.92 Proof:
% 90.15/11.92 coantidomain(codomain(X))
% 90.15/11.92 = { by axiom 2 (codomain4) }
% 90.15/11.92 coantidomain(coantidomain(coantidomain(X)))
% 90.15/11.92 = { by axiom 2 (codomain4) R->L }
% 90.15/11.92 codomain(coantidomain(X))
% 90.15/11.92
% 90.15/11.92 Lemma 44: multiplication(X, codomain(X)) = X.
% 90.15/11.92 Proof:
% 90.15/11.92 multiplication(X, codomain(X))
% 90.15/11.92 = { by axiom 5 (additive_identity) R->L }
% 90.15/11.92 addition(multiplication(X, codomain(X)), zero)
% 90.15/11.92 = { by axiom 9 (codomain1) R->L }
% 90.15/11.92 addition(multiplication(X, codomain(X)), multiplication(X, coantidomain(X)))
% 90.15/11.92 = { by axiom 19 (right_distributivity) R->L }
% 90.15/11.92 multiplication(X, addition(codomain(X), coantidomain(X)))
% 90.15/11.92 = { by lemma 37 }
% 90.15/11.92 multiplication(X, one)
% 90.15/11.92 = { by axiom 6 (multiplicative_right_identity) }
% 90.15/11.92 X
% 90.15/11.92
% 90.15/11.92 Lemma 45: codomain(domain(X)) = domain(X).
% 90.15/11.92 Proof:
% 90.15/11.92 codomain(domain(X))
% 90.15/11.92 = { by lemma 42 R->L }
% 90.15/11.92 codomain(coantidomain(antidomain(X)))
% 90.15/11.92 = { by lemma 43 R->L }
% 90.15/11.92 coantidomain(codomain(antidomain(X)))
% 90.15/11.92 = { by axiom 7 (multiplicative_left_identity) R->L }
% 90.15/11.92 multiplication(one, coantidomain(codomain(antidomain(X))))
% 90.15/11.92 = { by lemma 37 R->L }
% 90.15/11.92 multiplication(addition(codomain(antidomain(X)), coantidomain(antidomain(X))), coantidomain(codomain(antidomain(X))))
% 90.15/11.92 = { by lemma 38 }
% 90.15/11.92 multiplication(coantidomain(antidomain(X)), coantidomain(codomain(antidomain(X))))
% 90.15/11.92 = { by lemma 43 }
% 90.15/11.92 multiplication(coantidomain(antidomain(X)), codomain(coantidomain(antidomain(X))))
% 90.15/11.92 = { by lemma 44 }
% 90.15/11.92 coantidomain(antidomain(X))
% 90.15/11.92 = { by lemma 42 }
% 90.15/11.92 domain(X)
% 90.15/11.92
% 90.15/11.92 Lemma 46: addition(one, domain(X)) = one.
% 90.15/11.92 Proof:
% 90.15/11.92 addition(one, domain(X))
% 90.15/11.92 = { by axiom 4 (additive_commutativity) R->L }
% 90.15/11.92 addition(domain(X), one)
% 90.15/11.92 = { by lemma 24 R->L }
% 90.15/11.92 addition(domain(X), addition(domain(X), antidomain(X)))
% 90.15/11.92 = { by lemma 34 }
% 90.15/11.92 addition(domain(X), antidomain(X))
% 90.15/11.92 = { by lemma 24 }
% 90.15/11.92 one
% 90.15/11.92
% 90.15/11.92 Lemma 47: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y).
% 90.15/11.92 Proof:
% 90.15/11.92 multiplication(antidomain(X), addition(Y, X))
% 90.15/11.92 = { by axiom 4 (additive_commutativity) R->L }
% 90.15/11.92 multiplication(antidomain(X), addition(X, Y))
% 90.15/11.92 = { by lemma 29 }
% 90.15/11.92 multiplication(antidomain(X), Y)
% 90.15/11.92
% 90.15/11.92 Lemma 48: multiplication(antidomain(X), addition(domain(X), Y)) = multiplication(antidomain(X), Y).
% 90.15/11.92 Proof:
% 90.15/11.92 multiplication(antidomain(X), addition(domain(X), Y))
% 90.15/11.92 = { by axiom 4 (additive_commutativity) R->L }
% 90.15/11.92 multiplication(antidomain(X), addition(Y, domain(X)))
% 90.15/11.92 = { by lemma 32 R->L }
% 90.15/11.92 multiplication(domain(antidomain(X)), addition(Y, domain(X)))
% 90.15/11.92 = { by lemma 28 R->L }
% 90.15/11.92 multiplication(antidomain(domain(X)), addition(Y, domain(X)))
% 90.15/11.92 = { by lemma 47 }
% 90.15/11.92 multiplication(antidomain(domain(X)), Y)
% 90.15/11.92 = { by lemma 28 }
% 90.15/11.92 multiplication(domain(antidomain(X)), Y)
% 90.15/11.92 = { by lemma 32 }
% 90.15/11.92 multiplication(antidomain(X), Y)
% 90.15/11.92
% 90.15/11.92 Lemma 49: addition(multiplication(X, Y), addition(Z, multiplication(X, W))) = addition(Z, multiplication(X, addition(Y, W))).
% 90.15/11.92 Proof:
% 90.15/11.92 addition(multiplication(X, Y), addition(Z, multiplication(X, W)))
% 90.15/11.92 = { by axiom 4 (additive_commutativity) R->L }
% 90.15/11.92 addition(multiplication(X, Y), addition(multiplication(X, W), Z))
% 90.15/11.92 = { by axiom 13 (additive_associativity) }
% 90.15/11.92 addition(addition(multiplication(X, Y), multiplication(X, W)), Z)
% 90.15/11.92 = { by axiom 19 (right_distributivity) R->L }
% 90.15/11.92 addition(multiplication(X, addition(Y, W)), Z)
% 90.15/11.92 = { by axiom 4 (additive_commutativity) }
% 90.15/11.92 addition(Z, multiplication(X, addition(Y, W)))
% 90.15/11.92
% 90.15/11.92 Lemma 50: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 90.15/11.92 Proof:
% 90.15/11.92 multiplication(addition(one, Y), X)
% 90.15/11.92 = { by axiom 20 (left_distributivity) }
% 90.15/11.92 addition(multiplication(one, X), multiplication(Y, X))
% 90.15/11.92 = { by axiom 7 (multiplicative_left_identity) }
% 90.15/11.92 addition(X, multiplication(Y, X))
% 90.15/11.92
% 90.15/11.92 Lemma 51: addition(X, multiplication(domain(Y), X)) = X.
% 90.15/11.92 Proof:
% 90.15/11.92 addition(X, multiplication(domain(Y), X))
% 90.15/11.92 = { by lemma 50 R->L }
% 90.15/11.92 multiplication(addition(one, domain(Y)), X)
% 90.15/11.92 = { by lemma 46 }
% 90.15/11.92 multiplication(one, X)
% 90.15/11.92 = { by axiom 7 (multiplicative_left_identity) }
% 90.15/11.92 X
% 90.15/11.92
% 90.15/11.92 Lemma 52: addition(X, multiplication(domain(Y), addition(X, Z))) = addition(X, multiplication(domain(Y), Z)).
% 90.15/11.92 Proof:
% 90.15/11.92 addition(X, multiplication(domain(Y), addition(X, Z)))
% 90.15/11.92 = { by axiom 4 (additive_commutativity) R->L }
% 90.15/11.92 addition(X, multiplication(domain(Y), addition(Z, X)))
% 90.15/11.92 = { by lemma 49 R->L }
% 90.15/11.92 addition(multiplication(domain(Y), Z), addition(X, multiplication(domain(Y), X)))
% 90.15/11.92 = { by lemma 51 }
% 90.15/11.92 addition(multiplication(domain(Y), Z), X)
% 90.15/11.92 = { by axiom 4 (additive_commutativity) }
% 90.15/11.92 addition(X, multiplication(domain(Y), Z))
% 90.15/11.92
% 90.15/11.92 Lemma 53: multiplication(addition(domain(X), Y), antidomain(X)) = multiplication(Y, antidomain(X)).
% 90.15/11.92 Proof:
% 90.15/11.92 multiplication(addition(domain(X), Y), antidomain(X))
% 90.15/11.92 = { by axiom 20 (left_distributivity) }
% 90.15/11.92 addition(multiplication(domain(X), antidomain(X)), multiplication(Y, antidomain(X)))
% 90.15/11.92 = { by lemma 41 }
% 90.15/11.92 addition(zero, multiplication(Y, antidomain(X)))
% 90.15/11.92 = { by lemma 27 }
% 90.15/11.92 multiplication(Y, antidomain(X))
% 90.15/11.92
% 90.15/11.92 Lemma 54: addition(domain(X), multiplication(antidomain(X), domain(Y))) = addition(domain(X), domain(Y)).
% 90.15/11.92 Proof:
% 90.15/11.92 addition(domain(X), multiplication(antidomain(X), domain(Y)))
% 90.15/11.92 = { by axiom 4 (additive_commutativity) R->L }
% 90.15/11.92 addition(multiplication(antidomain(X), domain(Y)), domain(X))
% 90.15/11.92 = { by axiom 6 (multiplicative_right_identity) R->L }
% 90.15/11.92 addition(multiplication(antidomain(X), domain(Y)), multiplication(domain(X), one))
% 90.15/11.92 = { by lemma 46 R->L }
% 90.15/11.92 addition(multiplication(antidomain(X), domain(Y)), multiplication(domain(X), addition(one, domain(Y))))
% 90.15/11.92 = { by lemma 39 }
% 90.15/11.92 addition(multiplication(antidomain(X), domain(Y)), addition(domain(X), multiplication(domain(X), domain(Y))))
% 90.15/11.92 = { by axiom 4 (additive_commutativity) R->L }
% 90.15/11.92 addition(multiplication(antidomain(X), domain(Y)), addition(multiplication(domain(X), domain(Y)), domain(X)))
% 90.15/11.92 = { by axiom 13 (additive_associativity) }
% 90.15/11.92 addition(addition(multiplication(antidomain(X), domain(Y)), multiplication(domain(X), domain(Y))), domain(X))
% 90.15/11.92 = { by axiom 20 (left_distributivity) R->L }
% 90.15/11.92 addition(multiplication(addition(antidomain(X), domain(X)), domain(Y)), domain(X))
% 90.15/11.92 = { by axiom 4 (additive_commutativity) }
% 90.15/11.92 addition(domain(X), multiplication(addition(antidomain(X), domain(X)), domain(Y)))
% 90.15/11.92 = { by axiom 4 (additive_commutativity) }
% 90.15/11.92 addition(domain(X), multiplication(addition(domain(X), antidomain(X)), domain(Y)))
% 90.15/11.92 = { by lemma 24 }
% 90.15/11.92 addition(domain(X), multiplication(one, domain(Y)))
% 90.15/11.92 = { by axiom 7 (multiplicative_left_identity) }
% 90.15/11.92 addition(domain(X), domain(Y))
% 90.15/11.92
% 90.15/11.92 Lemma 55: multiplication(antidomain(X), domain(Y)) = multiplication(domain(Y), antidomain(X)).
% 90.15/11.92 Proof:
% 90.15/11.92 multiplication(antidomain(X), domain(Y))
% 90.15/11.92 = { by lemma 48 R->L }
% 90.15/11.92 multiplication(antidomain(X), addition(domain(X), domain(Y)))
% 90.15/11.92 = { by axiom 6 (multiplicative_right_identity) R->L }
% 90.15/11.92 multiplication(antidomain(X), addition(domain(X), multiplication(domain(Y), one)))
% 90.15/11.92 = { by lemma 24 R->L }
% 90.15/11.92 multiplication(antidomain(X), addition(domain(X), multiplication(domain(Y), addition(domain(X), antidomain(X)))))
% 90.15/11.92 = { by lemma 52 }
% 90.15/11.92 multiplication(antidomain(X), addition(domain(X), multiplication(domain(Y), antidomain(X))))
% 90.15/11.92 = { by lemma 48 }
% 90.15/11.92 multiplication(antidomain(X), multiplication(domain(Y), antidomain(X)))
% 90.15/11.92 = { by axiom 14 (multiplicative_associativity) }
% 90.15/11.92 multiplication(multiplication(antidomain(X), domain(Y)), antidomain(X))
% 90.15/11.92 = { by lemma 53 R->L }
% 90.15/11.92 multiplication(addition(domain(X), multiplication(antidomain(X), domain(Y))), antidomain(X))
% 90.15/11.92 = { by lemma 54 }
% 90.15/11.92 multiplication(addition(domain(X), domain(Y)), antidomain(X))
% 90.15/11.92 = { by lemma 53 }
% 90.15/11.92 multiplication(domain(Y), antidomain(X))
% 90.15/11.92
% 90.15/11.92 Lemma 56: multiplication(antidomain(X), multiplication(X, Y)) = zero.
% 90.15/11.92 Proof:
% 90.15/11.92 multiplication(antidomain(X), multiplication(X, Y))
% 90.15/11.92 = { by axiom 14 (multiplicative_associativity) }
% 90.15/11.92 multiplication(multiplication(antidomain(X), X), Y)
% 90.15/11.92 = { by axiom 10 (domain1) }
% 90.15/11.92 multiplication(zero, Y)
% 90.15/11.92 = { by axiom 8 (left_annihilation) }
% 90.15/11.92 zero
% 90.15/11.92
% 90.15/11.92 Lemma 57: addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))) = antidomain(multiplication(X, domain(Y))).
% 90.15/11.92 Proof:
% 90.15/11.92 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
% 90.15/11.92 = { by axiom 1 (domain4) }
% 90.15/11.92 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
% 90.15/11.92 = { by axiom 21 (domain2) }
% 90.15/11.92 antidomain(multiplication(X, antidomain(antidomain(Y))))
% 90.15/11.92 = { by axiom 1 (domain4) R->L }
% 90.15/11.92 antidomain(multiplication(X, domain(Y)))
% 90.15/11.92
% 90.15/11.92 Lemma 58: multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y))) = zero.
% 90.15/11.92 Proof:
% 90.15/11.92 multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y)))
% 90.15/11.92 = { by lemma 30 R->L }
% 90.15/11.92 multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))), multiplication(X, domain(Y)))
% 90.15/11.92 = { by lemma 57 }
% 90.15/11.92 multiplication(antidomain(multiplication(X, domain(Y))), multiplication(X, domain(Y)))
% 90.15/11.92 = { by axiom 10 (domain1) }
% 90.15/11.93 zero
% 90.15/11.93
% 90.15/11.93 Lemma 59: domain(multiplication(domain(X), antidomain(Y))) = domain(multiplication(antidomain(Y), X)).
% 90.15/11.93 Proof:
% 90.15/11.93 domain(multiplication(domain(X), antidomain(Y)))
% 90.15/11.93 = { by lemma 55 R->L }
% 90.15/11.93 domain(multiplication(antidomain(Y), domain(X)))
% 90.15/11.93 = { by axiom 1 (domain4) }
% 90.15/11.93 antidomain(antidomain(multiplication(antidomain(Y), domain(X))))
% 90.15/11.93 = { by lemma 57 R->L }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), antidomain(multiplication(antidomain(Y), domain(X)))))
% 90.15/11.93 = { by axiom 6 (multiplicative_right_identity) R->L }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), one)))
% 90.15/11.93 = { by lemma 24 R->L }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), addition(domain(multiplication(antidomain(Y), X)), antidomain(multiplication(antidomain(Y), X))))))
% 90.15/11.93 = { by lemma 29 R->L }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), addition(multiplication(antidomain(Y), domain(X)), addition(domain(multiplication(antidomain(Y), X)), antidomain(multiplication(antidomain(Y), X)))))))
% 90.15/11.93 = { by axiom 4 (additive_commutativity) R->L }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), addition(addition(domain(multiplication(antidomain(Y), X)), antidomain(multiplication(antidomain(Y), X))), multiplication(antidomain(Y), domain(X))))))
% 90.15/11.93 = { by axiom 13 (additive_associativity) R->L }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), addition(domain(multiplication(antidomain(Y), X)), addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(Y), domain(X)))))))
% 90.15/11.93 = { by axiom 4 (additive_commutativity) }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), addition(domain(multiplication(antidomain(Y), X)), addition(multiplication(antidomain(Y), domain(X)), antidomain(multiplication(antidomain(Y), X)))))))
% 90.15/11.93 = { by axiom 13 (additive_associativity) }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), addition(addition(domain(multiplication(antidomain(Y), X)), multiplication(antidomain(Y), domain(X))), antidomain(multiplication(antidomain(Y), X))))))
% 90.15/11.93 = { by lemma 49 R->L }
% 90.15/11.93 antidomain(addition(multiplication(antidomain(multiplication(antidomain(Y), domain(X))), addition(domain(multiplication(antidomain(Y), X)), multiplication(antidomain(Y), domain(X)))), addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), antidomain(multiplication(antidomain(Y), X))))))
% 90.15/11.93 = { by lemma 50 R->L }
% 90.15/11.93 antidomain(addition(multiplication(antidomain(multiplication(antidomain(Y), domain(X))), addition(domain(multiplication(antidomain(Y), X)), multiplication(antidomain(Y), domain(X)))), multiplication(addition(one, antidomain(multiplication(antidomain(Y), domain(X)))), antidomain(multiplication(antidomain(Y), X)))))
% 90.15/11.93 = { by lemma 47 }
% 90.15/11.93 antidomain(addition(multiplication(antidomain(multiplication(antidomain(Y), domain(X))), domain(multiplication(antidomain(Y), X))), multiplication(addition(one, antidomain(multiplication(antidomain(Y), domain(X)))), antidomain(multiplication(antidomain(Y), X)))))
% 90.15/11.93 = { by lemma 36 }
% 90.15/11.93 antidomain(addition(multiplication(antidomain(multiplication(antidomain(Y), domain(X))), domain(multiplication(antidomain(Y), X))), multiplication(one, antidomain(multiplication(antidomain(Y), X)))))
% 90.15/11.93 = { by axiom 7 (multiplicative_left_identity) }
% 90.15/11.93 antidomain(addition(multiplication(antidomain(multiplication(antidomain(Y), domain(X))), domain(multiplication(antidomain(Y), X))), antidomain(multiplication(antidomain(Y), X))))
% 90.15/11.93 = { by axiom 4 (additive_commutativity) }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), domain(multiplication(antidomain(Y), X)))))
% 90.15/11.93 = { by axiom 7 (multiplicative_left_identity) R->L }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(one, multiplication(antidomain(multiplication(antidomain(Y), domain(X))), domain(multiplication(antidomain(Y), X))))))
% 90.15/11.93 = { by lemma 25 R->L }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(zero), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), domain(multiplication(antidomain(Y), X))))))
% 90.15/11.93 = { by lemma 56 R->L }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(multiplication(antidomain(Y), domain(X))), multiplication(multiplication(antidomain(Y), domain(X)), X))), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), domain(multiplication(antidomain(Y), X))))))
% 90.15/11.93 = { by axiom 14 (multiplicative_associativity) R->L }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(multiplication(antidomain(Y), domain(X))), multiplication(antidomain(Y), multiplication(domain(X), X)))), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), domain(multiplication(antidomain(Y), X))))))
% 90.15/11.93 = { by lemma 31 }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), multiplication(antidomain(multiplication(antidomain(multiplication(antidomain(Y), domain(X))), multiplication(antidomain(Y), X))), multiplication(antidomain(multiplication(antidomain(Y), domain(X))), domain(multiplication(antidomain(Y), X))))))
% 90.15/11.93 = { by lemma 58 }
% 90.15/11.93 antidomain(addition(antidomain(multiplication(antidomain(Y), X)), zero))
% 90.15/11.93 = { by axiom 5 (additive_identity) }
% 90.15/11.93 antidomain(antidomain(multiplication(antidomain(Y), X)))
% 90.15/11.93 = { by axiom 1 (domain4) R->L }
% 90.15/11.93 domain(multiplication(antidomain(Y), X))
% 90.15/11.93
% 90.15/11.93 Lemma 60: addition(domain(X), domain(multiplication(X, Y))) = domain(X).
% 90.15/11.93 Proof:
% 90.15/11.93 addition(domain(X), domain(multiplication(X, Y)))
% 90.15/11.93 = { by lemma 54 R->L }
% 90.15/11.93 addition(domain(X), multiplication(antidomain(X), domain(multiplication(X, Y))))
% 90.15/11.93 = { by axiom 7 (multiplicative_left_identity) R->L }
% 90.15/11.93 addition(domain(X), multiplication(one, multiplication(antidomain(X), domain(multiplication(X, Y)))))
% 90.15/11.93 = { by lemma 25 R->L }
% 90.15/11.93 addition(domain(X), multiplication(antidomain(zero), multiplication(antidomain(X), domain(multiplication(X, Y)))))
% 90.15/11.93 = { by lemma 56 R->L }
% 90.15/11.93 addition(domain(X), multiplication(antidomain(multiplication(antidomain(X), multiplication(X, Y))), multiplication(antidomain(X), domain(multiplication(X, Y)))))
% 90.15/11.93 = { by lemma 58 }
% 90.15/11.93 addition(domain(X), zero)
% 90.15/11.93 = { by axiom 5 (additive_identity) }
% 90.15/11.93 domain(X)
% 90.15/11.93
% 90.15/11.93 Lemma 61: addition(domain(X), domain(multiplication(antidomain(X), Y))) = addition(domain(X), domain(Y)).
% 90.15/11.93 Proof:
% 90.15/11.93 addition(domain(X), domain(multiplication(antidomain(X), Y)))
% 90.15/11.93 = { by lemma 59 R->L }
% 90.15/11.93 addition(domain(X), domain(multiplication(domain(Y), antidomain(X))))
% 90.15/11.93 = { by lemma 51 R->L }
% 90.15/11.93 addition(domain(X), addition(domain(multiplication(domain(Y), antidomain(X))), multiplication(domain(domain(Y)), domain(multiplication(domain(Y), antidomain(X))))))
% 90.15/11.93 = { by lemma 45 R->L }
% 90.15/11.93 addition(domain(X), addition(domain(multiplication(domain(Y), antidomain(X))), multiplication(domain(domain(Y)), codomain(domain(multiplication(domain(Y), antidomain(X)))))))
% 90.15/11.93 = { by lemma 44 R->L }
% 90.15/11.93 addition(domain(X), addition(multiplication(domain(multiplication(domain(Y), antidomain(X))), codomain(domain(multiplication(domain(Y), antidomain(X))))), multiplication(domain(domain(Y)), codomain(domain(multiplication(domain(Y), antidomain(X)))))))
% 90.15/11.93 = { by axiom 20 (left_distributivity) R->L }
% 90.15/11.93 addition(domain(X), multiplication(addition(domain(multiplication(domain(Y), antidomain(X))), domain(domain(Y))), codomain(domain(multiplication(domain(Y), antidomain(X))))))
% 90.15/11.93 = { by axiom 4 (additive_commutativity) }
% 90.15/11.93 addition(domain(X), multiplication(addition(domain(domain(Y)), domain(multiplication(domain(Y), antidomain(X)))), codomain(domain(multiplication(domain(Y), antidomain(X))))))
% 90.15/11.93 = { by lemma 60 }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(Y)), codomain(domain(multiplication(domain(Y), antidomain(X))))))
% 90.15/11.93 = { by lemma 45 }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(Y)), domain(multiplication(domain(Y), antidomain(X)))))
% 90.15/11.93 = { by lemma 55 R->L }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(Y)), domain(multiplication(antidomain(X), domain(Y)))))
% 90.15/11.93 = { by lemma 33 R->L }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), domain(multiplication(antidomain(X), domain(Y)))))
% 90.15/11.93 = { by lemma 59 R->L }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), domain(multiplication(domain(domain(Y)), antidomain(X)))))
% 90.15/11.93 = { by axiom 6 (multiplicative_right_identity) R->L }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), multiplication(domain(multiplication(domain(domain(Y)), antidomain(X))), one)))
% 90.15/11.93 = { by lemma 24 R->L }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), multiplication(domain(multiplication(domain(domain(Y)), antidomain(X))), addition(domain(X), antidomain(X)))))
% 90.15/11.93 = { by lemma 51 R->L }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), multiplication(domain(multiplication(domain(domain(Y)), antidomain(X))), addition(domain(X), addition(antidomain(X), multiplication(domain(domain(Y)), antidomain(X)))))))
% 90.15/11.93 = { by axiom 13 (additive_associativity) }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), multiplication(domain(multiplication(domain(domain(Y)), antidomain(X))), addition(addition(domain(X), antidomain(X)), multiplication(domain(domain(Y)), antidomain(X))))))
% 90.15/11.93 = { by lemma 24 }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), multiplication(domain(multiplication(domain(domain(Y)), antidomain(X))), addition(one, multiplication(domain(domain(Y)), antidomain(X))))))
% 90.15/11.93 = { by axiom 4 (additive_commutativity) R->L }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), multiplication(domain(multiplication(domain(domain(Y)), antidomain(X))), addition(multiplication(domain(domain(Y)), antidomain(X)), one))))
% 90.15/11.93 = { by axiom 19 (right_distributivity) }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), addition(multiplication(domain(multiplication(domain(domain(Y)), antidomain(X))), multiplication(domain(domain(Y)), antidomain(X))), multiplication(domain(multiplication(domain(domain(Y)), antidomain(X))), one))))
% 90.15/11.93 = { by lemma 31 }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), addition(multiplication(domain(domain(Y)), antidomain(X)), multiplication(domain(multiplication(domain(domain(Y)), antidomain(X))), one))))
% 90.15/11.93 = { by axiom 6 (multiplicative_right_identity) }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), addition(multiplication(domain(domain(Y)), antidomain(X)), domain(multiplication(domain(domain(Y)), antidomain(X))))))
% 90.15/11.93 = { by lemma 59 }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(domain(Y))), addition(multiplication(domain(domain(Y)), antidomain(X)), domain(multiplication(antidomain(X), domain(Y))))))
% 90.15/11.93 = { by axiom 19 (right_distributivity) }
% 90.15/11.93 addition(domain(X), addition(multiplication(domain(domain(domain(Y))), multiplication(domain(domain(Y)), antidomain(X))), multiplication(domain(domain(domain(Y))), domain(multiplication(antidomain(X), domain(Y))))))
% 90.15/11.93 = { by axiom 14 (multiplicative_associativity) }
% 90.15/11.93 addition(domain(X), addition(multiplication(multiplication(domain(domain(domain(Y))), domain(domain(Y))), antidomain(X)), multiplication(domain(domain(domain(Y))), domain(multiplication(antidomain(X), domain(Y))))))
% 90.15/11.93 = { by lemma 31 }
% 90.15/11.93 addition(domain(X), addition(multiplication(domain(domain(Y)), antidomain(X)), multiplication(domain(domain(domain(Y))), domain(multiplication(antidomain(X), domain(Y))))))
% 90.15/11.93 = { by lemma 33 }
% 90.15/11.93 addition(domain(X), addition(multiplication(domain(domain(Y)), antidomain(X)), multiplication(domain(domain(Y)), domain(multiplication(antidomain(X), domain(Y))))))
% 90.15/11.93 = { by axiom 19 (right_distributivity) R->L }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(Y)), addition(antidomain(X), domain(multiplication(antidomain(X), domain(Y))))))
% 90.15/11.93 = { by lemma 32 R->L }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(Y)), addition(domain(antidomain(X)), domain(multiplication(antidomain(X), domain(Y))))))
% 90.15/11.93 = { by lemma 60 }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(Y)), domain(antidomain(X))))
% 90.15/11.93 = { by lemma 32 }
% 90.15/11.93 addition(domain(X), multiplication(domain(domain(Y)), antidomain(X)))
% 90.15/11.94 = { by lemma 33 }
% 90.15/11.94 addition(domain(X), multiplication(domain(Y), antidomain(X)))
% 90.15/11.94 = { by lemma 52 R->L }
% 90.15/11.94 addition(domain(X), multiplication(domain(Y), addition(domain(X), antidomain(X))))
% 90.15/11.94 = { by lemma 24 }
% 90.15/11.94 addition(domain(X), multiplication(domain(Y), one))
% 90.15/11.94 = { by axiom 6 (multiplicative_right_identity) }
% 90.15/11.94 addition(domain(X), domain(Y))
% 90.15/11.94
% 90.15/11.94 Goal 1 (goals): domain(addition(x0, x1)) = addition(domain(x0), domain(x1)).
% 90.15/11.94 Proof:
% 90.15/11.94 domain(addition(x0, x1))
% 90.15/11.94 = { by axiom 5 (additive_identity) R->L }
% 90.15/11.94 addition(domain(addition(x0, x1)), zero)
% 90.15/11.94 = { by lemma 58 R->L }
% 90.15/11.94 addition(domain(addition(x0, x1)), multiplication(antidomain(multiplication(antidomain(addition(x1, x0)), x1)), multiplication(antidomain(addition(x1, x0)), domain(x1))))
% 90.15/11.94 = { by lemma 47 R->L }
% 90.15/11.94 addition(domain(addition(x0, x1)), multiplication(antidomain(multiplication(antidomain(addition(x1, x0)), addition(x1, addition(x1, x0)))), multiplication(antidomain(addition(x1, x0)), domain(x1))))
% 90.15/11.94 = { by lemma 34 }
% 90.15/11.94 addition(domain(addition(x0, x1)), multiplication(antidomain(multiplication(antidomain(addition(x1, x0)), addition(x1, x0))), multiplication(antidomain(addition(x1, x0)), domain(x1))))
% 90.15/11.94 = { by axiom 10 (domain1) }
% 90.15/11.94 addition(domain(addition(x0, x1)), multiplication(antidomain(zero), multiplication(antidomain(addition(x1, x0)), domain(x1))))
% 90.15/11.94 = { by lemma 25 }
% 90.15/11.94 addition(domain(addition(x0, x1)), multiplication(one, multiplication(antidomain(addition(x1, x0)), domain(x1))))
% 90.15/11.94 = { by axiom 7 (multiplicative_left_identity) }
% 90.15/11.94 addition(domain(addition(x0, x1)), multiplication(antidomain(addition(x1, x0)), domain(x1)))
% 90.15/11.94 = { by axiom 4 (additive_commutativity) }
% 90.15/11.94 addition(domain(addition(x0, x1)), multiplication(antidomain(addition(x0, x1)), domain(x1)))
% 90.15/11.94 = { by lemma 54 }
% 90.15/11.94 addition(domain(addition(x0, x1)), domain(x1))
% 90.15/11.94 = { by axiom 4 (additive_commutativity) }
% 90.15/11.94 addition(domain(x1), domain(addition(x0, x1)))
% 90.15/11.94 = { by lemma 61 R->L }
% 90.15/11.94 addition(domain(x1), domain(multiplication(antidomain(x1), addition(x0, x1))))
% 90.15/11.94 = { by lemma 47 }
% 90.15/11.94 addition(domain(x1), domain(multiplication(antidomain(x1), x0)))
% 90.15/11.94 = { by lemma 61 }
% 90.15/11.94 addition(domain(x1), domain(x0))
% 90.15/11.94 = { by axiom 4 (additive_commutativity) }
% 90.15/11.94 addition(domain(x0), domain(x1))
% 90.15/11.94 % SZS output end Proof
% 90.15/11.94
% 90.15/11.94 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------