TSTP Solution File: KLE073+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE073+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 : n007.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 05:35:47 EDT 2023
% Result : Theorem 151.84s 19.47s
% Output : Proof 153.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : KLE073+1 : TPTP v8.1.2. Released v4.0.0.
% 0.03/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n007.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Tue Aug 29 11:16:28 EDT 2023
% 0.14/0.35 % CPUTime :
% 151.84/19.47 Command-line arguments: --no-flatten-goal
% 151.84/19.47
% 151.84/19.47 % SZS status Theorem
% 151.84/19.47
% 152.65/19.61 % SZS output start Proof
% 152.65/19.61 Take the following subset of the input axioms:
% 152.65/19.61 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 152.65/19.61 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 152.65/19.61 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 152.65/19.61 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 152.65/19.61 fof(codomain1, axiom, ![X0]: multiplication(X0, coantidomain(X0))=zero).
% 152.65/19.61 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))).
% 152.65/19.61 fof(codomain3, axiom, ![X0_2]: addition(coantidomain(coantidomain(X0_2)), coantidomain(X0_2))=one).
% 152.65/19.61 fof(codomain4, axiom, ![X0_2]: codomain(X0_2)=coantidomain(coantidomain(X0_2))).
% 152.65/19.61 fof(domain1, axiom, ![X0_2]: multiplication(antidomain(X0_2), X0_2)=zero).
% 152.65/19.61 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))))).
% 152.65/19.61 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 152.65/19.61 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 152.65/19.61 fof(goals, conjecture, ![X2, X0_2, X1_2]: domain(multiplication(X0_2, addition(domain(X1_2), domain(X2))))=addition(domain(multiplication(X0_2, domain(X1_2))), domain(multiplication(X0_2, domain(X2))))).
% 152.65/19.61 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 152.65/19.61 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 152.65/19.61 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 152.65/19.61 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 152.65/19.61 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 152.65/19.61 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 152.65/19.61 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 152.65/19.61
% 152.65/19.61 Now clausify the problem and encode Horn clauses using encoding 3 of
% 152.65/19.61 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 152.65/19.61 We repeatedly replace C & s=t => u=v by the two clauses:
% 152.65/19.61 fresh(y, y, x1...xn) = u
% 152.65/19.61 C => fresh(s, t, x1...xn) = v
% 152.65/19.61 where fresh is a fresh function symbol and x1..xn are the free
% 152.65/19.61 variables of u and v.
% 152.65/19.61 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 152.65/19.61 input problem has no model of domain size 1).
% 152.65/19.61
% 152.65/19.61 The encoding turns the above axioms into the following unit equations and goals:
% 152.65/19.61
% 152.65/19.61 Axiom 1 (domain4): domain(X) = antidomain(antidomain(X)).
% 152.65/19.61 Axiom 2 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 152.65/19.61 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 152.65/19.61 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 152.65/19.61 Axiom 5 (additive_identity): addition(X, zero) = X.
% 152.65/19.61 Axiom 6 (multiplicative_right_identity): multiplication(X, one) = X.
% 152.65/19.61 Axiom 7 (multiplicative_left_identity): multiplication(one, X) = X.
% 152.65/19.61 Axiom 8 (left_annihilation): multiplication(zero, X) = zero.
% 152.65/19.61 Axiom 9 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 152.65/19.61 Axiom 10 (domain1): multiplication(antidomain(X), X) = zero.
% 152.65/19.61 Axiom 11 (order): fresh(X, X, Y, Z) = true.
% 152.65/19.61 Axiom 12 (order_1): fresh2(X, X, Y, Z) = Z.
% 152.65/19.61 Axiom 13 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 152.65/19.61 Axiom 14 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 152.65/19.61 Axiom 15 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 152.65/19.61 Axiom 16 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 152.65/19.61 Axiom 17 (order): fresh(addition(X, Y), Y, X, Y) = leq(X, Y).
% 152.65/19.61 Axiom 18 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 152.65/19.61 Axiom 19 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 152.65/19.61 Axiom 20 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 152.65/19.61 Axiom 21 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 152.65/19.61 Axiom 22 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
% 152.65/19.61
% 152.65/19.61 Lemma 23: antidomain(one) = zero.
% 152.65/19.61 Proof:
% 152.65/19.61 antidomain(one)
% 152.65/19.61 = { by axiom 6 (multiplicative_right_identity) R->L }
% 152.65/19.61 multiplication(antidomain(one), one)
% 152.65/19.61 = { by axiom 10 (domain1) }
% 152.65/19.61 zero
% 152.65/19.61
% 152.65/19.61 Lemma 24: addition(domain(X), antidomain(X)) = one.
% 152.65/19.61 Proof:
% 152.65/19.61 addition(domain(X), antidomain(X))
% 152.65/19.61 = { by axiom 1 (domain4) }
% 152.65/19.61 addition(antidomain(antidomain(X)), antidomain(X))
% 152.65/19.61 = { by axiom 15 (domain3) }
% 152.65/19.61 one
% 152.65/19.61
% 152.65/19.61 Lemma 25: antidomain(zero) = one.
% 152.65/19.61 Proof:
% 152.65/19.61 antidomain(zero)
% 152.65/19.61 = { by lemma 23 R->L }
% 152.65/19.61 antidomain(antidomain(one))
% 152.65/19.61 = { by axiom 1 (domain4) R->L }
% 152.65/19.61 domain(one)
% 152.65/19.61 = { by axiom 5 (additive_identity) R->L }
% 152.65/19.61 addition(domain(one), zero)
% 152.65/19.61 = { by lemma 23 R->L }
% 152.65/19.61 addition(domain(one), antidomain(one))
% 152.65/19.61 = { by lemma 24 }
% 152.65/19.61 one
% 152.65/19.61
% 152.65/19.61 Lemma 26: coantidomain(one) = zero.
% 152.65/19.61 Proof:
% 152.65/19.61 coantidomain(one)
% 152.65/19.61 = { by axiom 7 (multiplicative_left_identity) R->L }
% 152.65/19.61 multiplication(one, coantidomain(one))
% 152.65/19.61 = { by axiom 9 (codomain1) }
% 152.65/19.61 zero
% 152.65/19.61
% 152.65/19.61 Lemma 27: addition(zero, X) = X.
% 152.65/19.61 Proof:
% 152.65/19.61 addition(zero, X)
% 152.65/19.61 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.61 addition(X, zero)
% 152.65/19.61 = { by axiom 5 (additive_identity) }
% 152.65/19.61 X
% 152.65/19.61
% 152.65/19.61 Lemma 28: antidomain(domain(X)) = domain(antidomain(X)).
% 152.65/19.61 Proof:
% 152.65/19.61 antidomain(domain(X))
% 152.65/19.62 = { by axiom 1 (domain4) }
% 152.65/19.62 antidomain(antidomain(antidomain(X)))
% 152.65/19.62 = { by axiom 1 (domain4) R->L }
% 152.65/19.62 domain(antidomain(X))
% 152.65/19.62
% 152.65/19.62 Lemma 29: addition(codomain(X), coantidomain(X)) = one.
% 152.65/19.62 Proof:
% 152.65/19.62 addition(codomain(X), coantidomain(X))
% 152.65/19.62 = { by axiom 2 (codomain4) }
% 152.65/19.62 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 152.65/19.62 = { by axiom 16 (codomain3) }
% 152.65/19.62 one
% 152.65/19.62
% 152.65/19.62 Lemma 30: addition(X, addition(X, Y)) = addition(X, Y).
% 152.65/19.62 Proof:
% 152.65/19.62 addition(X, addition(X, Y))
% 152.65/19.62 = { by axiom 13 (additive_associativity) }
% 152.65/19.62 addition(addition(X, X), Y)
% 152.65/19.62 = { by axiom 3 (additive_idempotence) }
% 152.65/19.62 addition(X, Y)
% 152.65/19.62
% 152.65/19.62 Lemma 31: leq(X, addition(Y, X)) = true.
% 152.65/19.62 Proof:
% 152.65/19.62 leq(X, addition(Y, X))
% 152.65/19.62 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.62 leq(X, addition(X, Y))
% 152.65/19.62 = { by axiom 17 (order) R->L }
% 152.65/19.62 fresh(addition(X, addition(X, Y)), addition(X, Y), X, addition(X, Y))
% 152.65/19.62 = { by lemma 30 }
% 152.65/19.62 fresh(addition(X, Y), addition(X, Y), X, addition(X, Y))
% 152.65/19.62 = { by axiom 11 (order) }
% 152.65/19.62 true
% 152.65/19.62
% 152.65/19.62 Lemma 32: addition(one, coantidomain(X)) = one.
% 152.65/19.62 Proof:
% 152.65/19.62 addition(one, coantidomain(X))
% 152.65/19.62 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.62 addition(coantidomain(X), one)
% 152.65/19.62 = { by axiom 18 (order_1) R->L }
% 152.65/19.62 fresh2(leq(coantidomain(X), one), true, coantidomain(X), one)
% 152.65/19.62 = { by lemma 29 R->L }
% 152.65/19.62 fresh2(leq(coantidomain(X), addition(codomain(X), coantidomain(X))), true, coantidomain(X), one)
% 152.65/19.62 = { by lemma 31 }
% 152.65/19.62 fresh2(true, true, coantidomain(X), one)
% 152.65/19.62 = { by axiom 12 (order_1) }
% 152.65/19.62 one
% 152.65/19.62
% 152.65/19.62 Lemma 33: addition(domain(X), addition(Y, antidomain(X))) = addition(Y, one).
% 152.65/19.62 Proof:
% 152.65/19.62 addition(domain(X), addition(Y, antidomain(X)))
% 152.65/19.62 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.62 addition(domain(X), addition(antidomain(X), Y))
% 152.65/19.62 = { by axiom 13 (additive_associativity) }
% 152.65/19.62 addition(addition(domain(X), antidomain(X)), Y)
% 152.65/19.62 = { by lemma 24 }
% 152.65/19.62 addition(one, Y)
% 152.65/19.62 = { by axiom 4 (additive_commutativity) }
% 152.65/19.62 addition(Y, one)
% 152.65/19.62
% 152.65/19.62 Lemma 34: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 152.65/19.62 Proof:
% 152.65/19.62 multiplication(antidomain(X), addition(X, Y))
% 152.65/19.62 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.62 multiplication(antidomain(X), addition(Y, X))
% 152.65/19.62 = { by axiom 19 (right_distributivity) }
% 152.65/19.62 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 152.65/19.62 = { by axiom 10 (domain1) }
% 152.65/19.62 addition(multiplication(antidomain(X), Y), zero)
% 152.65/19.62 = { by axiom 5 (additive_identity) }
% 152.65/19.62 multiplication(antidomain(X), Y)
% 152.65/19.62
% 152.65/19.62 Lemma 35: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
% 152.65/19.62 Proof:
% 152.65/19.62 multiplication(addition(X, one), Y)
% 152.65/19.62 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.62 multiplication(addition(one, X), Y)
% 152.65/19.62 = { by axiom 20 (left_distributivity) }
% 152.65/19.62 addition(multiplication(one, Y), multiplication(X, Y))
% 152.65/19.62 = { by axiom 7 (multiplicative_left_identity) }
% 152.65/19.62 addition(Y, multiplication(X, Y))
% 152.65/19.62
% 152.65/19.62 Lemma 36: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 152.65/19.62 Proof:
% 152.65/19.62 multiplication(addition(one, Y), X)
% 152.65/19.62 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.62 multiplication(addition(Y, one), X)
% 152.65/19.62 = { by lemma 35 }
% 152.65/19.62 addition(X, multiplication(Y, X))
% 152.65/19.62
% 152.65/19.62 Lemma 37: addition(one, antidomain(X)) = one.
% 152.65/19.62 Proof:
% 152.65/19.62 addition(one, antidomain(X))
% 152.65/19.62 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.62 addition(antidomain(X), one)
% 152.65/19.62 = { by axiom 18 (order_1) R->L }
% 152.65/19.62 fresh2(leq(antidomain(X), one), true, antidomain(X), one)
% 152.65/19.62 = { by lemma 24 R->L }
% 152.65/19.62 fresh2(leq(antidomain(X), addition(domain(X), antidomain(X))), true, antidomain(X), one)
% 152.65/19.62 = { by lemma 31 }
% 152.65/19.62 fresh2(true, true, antidomain(X), one)
% 152.65/19.62 = { by axiom 12 (order_1) }
% 152.65/19.62 one
% 152.65/19.62
% 152.65/19.62 Lemma 38: multiplication(domain(X), antidomain(X)) = zero.
% 152.65/19.62 Proof:
% 152.65/19.62 multiplication(domain(X), antidomain(X))
% 152.65/19.62 = { by axiom 1 (domain4) }
% 152.65/19.62 multiplication(antidomain(antidomain(X)), antidomain(X))
% 152.65/19.62 = { by axiom 10 (domain1) }
% 152.65/19.62 zero
% 152.65/19.62
% 152.65/19.62 Lemma 39: multiplication(addition(X, domain(Y)), antidomain(Y)) = multiplication(X, antidomain(Y)).
% 152.65/19.62 Proof:
% 152.65/19.62 multiplication(addition(X, domain(Y)), antidomain(Y))
% 152.65/19.62 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.62 multiplication(addition(domain(Y), X), antidomain(Y))
% 152.65/19.62 = { by axiom 20 (left_distributivity) }
% 152.65/19.62 addition(multiplication(domain(Y), antidomain(Y)), multiplication(X, antidomain(Y)))
% 152.65/19.62 = { by lemma 38 }
% 152.65/19.62 addition(zero, multiplication(X, antidomain(Y)))
% 152.65/19.62 = { by lemma 27 }
% 152.65/19.62 multiplication(X, antidomain(Y))
% 152.65/19.62
% 152.65/19.62 Lemma 40: addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))) = antidomain(multiplication(X, domain(Y))).
% 152.65/19.62 Proof:
% 152.65/19.62 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
% 152.65/19.62 = { by axiom 1 (domain4) }
% 152.65/19.62 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
% 152.65/19.62 = { by axiom 21 (domain2) }
% 152.65/19.62 antidomain(multiplication(X, antidomain(antidomain(Y))))
% 152.65/19.62 = { by axiom 1 (domain4) R->L }
% 152.65/19.62 antidomain(multiplication(X, domain(Y)))
% 152.65/19.62
% 152.65/19.62 Lemma 41: coantidomain(codomain(X)) = codomain(coantidomain(X)).
% 152.65/19.62 Proof:
% 152.65/19.62 coantidomain(codomain(X))
% 152.65/19.62 = { by axiom 2 (codomain4) }
% 152.65/19.62 coantidomain(coantidomain(coantidomain(X)))
% 152.65/19.62 = { by axiom 2 (codomain4) R->L }
% 152.65/19.62 codomain(coantidomain(X))
% 152.65/19.62
% 152.65/19.62 Lemma 42: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
% 152.65/19.62 Proof:
% 152.65/19.62 multiplication(addition(X, Y), coantidomain(X))
% 152.65/19.62 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.62 multiplication(addition(Y, X), coantidomain(X))
% 152.65/19.62 = { by axiom 20 (left_distributivity) }
% 152.65/19.62 addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
% 152.65/19.62 = { by axiom 9 (codomain1) }
% 152.65/19.62 addition(multiplication(Y, coantidomain(X)), zero)
% 152.65/19.62 = { by axiom 5 (additive_identity) }
% 152.65/19.62 multiplication(Y, coantidomain(X))
% 152.65/19.62
% 152.65/19.62 Lemma 43: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y).
% 152.65/19.62 Proof:
% 152.65/19.62 multiplication(X, addition(Y, coantidomain(X)))
% 152.65/19.62 = { by axiom 19 (right_distributivity) }
% 152.65/19.62 addition(multiplication(X, Y), multiplication(X, coantidomain(X)))
% 152.65/19.62 = { by axiom 9 (codomain1) }
% 152.65/19.62 addition(multiplication(X, Y), zero)
% 152.65/19.62 = { by axiom 5 (additive_identity) }
% 152.65/19.62 multiplication(X, Y)
% 152.65/19.62
% 152.65/19.62 Lemma 44: multiplication(X, codomain(X)) = X.
% 152.65/19.62 Proof:
% 152.65/19.62 multiplication(X, codomain(X))
% 152.65/19.62 = { by lemma 43 R->L }
% 152.65/19.62 multiplication(X, addition(codomain(X), coantidomain(X)))
% 152.65/19.62 = { by lemma 29 }
% 152.65/19.62 multiplication(X, one)
% 152.65/19.62 = { by axiom 6 (multiplicative_right_identity) }
% 152.65/19.62 X
% 152.65/19.62
% 152.65/19.62 Lemma 45: codomain(coantidomain(X)) = coantidomain(X).
% 152.65/19.62 Proof:
% 152.65/19.62 codomain(coantidomain(X))
% 152.65/19.62 = { by lemma 41 R->L }
% 152.65/19.62 coantidomain(codomain(X))
% 152.65/19.62 = { by axiom 7 (multiplicative_left_identity) R->L }
% 152.65/19.62 multiplication(one, coantidomain(codomain(X)))
% 152.65/19.62 = { by lemma 29 R->L }
% 152.65/19.62 multiplication(addition(codomain(X), coantidomain(X)), coantidomain(codomain(X)))
% 152.65/19.62 = { by lemma 42 }
% 152.65/19.62 multiplication(coantidomain(X), coantidomain(codomain(X)))
% 152.65/19.62 = { by lemma 41 }
% 152.65/19.62 multiplication(coantidomain(X), codomain(coantidomain(X)))
% 152.65/19.62 = { by lemma 44 }
% 152.65/19.62 coantidomain(X)
% 152.65/19.62
% 152.65/19.62 Lemma 46: multiplication(coantidomain(X), codomain(X)) = zero.
% 152.65/19.62 Proof:
% 152.65/19.62 multiplication(coantidomain(X), codomain(X))
% 152.65/19.62 = { by axiom 2 (codomain4) }
% 152.65/19.62 multiplication(coantidomain(X), coantidomain(coantidomain(X)))
% 152.65/19.62 = { by axiom 9 (codomain1) }
% 152.65/19.62 zero
% 152.65/19.62
% 152.65/19.62 Lemma 47: antidomain(codomain(X)) = coantidomain(X).
% 152.65/19.62 Proof:
% 152.65/19.62 antidomain(codomain(X))
% 152.65/19.62 = { by axiom 7 (multiplicative_left_identity) R->L }
% 152.65/19.62 multiplication(one, antidomain(codomain(X)))
% 152.65/19.62 = { by lemma 32 R->L }
% 152.65/19.62 multiplication(addition(one, coantidomain(X)), antidomain(codomain(X)))
% 152.65/19.62 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.62 multiplication(addition(coantidomain(X), one), antidomain(codomain(X)))
% 152.65/19.62 = { by lemma 33 R->L }
% 152.65/19.62 multiplication(addition(domain(codomain(X)), addition(coantidomain(X), antidomain(codomain(X)))), antidomain(codomain(X)))
% 152.65/19.62 = { by axiom 6 (multiplicative_right_identity) R->L }
% 152.65/19.62 multiplication(addition(domain(codomain(X)), addition(coantidomain(X), multiplication(antidomain(codomain(X)), one))), antidomain(codomain(X)))
% 152.65/19.62 = { by lemma 29 R->L }
% 152.65/19.62 multiplication(addition(domain(codomain(X)), addition(coantidomain(X), multiplication(antidomain(codomain(X)), addition(codomain(X), coantidomain(X))))), antidomain(codomain(X)))
% 152.65/19.62 = { by lemma 34 }
% 152.65/19.62 multiplication(addition(domain(codomain(X)), addition(coantidomain(X), multiplication(antidomain(codomain(X)), coantidomain(X)))), antidomain(codomain(X)))
% 152.65/19.62 = { by lemma 36 R->L }
% 152.65/19.62 multiplication(addition(domain(codomain(X)), multiplication(addition(one, antidomain(codomain(X))), coantidomain(X))), antidomain(codomain(X)))
% 152.65/19.62 = { by lemma 37 }
% 152.65/19.62 multiplication(addition(domain(codomain(X)), multiplication(one, coantidomain(X))), antidomain(codomain(X)))
% 152.65/19.62 = { by axiom 7 (multiplicative_left_identity) }
% 152.65/19.62 multiplication(addition(domain(codomain(X)), coantidomain(X)), antidomain(codomain(X)))
% 152.65/19.62 = { by axiom 4 (additive_commutativity) }
% 152.65/19.63 multiplication(addition(coantidomain(X), domain(codomain(X))), antidomain(codomain(X)))
% 152.65/19.63 = { by lemma 39 }
% 152.65/19.63 multiplication(coantidomain(X), antidomain(codomain(X)))
% 152.65/19.63 = { by axiom 2 (codomain4) }
% 152.65/19.63 multiplication(coantidomain(X), antidomain(coantidomain(coantidomain(X))))
% 152.65/19.63 = { by axiom 5 (additive_identity) R->L }
% 152.65/19.63 multiplication(coantidomain(X), addition(antidomain(coantidomain(coantidomain(X))), zero))
% 152.65/19.63 = { by axiom 10 (domain1) R->L }
% 152.65/19.63 multiplication(coantidomain(X), addition(antidomain(coantidomain(coantidomain(X))), multiplication(antidomain(multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X))))), multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))))
% 152.65/19.63 = { by lemma 40 R->L }
% 152.65/19.63 multiplication(coantidomain(X), addition(antidomain(coantidomain(coantidomain(X))), multiplication(addition(antidomain(multiplication(codomain(coantidomain(X)), coantidomain(coantidomain(X)))), antidomain(multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))), multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))))
% 152.65/19.63 = { by axiom 2 (codomain4) }
% 152.65/19.63 multiplication(coantidomain(X), addition(antidomain(coantidomain(coantidomain(X))), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(coantidomain(X))), coantidomain(coantidomain(X)))), antidomain(multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))), multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))))
% 152.65/19.63 = { by lemma 45 R->L }
% 152.65/19.63 multiplication(coantidomain(X), addition(antidomain(coantidomain(coantidomain(X))), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(coantidomain(X))), codomain(coantidomain(coantidomain(X))))), antidomain(multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))), multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))))
% 152.65/19.63 = { by lemma 46 }
% 152.65/19.63 multiplication(coantidomain(X), addition(antidomain(coantidomain(coantidomain(X))), multiplication(addition(antidomain(zero), antidomain(multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))), multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))))
% 152.65/19.63 = { by lemma 25 }
% 152.65/19.63 multiplication(coantidomain(X), addition(antidomain(coantidomain(coantidomain(X))), multiplication(addition(one, antidomain(multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))), multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))))
% 152.65/19.63 = { by lemma 37 }
% 152.65/19.63 multiplication(coantidomain(X), addition(antidomain(coantidomain(coantidomain(X))), multiplication(one, multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))))))
% 152.65/19.63 = { by axiom 7 (multiplicative_left_identity) }
% 152.65/19.63 multiplication(coantidomain(X), addition(antidomain(coantidomain(coantidomain(X))), multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X))))))
% 152.65/19.63 = { by axiom 4 (additive_commutativity) R->L }
% 152.65/19.63 multiplication(coantidomain(X), addition(multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X)))), antidomain(coantidomain(coantidomain(X)))))
% 152.65/19.63 = { by axiom 19 (right_distributivity) }
% 152.65/19.63 addition(multiplication(coantidomain(X), multiplication(codomain(coantidomain(X)), domain(coantidomain(coantidomain(X))))), multiplication(coantidomain(X), antidomain(coantidomain(coantidomain(X)))))
% 152.65/19.63 = { by axiom 14 (multiplicative_associativity) }
% 152.65/19.63 addition(multiplication(multiplication(coantidomain(X), codomain(coantidomain(X))), domain(coantidomain(coantidomain(X)))), multiplication(coantidomain(X), antidomain(coantidomain(coantidomain(X)))))
% 152.65/19.63 = { by lemma 44 }
% 152.65/19.63 addition(multiplication(coantidomain(X), domain(coantidomain(coantidomain(X)))), multiplication(coantidomain(X), antidomain(coantidomain(coantidomain(X)))))
% 152.65/19.63 = { by axiom 19 (right_distributivity) R->L }
% 152.65/19.63 multiplication(coantidomain(X), addition(domain(coantidomain(coantidomain(X))), antidomain(coantidomain(coantidomain(X)))))
% 152.65/19.63 = { by lemma 24 }
% 152.65/19.63 multiplication(coantidomain(X), one)
% 152.65/19.63 = { by axiom 6 (multiplicative_right_identity) }
% 152.65/19.63 coantidomain(X)
% 152.65/19.63
% 152.65/19.63 Lemma 48: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
% 152.65/19.63 Proof:
% 152.65/19.63 multiplication(addition(X, antidomain(Y)), Y)
% 152.65/19.63 = { by axiom 20 (left_distributivity) }
% 152.65/19.63 addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
% 152.65/19.63 = { by axiom 10 (domain1) }
% 152.65/19.63 addition(multiplication(X, Y), zero)
% 152.65/19.63 = { by axiom 5 (additive_identity) }
% 152.65/19.63 multiplication(X, Y)
% 152.65/19.63
% 152.65/19.63 Lemma 49: multiplication(domain(X), X) = X.
% 152.65/19.63 Proof:
% 152.65/19.63 multiplication(domain(X), X)
% 152.65/19.63 = { by lemma 48 R->L }
% 152.65/19.63 multiplication(addition(domain(X), antidomain(X)), X)
% 152.65/19.63 = { by lemma 24 }
% 152.65/19.63 multiplication(one, X)
% 152.65/19.63 = { by axiom 7 (multiplicative_left_identity) }
% 153.12/19.63 X
% 153.12/19.63
% 153.12/19.63 Lemma 50: domain(antidomain(X)) = antidomain(X).
% 153.12/19.63 Proof:
% 153.12/19.63 domain(antidomain(X))
% 153.12/19.63 = { by lemma 28 R->L }
% 153.12/19.63 antidomain(domain(X))
% 153.12/19.63 = { by axiom 6 (multiplicative_right_identity) R->L }
% 153.12/19.63 multiplication(antidomain(domain(X)), one)
% 153.12/19.63 = { by lemma 24 R->L }
% 153.12/19.63 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 153.12/19.63 = { by lemma 34 }
% 153.12/19.63 multiplication(antidomain(domain(X)), antidomain(X))
% 153.12/19.63 = { by lemma 28 }
% 153.12/19.63 multiplication(domain(antidomain(X)), antidomain(X))
% 153.12/19.63 = { by lemma 49 }
% 153.12/19.63 antidomain(X)
% 153.12/19.63
% 153.12/19.63 Lemma 51: domain(domain(X)) = domain(X).
% 153.12/19.63 Proof:
% 153.12/19.63 domain(domain(X))
% 153.12/19.63 = { by axiom 1 (domain4) }
% 153.12/19.63 domain(antidomain(antidomain(X)))
% 153.12/19.63 = { by lemma 50 }
% 153.12/19.63 antidomain(antidomain(X))
% 153.12/19.63 = { by axiom 1 (domain4) R->L }
% 153.12/19.63 domain(X)
% 153.12/19.63
% 153.12/19.63 Lemma 52: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
% 153.12/19.63 Proof:
% 153.12/19.63 multiplication(X, addition(Y, one))
% 153.12/19.63 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.63 multiplication(X, addition(one, Y))
% 153.12/19.63 = { by axiom 19 (right_distributivity) }
% 153.12/19.63 addition(multiplication(X, one), multiplication(X, Y))
% 153.12/19.63 = { by axiom 6 (multiplicative_right_identity) }
% 153.12/19.63 addition(X, multiplication(X, Y))
% 153.12/19.63
% 153.12/19.63 Lemma 53: multiplication(X, addition(one, Y)) = addition(X, multiplication(X, Y)).
% 153.12/19.63 Proof:
% 153.12/19.63 multiplication(X, addition(one, Y))
% 153.12/19.63 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.63 multiplication(X, addition(Y, one))
% 153.12/19.63 = { by lemma 52 }
% 153.12/19.63 addition(X, multiplication(X, Y))
% 153.12/19.63
% 153.12/19.63 Lemma 54: addition(antidomain(X), codomain(domain(X))) = one.
% 153.12/19.63 Proof:
% 153.12/19.63 addition(antidomain(X), codomain(domain(X)))
% 153.12/19.63 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.63 addition(codomain(domain(X)), antidomain(X))
% 153.12/19.63 = { by axiom 6 (multiplicative_right_identity) R->L }
% 153.12/19.63 addition(codomain(domain(X)), multiplication(antidomain(X), one))
% 153.12/19.63 = { by lemma 32 R->L }
% 153.12/19.63 addition(codomain(domain(X)), multiplication(antidomain(X), addition(one, coantidomain(domain(X)))))
% 153.12/19.63 = { by lemma 53 }
% 153.12/19.63 addition(codomain(domain(X)), addition(antidomain(X), multiplication(antidomain(X), coantidomain(domain(X)))))
% 153.12/19.63 = { by lemma 42 R->L }
% 153.12/19.63 addition(codomain(domain(X)), addition(antidomain(X), multiplication(addition(domain(X), antidomain(X)), coantidomain(domain(X)))))
% 153.12/19.63 = { by lemma 24 }
% 153.12/19.63 addition(codomain(domain(X)), addition(antidomain(X), multiplication(one, coantidomain(domain(X)))))
% 153.12/19.63 = { by axiom 7 (multiplicative_left_identity) }
% 153.12/19.63 addition(codomain(domain(X)), addition(antidomain(X), coantidomain(domain(X))))
% 153.12/19.63 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.63 addition(codomain(domain(X)), addition(coantidomain(domain(X)), antidomain(X)))
% 153.12/19.63 = { by axiom 13 (additive_associativity) }
% 153.12/19.63 addition(addition(codomain(domain(X)), coantidomain(domain(X))), antidomain(X))
% 153.12/19.63 = { by lemma 29 }
% 153.12/19.63 addition(one, antidomain(X))
% 153.12/19.63 = { by lemma 37 }
% 153.12/19.63 one
% 153.12/19.63
% 153.12/19.63 Lemma 55: multiplication(coantidomain(X), addition(Y, codomain(X))) = multiplication(coantidomain(X), Y).
% 153.12/19.63 Proof:
% 153.12/19.63 multiplication(coantidomain(X), addition(Y, codomain(X)))
% 153.12/19.63 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.63 multiplication(coantidomain(X), addition(codomain(X), Y))
% 153.12/19.63 = { by axiom 19 (right_distributivity) }
% 153.12/19.63 addition(multiplication(coantidomain(X), codomain(X)), multiplication(coantidomain(X), Y))
% 153.12/19.63 = { by lemma 46 }
% 153.12/19.63 addition(zero, multiplication(coantidomain(X), Y))
% 153.12/19.63 = { by lemma 27 }
% 153.12/19.63 multiplication(coantidomain(X), Y)
% 153.12/19.63
% 153.12/19.63 Lemma 56: multiplication(coantidomain(antidomain(X)), X) = X.
% 153.12/19.63 Proof:
% 153.12/19.63 multiplication(coantidomain(antidomain(X)), X)
% 153.12/19.63 = { by axiom 5 (additive_identity) R->L }
% 153.12/19.63 multiplication(addition(coantidomain(antidomain(X)), zero), X)
% 153.12/19.63 = { by axiom 9 (codomain1) R->L }
% 153.12/19.63 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), coantidomain(multiplication(codomain(antidomain(X)), domain(X))))), X)
% 153.12/19.63 = { by axiom 2 (codomain4) }
% 153.12/19.63 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), domain(X))))), X)
% 153.12/19.63 = { by axiom 22 (codomain2) R->L }
% 153.12/19.63 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), addition(coantidomain(multiplication(antidomain(X), domain(X))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(X))), domain(X)))))), X)
% 153.12/19.63 = { by axiom 2 (codomain4) R->L }
% 153.12/19.63 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), addition(coantidomain(multiplication(antidomain(X), domain(X))), coantidomain(multiplication(codomain(antidomain(X)), domain(X)))))), X)
% 153.12/19.64 = { by axiom 1 (domain4) }
% 153.12/19.64 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), addition(coantidomain(multiplication(antidomain(X), antidomain(antidomain(X)))), coantidomain(multiplication(codomain(antidomain(X)), domain(X)))))), X)
% 153.12/19.64 = { by lemma 50 R->L }
% 153.12/19.64 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), addition(coantidomain(multiplication(domain(antidomain(X)), antidomain(antidomain(X)))), coantidomain(multiplication(codomain(antidomain(X)), domain(X)))))), X)
% 153.12/19.64 = { by lemma 38 }
% 153.12/19.64 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), addition(coantidomain(zero), coantidomain(multiplication(codomain(antidomain(X)), domain(X)))))), X)
% 153.12/19.64 = { by lemma 26 R->L }
% 153.12/19.64 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), addition(coantidomain(coantidomain(one)), coantidomain(multiplication(codomain(antidomain(X)), domain(X)))))), X)
% 153.12/19.64 = { by axiom 2 (codomain4) R->L }
% 153.12/19.64 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), addition(codomain(one), coantidomain(multiplication(codomain(antidomain(X)), domain(X)))))), X)
% 153.12/19.64 = { by axiom 5 (additive_identity) R->L }
% 153.12/19.64 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), addition(addition(codomain(one), zero), coantidomain(multiplication(codomain(antidomain(X)), domain(X)))))), X)
% 153.12/19.64 = { by lemma 26 R->L }
% 153.12/19.64 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), addition(addition(codomain(one), coantidomain(one)), coantidomain(multiplication(codomain(antidomain(X)), domain(X)))))), X)
% 153.12/19.64 = { by lemma 29 }
% 153.12/19.64 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), addition(one, coantidomain(multiplication(codomain(antidomain(X)), domain(X)))))), X)
% 153.12/19.64 = { by lemma 32 }
% 153.12/19.64 multiplication(addition(coantidomain(antidomain(X)), multiplication(multiplication(codomain(antidomain(X)), domain(X)), one)), X)
% 153.12/19.64 = { by axiom 14 (multiplicative_associativity) R->L }
% 153.12/19.64 multiplication(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), multiplication(domain(X), one))), X)
% 153.12/19.64 = { by axiom 6 (multiplicative_right_identity) }
% 153.12/19.64 multiplication(addition(coantidomain(antidomain(X)), multiplication(codomain(antidomain(X)), domain(X))), X)
% 153.12/19.64 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.64 multiplication(addition(multiplication(codomain(antidomain(X)), domain(X)), coantidomain(antidomain(X))), X)
% 153.12/19.64 = { by axiom 20 (left_distributivity) }
% 153.12/19.64 addition(multiplication(multiplication(codomain(antidomain(X)), domain(X)), X), multiplication(coantidomain(antidomain(X)), X))
% 153.12/19.64 = { by axiom 14 (multiplicative_associativity) R->L }
% 153.12/19.64 addition(multiplication(codomain(antidomain(X)), multiplication(domain(X), X)), multiplication(coantidomain(antidomain(X)), X))
% 153.12/19.64 = { by lemma 49 }
% 153.12/19.64 addition(multiplication(codomain(antidomain(X)), X), multiplication(coantidomain(antidomain(X)), X))
% 153.12/19.64 = { by axiom 20 (left_distributivity) R->L }
% 153.12/19.64 multiplication(addition(codomain(antidomain(X)), coantidomain(antidomain(X))), X)
% 153.12/19.64 = { by lemma 29 }
% 153.12/19.64 multiplication(one, X)
% 153.12/19.64 = { by axiom 7 (multiplicative_left_identity) }
% 153.12/19.64 X
% 153.12/19.64
% 153.12/19.64 Lemma 57: coantidomain(antidomain(X)) = domain(X).
% 153.12/19.64 Proof:
% 153.12/19.64 coantidomain(antidomain(X))
% 153.12/19.64 = { by axiom 6 (multiplicative_right_identity) R->L }
% 153.12/19.64 multiplication(coantidomain(antidomain(X)), one)
% 153.12/19.64 = { by lemma 54 R->L }
% 153.12/19.64 multiplication(coantidomain(antidomain(X)), addition(antidomain(antidomain(X)), codomain(domain(antidomain(X)))))
% 153.12/19.64 = { by axiom 1 (domain4) R->L }
% 153.12/19.64 multiplication(coantidomain(antidomain(X)), addition(domain(X), codomain(domain(antidomain(X)))))
% 153.12/19.64 = { by lemma 50 }
% 153.12/19.64 multiplication(coantidomain(antidomain(X)), addition(domain(X), codomain(antidomain(X))))
% 153.12/19.64 = { by lemma 55 }
% 153.12/19.64 multiplication(coantidomain(antidomain(X)), domain(X))
% 153.12/19.64 = { by lemma 50 R->L }
% 153.12/19.64 multiplication(coantidomain(domain(antidomain(X))), domain(X))
% 153.12/19.64 = { by axiom 1 (domain4) }
% 153.12/19.64 multiplication(coantidomain(antidomain(antidomain(antidomain(X)))), domain(X))
% 153.12/19.64 = { by axiom 1 (domain4) R->L }
% 153.12/19.64 multiplication(coantidomain(antidomain(domain(X))), domain(X))
% 153.12/19.64 = { by lemma 56 }
% 153.12/19.64 domain(X)
% 153.12/19.64
% 153.12/19.64 Lemma 58: coantidomain(domain(X)) = antidomain(X).
% 153.12/19.64 Proof:
% 153.12/19.64 coantidomain(domain(X))
% 153.12/19.64 = { by axiom 6 (multiplicative_right_identity) R->L }
% 153.12/19.64 multiplication(coantidomain(domain(X)), one)
% 153.12/19.64 = { by lemma 54 R->L }
% 153.12/19.64 multiplication(coantidomain(domain(X)), addition(antidomain(X), codomain(domain(X))))
% 153.12/19.64 = { by lemma 55 }
% 153.12/19.64 multiplication(coantidomain(domain(X)), antidomain(X))
% 153.12/19.64 = { by axiom 1 (domain4) }
% 153.12/19.64 multiplication(coantidomain(antidomain(antidomain(X))), antidomain(X))
% 153.12/19.64 = { by lemma 56 }
% 153.12/19.64 antidomain(X)
% 153.12/19.64
% 153.12/19.64 Lemma 59: codomain(domain(X)) = domain(X).
% 153.12/19.64 Proof:
% 153.12/19.64 codomain(domain(X))
% 153.12/19.64 = { by lemma 57 R->L }
% 153.12/19.64 codomain(coantidomain(antidomain(X)))
% 153.12/19.64 = { by lemma 45 }
% 153.12/19.64 coantidomain(antidomain(X))
% 153.12/19.64 = { by lemma 57 }
% 153.12/19.64 domain(X)
% 153.12/19.64
% 153.12/19.64 Lemma 60: addition(one, domain(X)) = one.
% 153.12/19.64 Proof:
% 153.12/19.64 addition(one, domain(X))
% 153.12/19.64 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.64 addition(domain(X), one)
% 153.12/19.64 = { by lemma 24 R->L }
% 153.12/19.64 addition(domain(X), addition(domain(X), antidomain(X)))
% 153.12/19.64 = { by lemma 30 }
% 153.12/19.64 addition(domain(X), antidomain(X))
% 153.12/19.64 = { by lemma 24 }
% 153.12/19.64 one
% 153.12/19.64
% 153.12/19.64 Lemma 61: addition(X, addition(Y, Z)) = addition(Y, addition(X, Z)).
% 153.12/19.64 Proof:
% 153.12/19.64 addition(X, addition(Y, Z))
% 153.12/19.64 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.64 addition(addition(Y, Z), X)
% 153.12/19.64 = { by axiom 13 (additive_associativity) R->L }
% 153.12/19.64 addition(Y, addition(Z, X))
% 153.12/19.64 = { by axiom 4 (additive_commutativity) }
% 153.12/19.64 addition(Y, addition(X, Z))
% 153.12/19.64
% 153.12/19.64 Lemma 62: multiplication(addition(codomain(X), Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
% 153.12/19.64 Proof:
% 153.12/19.64 multiplication(addition(codomain(X), Y), coantidomain(X))
% 153.12/19.64 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.64 multiplication(addition(Y, codomain(X)), coantidomain(X))
% 153.12/19.64 = { by lemma 45 R->L }
% 153.12/19.64 multiplication(addition(Y, codomain(X)), codomain(coantidomain(X)))
% 153.12/19.64 = { by lemma 41 R->L }
% 153.12/19.64 multiplication(addition(Y, codomain(X)), coantidomain(codomain(X)))
% 153.12/19.64 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.64 multiplication(addition(codomain(X), Y), coantidomain(codomain(X)))
% 153.12/19.64 = { by lemma 42 }
% 153.12/19.64 multiplication(Y, coantidomain(codomain(X)))
% 153.12/19.64 = { by lemma 41 }
% 153.12/19.64 multiplication(Y, codomain(coantidomain(X)))
% 153.12/19.64 = { by lemma 45 }
% 153.12/19.64 multiplication(Y, coantidomain(X))
% 153.12/19.64
% 153.12/19.64 Lemma 63: addition(X, multiplication(addition(X, Y), domain(Z))) = addition(X, multiplication(Y, domain(Z))).
% 153.12/19.64 Proof:
% 153.12/19.64 addition(X, multiplication(addition(X, Y), domain(Z)))
% 153.12/19.64 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.64 addition(X, multiplication(addition(Y, X), domain(Z)))
% 153.12/19.64 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.64 addition(multiplication(addition(Y, X), domain(Z)), X)
% 153.12/19.64 = { by axiom 20 (left_distributivity) }
% 153.12/19.64 addition(addition(multiplication(Y, domain(Z)), multiplication(X, domain(Z))), X)
% 153.12/19.64 = { by axiom 13 (additive_associativity) R->L }
% 153.12/19.64 addition(multiplication(Y, domain(Z)), addition(multiplication(X, domain(Z)), X))
% 153.12/19.64 = { by axiom 4 (additive_commutativity) }
% 153.12/19.64 addition(multiplication(Y, domain(Z)), addition(X, multiplication(X, domain(Z))))
% 153.12/19.64 = { by lemma 53 R->L }
% 153.12/19.64 addition(multiplication(Y, domain(Z)), multiplication(X, addition(one, domain(Z))))
% 153.12/19.64 = { by lemma 60 }
% 153.12/19.64 addition(multiplication(Y, domain(Z)), multiplication(X, one))
% 153.12/19.64 = { by axiom 6 (multiplicative_right_identity) }
% 153.12/19.64 addition(multiplication(Y, domain(Z)), X)
% 153.12/19.64 = { by axiom 4 (additive_commutativity) }
% 153.12/19.64 addition(X, multiplication(Y, domain(Z)))
% 153.12/19.64
% 153.12/19.64 Lemma 64: multiplication(coantidomain(X), addition(codomain(X), Y)) = multiplication(coantidomain(X), Y).
% 153.12/19.64 Proof:
% 153.12/19.64 multiplication(coantidomain(X), addition(codomain(X), Y))
% 153.12/19.64 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.64 multiplication(coantidomain(X), addition(Y, codomain(X)))
% 153.12/19.64 = { by lemma 55 }
% 153.12/19.64 multiplication(coantidomain(X), Y)
% 153.12/19.64
% 153.12/19.64 Lemma 65: addition(multiplication(X, Y), addition(Z, multiplication(X, W))) = addition(Z, multiplication(X, addition(Y, W))).
% 153.12/19.64 Proof:
% 153.12/19.64 addition(multiplication(X, Y), addition(Z, multiplication(X, W)))
% 153.12/19.64 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.64 addition(multiplication(X, Y), addition(multiplication(X, W), Z))
% 153.12/19.64 = { by axiom 13 (additive_associativity) }
% 153.12/19.64 addition(addition(multiplication(X, Y), multiplication(X, W)), Z)
% 153.12/19.64 = { by axiom 19 (right_distributivity) R->L }
% 153.12/19.64 addition(multiplication(X, addition(Y, W)), Z)
% 153.12/19.64 = { by axiom 4 (additive_commutativity) }
% 153.12/19.64 addition(Z, multiplication(X, addition(Y, W)))
% 153.12/19.64
% 153.12/19.64 Lemma 66: addition(X, multiplication(domain(Y), X)) = X.
% 153.12/19.64 Proof:
% 153.12/19.64 addition(X, multiplication(domain(Y), X))
% 153.12/19.64 = { by lemma 36 R->L }
% 153.12/19.64 multiplication(addition(one, domain(Y)), X)
% 153.12/19.64 = { by lemma 60 }
% 153.12/19.64 multiplication(one, X)
% 153.12/19.64 = { by axiom 7 (multiplicative_left_identity) }
% 153.12/19.64 X
% 153.12/19.64
% 153.12/19.64 Lemma 67: addition(X, multiplication(domain(Y), addition(X, Z))) = addition(X, multiplication(domain(Y), Z)).
% 153.12/19.64 Proof:
% 153.12/19.64 addition(X, multiplication(domain(Y), addition(X, Z)))
% 153.12/19.64 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.64 addition(X, multiplication(domain(Y), addition(Z, X)))
% 153.12/19.64 = { by lemma 65 R->L }
% 153.12/19.64 addition(multiplication(domain(Y), Z), addition(X, multiplication(domain(Y), X)))
% 153.12/19.64 = { by lemma 66 }
% 153.12/19.64 addition(multiplication(domain(Y), Z), X)
% 153.12/19.64 = { by axiom 4 (additive_commutativity) }
% 153.12/19.64 addition(X, multiplication(domain(Y), Z))
% 153.12/19.64
% 153.12/19.64 Lemma 68: multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y))) = zero.
% 153.12/19.64 Proof:
% 153.12/19.64 multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y)))
% 153.12/19.64 = { by lemma 48 R->L }
% 153.12/19.64 multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))), multiplication(X, domain(Y)))
% 153.12/19.64 = { by lemma 40 }
% 153.12/19.64 multiplication(antidomain(multiplication(X, domain(Y))), multiplication(X, domain(Y)))
% 153.12/19.64 = { by axiom 10 (domain1) }
% 153.12/19.64 zero
% 153.12/19.64
% 153.12/19.64 Lemma 69: multiplication(X, domain(multiplication(coantidomain(X), Y))) = zero.
% 153.12/19.64 Proof:
% 153.12/19.64 multiplication(X, domain(multiplication(coantidomain(X), Y)))
% 153.12/19.64 = { by axiom 7 (multiplicative_left_identity) R->L }
% 153.12/19.64 multiplication(one, multiplication(X, domain(multiplication(coantidomain(X), Y))))
% 153.12/19.64 = { by lemma 25 R->L }
% 153.12/19.64 multiplication(antidomain(zero), multiplication(X, domain(multiplication(coantidomain(X), Y))))
% 153.12/19.64 = { by axiom 8 (left_annihilation) R->L }
% 153.12/19.64 multiplication(antidomain(multiplication(zero, Y)), multiplication(X, domain(multiplication(coantidomain(X), Y))))
% 153.12/19.64 = { by axiom 9 (codomain1) R->L }
% 153.12/19.64 multiplication(antidomain(multiplication(multiplication(X, coantidomain(X)), Y)), multiplication(X, domain(multiplication(coantidomain(X), Y))))
% 153.12/19.65 = { by axiom 14 (multiplicative_associativity) R->L }
% 153.12/19.65 multiplication(antidomain(multiplication(X, multiplication(coantidomain(X), Y))), multiplication(X, domain(multiplication(coantidomain(X), Y))))
% 153.12/19.65 = { by lemma 68 }
% 153.12/19.65 zero
% 153.12/19.65
% 153.12/19.65 Lemma 70: addition(domain(X), multiplication(antidomain(X), domain(Y))) = addition(domain(X), domain(Y)).
% 153.12/19.65 Proof:
% 153.12/19.65 addition(domain(X), multiplication(antidomain(X), domain(Y)))
% 153.12/19.65 = { by lemma 63 R->L }
% 153.12/19.65 addition(domain(X), multiplication(addition(domain(X), antidomain(X)), domain(Y)))
% 153.12/19.65 = { by lemma 24 }
% 153.12/19.65 addition(domain(X), multiplication(one, domain(Y)))
% 153.12/19.65 = { by axiom 7 (multiplicative_left_identity) }
% 153.12/19.65 addition(domain(X), domain(Y))
% 153.12/19.65
% 153.12/19.65 Lemma 71: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y).
% 153.12/19.65 Proof:
% 153.12/19.65 multiplication(antidomain(X), addition(Y, X))
% 153.12/19.65 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.65 multiplication(antidomain(X), addition(X, Y))
% 153.12/19.65 = { by lemma 34 }
% 153.12/19.65 multiplication(antidomain(X), Y)
% 153.12/19.65
% 153.12/19.65 Lemma 72: addition(domain(X), domain(addition(X, Y))) = domain(addition(X, Y)).
% 153.12/19.65 Proof:
% 153.12/19.65 addition(domain(X), domain(addition(X, Y)))
% 153.12/19.65 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.65 addition(domain(X), domain(addition(Y, X)))
% 153.12/19.65 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.65 addition(domain(addition(Y, X)), domain(X))
% 153.12/19.65 = { by lemma 70 R->L }
% 153.12/19.65 addition(domain(addition(Y, X)), multiplication(antidomain(addition(Y, X)), domain(X)))
% 153.12/19.65 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.65 addition(domain(addition(Y, X)), multiplication(antidomain(addition(X, Y)), domain(X)))
% 153.12/19.65 = { by axiom 7 (multiplicative_left_identity) R->L }
% 153.12/19.65 addition(domain(addition(Y, X)), multiplication(one, multiplication(antidomain(addition(X, Y)), domain(X))))
% 153.12/19.65 = { by lemma 25 R->L }
% 153.12/19.65 addition(domain(addition(Y, X)), multiplication(antidomain(zero), multiplication(antidomain(addition(X, Y)), domain(X))))
% 153.12/19.65 = { by axiom 10 (domain1) R->L }
% 153.12/19.65 addition(domain(addition(Y, X)), multiplication(antidomain(multiplication(antidomain(addition(X, Y)), addition(X, Y))), multiplication(antidomain(addition(X, Y)), domain(X))))
% 153.12/19.65 = { by lemma 30 R->L }
% 153.12/19.65 addition(domain(addition(Y, X)), multiplication(antidomain(multiplication(antidomain(addition(X, Y)), addition(X, addition(X, Y)))), multiplication(antidomain(addition(X, Y)), domain(X))))
% 153.12/19.65 = { by lemma 71 }
% 153.12/19.65 addition(domain(addition(Y, X)), multiplication(antidomain(multiplication(antidomain(addition(X, Y)), X)), multiplication(antidomain(addition(X, Y)), domain(X))))
% 153.12/19.65 = { by lemma 68 }
% 153.12/19.65 addition(domain(addition(Y, X)), zero)
% 153.12/19.65 = { by axiom 5 (additive_identity) }
% 153.12/19.65 domain(addition(Y, X))
% 153.12/19.65 = { by axiom 4 (additive_commutativity) }
% 153.12/19.65 domain(addition(X, Y))
% 153.12/19.65
% 153.12/19.65 Lemma 73: multiplication(antidomain(X), multiplication(X, Y)) = zero.
% 153.12/19.65 Proof:
% 153.12/19.65 multiplication(antidomain(X), multiplication(X, Y))
% 153.12/19.65 = { by axiom 14 (multiplicative_associativity) }
% 153.12/19.65 multiplication(multiplication(antidomain(X), X), Y)
% 153.12/19.65 = { by axiom 10 (domain1) }
% 153.12/19.65 multiplication(zero, Y)
% 153.12/19.65 = { by axiom 8 (left_annihilation) }
% 153.12/19.65 zero
% 153.12/19.65
% 153.12/19.65 Lemma 74: addition(domain(X), domain(multiplication(X, Y))) = domain(X).
% 153.12/19.65 Proof:
% 153.12/19.65 addition(domain(X), domain(multiplication(X, Y)))
% 153.12/19.65 = { by lemma 70 R->L }
% 153.12/19.65 addition(domain(X), multiplication(antidomain(X), domain(multiplication(X, Y))))
% 153.12/19.65 = { by axiom 7 (multiplicative_left_identity) R->L }
% 153.12/19.65 addition(domain(X), multiplication(one, multiplication(antidomain(X), domain(multiplication(X, Y)))))
% 153.12/19.65 = { by lemma 25 R->L }
% 153.12/19.65 addition(domain(X), multiplication(antidomain(zero), multiplication(antidomain(X), domain(multiplication(X, Y)))))
% 153.12/19.65 = { by lemma 73 R->L }
% 153.12/19.65 addition(domain(X), multiplication(antidomain(multiplication(antidomain(X), multiplication(X, Y))), multiplication(antidomain(X), domain(multiplication(X, Y)))))
% 153.12/19.65 = { by lemma 68 }
% 153.12/19.65 addition(domain(X), zero)
% 153.12/19.65 = { by axiom 5 (additive_identity) }
% 153.12/19.65 domain(X)
% 153.12/19.65
% 153.12/19.65 Lemma 75: addition(domain(X), addition(antidomain(X), Y)) = addition(Y, one).
% 153.12/19.65 Proof:
% 153.12/19.65 addition(domain(X), addition(antidomain(X), Y))
% 153.12/19.65 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.65 addition(domain(X), addition(Y, antidomain(X)))
% 153.12/19.65 = { by lemma 33 }
% 153.12/19.65 addition(Y, one)
% 153.12/19.65
% 153.12/19.65 Lemma 76: domain(multiplication(coantidomain(X), addition(Y, addition(Z, one)))) = coantidomain(X).
% 153.12/19.65 Proof:
% 153.12/19.65 domain(multiplication(coantidomain(X), addition(Y, addition(Z, one))))
% 153.12/19.65 = { by axiom 5 (additive_identity) R->L }
% 153.12/19.65 addition(domain(multiplication(coantidomain(X), addition(Y, addition(Z, one)))), zero)
% 153.12/19.65 = { by lemma 69 R->L }
% 153.12/19.65 addition(domain(multiplication(coantidomain(X), addition(Y, addition(Z, one)))), multiplication(X, domain(multiplication(coantidomain(X), addition(Y, addition(Z, one))))))
% 153.12/19.65 = { by lemma 35 R->L }
% 153.12/19.65 multiplication(addition(X, one), domain(multiplication(coantidomain(X), addition(Y, addition(Z, one)))))
% 153.12/19.65 = { by axiom 13 (additive_associativity) }
% 153.12/19.65 multiplication(addition(X, one), domain(multiplication(coantidomain(X), addition(addition(Y, Z), one))))
% 153.12/19.65 = { by lemma 52 }
% 153.12/19.65 multiplication(addition(X, one), domain(addition(coantidomain(X), multiplication(coantidomain(X), addition(Y, Z)))))
% 153.12/19.65 = { by lemma 72 R->L }
% 153.12/19.65 multiplication(addition(X, one), addition(domain(coantidomain(X)), domain(addition(coantidomain(X), multiplication(coantidomain(X), addition(Y, Z))))))
% 153.12/19.65 = { by lemma 52 R->L }
% 153.12/19.65 multiplication(addition(X, one), addition(domain(coantidomain(X)), domain(multiplication(coantidomain(X), addition(addition(Y, Z), one)))))
% 153.12/19.65 = { by lemma 74 }
% 153.12/19.65 multiplication(addition(X, one), domain(coantidomain(X)))
% 153.12/19.65 = { by lemma 47 R->L }
% 153.12/19.65 multiplication(addition(X, one), domain(antidomain(codomain(X))))
% 153.12/19.65 = { by lemma 50 }
% 153.12/19.65 multiplication(addition(X, one), antidomain(codomain(X)))
% 153.12/19.65 = { by lemma 47 }
% 153.12/19.65 multiplication(addition(X, one), coantidomain(X))
% 153.12/19.65 = { by lemma 75 R->L }
% 153.12/19.65 multiplication(addition(domain(W), addition(antidomain(W), X)), coantidomain(X))
% 153.12/19.65 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.65 multiplication(addition(domain(W), addition(X, antidomain(W))), coantidomain(X))
% 153.12/19.65 = { by lemma 61 R->L }
% 153.12/19.65 multiplication(addition(X, addition(domain(W), antidomain(W))), coantidomain(X))
% 153.12/19.65 = { by lemma 42 }
% 153.12/19.65 multiplication(addition(domain(W), antidomain(W)), coantidomain(X))
% 153.12/19.65 = { by lemma 24 }
% 153.12/19.65 multiplication(one, coantidomain(X))
% 153.12/19.65 = { by axiom 7 (multiplicative_left_identity) }
% 153.12/19.65 coantidomain(X)
% 153.12/19.65
% 153.12/19.65 Lemma 77: multiplication(antidomain(X), addition(domain(X), Y)) = multiplication(antidomain(X), Y).
% 153.12/19.65 Proof:
% 153.12/19.65 multiplication(antidomain(X), addition(domain(X), Y))
% 153.12/19.65 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.65 multiplication(antidomain(X), addition(Y, domain(X)))
% 153.12/19.65 = { by lemma 50 R->L }
% 153.12/19.65 multiplication(domain(antidomain(X)), addition(Y, domain(X)))
% 153.12/19.65 = { by lemma 28 R->L }
% 153.12/19.65 multiplication(antidomain(domain(X)), addition(Y, domain(X)))
% 153.12/19.65 = { by lemma 71 }
% 153.12/19.65 multiplication(antidomain(domain(X)), Y)
% 153.12/19.65 = { by lemma 28 }
% 153.12/19.65 multiplication(domain(antidomain(X)), Y)
% 153.12/19.65 = { by lemma 50 }
% 153.12/19.65 multiplication(antidomain(X), Y)
% 153.12/19.65
% 153.12/19.65 Lemma 78: addition(domain(X), multiplication(domain(Y), antidomain(X))) = addition(domain(X), domain(Y)).
% 153.12/19.65 Proof:
% 153.12/19.65 addition(domain(X), multiplication(domain(Y), antidomain(X)))
% 153.12/19.65 = { by lemma 67 R->L }
% 153.12/19.65 addition(domain(X), multiplication(domain(Y), addition(domain(X), antidomain(X))))
% 153.12/19.65 = { by lemma 24 }
% 153.12/19.65 addition(domain(X), multiplication(domain(Y), one))
% 153.12/19.65 = { by axiom 6 (multiplicative_right_identity) }
% 153.12/19.65 addition(domain(X), domain(Y))
% 153.12/19.65
% 153.12/19.65 Lemma 79: multiplication(addition(domain(X), Y), antidomain(X)) = multiplication(Y, antidomain(X)).
% 153.12/19.65 Proof:
% 153.12/19.65 multiplication(addition(domain(X), Y), antidomain(X))
% 153.12/19.65 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.65 multiplication(addition(Y, domain(X)), antidomain(X))
% 153.12/19.65 = { by lemma 39 }
% 153.12/19.65 multiplication(Y, antidomain(X))
% 153.12/19.65
% 153.12/19.65 Lemma 80: multiplication(antidomain(X), domain(Y)) = multiplication(domain(Y), antidomain(X)).
% 153.12/19.65 Proof:
% 153.12/19.65 multiplication(antidomain(X), domain(Y))
% 153.12/19.65 = { by lemma 77 R->L }
% 153.12/19.65 multiplication(antidomain(X), addition(domain(X), domain(Y)))
% 153.12/19.65 = { by lemma 78 R->L }
% 153.12/19.65 multiplication(antidomain(X), addition(domain(X), multiplication(domain(Y), antidomain(X))))
% 153.12/19.65 = { by lemma 77 }
% 153.12/19.65 multiplication(antidomain(X), multiplication(domain(Y), antidomain(X)))
% 153.12/19.65 = { by axiom 14 (multiplicative_associativity) }
% 153.12/19.65 multiplication(multiplication(antidomain(X), domain(Y)), antidomain(X))
% 153.12/19.65 = { by lemma 79 R->L }
% 153.12/19.65 multiplication(addition(domain(X), multiplication(antidomain(X), domain(Y))), antidomain(X))
% 153.12/19.65 = { by lemma 70 }
% 153.12/19.65 multiplication(addition(domain(X), domain(Y)), antidomain(X))
% 153.12/19.65 = { by lemma 79 }
% 153.12/19.65 multiplication(domain(Y), antidomain(X))
% 153.12/19.65
% 153.12/19.65 Lemma 81: addition(antidomain(X), multiplication(domain(X), antidomain(Y))) = addition(antidomain(X), antidomain(Y)).
% 153.12/19.65 Proof:
% 153.12/19.65 addition(antidomain(X), multiplication(domain(X), antidomain(Y)))
% 153.12/19.65 = { by lemma 80 R->L }
% 153.12/19.65 addition(antidomain(X), multiplication(antidomain(Y), domain(X)))
% 153.12/19.65 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.65 addition(multiplication(antidomain(Y), domain(X)), antidomain(X))
% 153.12/19.65 = { by axiom 7 (multiplicative_left_identity) R->L }
% 153.12/19.65 addition(multiplication(antidomain(Y), domain(X)), multiplication(one, antidomain(X)))
% 153.12/19.65 = { by lemma 37 R->L }
% 153.12/19.65 addition(multiplication(antidomain(Y), domain(X)), multiplication(addition(one, antidomain(Y)), antidomain(X)))
% 153.12/19.65 = { by lemma 71 R->L }
% 153.12/19.65 addition(multiplication(antidomain(Y), addition(domain(X), Y)), multiplication(addition(one, antidomain(Y)), antidomain(X)))
% 153.12/19.65 = { by lemma 36 }
% 153.12/19.65 addition(multiplication(antidomain(Y), addition(domain(X), Y)), addition(antidomain(X), multiplication(antidomain(Y), antidomain(X))))
% 153.12/19.65 = { by lemma 65 }
% 153.12/19.65 addition(antidomain(X), multiplication(antidomain(Y), addition(addition(domain(X), Y), antidomain(X))))
% 153.12/19.65 = { by axiom 13 (additive_associativity) R->L }
% 153.12/19.65 addition(antidomain(X), multiplication(antidomain(Y), addition(domain(X), addition(Y, antidomain(X)))))
% 153.12/19.65 = { by lemma 61 R->L }
% 153.12/19.65 addition(antidomain(X), multiplication(antidomain(Y), addition(Y, addition(domain(X), antidomain(X)))))
% 153.12/19.65 = { by lemma 34 }
% 153.12/19.65 addition(antidomain(X), multiplication(antidomain(Y), addition(domain(X), antidomain(X))))
% 153.12/19.65 = { by lemma 24 }
% 153.12/19.65 addition(antidomain(X), multiplication(antidomain(Y), one))
% 153.12/19.65 = { by axiom 6 (multiplicative_right_identity) }
% 153.12/19.65 addition(antidomain(X), antidomain(Y))
% 153.12/19.65
% 153.12/19.65 Lemma 82: domain(multiplication(X, domain(Y))) = domain(multiplication(X, Y)).
% 153.12/19.65 Proof:
% 153.12/19.65 domain(multiplication(X, domain(Y)))
% 153.12/19.66 = { by axiom 1 (domain4) }
% 153.12/19.66 antidomain(antidomain(multiplication(X, domain(Y))))
% 153.12/19.66 = { by lemma 40 R->L }
% 153.12/19.66 antidomain(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))))
% 153.12/19.66 = { by lemma 81 R->L }
% 153.12/19.66 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(domain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))))
% 153.12/19.66 = { by lemma 80 R->L }
% 153.12/19.66 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y)))))
% 153.12/19.66 = { by axiom 7 (multiplicative_left_identity) R->L }
% 153.12/19.66 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(one, multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 153.12/19.66 = { by lemma 25 R->L }
% 153.12/19.66 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(zero), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 153.12/19.66 = { by lemma 73 R->L }
% 153.12/19.66 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(antidomain(multiplication(X, domain(Y))), multiplication(multiplication(X, domain(Y)), Y))), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 153.12/19.66 = { by axiom 14 (multiplicative_associativity) R->L }
% 153.12/19.66 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(antidomain(multiplication(X, domain(Y))), multiplication(X, multiplication(domain(Y), Y)))), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 153.12/19.66 = { by lemma 49 }
% 153.12/19.66 antidomain(addition(antidomain(multiplication(X, Y)), multiplication(antidomain(multiplication(antidomain(multiplication(X, domain(Y))), multiplication(X, Y))), multiplication(antidomain(multiplication(X, domain(Y))), domain(multiplication(X, Y))))))
% 153.12/19.66 = { by lemma 68 }
% 153.12/19.66 antidomain(addition(antidomain(multiplication(X, Y)), zero))
% 153.12/19.66 = { by axiom 5 (additive_identity) }
% 153.12/19.66 antidomain(antidomain(multiplication(X, Y)))
% 153.12/19.66 = { by axiom 1 (domain4) R->L }
% 153.12/19.66 domain(multiplication(X, Y))
% 153.12/19.66
% 153.12/19.66 Lemma 83: domain(multiplication(domain(X), antidomain(Y))) = domain(multiplication(antidomain(Y), X)).
% 153.12/19.66 Proof:
% 153.12/19.66 domain(multiplication(domain(X), antidomain(Y)))
% 153.12/19.66 = { by lemma 80 R->L }
% 153.12/19.66 domain(multiplication(antidomain(Y), domain(X)))
% 153.12/19.66 = { by lemma 82 }
% 153.12/19.66 domain(multiplication(antidomain(Y), X))
% 153.12/19.66
% 153.12/19.66 Lemma 84: multiplication(domain(X), antidomain(Y)) = domain(multiplication(antidomain(Y), X)).
% 153.12/19.66 Proof:
% 153.12/19.66 multiplication(domain(X), antidomain(Y))
% 153.12/19.66 = { by lemma 51 R->L }
% 153.12/19.66 multiplication(domain(domain(X)), antidomain(Y))
% 153.12/19.66 = { by lemma 50 R->L }
% 153.12/19.66 multiplication(domain(domain(X)), domain(antidomain(Y)))
% 153.12/19.66 = { by lemma 74 R->L }
% 153.12/19.66 multiplication(domain(domain(X)), addition(domain(antidomain(Y)), domain(multiplication(antidomain(Y), domain(X)))))
% 153.12/19.66 = { by lemma 50 }
% 153.12/19.66 multiplication(domain(domain(X)), addition(antidomain(Y), domain(multiplication(antidomain(Y), domain(X)))))
% 153.12/19.66 = { by axiom 19 (right_distributivity) }
% 153.12/19.66 addition(multiplication(domain(domain(X)), antidomain(Y)), multiplication(domain(domain(X)), domain(multiplication(antidomain(Y), domain(X)))))
% 153.12/19.66 = { by lemma 51 R->L }
% 153.12/19.66 addition(multiplication(domain(domain(X)), antidomain(Y)), multiplication(domain(domain(domain(X))), domain(multiplication(antidomain(Y), domain(X)))))
% 153.12/19.66 = { by lemma 49 R->L }
% 153.12/19.66 addition(multiplication(multiplication(domain(domain(domain(X))), domain(domain(X))), antidomain(Y)), multiplication(domain(domain(domain(X))), domain(multiplication(antidomain(Y), domain(X)))))
% 153.12/19.66 = { by axiom 14 (multiplicative_associativity) R->L }
% 153.12/19.66 addition(multiplication(domain(domain(domain(X))), multiplication(domain(domain(X)), antidomain(Y))), multiplication(domain(domain(domain(X))), domain(multiplication(antidomain(Y), domain(X)))))
% 153.12/19.66 = { by axiom 19 (right_distributivity) R->L }
% 153.12/19.66 multiplication(domain(domain(domain(X))), addition(multiplication(domain(domain(X)), antidomain(Y)), domain(multiplication(antidomain(Y), domain(X)))))
% 153.12/19.66 = { by lemma 83 R->L }
% 153.12/19.66 multiplication(domain(domain(domain(X))), addition(multiplication(domain(domain(X)), antidomain(Y)), domain(multiplication(domain(domain(X)), antidomain(Y)))))
% 153.12/19.66 = { by axiom 6 (multiplicative_right_identity) R->L }
% 153.12/19.66 multiplication(domain(domain(domain(X))), addition(multiplication(domain(domain(X)), antidomain(Y)), multiplication(domain(multiplication(domain(domain(X)), antidomain(Y))), one)))
% 153.12/19.66 = { by lemma 49 R->L }
% 153.12/19.66 multiplication(domain(domain(domain(X))), addition(multiplication(domain(multiplication(domain(domain(X)), antidomain(Y))), multiplication(domain(domain(X)), antidomain(Y))), multiplication(domain(multiplication(domain(domain(X)), antidomain(Y))), one)))
% 153.12/19.66 = { by axiom 19 (right_distributivity) R->L }
% 153.12/19.66 multiplication(domain(domain(domain(X))), multiplication(domain(multiplication(domain(domain(X)), antidomain(Y))), addition(multiplication(domain(domain(X)), antidomain(Y)), one)))
% 153.12/19.66 = { by lemma 75 R->L }
% 153.12/19.66 multiplication(domain(domain(domain(X))), multiplication(domain(multiplication(domain(domain(X)), antidomain(Y))), addition(domain(Y), addition(antidomain(Y), multiplication(domain(domain(X)), antidomain(Y))))))
% 153.12/19.66 = { by lemma 66 }
% 153.12/19.66 multiplication(domain(domain(domain(X))), multiplication(domain(multiplication(domain(domain(X)), antidomain(Y))), addition(domain(Y), antidomain(Y))))
% 153.12/19.66 = { by lemma 24 }
% 153.12/19.66 multiplication(domain(domain(domain(X))), multiplication(domain(multiplication(domain(domain(X)), antidomain(Y))), one))
% 153.12/19.66 = { by axiom 6 (multiplicative_right_identity) }
% 153.12/19.66 multiplication(domain(domain(domain(X))), domain(multiplication(domain(domain(X)), antidomain(Y))))
% 153.12/19.66 = { by lemma 83 }
% 153.12/19.66 multiplication(domain(domain(domain(X))), domain(multiplication(antidomain(Y), domain(X))))
% 153.12/19.66 = { by lemma 51 }
% 153.12/19.66 multiplication(domain(domain(X)), domain(multiplication(antidomain(Y), domain(X))))
% 153.12/19.66 = { by lemma 80 }
% 153.12/19.66 multiplication(domain(domain(X)), domain(multiplication(domain(X), antidomain(Y))))
% 153.12/19.66 = { by lemma 59 R->L }
% 153.12/19.66 multiplication(domain(domain(X)), codomain(domain(multiplication(domain(X), antidomain(Y)))))
% 153.12/19.66 = { by lemma 74 R->L }
% 153.12/19.66 multiplication(addition(domain(domain(X)), domain(multiplication(domain(X), antidomain(Y)))), codomain(domain(multiplication(domain(X), antidomain(Y)))))
% 153.12/19.66 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.66 multiplication(addition(domain(multiplication(domain(X), antidomain(Y))), domain(domain(X))), codomain(domain(multiplication(domain(X), antidomain(Y)))))
% 153.12/19.66 = { by axiom 20 (left_distributivity) }
% 153.12/19.66 addition(multiplication(domain(multiplication(domain(X), antidomain(Y))), codomain(domain(multiplication(domain(X), antidomain(Y))))), multiplication(domain(domain(X)), codomain(domain(multiplication(domain(X), antidomain(Y))))))
% 153.12/19.66 = { by lemma 44 }
% 153.12/19.66 addition(domain(multiplication(domain(X), antidomain(Y))), multiplication(domain(domain(X)), codomain(domain(multiplication(domain(X), antidomain(Y))))))
% 153.12/19.66 = { by lemma 59 }
% 153.12/19.66 addition(domain(multiplication(domain(X), antidomain(Y))), multiplication(domain(domain(X)), domain(multiplication(domain(X), antidomain(Y)))))
% 153.12/19.66 = { by lemma 66 }
% 153.12/19.66 domain(multiplication(domain(X), antidomain(Y)))
% 153.12/19.66 = { by lemma 83 }
% 153.12/19.66 domain(multiplication(antidomain(Y), X))
% 153.12/19.66
% 153.12/19.66 Lemma 85: multiplication(domain(X), coantidomain(Y)) = domain(multiplication(coantidomain(Y), X)).
% 153.12/19.66 Proof:
% 153.12/19.66 multiplication(domain(X), coantidomain(Y))
% 153.12/19.66 = { by lemma 62 R->L }
% 153.12/19.66 multiplication(addition(codomain(Y), domain(X)), coantidomain(Y))
% 153.12/19.66 = { by axiom 7 (multiplicative_left_identity) R->L }
% 153.12/19.66 multiplication(addition(codomain(Y), multiplication(one, domain(X))), coantidomain(Y))
% 153.12/19.66 = { by lemma 29 R->L }
% 153.12/19.66 multiplication(addition(codomain(Y), multiplication(addition(codomain(Y), coantidomain(Y)), domain(X))), coantidomain(Y))
% 153.12/19.66 = { by lemma 63 }
% 153.12/19.66 multiplication(addition(codomain(Y), multiplication(coantidomain(Y), domain(X))), coantidomain(Y))
% 153.12/19.66 = { by lemma 62 }
% 153.12/19.66 multiplication(multiplication(coantidomain(Y), domain(X)), coantidomain(Y))
% 153.12/19.66 = { by axiom 14 (multiplicative_associativity) R->L }
% 153.12/19.66 multiplication(coantidomain(Y), multiplication(domain(X), coantidomain(Y)))
% 153.12/19.66 = { by lemma 64 R->L }
% 153.12/19.66 multiplication(coantidomain(Y), addition(codomain(Y), multiplication(domain(X), coantidomain(Y))))
% 153.12/19.66 = { by lemma 67 R->L }
% 153.12/19.66 multiplication(coantidomain(Y), addition(codomain(Y), multiplication(domain(X), addition(codomain(Y), coantidomain(Y)))))
% 153.12/19.66 = { by lemma 29 }
% 153.12/19.66 multiplication(coantidomain(Y), addition(codomain(Y), multiplication(domain(X), one)))
% 153.12/19.66 = { by axiom 6 (multiplicative_right_identity) }
% 153.12/19.66 multiplication(coantidomain(Y), addition(codomain(Y), domain(X)))
% 153.12/19.66 = { by lemma 64 }
% 153.12/19.66 multiplication(coantidomain(Y), domain(X))
% 153.12/19.66 = { by lemma 76 R->L }
% 153.12/19.66 multiplication(domain(multiplication(coantidomain(Y), addition(Z, addition(W, one)))), domain(X))
% 153.12/19.66 = { by axiom 1 (domain4) }
% 153.12/19.66 multiplication(domain(multiplication(coantidomain(Y), addition(Z, addition(W, one)))), antidomain(antidomain(X)))
% 153.12/19.66 = { by lemma 84 }
% 153.12/19.66 domain(multiplication(antidomain(antidomain(X)), multiplication(coantidomain(Y), addition(Z, addition(W, one)))))
% 153.12/19.66 = { by axiom 1 (domain4) R->L }
% 153.12/19.66 domain(multiplication(domain(X), multiplication(coantidomain(Y), addition(Z, addition(W, one)))))
% 153.12/19.66 = { by lemma 82 R->L }
% 153.12/19.66 domain(multiplication(domain(X), domain(multiplication(coantidomain(Y), addition(Z, addition(W, one))))))
% 153.12/19.66 = { by axiom 1 (domain4) }
% 153.12/19.66 domain(multiplication(domain(X), antidomain(antidomain(multiplication(coantidomain(Y), addition(Z, addition(W, one)))))))
% 153.12/19.66 = { by lemma 80 R->L }
% 153.12/19.66 domain(multiplication(antidomain(antidomain(multiplication(coantidomain(Y), addition(Z, addition(W, one))))), domain(X)))
% 153.12/19.66 = { by axiom 1 (domain4) R->L }
% 153.12/19.66 domain(multiplication(domain(multiplication(coantidomain(Y), addition(Z, addition(W, one)))), domain(X)))
% 153.12/19.66 = { by lemma 82 }
% 153.12/19.66 domain(multiplication(domain(multiplication(coantidomain(Y), addition(Z, addition(W, one)))), X))
% 153.12/19.66 = { by lemma 76 }
% 153.12/19.66 domain(multiplication(coantidomain(Y), X))
% 153.12/19.66
% 153.12/19.66 Lemma 86: addition(antidomain(X), antidomain(multiplication(antidomain(X), Y))) = one.
% 153.12/19.66 Proof:
% 153.12/19.66 addition(antidomain(X), antidomain(multiplication(antidomain(X), Y)))
% 153.12/19.66 = { by lemma 58 R->L }
% 153.12/19.66 addition(antidomain(X), antidomain(multiplication(coantidomain(domain(X)), Y)))
% 153.12/19.66 = { by lemma 81 R->L }
% 153.12/19.66 addition(antidomain(X), multiplication(domain(X), antidomain(multiplication(coantidomain(domain(X)), Y))))
% 153.12/19.66 = { by lemma 27 R->L }
% 153.12/19.66 addition(antidomain(X), addition(zero, multiplication(domain(X), antidomain(multiplication(coantidomain(domain(X)), Y)))))
% 153.12/19.66 = { by lemma 69 R->L }
% 153.12/19.66 addition(antidomain(X), addition(multiplication(domain(X), domain(multiplication(coantidomain(domain(X)), Y))), multiplication(domain(X), antidomain(multiplication(coantidomain(domain(X)), Y)))))
% 153.12/19.67 = { by axiom 19 (right_distributivity) R->L }
% 153.12/19.67 addition(antidomain(X), multiplication(domain(X), addition(domain(multiplication(coantidomain(domain(X)), Y)), antidomain(multiplication(coantidomain(domain(X)), Y)))))
% 153.12/19.67 = { by lemma 24 }
% 153.12/19.67 addition(antidomain(X), multiplication(domain(X), one))
% 153.12/19.67 = { by axiom 6 (multiplicative_right_identity) }
% 153.12/19.67 addition(antidomain(X), domain(X))
% 153.12/19.67 = { by axiom 4 (additive_commutativity) }
% 153.12/19.67 addition(domain(X), antidomain(X))
% 153.12/19.67 = { by lemma 24 }
% 153.12/19.67 one
% 153.12/19.67
% 153.12/19.67 Lemma 87: addition(domain(X), domain(multiplication(antidomain(X), Y))) = addition(domain(X), domain(Y)).
% 153.12/19.67 Proof:
% 153.12/19.67 addition(domain(X), domain(multiplication(antidomain(X), Y)))
% 153.12/19.67 = { by lemma 49 R->L }
% 153.12/19.67 addition(domain(X), domain(multiplication(multiplication(domain(antidomain(X)), antidomain(X)), Y)))
% 153.12/19.67 = { by lemma 50 }
% 153.12/19.67 addition(domain(X), domain(multiplication(multiplication(antidomain(X), antidomain(X)), Y)))
% 153.12/19.67 = { by axiom 14 (multiplicative_associativity) R->L }
% 153.12/19.67 addition(domain(X), domain(multiplication(antidomain(X), multiplication(antidomain(X), Y))))
% 153.12/19.67 = { by lemma 58 R->L }
% 153.12/19.67 addition(domain(X), domain(multiplication(coantidomain(domain(X)), multiplication(antidomain(X), Y))))
% 153.12/19.67 = { by lemma 85 R->L }
% 153.12/19.67 addition(domain(X), multiplication(domain(multiplication(antidomain(X), Y)), coantidomain(domain(X))))
% 153.12/19.67 = { by axiom 7 (multiplicative_left_identity) R->L }
% 153.12/19.67 addition(domain(multiplication(one, X)), multiplication(domain(multiplication(antidomain(X), Y)), coantidomain(domain(X))))
% 153.12/19.67 = { by lemma 86 R->L }
% 153.12/19.67 addition(domain(multiplication(addition(antidomain(X), antidomain(multiplication(antidomain(X), Y))), X)), multiplication(domain(multiplication(antidomain(X), Y)), coantidomain(domain(X))))
% 153.12/19.67 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.67 addition(domain(multiplication(addition(antidomain(multiplication(antidomain(X), Y)), antidomain(X)), X)), multiplication(domain(multiplication(antidomain(X), Y)), coantidomain(domain(X))))
% 153.12/19.67 = { by lemma 48 }
% 153.12/19.67 addition(domain(multiplication(antidomain(multiplication(antidomain(X), Y)), X)), multiplication(domain(multiplication(antidomain(X), Y)), coantidomain(domain(X))))
% 153.12/19.67 = { by lemma 58 R->L }
% 153.12/19.67 addition(domain(multiplication(coantidomain(domain(multiplication(antidomain(X), Y))), X)), multiplication(domain(multiplication(antidomain(X), Y)), coantidomain(domain(X))))
% 153.12/19.67 = { by lemma 84 R->L }
% 153.12/19.67 addition(domain(multiplication(coantidomain(multiplication(domain(Y), antidomain(X))), X)), multiplication(domain(multiplication(antidomain(X), Y)), coantidomain(domain(X))))
% 153.12/19.67 = { by lemma 84 R->L }
% 153.12/19.67 addition(domain(multiplication(coantidomain(multiplication(domain(Y), antidomain(X))), X)), multiplication(multiplication(domain(Y), antidomain(X)), coantidomain(domain(X))))
% 153.12/19.67 = { by lemma 85 R->L }
% 153.12/19.67 addition(multiplication(domain(X), coantidomain(multiplication(domain(Y), antidomain(X)))), multiplication(multiplication(domain(Y), antidomain(X)), coantidomain(domain(X))))
% 153.12/19.67 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.67 addition(multiplication(multiplication(domain(Y), antidomain(X)), coantidomain(domain(X))), multiplication(domain(X), coantidomain(multiplication(domain(Y), antidomain(X)))))
% 153.12/19.67 = { by lemma 43 R->L }
% 153.12/19.67 addition(multiplication(multiplication(domain(Y), antidomain(X)), coantidomain(domain(X))), multiplication(domain(X), addition(coantidomain(multiplication(domain(Y), antidomain(X))), coantidomain(domain(X)))))
% 153.12/19.67 = { by axiom 4 (additive_commutativity) }
% 153.12/19.67 addition(multiplication(multiplication(domain(Y), antidomain(X)), coantidomain(domain(X))), multiplication(domain(X), addition(coantidomain(domain(X)), coantidomain(multiplication(domain(Y), antidomain(X))))))
% 153.12/19.67 = { by lemma 43 R->L }
% 153.12/19.67 addition(multiplication(multiplication(domain(Y), antidomain(X)), addition(coantidomain(domain(X)), coantidomain(multiplication(domain(Y), antidomain(X))))), multiplication(domain(X), addition(coantidomain(domain(X)), coantidomain(multiplication(domain(Y), antidomain(X))))))
% 153.12/19.67 = { by axiom 20 (left_distributivity) R->L }
% 153.12/19.67 multiplication(addition(multiplication(domain(Y), antidomain(X)), domain(X)), addition(coantidomain(domain(X)), coantidomain(multiplication(domain(Y), antidomain(X)))))
% 153.12/19.67 = { by axiom 4 (additive_commutativity) }
% 153.12/19.67 multiplication(addition(domain(X), multiplication(domain(Y), antidomain(X))), addition(coantidomain(domain(X)), coantidomain(multiplication(domain(Y), antidomain(X)))))
% 153.12/19.67 = { by lemma 78 }
% 153.12/19.67 multiplication(addition(domain(X), domain(Y)), addition(coantidomain(domain(X)), coantidomain(multiplication(domain(Y), antidomain(X)))))
% 153.12/19.67 = { by lemma 58 }
% 153.12/19.67 multiplication(addition(domain(X), domain(Y)), addition(antidomain(X), coantidomain(multiplication(domain(Y), antidomain(X)))))
% 153.12/19.67 = { by lemma 84 }
% 153.12/19.67 multiplication(addition(domain(X), domain(Y)), addition(antidomain(X), coantidomain(domain(multiplication(antidomain(X), Y)))))
% 153.12/19.67 = { by lemma 58 }
% 153.12/19.67 multiplication(addition(domain(X), domain(Y)), addition(antidomain(X), antidomain(multiplication(antidomain(X), Y))))
% 153.12/19.67 = { by lemma 86 }
% 153.12/19.67 multiplication(addition(domain(X), domain(Y)), one)
% 153.12/19.67 = { by axiom 6 (multiplicative_right_identity) }
% 153.12/19.67 addition(domain(X), domain(Y))
% 153.12/19.67
% 153.12/19.67 Lemma 88: addition(domain(X), domain(Y)) = domain(addition(X, Y)).
% 153.12/19.67 Proof:
% 153.12/19.67 addition(domain(X), domain(Y))
% 153.12/19.67 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.67 addition(domain(Y), domain(X))
% 153.12/19.67 = { by lemma 87 R->L }
% 153.12/19.67 addition(domain(Y), domain(multiplication(antidomain(Y), X)))
% 153.12/19.67 = { by lemma 71 R->L }
% 153.12/19.67 addition(domain(Y), domain(multiplication(antidomain(Y), addition(X, Y))))
% 153.12/19.67 = { by lemma 87 }
% 153.12/19.67 addition(domain(Y), domain(addition(X, Y)))
% 153.12/19.67 = { by axiom 4 (additive_commutativity) R->L }
% 153.12/19.67 addition(domain(Y), domain(addition(Y, X)))
% 153.12/19.67 = { by lemma 72 }
% 153.12/19.67 domain(addition(Y, X))
% 153.12/19.67 = { by axiom 4 (additive_commutativity) }
% 153.12/19.67 domain(addition(X, Y))
% 153.12/19.67
% 153.12/19.67 Goal 1 (goals): domain(multiplication(x0, addition(domain(x1), domain(x2)))) = addition(domain(multiplication(x0, domain(x1))), domain(multiplication(x0, domain(x2)))).
% 153.12/19.67 Proof:
% 153.12/19.67 domain(multiplication(x0, addition(domain(x1), domain(x2))))
% 153.12/19.67 = { by lemma 88 }
% 153.12/19.67 domain(multiplication(x0, domain(addition(x1, x2))))
% 153.12/19.67 = { by lemma 82 }
% 153.12/19.67 domain(multiplication(x0, addition(x1, x2)))
% 153.12/19.67 = { by axiom 19 (right_distributivity) }
% 153.12/19.67 domain(addition(multiplication(x0, x1), multiplication(x0, x2)))
% 153.12/19.67 = { by lemma 88 R->L }
% 153.12/19.67 addition(domain(multiplication(x0, x1)), domain(multiplication(x0, x2)))
% 153.12/19.67 = { by lemma 82 R->L }
% 153.12/19.67 addition(domain(multiplication(x0, x1)), domain(multiplication(x0, domain(x2))))
% 153.12/19.67 = { by lemma 82 R->L }
% 153.12/19.67 addition(domain(multiplication(x0, domain(x1))), domain(multiplication(x0, domain(x2))))
% 153.12/19.67 % SZS output end Proof
% 153.12/19.67
% 153.12/19.67 RESULT: Theorem (the conjecture is true).
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