TSTP Solution File: KLE102+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE102+1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n026.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 05:35:53 EDT 2023
% Result : Theorem 140.42s 18.63s
% Output : Proof 142.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.13 % Problem : KLE102+1 : TPTP v8.1.2. Released v4.0.0.
% 0.11/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n026.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Tue Aug 29 12:45:36 EDT 2023
% 0.14/0.35 % CPUTime :
% 140.42/18.63 Command-line arguments: --no-flatten-goal
% 140.42/18.63
% 140.42/18.63 % SZS status Theorem
% 140.42/18.63
% 141.57/18.80 % SZS output start Proof
% 141.57/18.80 Take the following subset of the input axioms:
% 141.57/18.80 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 141.57/18.80 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 141.57/18.80 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 141.57/18.80 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 141.57/18.80 fof(backward_box, axiom, ![X0, X1]: backward_box(X0, X1)=c(backward_diamond(X0, c(X1)))).
% 141.57/18.80 fof(backward_diamond, axiom, ![X0_2, X1_2]: backward_diamond(X0_2, X1_2)=codomain(multiplication(codomain(X1_2), X0_2))).
% 141.57/18.80 fof(codomain1, axiom, ![X0_2]: multiplication(X0_2, coantidomain(X0_2))=zero).
% 141.57/18.80 fof(codomain3, axiom, ![X0_2]: addition(coantidomain(coantidomain(X0_2)), coantidomain(X0_2))=one).
% 141.57/18.80 fof(codomain4, axiom, ![X0_2]: codomain(X0_2)=coantidomain(coantidomain(X0_2))).
% 141.57/18.80 fof(complement, axiom, ![X0_2]: c(X0_2)=antidomain(domain(X0_2))).
% 141.57/18.80 fof(domain1, axiom, ![X0_2]: multiplication(antidomain(X0_2), X0_2)=zero).
% 141.57/18.80 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))))).
% 141.57/18.80 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 141.57/18.80 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 141.57/18.80 fof(domain_difference, axiom, ![X0_2, X1_2]: domain_difference(X0_2, X1_2)=multiplication(domain(X0_2), antidomain(X1_2))).
% 141.57/18.80 fof(forward_box, axiom, ![X0_2, X1_2]: forward_box(X0_2, X1_2)=c(forward_diamond(X0_2, c(X1_2)))).
% 141.57/18.80 fof(forward_diamond, axiom, ![X0_2, X1_2]: forward_diamond(X0_2, X1_2)=domain(multiplication(X0_2, domain(X1_2)))).
% 141.57/18.80 fof(goals, conjecture, ![X2, X0_2, X1_2]: (multiplication(forward_diamond(X0_2, domain(X1_2)), domain(X2))=zero <= multiplication(domain(X1_2), backward_diamond(X0_2, domain(X2)))=zero)).
% 141.57/18.80 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 141.57/18.80 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 141.57/18.80 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 141.57/18.80 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 141.57/18.80 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 141.57/18.80 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 141.57/18.80 fof(right_annihilation, axiom, ![A3]: multiplication(A3, zero)=zero).
% 141.57/18.80 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 141.57/18.80
% 141.57/18.80 Now clausify the problem and encode Horn clauses using encoding 3 of
% 141.57/18.80 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 141.57/18.80 We repeatedly replace C & s=t => u=v by the two clauses:
% 141.57/18.80 fresh(y, y, x1...xn) = u
% 141.57/18.80 C => fresh(s, t, x1...xn) = v
% 141.57/18.80 where fresh is a fresh function symbol and x1..xn are the free
% 141.57/18.80 variables of u and v.
% 141.57/18.80 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 141.57/18.80 input problem has no model of domain size 1).
% 141.57/18.80
% 141.57/18.80 The encoding turns the above axioms into the following unit equations and goals:
% 141.57/18.80
% 141.57/18.80 Axiom 1 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 141.57/18.80 Axiom 2 (complement): c(X) = antidomain(domain(X)).
% 141.57/18.80 Axiom 3 (domain4): domain(X) = antidomain(antidomain(X)).
% 141.57/18.80 Axiom 4 (additive_idempotence): addition(X, X) = X.
% 141.57/18.80 Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 141.57/18.80 Axiom 6 (additive_identity): addition(X, zero) = X.
% 141.57/18.80 Axiom 7 (multiplicative_right_identity): multiplication(X, one) = X.
% 141.57/18.80 Axiom 8 (right_annihilation): multiplication(X, zero) = zero.
% 141.57/18.80 Axiom 9 (multiplicative_left_identity): multiplication(one, X) = X.
% 141.57/18.80 Axiom 10 (left_annihilation): multiplication(zero, X) = zero.
% 141.57/18.80 Axiom 11 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 141.57/18.80 Axiom 12 (domain1): multiplication(antidomain(X), X) = zero.
% 141.57/18.80 Axiom 13 (order): fresh(X, X, Y, Z) = true.
% 141.57/18.80 Axiom 14 (order_1): fresh2(X, X, Y, Z) = Z.
% 141.57/18.80 Axiom 15 (backward_diamond): backward_diamond(X, Y) = codomain(multiplication(codomain(Y), X)).
% 141.57/18.80 Axiom 16 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
% 141.57/18.80 Axiom 17 (backward_box): backward_box(X, Y) = c(backward_diamond(X, c(Y))).
% 141.57/18.80 Axiom 18 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 141.57/18.80 Axiom 19 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 141.57/18.80 Axiom 20 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 141.57/18.80 Axiom 21 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 141.57/18.80 Axiom 22 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 141.57/18.80 Axiom 23 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 141.57/18.80 Axiom 24 (order): fresh(addition(X, Y), Y, X, Y) = leq(X, Y).
% 141.57/18.80 Axiom 25 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 141.57/18.80 Axiom 26 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 141.57/18.80 Axiom 27 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 141.57/18.80 Axiom 28 (goals): multiplication(domain(x1), backward_diamond(x0, domain(x2))) = zero.
% 141.57/18.80 Axiom 29 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 141.57/18.80
% 141.57/18.80 Lemma 30: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
% 141.57/18.80 Proof:
% 141.57/18.80 multiplication(addition(X, antidomain(Y)), Y)
% 141.57/18.80 = { by axiom 27 (left_distributivity) }
% 141.57/18.80 addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
% 141.57/18.80 = { by axiom 12 (domain1) }
% 141.57/18.80 addition(multiplication(X, Y), zero)
% 141.57/18.80 = { by axiom 6 (additive_identity) }
% 141.57/18.80 multiplication(X, Y)
% 141.57/18.80
% 141.57/18.80 Lemma 31: addition(domain(X), antidomain(X)) = one.
% 141.57/18.80 Proof:
% 141.57/18.80 addition(domain(X), antidomain(X))
% 141.57/18.80 = { by axiom 3 (domain4) }
% 141.57/18.80 addition(antidomain(antidomain(X)), antidomain(X))
% 141.57/18.80 = { by axiom 23 (domain3) }
% 141.57/18.80 one
% 141.57/18.80
% 141.57/18.80 Lemma 32: multiplication(domain(X), X) = X.
% 141.57/18.80 Proof:
% 141.57/18.80 multiplication(domain(X), X)
% 141.57/18.80 = { by lemma 30 R->L }
% 141.57/18.80 multiplication(addition(domain(X), antidomain(X)), X)
% 141.57/18.80 = { by lemma 31 }
% 141.57/18.80 multiplication(one, X)
% 141.57/18.80 = { by axiom 9 (multiplicative_left_identity) }
% 141.57/18.80 X
% 141.57/18.80
% 141.57/18.80 Lemma 33: domain(antidomain(X)) = c(X).
% 141.57/18.80 Proof:
% 141.57/18.80 domain(antidomain(X))
% 141.57/18.80 = { by axiom 3 (domain4) }
% 141.57/18.80 antidomain(antidomain(antidomain(X)))
% 141.57/18.80 = { by axiom 3 (domain4) R->L }
% 141.57/18.80 antidomain(domain(X))
% 141.57/18.80 = { by axiom 2 (complement) R->L }
% 141.57/18.80 c(X)
% 141.57/18.80
% 141.57/18.80 Lemma 34: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 141.57/18.80 Proof:
% 141.57/18.80 multiplication(antidomain(X), addition(X, Y))
% 141.57/18.80 = { by axiom 5 (additive_commutativity) R->L }
% 141.57/18.80 multiplication(antidomain(X), addition(Y, X))
% 141.57/18.80 = { by axiom 26 (right_distributivity) }
% 141.57/18.80 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 141.57/18.80 = { by axiom 12 (domain1) }
% 141.57/18.80 addition(multiplication(antidomain(X), Y), zero)
% 141.57/18.80 = { by axiom 6 (additive_identity) }
% 141.57/18.80 multiplication(antidomain(X), Y)
% 141.57/18.80
% 141.57/18.80 Lemma 35: antidomain(X) = c(X).
% 141.57/18.80 Proof:
% 141.57/18.80 antidomain(X)
% 141.57/18.80 = { by lemma 32 R->L }
% 141.57/18.80 multiplication(domain(antidomain(X)), antidomain(X))
% 141.57/18.80 = { by lemma 33 }
% 141.57/18.80 multiplication(c(X), antidomain(X))
% 141.57/18.80 = { by axiom 2 (complement) }
% 141.57/18.80 multiplication(antidomain(domain(X)), antidomain(X))
% 141.57/18.80 = { by lemma 34 R->L }
% 141.57/18.80 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 141.57/18.80 = { by lemma 31 }
% 141.57/18.80 multiplication(antidomain(domain(X)), one)
% 141.57/18.80 = { by axiom 7 (multiplicative_right_identity) }
% 141.57/18.80 antidomain(domain(X))
% 141.57/18.80 = { by axiom 2 (complement) R->L }
% 141.57/18.80 c(X)
% 141.57/18.80
% 141.57/18.80 Lemma 36: antidomain(one) = zero.
% 141.57/18.80 Proof:
% 141.57/18.80 antidomain(one)
% 141.57/18.80 = { by axiom 7 (multiplicative_right_identity) R->L }
% 141.57/18.80 multiplication(antidomain(one), one)
% 141.57/18.80 = { by axiom 12 (domain1) }
% 141.57/18.80 zero
% 141.57/18.80
% 141.57/18.80 Lemma 37: domain(one) = one.
% 141.57/18.80 Proof:
% 141.57/18.80 domain(one)
% 141.57/18.80 = { by axiom 6 (additive_identity) R->L }
% 141.57/18.80 addition(domain(one), zero)
% 141.57/18.80 = { by lemma 36 R->L }
% 141.57/18.80 addition(domain(one), antidomain(one))
% 141.57/18.80 = { by lemma 31 }
% 141.57/18.80 one
% 141.57/18.80
% 141.57/18.80 Lemma 38: coantidomain(one) = zero.
% 141.57/18.80 Proof:
% 141.57/18.80 coantidomain(one)
% 141.57/18.80 = { by axiom 9 (multiplicative_left_identity) R->L }
% 141.57/18.80 multiplication(one, coantidomain(one))
% 141.57/18.80 = { by axiom 11 (codomain1) }
% 141.57/18.80 zero
% 141.57/18.80
% 141.57/18.80 Lemma 39: addition(codomain(X), coantidomain(X)) = one.
% 141.57/18.80 Proof:
% 141.57/18.80 addition(codomain(X), coantidomain(X))
% 141.57/18.80 = { by axiom 1 (codomain4) }
% 141.57/18.80 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 141.57/18.80 = { by axiom 22 (codomain3) }
% 141.57/18.80 one
% 141.57/18.80
% 141.57/18.80 Lemma 40: codomain(one) = one.
% 141.57/18.80 Proof:
% 141.57/18.80 codomain(one)
% 141.57/18.80 = { by axiom 6 (additive_identity) R->L }
% 141.57/18.80 addition(codomain(one), zero)
% 141.57/18.80 = { by lemma 38 R->L }
% 141.57/18.80 addition(codomain(one), coantidomain(one))
% 141.57/18.80 = { by lemma 39 }
% 141.57/18.80 one
% 141.57/18.80
% 141.57/18.80 Lemma 41: c(zero) = one.
% 141.57/18.80 Proof:
% 141.57/18.80 c(zero)
% 141.57/18.80 = { by lemma 33 R->L }
% 141.57/18.80 domain(antidomain(zero))
% 141.57/18.80 = { by lemma 36 R->L }
% 141.57/18.80 domain(antidomain(antidomain(one)))
% 141.57/18.80 = { by axiom 3 (domain4) R->L }
% 141.57/18.80 domain(domain(one))
% 141.57/18.80 = { by lemma 37 }
% 141.57/18.80 domain(one)
% 141.57/18.80 = { by lemma 37 }
% 141.57/18.80 one
% 141.57/18.80
% 141.57/18.80 Lemma 42: addition(zero, X) = X.
% 141.57/18.80 Proof:
% 141.57/18.80 addition(zero, X)
% 141.57/18.80 = { by axiom 5 (additive_commutativity) R->L }
% 141.57/18.80 addition(X, zero)
% 141.57/18.80 = { by axiom 6 (additive_identity) }
% 141.57/18.80 X
% 141.57/18.80
% 141.57/18.80 Lemma 43: antidomain(c(X)) = domain(domain(X)).
% 141.57/18.80 Proof:
% 141.57/18.80 antidomain(c(X))
% 141.57/18.80 = { by axiom 2 (complement) }
% 141.57/18.80 antidomain(antidomain(domain(X)))
% 141.57/18.80 = { by axiom 3 (domain4) R->L }
% 141.57/18.80 domain(domain(X))
% 141.57/18.80
% 141.57/18.80 Lemma 44: antidomain(c(X)) = c(antidomain(X)).
% 141.57/18.80 Proof:
% 141.57/18.80 antidomain(c(X))
% 141.57/18.80 = { by lemma 33 R->L }
% 141.57/18.80 antidomain(domain(antidomain(X)))
% 141.57/18.80 = { by axiom 2 (complement) R->L }
% 141.57/18.80 c(antidomain(X))
% 141.57/18.81
% 141.57/18.81 Lemma 45: domain(domain(X)) = forward_diamond(one, X).
% 141.57/18.81 Proof:
% 141.57/18.81 domain(domain(X))
% 141.57/18.81 = { by axiom 9 (multiplicative_left_identity) R->L }
% 141.57/18.81 domain(multiplication(one, domain(X)))
% 141.57/18.81 = { by axiom 18 (forward_diamond) R->L }
% 141.57/18.81 forward_diamond(one, X)
% 141.57/18.81
% 141.57/18.81 Lemma 46: addition(X, addition(X, Y)) = addition(X, Y).
% 141.57/18.81 Proof:
% 141.57/18.81 addition(X, addition(X, Y))
% 141.57/18.81 = { by axiom 19 (additive_associativity) }
% 141.57/18.81 addition(addition(X, X), Y)
% 141.57/18.81 = { by axiom 4 (additive_idempotence) }
% 141.57/18.81 addition(X, Y)
% 141.57/18.81
% 141.57/18.81 Lemma 47: leq(X, addition(Y, X)) = true.
% 141.57/18.81 Proof:
% 141.57/18.81 leq(X, addition(Y, X))
% 141.57/18.81 = { by axiom 5 (additive_commutativity) R->L }
% 141.57/18.81 leq(X, addition(X, Y))
% 141.57/18.81 = { by axiom 24 (order) R->L }
% 141.57/18.81 fresh(addition(X, addition(X, Y)), addition(X, Y), X, addition(X, Y))
% 141.57/18.81 = { by lemma 46 }
% 141.57/18.81 fresh(addition(X, Y), addition(X, Y), X, addition(X, Y))
% 141.57/18.81 = { by axiom 13 (order) }
% 141.57/18.81 true
% 141.57/18.81
% 141.57/18.81 Lemma 48: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 141.57/18.81 Proof:
% 141.57/18.81 multiplication(addition(one, Y), X)
% 141.57/18.81 = { by axiom 27 (left_distributivity) }
% 141.57/18.81 addition(multiplication(one, X), multiplication(Y, X))
% 141.57/18.81 = { by axiom 9 (multiplicative_left_identity) }
% 141.57/18.81 addition(X, multiplication(Y, X))
% 141.57/18.81
% 141.57/18.81 Lemma 49: multiplication(c(multiplication(X, Y)), multiplication(X, domain(Y))) = zero.
% 141.57/18.81 Proof:
% 141.57/18.81 multiplication(c(multiplication(X, Y)), multiplication(X, domain(Y)))
% 141.57/18.81 = { by lemma 35 R->L }
% 141.57/18.81 multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y)))
% 141.57/18.81 = { by lemma 30 R->L }
% 141.57/18.81 multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))), multiplication(X, domain(Y)))
% 141.57/18.81 = { by axiom 3 (domain4) }
% 141.57/18.81 multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))), multiplication(X, domain(Y)))
% 141.57/18.81 = { by axiom 29 (domain2) }
% 141.57/18.81 multiplication(antidomain(multiplication(X, antidomain(antidomain(Y)))), multiplication(X, domain(Y)))
% 141.57/18.81 = { by axiom 3 (domain4) R->L }
% 141.57/18.81 multiplication(antidomain(multiplication(X, domain(Y))), multiplication(X, domain(Y)))
% 141.57/18.81 = { by axiom 12 (domain1) }
% 141.57/18.81 zero
% 141.57/18.81
% 141.57/18.81 Lemma 50: addition(c(X), domain(X)) = one.
% 141.57/18.81 Proof:
% 141.57/18.81 addition(c(X), domain(X))
% 142.23/18.81 = { by axiom 3 (domain4) }
% 142.23/18.81 addition(c(X), antidomain(antidomain(X)))
% 142.23/18.81 = { by lemma 33 R->L }
% 142.23/18.81 addition(domain(antidomain(X)), antidomain(antidomain(X)))
% 142.23/18.81 = { by lemma 31 }
% 142.23/18.81 one
% 142.23/18.81
% 142.23/18.81 Lemma 51: multiplication(X, c(coantidomain(X))) = X.
% 142.23/18.81 Proof:
% 142.23/18.81 multiplication(X, c(coantidomain(X)))
% 142.23/18.81 = { by lemma 42 R->L }
% 142.23/18.81 addition(zero, multiplication(X, c(coantidomain(X))))
% 142.23/18.81 = { by lemma 49 R->L }
% 142.23/18.81 addition(multiplication(c(multiplication(X, coantidomain(X))), multiplication(X, domain(coantidomain(X)))), multiplication(X, c(coantidomain(X))))
% 142.23/18.81 = { by axiom 11 (codomain1) }
% 142.23/18.81 addition(multiplication(c(zero), multiplication(X, domain(coantidomain(X)))), multiplication(X, c(coantidomain(X))))
% 142.23/18.81 = { by lemma 41 }
% 142.23/18.81 addition(multiplication(one, multiplication(X, domain(coantidomain(X)))), multiplication(X, c(coantidomain(X))))
% 142.23/18.81 = { by axiom 9 (multiplicative_left_identity) }
% 142.23/18.81 addition(multiplication(X, domain(coantidomain(X))), multiplication(X, c(coantidomain(X))))
% 142.23/18.81 = { by axiom 26 (right_distributivity) R->L }
% 142.23/18.81 multiplication(X, addition(domain(coantidomain(X)), c(coantidomain(X))))
% 142.23/18.81 = { by axiom 5 (additive_commutativity) }
% 142.23/18.81 multiplication(X, addition(c(coantidomain(X)), domain(coantidomain(X))))
% 142.23/18.81 = { by lemma 50 }
% 142.23/18.81 multiplication(X, one)
% 142.23/18.81 = { by axiom 7 (multiplicative_right_identity) }
% 142.23/18.81 X
% 142.23/18.81
% 142.23/18.81 Lemma 52: coantidomain(codomain(X)) = codomain(coantidomain(X)).
% 142.23/18.81 Proof:
% 142.23/18.81 coantidomain(codomain(X))
% 142.23/18.81 = { by axiom 1 (codomain4) }
% 142.23/18.81 coantidomain(coantidomain(coantidomain(X)))
% 142.23/18.81 = { by axiom 1 (codomain4) R->L }
% 142.23/18.81 codomain(coantidomain(X))
% 142.23/18.81
% 142.23/18.81 Lemma 53: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
% 142.23/18.81 Proof:
% 142.23/18.81 multiplication(addition(X, Y), coantidomain(X))
% 142.23/18.81 = { by axiom 5 (additive_commutativity) R->L }
% 142.23/18.81 multiplication(addition(Y, X), coantidomain(X))
% 142.23/18.81 = { by axiom 27 (left_distributivity) }
% 142.23/18.81 addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
% 142.23/18.81 = { by axiom 11 (codomain1) }
% 142.23/18.81 addition(multiplication(Y, coantidomain(X)), zero)
% 142.23/18.81 = { by axiom 6 (additive_identity) }
% 142.23/18.81 multiplication(Y, coantidomain(X))
% 142.23/18.81
% 142.23/18.81 Lemma 54: multiplication(X, codomain(X)) = X.
% 142.23/18.81 Proof:
% 142.23/18.81 multiplication(X, codomain(X))
% 142.23/18.81 = { by axiom 6 (additive_identity) R->L }
% 142.23/18.81 addition(multiplication(X, codomain(X)), zero)
% 142.23/18.81 = { by axiom 11 (codomain1) R->L }
% 142.23/18.81 addition(multiplication(X, codomain(X)), multiplication(X, coantidomain(X)))
% 142.23/18.81 = { by axiom 26 (right_distributivity) R->L }
% 142.23/18.81 multiplication(X, addition(codomain(X), coantidomain(X)))
% 142.23/18.81 = { by lemma 39 }
% 142.23/18.81 multiplication(X, one)
% 142.23/18.81 = { by axiom 7 (multiplicative_right_identity) }
% 142.23/18.81 X
% 142.23/18.81
% 142.23/18.81 Lemma 55: codomain(coantidomain(X)) = coantidomain(X).
% 142.23/18.81 Proof:
% 142.23/18.81 codomain(coantidomain(X))
% 142.23/18.81 = { by lemma 52 R->L }
% 142.23/18.81 coantidomain(codomain(X))
% 142.23/18.81 = { by axiom 9 (multiplicative_left_identity) R->L }
% 142.23/18.81 multiplication(one, coantidomain(codomain(X)))
% 142.23/18.81 = { by lemma 39 R->L }
% 142.23/18.81 multiplication(addition(codomain(X), coantidomain(X)), coantidomain(codomain(X)))
% 142.23/18.81 = { by lemma 53 }
% 142.23/18.81 multiplication(coantidomain(X), coantidomain(codomain(X)))
% 142.23/18.81 = { by lemma 52 }
% 142.23/18.81 multiplication(coantidomain(X), codomain(coantidomain(X)))
% 142.23/18.81 = { by lemma 54 }
% 142.23/18.81 coantidomain(X)
% 142.23/18.81
% 142.23/18.81 Lemma 56: coantidomain(codomain(X)) = coantidomain(X).
% 142.23/18.81 Proof:
% 142.23/18.81 coantidomain(codomain(X))
% 142.23/18.81 = { by axiom 1 (codomain4) }
% 142.23/18.81 coantidomain(coantidomain(coantidomain(X)))
% 142.23/18.81 = { by axiom 1 (codomain4) R->L }
% 142.23/18.81 codomain(coantidomain(X))
% 142.23/18.81 = { by lemma 55 }
% 142.23/18.81 coantidomain(X)
% 142.23/18.81
% 142.23/18.81 Lemma 57: backward_diamond(X, one) = codomain(X).
% 142.23/18.81 Proof:
% 142.23/18.81 backward_diamond(X, one)
% 142.23/18.81 = { by axiom 15 (backward_diamond) }
% 142.23/18.81 codomain(multiplication(codomain(one), X))
% 142.23/18.81 = { by lemma 40 }
% 142.23/18.81 codomain(multiplication(one, X))
% 142.23/18.81 = { by axiom 9 (multiplicative_left_identity) }
% 142.23/18.81 codomain(X)
% 142.23/18.81
% 142.23/18.81 Lemma 58: c(codomain(X)) = backward_box(X, zero).
% 142.23/18.81 Proof:
% 142.23/18.81 c(codomain(X))
% 142.23/18.81 = { by lemma 57 R->L }
% 142.23/18.81 c(backward_diamond(X, one))
% 142.23/18.81 = { by lemma 41 R->L }
% 142.23/18.81 c(backward_diamond(X, c(zero)))
% 142.23/18.81 = { by axiom 17 (backward_box) R->L }
% 142.23/18.81 backward_box(X, zero)
% 142.23/18.81
% 142.23/18.81 Lemma 59: addition(one, domain(X)) = one.
% 142.23/18.81 Proof:
% 142.23/18.81 addition(one, domain(X))
% 142.23/18.81 = { by axiom 5 (additive_commutativity) R->L }
% 142.23/18.81 addition(domain(X), one)
% 142.23/18.81 = { by lemma 31 R->L }
% 142.23/18.81 addition(domain(X), addition(domain(X), antidomain(X)))
% 142.23/18.81 = { by lemma 46 }
% 142.23/18.81 addition(domain(X), antidomain(X))
% 142.23/18.81 = { by lemma 31 }
% 142.23/18.81 one
% 142.23/18.81
% 142.23/18.81 Lemma 60: addition(one, c(X)) = one.
% 142.23/18.81 Proof:
% 142.23/18.81 addition(one, c(X))
% 142.23/18.81 = { by lemma 33 R->L }
% 142.23/18.81 addition(one, domain(antidomain(X)))
% 142.23/18.81 = { by lemma 59 }
% 142.23/18.81 one
% 142.23/18.81
% 142.23/18.81 Lemma 61: c(coantidomain(X)) = codomain(X).
% 142.23/18.81 Proof:
% 142.23/18.81 c(coantidomain(X))
% 142.23/18.81 = { by axiom 14 (order_1) R->L }
% 142.23/18.81 fresh2(true, true, codomain(X), c(coantidomain(X)))
% 142.23/18.81 = { by lemma 47 R->L }
% 142.23/18.81 fresh2(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), addition(c(coantidomain(codomain(X))), multiplication(codomain(X), c(coantidomain(codomain(X)))))), true, codomain(X), c(coantidomain(X)))
% 142.23/18.81 = { by lemma 48 R->L }
% 142.23/18.81 fresh2(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), multiplication(addition(one, codomain(X)), c(coantidomain(codomain(X))))), true, codomain(X), c(coantidomain(X)))
% 142.23/18.81 = { by axiom 5 (additive_commutativity) R->L }
% 142.23/18.81 fresh2(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), multiplication(addition(codomain(X), one), c(coantidomain(codomain(X))))), true, codomain(X), c(coantidomain(X)))
% 142.23/18.81 = { by lemma 39 R->L }
% 142.23/18.81 fresh2(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), multiplication(addition(codomain(X), addition(codomain(X), coantidomain(X))), c(coantidomain(codomain(X))))), true, codomain(X), c(coantidomain(X)))
% 142.23/18.81 = { by lemma 46 }
% 142.23/18.81 fresh2(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), multiplication(addition(codomain(X), coantidomain(X)), c(coantidomain(codomain(X))))), true, codomain(X), c(coantidomain(X)))
% 142.23/18.81 = { by lemma 39 }
% 142.23/18.81 fresh2(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), multiplication(one, c(coantidomain(codomain(X))))), true, codomain(X), c(coantidomain(X)))
% 142.23/18.81 = { by axiom 9 (multiplicative_left_identity) }
% 142.23/18.81 fresh2(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), c(coantidomain(codomain(X)))), true, codomain(X), c(coantidomain(X)))
% 142.23/18.81 = { by lemma 51 }
% 142.23/18.81 fresh2(leq(codomain(X), c(coantidomain(codomain(X)))), true, codomain(X), c(coantidomain(X)))
% 142.23/18.81 = { by lemma 56 }
% 142.23/18.81 fresh2(leq(codomain(X), c(coantidomain(X))), true, codomain(X), c(coantidomain(X)))
% 142.23/18.81 = { by axiom 25 (order_1) }
% 142.23/18.81 addition(codomain(X), c(coantidomain(X)))
% 142.23/18.81 = { by lemma 55 R->L }
% 142.23/18.81 addition(codomain(X), c(codomain(coantidomain(X))))
% 142.23/18.81 = { by lemma 58 }
% 142.23/18.81 addition(codomain(X), backward_box(coantidomain(X), zero))
% 142.23/18.81 = { by axiom 1 (codomain4) }
% 142.23/18.81 addition(coantidomain(coantidomain(X)), backward_box(coantidomain(X), zero))
% 142.23/18.81 = { by lemma 58 R->L }
% 142.23/18.81 addition(coantidomain(coantidomain(X)), c(codomain(coantidomain(X))))
% 142.23/18.81 = { by lemma 35 R->L }
% 142.23/18.81 addition(coantidomain(coantidomain(X)), antidomain(codomain(coantidomain(X))))
% 142.23/18.81 = { by axiom 7 (multiplicative_right_identity) R->L }
% 142.23/18.81 addition(coantidomain(coantidomain(X)), multiplication(antidomain(codomain(coantidomain(X))), one))
% 142.23/18.81 = { by lemma 39 R->L }
% 142.23/18.81 addition(coantidomain(coantidomain(X)), multiplication(antidomain(codomain(coantidomain(X))), addition(codomain(coantidomain(X)), coantidomain(coantidomain(X)))))
% 142.23/18.81 = { by lemma 34 }
% 142.23/18.81 addition(coantidomain(coantidomain(X)), multiplication(antidomain(codomain(coantidomain(X))), coantidomain(coantidomain(X))))
% 142.23/18.81 = { by lemma 35 }
% 142.23/18.81 addition(coantidomain(coantidomain(X)), multiplication(c(codomain(coantidomain(X))), coantidomain(coantidomain(X))))
% 142.23/18.81 = { by lemma 58 }
% 142.23/18.81 addition(coantidomain(coantidomain(X)), multiplication(backward_box(coantidomain(X), zero), coantidomain(coantidomain(X))))
% 142.23/18.81 = { by lemma 48 R->L }
% 142.23/18.81 multiplication(addition(one, backward_box(coantidomain(X), zero)), coantidomain(coantidomain(X)))
% 142.23/18.81 = { by axiom 17 (backward_box) }
% 142.23/18.81 multiplication(addition(one, c(backward_diamond(coantidomain(X), c(zero)))), coantidomain(coantidomain(X)))
% 142.23/18.81 = { by lemma 60 }
% 142.23/18.81 multiplication(one, coantidomain(coantidomain(X)))
% 142.23/18.81 = { by axiom 9 (multiplicative_left_identity) }
% 142.23/18.81 coantidomain(coantidomain(X))
% 142.23/18.81 = { by axiom 1 (codomain4) R->L }
% 142.23/18.81 codomain(X)
% 142.23/18.81
% 142.23/18.81 Lemma 62: domain(c(X)) = c(X).
% 142.23/18.81 Proof:
% 142.23/18.81 domain(c(X))
% 142.23/18.81 = { by lemma 35 R->L }
% 142.23/18.81 domain(antidomain(X))
% 142.23/18.81 = { by lemma 33 }
% 142.23/18.81 c(X)
% 142.23/18.81
% 142.23/18.81 Lemma 63: domain(codomain(X)) = codomain(X).
% 142.23/18.81 Proof:
% 142.23/18.81 domain(codomain(X))
% 142.23/18.81 = { by lemma 61 R->L }
% 142.23/18.81 domain(c(coantidomain(X)))
% 142.23/18.81 = { by lemma 62 }
% 142.23/18.81 c(coantidomain(X))
% 142.23/18.81 = { by lemma 61 }
% 142.23/18.81 codomain(X)
% 142.23/18.81
% 142.23/18.81 Lemma 64: backward_box(X, zero) = coantidomain(X).
% 142.23/18.81 Proof:
% 142.23/18.81 backward_box(X, zero)
% 142.23/18.81 = { by lemma 58 R->L }
% 142.23/18.81 c(codomain(X))
% 142.23/18.81 = { by axiom 1 (codomain4) }
% 142.23/18.81 c(coantidomain(coantidomain(X)))
% 142.23/18.81 = { by lemma 61 }
% 142.23/18.81 codomain(coantidomain(X))
% 142.23/18.81 = { by lemma 55 }
% 142.23/18.81 coantidomain(X)
% 142.23/18.81
% 142.23/18.81 Lemma 65: codomain(codomain(X)) = codomain(X).
% 142.23/18.81 Proof:
% 142.23/18.81 codomain(codomain(X))
% 142.23/18.81 = { by axiom 1 (codomain4) }
% 142.23/18.81 codomain(coantidomain(coantidomain(X)))
% 142.23/18.81 = { by lemma 55 }
% 142.23/18.81 coantidomain(coantidomain(X))
% 142.23/18.81 = { by axiom 1 (codomain4) R->L }
% 142.23/18.81 codomain(X)
% 142.23/18.81
% 142.23/18.81 Lemma 66: backward_box(c(X), zero) = backward_box(one, X).
% 142.23/18.81 Proof:
% 142.23/18.81 backward_box(c(X), zero)
% 142.23/18.81 = { by lemma 58 R->L }
% 142.23/18.81 c(codomain(c(X)))
% 142.23/18.81 = { by lemma 65 R->L }
% 142.23/18.81 c(codomain(codomain(c(X))))
% 142.23/18.81 = { by axiom 7 (multiplicative_right_identity) R->L }
% 142.23/18.81 c(codomain(multiplication(codomain(c(X)), one)))
% 142.23/18.81 = { by axiom 15 (backward_diamond) R->L }
% 142.23/18.81 c(backward_diamond(one, c(X)))
% 142.23/18.81 = { by axiom 17 (backward_box) R->L }
% 142.23/18.81 backward_box(one, X)
% 142.23/18.81
% 142.23/18.81 Lemma 67: coantidomain(c(X)) = backward_box(one, X).
% 142.23/18.81 Proof:
% 142.23/18.81 coantidomain(c(X))
% 142.23/18.81 = { by lemma 64 R->L }
% 142.23/18.81 backward_box(c(X), zero)
% 142.23/18.81 = { by lemma 66 }
% 142.23/18.81 backward_box(one, X)
% 142.23/18.81
% 142.23/18.81 Lemma 68: c(antidomain(X)) = forward_diamond(one, X).
% 142.23/18.81 Proof:
% 142.23/18.81 c(antidomain(X))
% 142.23/18.81 = { by lemma 33 R->L }
% 142.23/18.81 domain(antidomain(antidomain(X)))
% 142.23/18.81 = { by axiom 3 (domain4) R->L }
% 142.23/18.81 domain(domain(X))
% 142.23/18.81 = { by lemma 45 }
% 142.23/18.81 forward_diamond(one, X)
% 142.23/18.81
% 142.23/18.81 Lemma 69: c(domain(X)) = c(X).
% 142.23/18.81 Proof:
% 142.23/18.81 c(domain(X))
% 142.23/18.81 = { by lemma 35 R->L }
% 142.23/18.81 antidomain(domain(X))
% 142.23/18.81 = { by axiom 2 (complement) R->L }
% 142.23/18.81 c(X)
% 142.23/18.81
% 142.23/18.81 Lemma 70: c(c(X)) = domain(X).
% 142.23/18.81 Proof:
% 142.23/18.81 c(c(X))
% 142.23/18.81 = { by lemma 35 R->L }
% 142.23/18.81 c(antidomain(X))
% 142.23/18.81 = { by lemma 35 R->L }
% 142.23/18.81 antidomain(antidomain(X))
% 142.23/18.81 = { by axiom 3 (domain4) R->L }
% 142.23/18.81 domain(X)
% 142.23/18.81
% 142.23/18.81 Lemma 71: multiplication(domain(X), domain(Y)) = domain_difference(X, antidomain(Y)).
% 142.23/18.81 Proof:
% 142.23/18.81 multiplication(domain(X), domain(Y))
% 142.23/18.81 = { by axiom 3 (domain4) }
% 142.23/18.81 multiplication(domain(X), antidomain(antidomain(Y)))
% 142.23/18.81 = { by axiom 20 (domain_difference) R->L }
% 142.23/18.81 domain_difference(X, antidomain(Y))
% 142.23/18.81
% 142.23/18.81 Lemma 72: multiplication(c(X), domain(Y)) = domain_difference(antidomain(X), antidomain(Y)).
% 142.23/18.81 Proof:
% 142.23/18.81 multiplication(c(X), domain(Y))
% 142.23/18.81 = { by lemma 33 R->L }
% 142.23/18.81 multiplication(domain(antidomain(X)), domain(Y))
% 142.23/18.81 = { by lemma 71 }
% 142.23/18.81 domain_difference(antidomain(X), antidomain(Y))
% 142.23/18.81
% 142.23/18.81 Lemma 73: forward_diamond(one, X) = domain(X).
% 142.23/18.81 Proof:
% 142.23/18.81 forward_diamond(one, X)
% 142.23/18.81 = { by lemma 68 R->L }
% 142.23/18.81 c(antidomain(X))
% 142.23/18.81 = { by lemma 44 R->L }
% 142.23/18.81 antidomain(c(X))
% 142.23/18.81 = { by axiom 7 (multiplicative_right_identity) R->L }
% 142.23/18.81 multiplication(antidomain(c(X)), one)
% 142.23/18.81 = { by lemma 50 R->L }
% 142.23/18.81 multiplication(antidomain(c(X)), addition(c(X), domain(X)))
% 142.23/18.81 = { by lemma 34 }
% 142.23/18.81 multiplication(antidomain(c(X)), domain(X))
% 142.23/18.81 = { by lemma 44 }
% 142.23/18.81 multiplication(c(antidomain(X)), domain(X))
% 142.23/18.81 = { by lemma 72 }
% 142.23/18.81 domain_difference(antidomain(antidomain(X)), antidomain(X))
% 142.23/18.81 = { by axiom 20 (domain_difference) }
% 142.23/18.81 multiplication(domain(antidomain(antidomain(X))), antidomain(antidomain(X)))
% 142.23/18.81 = { by lemma 32 }
% 142.23/18.81 antidomain(antidomain(X))
% 142.23/18.81 = { by axiom 3 (domain4) R->L }
% 142.23/18.82 domain(X)
% 142.23/18.82
% 142.23/18.82 Lemma 74: domain(multiplication(X, c(Y))) = forward_diamond(X, antidomain(Y)).
% 142.23/18.82 Proof:
% 142.23/18.82 domain(multiplication(X, c(Y)))
% 142.23/18.82 = { by lemma 33 R->L }
% 142.23/18.82 domain(multiplication(X, domain(antidomain(Y))))
% 142.23/18.82 = { by axiom 18 (forward_diamond) R->L }
% 142.23/18.82 forward_diamond(X, antidomain(Y))
% 142.23/18.82
% 142.23/18.82 Lemma 75: multiplication(domain(X), c(Y)) = domain_difference(X, domain(Y)).
% 142.23/18.82 Proof:
% 142.23/18.82 multiplication(domain(X), c(Y))
% 142.23/18.82 = { by axiom 2 (complement) }
% 142.23/18.82 multiplication(domain(X), antidomain(domain(Y)))
% 142.23/18.82 = { by axiom 20 (domain_difference) R->L }
% 142.23/18.82 domain_difference(X, domain(Y))
% 142.23/18.82
% 142.23/18.82 Lemma 76: domain_difference(X, domain(Y)) = domain_difference(X, Y).
% 142.23/18.82 Proof:
% 142.23/18.82 domain_difference(X, domain(Y))
% 142.23/18.82 = { by lemma 75 R->L }
% 142.23/18.82 multiplication(domain(X), c(Y))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 multiplication(domain(X), antidomain(Y))
% 142.23/18.82 = { by axiom 20 (domain_difference) R->L }
% 142.23/18.82 domain_difference(X, Y)
% 142.23/18.82
% 142.23/18.82 Lemma 77: forward_diamond(domain(X), c(Y)) = domain(domain_difference(X, Y)).
% 142.23/18.82 Proof:
% 142.23/18.82 forward_diamond(domain(X), c(Y))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 forward_diamond(domain(X), antidomain(Y))
% 142.23/18.82 = { by lemma 74 R->L }
% 142.23/18.82 domain(multiplication(domain(X), c(Y)))
% 142.23/18.82 = { by lemma 75 }
% 142.23/18.82 domain(domain_difference(X, domain(Y)))
% 142.23/18.82 = { by lemma 76 }
% 142.23/18.82 domain(domain_difference(X, Y))
% 142.23/18.82
% 142.23/18.82 Lemma 78: domain(domain_difference(X, antidomain(Y))) = forward_diamond(domain(X), Y).
% 142.23/18.82 Proof:
% 142.23/18.82 domain(domain_difference(X, antidomain(Y)))
% 142.23/18.82 = { by lemma 71 R->L }
% 142.23/18.82 domain(multiplication(domain(X), domain(Y)))
% 142.23/18.82 = { by axiom 18 (forward_diamond) R->L }
% 142.23/18.82 forward_diamond(domain(X), Y)
% 142.23/18.82
% 142.23/18.82 Lemma 79: c(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)).
% 142.23/18.82 Proof:
% 142.23/18.82 c(multiplication(X, domain(Y)))
% 142.23/18.82 = { by axiom 2 (complement) }
% 142.23/18.82 antidomain(domain(multiplication(X, domain(Y))))
% 142.23/18.82 = { by axiom 18 (forward_diamond) R->L }
% 142.23/18.82 antidomain(forward_diamond(X, Y))
% 142.23/18.82
% 142.23/18.82 Lemma 80: c(multiplication(X, c(Y))) = forward_box(X, Y).
% 142.23/18.82 Proof:
% 142.23/18.82 c(multiplication(X, c(Y)))
% 142.23/18.82 = { by lemma 33 R->L }
% 142.23/18.82 c(multiplication(X, domain(antidomain(Y))))
% 142.23/18.82 = { by lemma 79 }
% 142.23/18.82 antidomain(forward_diamond(X, antidomain(Y)))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 c(forward_diamond(X, antidomain(Y)))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 c(forward_diamond(X, c(Y)))
% 142.23/18.82 = { by axiom 16 (forward_box) R->L }
% 142.23/18.82 forward_box(X, Y)
% 142.23/18.82
% 142.23/18.82 Lemma 81: c(forward_box(X, Y)) = forward_diamond(X, c(Y)).
% 142.23/18.82 Proof:
% 142.23/18.82 c(forward_box(X, Y))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 antidomain(forward_box(X, Y))
% 142.23/18.82 = { by lemma 80 R->L }
% 142.23/18.82 antidomain(c(multiplication(X, c(Y))))
% 142.23/18.82 = { by lemma 43 }
% 142.23/18.82 domain(domain(multiplication(X, c(Y))))
% 142.23/18.82 = { by lemma 45 }
% 142.23/18.82 forward_diamond(one, multiplication(X, c(Y)))
% 142.23/18.82 = { by lemma 73 }
% 142.23/18.82 domain(multiplication(X, c(Y)))
% 142.23/18.82 = { by lemma 74 }
% 142.23/18.82 forward_diamond(X, antidomain(Y))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 forward_diamond(X, c(Y))
% 142.23/18.82
% 142.23/18.82 Lemma 82: addition(c(X), domain_difference(Y, X)) = c(X).
% 142.23/18.82 Proof:
% 142.23/18.82 addition(c(X), domain_difference(Y, X))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 addition(antidomain(X), domain_difference(Y, X))
% 142.23/18.82 = { by axiom 20 (domain_difference) }
% 142.23/18.82 addition(antidomain(X), multiplication(domain(Y), antidomain(X)))
% 142.23/18.82 = { by lemma 48 R->L }
% 142.23/18.82 multiplication(addition(one, domain(Y)), antidomain(X))
% 142.23/18.82 = { by lemma 59 }
% 142.23/18.82 multiplication(one, antidomain(X))
% 142.23/18.82 = { by axiom 9 (multiplicative_left_identity) }
% 142.23/18.82 antidomain(X)
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 c(X)
% 142.23/18.82
% 142.23/18.82 Lemma 83: multiplication(domain(X), domain_difference(Y, X)) = zero.
% 142.23/18.82 Proof:
% 142.23/18.82 multiplication(domain(X), domain_difference(Y, X))
% 142.23/18.82 = { by lemma 73 R->L }
% 142.23/18.82 multiplication(forward_diamond(one, X), domain_difference(Y, X))
% 142.23/18.82 = { by lemma 68 R->L }
% 142.23/18.82 multiplication(c(antidomain(X)), domain_difference(Y, X))
% 142.23/18.82 = { by lemma 44 R->L }
% 142.23/18.82 multiplication(antidomain(c(X)), domain_difference(Y, X))
% 142.23/18.82 = { by lemma 34 R->L }
% 142.23/18.82 multiplication(antidomain(c(X)), addition(c(X), domain_difference(Y, X)))
% 142.23/18.82 = { by lemma 82 }
% 142.23/18.82 multiplication(antidomain(c(X)), c(X))
% 142.23/18.82 = { by axiom 12 (domain1) }
% 142.23/18.82 zero
% 142.23/18.82
% 142.23/18.82 Lemma 84: c(domain_difference(X, Y)) = forward_box(domain(X), Y).
% 142.23/18.82 Proof:
% 142.23/18.82 c(domain_difference(X, Y))
% 142.23/18.82 = { by lemma 76 R->L }
% 142.23/18.82 c(domain_difference(X, domain(Y)))
% 142.23/18.82 = { by lemma 75 R->L }
% 142.23/18.82 c(multiplication(domain(X), c(Y)))
% 142.23/18.82 = { by lemma 80 }
% 142.23/18.82 forward_box(domain(X), Y)
% 142.23/18.82
% 142.23/18.82 Lemma 85: addition(domain_difference(X, Y), domain_difference(X, c(Y))) = domain(X).
% 142.23/18.82 Proof:
% 142.23/18.82 addition(domain_difference(X, Y), domain_difference(X, c(Y)))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 addition(domain_difference(X, Y), domain_difference(X, antidomain(Y)))
% 142.23/18.82 = { by axiom 20 (domain_difference) }
% 142.23/18.82 addition(domain_difference(X, Y), multiplication(domain(X), antidomain(antidomain(Y))))
% 142.23/18.82 = { by axiom 3 (domain4) R->L }
% 142.23/18.82 addition(domain_difference(X, Y), multiplication(domain(X), domain(Y)))
% 142.23/18.82 = { by axiom 20 (domain_difference) }
% 142.23/18.82 addition(multiplication(domain(X), antidomain(Y)), multiplication(domain(X), domain(Y)))
% 142.23/18.82 = { by axiom 26 (right_distributivity) R->L }
% 142.23/18.82 multiplication(domain(X), addition(antidomain(Y), domain(Y)))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 multiplication(domain(X), addition(c(Y), domain(Y)))
% 142.23/18.82 = { by lemma 50 }
% 142.23/18.82 multiplication(domain(X), one)
% 142.23/18.82 = { by axiom 7 (multiplicative_right_identity) }
% 142.23/18.82 domain(X)
% 142.23/18.82
% 142.23/18.82 Lemma 86: domain_difference(X, domain_difference(Y, X)) = domain(X).
% 142.23/18.82 Proof:
% 142.23/18.82 domain_difference(X, domain_difference(Y, X))
% 142.23/18.82 = { by lemma 76 R->L }
% 142.23/18.82 domain_difference(X, domain_difference(Y, domain(X)))
% 142.23/18.82 = { by lemma 70 R->L }
% 142.23/18.82 domain_difference(X, domain_difference(Y, c(c(X))))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 domain_difference(X, domain_difference(Y, antidomain(c(X))))
% 142.23/18.82 = { by lemma 76 R->L }
% 142.23/18.82 domain_difference(X, domain(domain_difference(Y, antidomain(c(X)))))
% 142.23/18.82 = { by lemma 78 }
% 142.23/18.82 domain_difference(X, forward_diamond(domain(Y), c(X)))
% 142.23/18.82 = { by lemma 81 R->L }
% 142.23/18.82 domain_difference(X, c(forward_box(domain(Y), X)))
% 142.23/18.82 = { by lemma 42 R->L }
% 142.23/18.82 addition(zero, domain_difference(X, c(forward_box(domain(Y), X))))
% 142.23/18.82 = { by lemma 49 R->L }
% 142.23/18.82 addition(multiplication(c(multiplication(domain(X), domain_difference(Y, X))), multiplication(domain(X), domain(domain_difference(Y, X)))), domain_difference(X, c(forward_box(domain(Y), X))))
% 142.23/18.82 = { by lemma 83 }
% 142.23/18.82 addition(multiplication(c(zero), multiplication(domain(X), domain(domain_difference(Y, X)))), domain_difference(X, c(forward_box(domain(Y), X))))
% 142.23/18.82 = { by lemma 41 }
% 142.23/18.82 addition(multiplication(one, multiplication(domain(X), domain(domain_difference(Y, X)))), domain_difference(X, c(forward_box(domain(Y), X))))
% 142.23/18.82 = { by axiom 9 (multiplicative_left_identity) }
% 142.23/18.82 addition(multiplication(domain(X), domain(domain_difference(Y, X))), domain_difference(X, c(forward_box(domain(Y), X))))
% 142.23/18.82 = { by lemma 71 }
% 142.23/18.82 addition(domain_difference(X, antidomain(domain_difference(Y, X))), domain_difference(X, c(forward_box(domain(Y), X))))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 addition(domain_difference(X, c(domain_difference(Y, X))), domain_difference(X, c(forward_box(domain(Y), X))))
% 142.23/18.82 = { by lemma 84 }
% 142.23/18.82 addition(domain_difference(X, forward_box(domain(Y), X)), domain_difference(X, c(forward_box(domain(Y), X))))
% 142.23/18.82 = { by lemma 85 }
% 142.23/18.82 domain(X)
% 142.23/18.82
% 142.23/18.82 Lemma 87: multiplication(c(X), c(Y)) = domain_difference(c(X), Y).
% 142.23/18.82 Proof:
% 142.23/18.82 multiplication(c(X), c(Y))
% 142.23/18.82 = { by lemma 33 R->L }
% 142.23/18.82 multiplication(domain(antidomain(X)), c(Y))
% 142.23/18.82 = { by lemma 75 }
% 142.23/18.82 domain_difference(antidomain(X), domain(Y))
% 142.23/18.82 = { by lemma 76 }
% 142.23/18.82 domain_difference(antidomain(X), Y)
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 domain_difference(c(X), Y)
% 142.23/18.82
% 142.23/18.82 Lemma 88: multiplication(domain_difference(X, c(Y)), Y) = multiplication(domain(X), Y).
% 142.23/18.82 Proof:
% 142.23/18.82 multiplication(domain_difference(X, c(Y)), Y)
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 multiplication(domain_difference(X, antidomain(Y)), Y)
% 142.23/18.82 = { by lemma 71 R->L }
% 142.23/18.82 multiplication(multiplication(domain(X), domain(Y)), Y)
% 142.23/18.82 = { by axiom 21 (multiplicative_associativity) R->L }
% 142.23/18.82 multiplication(domain(X), multiplication(domain(Y), Y))
% 142.23/18.82 = { by lemma 32 }
% 142.23/18.82 multiplication(domain(X), Y)
% 142.23/18.82
% 142.23/18.82 Lemma 89: domain_difference(c(X), multiplication(X, Y)) = c(X).
% 142.23/18.82 Proof:
% 142.23/18.82 domain_difference(c(X), multiplication(X, Y))
% 142.23/18.82 = { by lemma 76 R->L }
% 142.23/18.82 domain_difference(c(X), domain(multiplication(X, Y)))
% 142.23/18.82 = { by lemma 70 R->L }
% 142.23/18.82 domain_difference(c(X), c(c(multiplication(X, Y))))
% 142.23/18.82 = { by lemma 42 R->L }
% 142.23/18.82 addition(zero, domain_difference(c(X), c(c(multiplication(X, Y)))))
% 142.23/18.82 = { by lemma 49 R->L }
% 142.23/18.82 addition(multiplication(c(multiplication(antidomain(X), multiplication(X, Y))), multiplication(antidomain(X), domain(multiplication(X, Y)))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 142.23/18.82 = { by axiom 21 (multiplicative_associativity) }
% 142.23/18.82 addition(multiplication(c(multiplication(multiplication(antidomain(X), X), Y)), multiplication(antidomain(X), domain(multiplication(X, Y)))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 142.23/18.82 = { by axiom 12 (domain1) }
% 142.23/18.82 addition(multiplication(c(multiplication(zero, Y)), multiplication(antidomain(X), domain(multiplication(X, Y)))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 142.23/18.82 = { by axiom 10 (left_annihilation) }
% 142.23/18.82 addition(multiplication(c(zero), multiplication(antidomain(X), domain(multiplication(X, Y)))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 142.23/18.82 = { by lemma 41 }
% 142.23/18.82 addition(multiplication(one, multiplication(antidomain(X), domain(multiplication(X, Y)))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 142.23/18.82 = { by axiom 9 (multiplicative_left_identity) }
% 142.23/18.82 addition(multiplication(antidomain(X), domain(multiplication(X, Y))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 addition(multiplication(c(X), domain(multiplication(X, Y))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 142.23/18.82 = { by lemma 72 }
% 142.23/18.82 addition(domain_difference(antidomain(X), antidomain(multiplication(X, Y))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 addition(domain_difference(c(X), antidomain(multiplication(X, Y))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 addition(domain_difference(c(X), c(multiplication(X, Y))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 142.23/18.82 = { by lemma 85 }
% 142.23/18.82 domain(c(X))
% 142.23/18.82 = { by lemma 62 }
% 142.23/18.82 c(X)
% 142.23/18.82
% 142.23/18.82 Lemma 90: c(forward_diamond(X, Y)) = forward_box(X, c(Y)).
% 142.23/18.82 Proof:
% 142.23/18.82 c(forward_diamond(X, Y))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 antidomain(forward_diamond(X, Y))
% 142.23/18.82 = { by lemma 79 R->L }
% 142.23/18.82 c(multiplication(X, domain(Y)))
% 142.23/18.82 = { by lemma 73 R->L }
% 142.23/18.82 c(multiplication(X, forward_diamond(one, Y)))
% 142.23/18.82 = { by lemma 68 R->L }
% 142.23/18.82 c(multiplication(X, c(antidomain(Y))))
% 142.23/18.82 = { by lemma 80 }
% 142.23/18.82 forward_box(X, antidomain(Y))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 forward_box(X, c(Y))
% 142.23/18.82
% 142.23/18.82 Lemma 91: domain(domain_difference(X, c(Y))) = forward_diamond(domain(X), Y).
% 142.23/18.82 Proof:
% 142.23/18.82 domain(domain_difference(X, c(Y)))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 domain(domain_difference(X, antidomain(Y)))
% 142.23/18.82 = { by lemma 78 }
% 142.23/18.82 forward_diamond(domain(X), Y)
% 142.23/18.82
% 142.23/18.82 Lemma 92: domain(multiplication(c(X), Y)) = forward_diamond(c(X), Y).
% 142.23/18.82 Proof:
% 142.23/18.82 domain(multiplication(c(X), Y))
% 142.23/18.82 = { by lemma 62 R->L }
% 142.23/18.82 domain(multiplication(domain(c(X)), Y))
% 142.23/18.82 = { by lemma 70 R->L }
% 142.23/18.82 c(c(multiplication(domain(c(X)), Y)))
% 142.23/18.82 = { by lemma 88 R->L }
% 142.23/18.82 c(c(multiplication(domain_difference(c(X), c(Y)), Y)))
% 142.23/18.82 = { by lemma 82 R->L }
% 142.23/18.82 c(addition(c(multiplication(domain_difference(c(X), c(Y)), Y)), domain_difference(c(domain_difference(c(X), c(Y))), multiplication(domain_difference(c(X), c(Y)), Y))))
% 142.23/18.82 = { by lemma 89 }
% 142.23/18.82 c(addition(c(multiplication(domain_difference(c(X), c(Y)), Y)), c(domain_difference(c(X), c(Y)))))
% 142.23/18.82 = { by axiom 5 (additive_commutativity) }
% 142.23/18.82 c(addition(c(domain_difference(c(X), c(Y))), c(multiplication(domain_difference(c(X), c(Y)), Y))))
% 142.23/18.82 = { by lemma 88 }
% 142.23/18.82 c(addition(c(domain_difference(c(X), c(Y))), c(multiplication(domain(c(X)), Y))))
% 142.23/18.82 = { by lemma 84 }
% 142.23/18.82 c(addition(forward_box(domain(c(X)), c(Y)), c(multiplication(domain(c(X)), Y))))
% 142.23/18.82 = { by axiom 5 (additive_commutativity) R->L }
% 142.23/18.82 c(addition(c(multiplication(domain(c(X)), Y)), forward_box(domain(c(X)), c(Y))))
% 142.23/18.82 = { by lemma 90 R->L }
% 142.23/18.82 c(addition(c(multiplication(domain(c(X)), Y)), c(forward_diamond(domain(c(X)), Y))))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 c(addition(c(multiplication(domain(c(X)), Y)), antidomain(forward_diamond(domain(c(X)), Y))))
% 142.23/18.82 = { by lemma 79 R->L }
% 142.23/18.82 c(addition(c(multiplication(domain(c(X)), Y)), c(multiplication(domain(c(X)), domain(Y)))))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 c(addition(c(multiplication(domain(c(X)), Y)), antidomain(multiplication(domain(c(X)), domain(Y)))))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 c(addition(antidomain(multiplication(domain(c(X)), Y)), antidomain(multiplication(domain(c(X)), domain(Y)))))
% 142.23/18.82 = { by axiom 3 (domain4) }
% 142.23/18.82 c(addition(antidomain(multiplication(domain(c(X)), Y)), antidomain(multiplication(domain(c(X)), antidomain(antidomain(Y))))))
% 142.23/18.82 = { by axiom 29 (domain2) }
% 142.23/18.82 c(antidomain(multiplication(domain(c(X)), antidomain(antidomain(Y)))))
% 142.23/18.82 = { by axiom 3 (domain4) R->L }
% 142.23/18.82 c(antidomain(multiplication(domain(c(X)), domain(Y))))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 c(c(multiplication(domain(c(X)), domain(Y))))
% 142.23/18.82 = { by lemma 79 }
% 142.23/18.82 c(antidomain(forward_diamond(domain(c(X)), Y)))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 c(c(forward_diamond(domain(c(X)), Y)))
% 142.23/18.82 = { by lemma 90 }
% 142.23/18.82 c(forward_box(domain(c(X)), c(Y)))
% 142.23/18.82 = { by lemma 81 }
% 142.23/18.82 forward_diamond(domain(c(X)), c(c(Y)))
% 142.23/18.82 = { by lemma 77 }
% 142.23/18.82 domain(domain_difference(c(X), c(Y)))
% 142.23/18.82 = { by lemma 91 }
% 142.23/18.82 forward_diamond(domain(c(X)), Y)
% 142.23/18.82 = { by lemma 62 }
% 142.23/18.82 forward_diamond(c(X), Y)
% 142.23/18.82
% 142.23/18.82 Lemma 93: domain_difference(multiplication(X, Y), X) = zero.
% 142.23/18.82 Proof:
% 142.23/18.82 domain_difference(multiplication(X, Y), X)
% 142.23/18.82 = { by lemma 76 R->L }
% 142.23/18.82 domain_difference(multiplication(X, Y), domain(X))
% 142.23/18.82 = { by axiom 20 (domain_difference) }
% 142.23/18.82 multiplication(domain(multiplication(X, Y)), antidomain(domain(X)))
% 142.23/18.82 = { by axiom 2 (complement) R->L }
% 142.23/18.82 multiplication(domain(multiplication(X, Y)), c(X))
% 142.23/18.82 = { by lemma 89 R->L }
% 142.23/18.82 multiplication(domain(multiplication(X, Y)), domain_difference(c(X), multiplication(X, Y)))
% 142.23/18.82 = { by lemma 83 }
% 142.23/18.82 zero
% 142.23/18.82
% 142.23/18.82 Lemma 94: multiplication(domain(X), multiplication(c(Y), Z)) = multiplication(domain_difference(X, Y), Z).
% 142.23/18.82 Proof:
% 142.23/18.82 multiplication(domain(X), multiplication(c(Y), Z))
% 142.23/18.82 = { by lemma 35 R->L }
% 142.23/18.82 multiplication(domain(X), multiplication(antidomain(Y), Z))
% 142.23/18.82 = { by axiom 21 (multiplicative_associativity) }
% 142.23/18.82 multiplication(multiplication(domain(X), antidomain(Y)), Z)
% 142.23/18.82 = { by axiom 20 (domain_difference) R->L }
% 142.23/18.82 multiplication(domain_difference(X, Y), Z)
% 142.23/18.82
% 142.23/18.82 Lemma 95: forward_diamond(c(X), c(Y)) = domain_difference(c(X), Y).
% 142.23/18.82 Proof:
% 142.23/18.82 forward_diamond(c(X), c(Y))
% 142.23/18.82 = { by lemma 33 R->L }
% 142.23/18.82 forward_diamond(domain(antidomain(X)), c(Y))
% 142.23/18.82 = { by lemma 77 }
% 142.23/18.82 domain(domain_difference(antidomain(X), Y))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 domain(domain_difference(c(X), Y))
% 142.23/18.82 = { by lemma 86 R->L }
% 142.23/18.82 domain_difference(domain_difference(c(X), Y), domain_difference(Y, domain_difference(c(X), Y)))
% 142.23/18.82 = { by lemma 86 }
% 142.23/18.82 domain_difference(domain_difference(c(X), Y), domain(Y))
% 142.23/18.82 = { by lemma 76 }
% 142.23/18.82 domain_difference(domain_difference(c(X), Y), Y)
% 142.23/18.82 = { by lemma 87 R->L }
% 142.23/18.82 domain_difference(multiplication(c(X), c(Y)), Y)
% 142.23/18.82 = { by axiom 20 (domain_difference) }
% 142.23/18.82 multiplication(domain(multiplication(c(X), c(Y))), antidomain(Y))
% 142.23/18.82 = { by lemma 74 }
% 142.23/18.82 multiplication(forward_diamond(c(X), antidomain(Y)), antidomain(Y))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 multiplication(forward_diamond(c(X), c(Y)), antidomain(Y))
% 142.23/18.82 = { by lemma 35 }
% 142.23/18.82 multiplication(forward_diamond(c(X), c(Y)), c(Y))
% 142.23/18.83 = { by lemma 92 R->L }
% 142.23/18.83 multiplication(domain(multiplication(c(X), c(Y))), c(Y))
% 142.23/18.83 = { by lemma 85 R->L }
% 142.23/18.83 multiplication(addition(domain_difference(multiplication(c(X), c(Y)), X), domain_difference(multiplication(c(X), c(Y)), c(X))), c(Y))
% 142.23/18.83 = { by lemma 93 }
% 142.23/18.83 multiplication(addition(domain_difference(multiplication(c(X), c(Y)), X), zero), c(Y))
% 142.23/18.83 = { by axiom 6 (additive_identity) }
% 142.23/18.83 multiplication(domain_difference(multiplication(c(X), c(Y)), X), c(Y))
% 142.23/18.83 = { by lemma 94 R->L }
% 142.23/18.83 multiplication(domain(multiplication(c(X), c(Y))), multiplication(c(X), c(Y)))
% 142.23/18.83 = { by lemma 32 }
% 142.23/18.83 multiplication(c(X), c(Y))
% 142.23/18.83 = { by lemma 87 }
% 142.23/18.83 domain_difference(c(X), Y)
% 142.23/18.83
% 142.23/18.83 Lemma 96: domain_difference(domain(X), Y) = domain_difference(X, Y).
% 142.23/18.83 Proof:
% 142.23/18.83 domain_difference(domain(X), Y)
% 142.23/18.83 = { by lemma 73 R->L }
% 142.23/18.83 domain_difference(forward_diamond(one, X), Y)
% 142.23/18.83 = { by lemma 68 R->L }
% 142.23/18.83 domain_difference(c(antidomain(X)), Y)
% 142.23/18.83 = { by lemma 44 R->L }
% 142.23/18.83 domain_difference(antidomain(c(X)), Y)
% 142.23/18.83 = { by axiom 20 (domain_difference) }
% 142.23/18.83 multiplication(domain(antidomain(c(X))), antidomain(Y))
% 142.23/18.83 = { by lemma 33 }
% 142.23/18.83 multiplication(c(c(X)), antidomain(Y))
% 142.23/18.83 = { by lemma 70 }
% 142.23/18.83 multiplication(domain(X), antidomain(Y))
% 142.23/18.83 = { by axiom 20 (domain_difference) R->L }
% 142.23/18.83 domain_difference(X, Y)
% 142.23/18.83
% 142.23/18.83 Lemma 97: domain(domain_difference(X, Y)) = domain_difference(X, Y).
% 142.23/18.83 Proof:
% 142.23/18.83 domain(domain_difference(X, Y))
% 142.23/18.83 = { by lemma 77 R->L }
% 142.23/18.83 forward_diamond(domain(X), c(Y))
% 142.23/18.83 = { by lemma 70 R->L }
% 142.23/18.83 forward_diamond(c(c(X)), c(Y))
% 142.23/18.83 = { by lemma 95 }
% 142.23/18.83 domain_difference(c(c(X)), Y)
% 142.23/18.83 = { by lemma 70 }
% 142.23/18.83 domain_difference(domain(X), Y)
% 142.23/18.83 = { by lemma 96 }
% 142.23/18.83 domain_difference(X, Y)
% 142.23/18.83
% 142.23/18.83 Lemma 98: domain(forward_diamond(X, Y)) = forward_diamond(X, Y).
% 142.23/18.83 Proof:
% 142.23/18.83 domain(forward_diamond(X, Y))
% 142.23/18.83 = { by lemma 73 R->L }
% 142.23/18.83 forward_diamond(one, forward_diamond(X, Y))
% 142.23/18.83 = { by lemma 68 R->L }
% 142.23/18.83 c(antidomain(forward_diamond(X, Y)))
% 142.23/18.83 = { by lemma 79 R->L }
% 142.23/18.83 c(c(multiplication(X, domain(Y))))
% 142.23/18.83 = { by lemma 70 }
% 142.23/18.83 domain(multiplication(X, domain(Y)))
% 142.23/18.83 = { by axiom 18 (forward_diamond) R->L }
% 142.23/18.83 forward_diamond(X, Y)
% 142.23/18.83
% 142.23/18.83 Lemma 99: domain(backward_box(X, Y)) = backward_box(X, Y).
% 142.23/18.83 Proof:
% 142.23/18.83 domain(backward_box(X, Y))
% 142.23/18.83 = { by axiom 17 (backward_box) }
% 142.23/18.83 domain(c(backward_diamond(X, c(Y))))
% 142.23/18.83 = { by lemma 62 }
% 142.23/18.83 c(backward_diamond(X, c(Y)))
% 142.23/18.83 = { by axiom 17 (backward_box) R->L }
% 142.23/18.83 backward_box(X, Y)
% 142.23/18.83
% 142.23/18.83 Lemma 100: coantidomain(multiplication(codomain(X), Y)) = coantidomain(backward_diamond(Y, X)).
% 142.23/18.83 Proof:
% 142.23/18.83 coantidomain(multiplication(codomain(X), Y))
% 142.23/18.83 = { by lemma 55 R->L }
% 142.23/18.83 codomain(coantidomain(multiplication(codomain(X), Y)))
% 142.23/18.83 = { by lemma 52 R->L }
% 142.23/18.83 coantidomain(codomain(multiplication(codomain(X), Y)))
% 142.23/18.83 = { by axiom 15 (backward_diamond) R->L }
% 142.23/18.83 coantidomain(backward_diamond(Y, X))
% 142.23/18.83
% 142.23/18.83 Lemma 101: forward_box(X, coantidomain(X)) = c(X).
% 142.23/18.83 Proof:
% 142.23/18.83 forward_box(X, coantidomain(X))
% 142.23/18.83 = { by lemma 80 R->L }
% 142.23/18.83 c(multiplication(X, c(coantidomain(X))))
% 142.23/18.83 = { by lemma 51 }
% 142.23/18.83 c(X)
% 142.23/18.83
% 142.23/18.83 Lemma 102: backward_diamond(X, codomain(Y)) = backward_diamond(X, Y).
% 142.23/18.83 Proof:
% 142.23/18.83 backward_diamond(X, codomain(Y))
% 142.23/18.83 = { by axiom 15 (backward_diamond) }
% 142.23/18.83 codomain(multiplication(codomain(codomain(Y)), X))
% 142.23/18.83 = { by lemma 65 }
% 142.23/18.83 codomain(multiplication(codomain(Y), X))
% 142.23/18.83 = { by axiom 15 (backward_diamond) R->L }
% 142.23/18.83 backward_diamond(X, Y)
% 142.23/18.83
% 142.23/18.83 Lemma 103: c(backward_diamond(X, Y)) = backward_box(X, coantidomain(Y)).
% 142.23/18.83 Proof:
% 142.23/18.83 c(backward_diamond(X, Y))
% 142.23/18.83 = { by lemma 102 R->L }
% 142.23/18.83 c(backward_diamond(X, codomain(Y)))
% 142.23/18.83 = { by lemma 61 R->L }
% 142.23/18.83 c(backward_diamond(X, c(coantidomain(Y))))
% 142.23/18.83 = { by axiom 17 (backward_box) R->L }
% 142.23/18.83 backward_box(X, coantidomain(Y))
% 142.23/18.83
% 142.23/18.83 Lemma 104: coantidomain(backward_diamond(X, Y)) = backward_box(X, coantidomain(Y)).
% 142.23/18.83 Proof:
% 142.23/18.83 coantidomain(backward_diamond(X, Y))
% 142.23/18.83 = { by lemma 100 R->L }
% 142.23/18.83 coantidomain(multiplication(codomain(Y), X))
% 142.23/18.83 = { by lemma 64 R->L }
% 142.23/18.83 backward_box(multiplication(codomain(Y), X), zero)
% 142.23/18.83 = { by lemma 58 R->L }
% 142.23/18.83 c(codomain(multiplication(codomain(Y), X)))
% 142.23/18.83 = { by lemma 101 R->L }
% 142.23/18.83 forward_box(codomain(multiplication(codomain(Y), X)), coantidomain(codomain(multiplication(codomain(Y), X))))
% 142.23/18.83 = { by lemma 56 }
% 142.23/18.83 forward_box(codomain(multiplication(codomain(Y), X)), coantidomain(multiplication(codomain(Y), X)))
% 142.23/18.83 = { by axiom 15 (backward_diamond) R->L }
% 142.23/18.83 forward_box(backward_diamond(X, Y), coantidomain(multiplication(codomain(Y), X)))
% 142.23/18.83 = { by lemma 100 }
% 142.23/18.83 forward_box(backward_diamond(X, Y), coantidomain(backward_diamond(X, Y)))
% 142.23/18.83 = { by lemma 101 }
% 142.23/18.83 c(backward_diamond(X, Y))
% 142.23/18.83 = { by lemma 103 }
% 142.23/18.83 backward_box(X, coantidomain(Y))
% 142.23/18.83
% 142.23/18.83 Lemma 105: backward_box(X, domain(Y)) = backward_box(X, Y).
% 142.23/18.83 Proof:
% 142.23/18.83 backward_box(X, domain(Y))
% 142.23/18.83 = { by axiom 17 (backward_box) }
% 142.23/18.83 c(backward_diamond(X, c(domain(Y))))
% 142.23/18.83 = { by lemma 69 }
% 142.23/18.83 c(backward_diamond(X, c(Y)))
% 142.23/18.83 = { by axiom 17 (backward_box) R->L }
% 142.23/18.83 backward_box(X, Y)
% 142.23/18.83
% 142.23/18.83 Lemma 106: forward_box(c(X), X) = domain(X).
% 142.23/18.83 Proof:
% 142.23/18.83 forward_box(c(X), X)
% 142.23/18.83 = { by lemma 62 R->L }
% 142.23/18.83 forward_box(domain(c(X)), X)
% 142.23/18.83 = { by lemma 80 R->L }
% 142.23/18.83 c(multiplication(domain(c(X)), c(X)))
% 142.23/18.83 = { by lemma 32 }
% 142.23/18.83 c(c(X))
% 142.23/18.83 = { by lemma 70 }
% 142.23/18.83 domain(X)
% 142.23/18.83
% 142.23/18.83 Lemma 107: forward_box(X, domain(Y)) = forward_box(X, Y).
% 142.23/18.83 Proof:
% 142.23/18.83 forward_box(X, domain(Y))
% 142.23/18.83 = { by axiom 16 (forward_box) }
% 142.23/18.83 c(forward_diamond(X, c(domain(Y))))
% 142.23/18.83 = { by lemma 69 }
% 142.23/18.83 c(forward_diamond(X, c(Y)))
% 142.23/18.83 = { by axiom 16 (forward_box) R->L }
% 142.23/18.83 forward_box(X, Y)
% 142.23/18.83
% 142.23/18.83 Lemma 108: c(multiplication(X, domain(Y))) = forward_box(X, c(Y)).
% 142.23/18.83 Proof:
% 142.23/18.83 c(multiplication(X, domain(Y)))
% 142.23/18.83 = { by axiom 2 (complement) }
% 142.23/18.83 antidomain(domain(multiplication(X, domain(Y))))
% 142.23/18.83 = { by axiom 18 (forward_diamond) R->L }
% 142.23/18.83 antidomain(forward_diamond(X, Y))
% 142.23/18.83 = { by lemma 35 }
% 142.23/18.83 c(forward_diamond(X, Y))
% 142.23/18.83 = { by lemma 90 }
% 142.23/18.83 forward_box(X, c(Y))
% 142.23/18.83
% 142.23/18.83 Lemma 109: c(backward_diamond(X, domain(Y))) = backward_box(X, c(Y)).
% 142.23/18.83 Proof:
% 142.23/18.83 c(backward_diamond(X, domain(Y)))
% 142.23/18.83 = { by lemma 70 R->L }
% 142.23/18.83 c(backward_diamond(X, c(c(Y))))
% 142.23/18.83 = { by axiom 17 (backward_box) R->L }
% 142.23/18.83 backward_box(X, c(Y))
% 142.23/18.83
% 142.23/18.83 Lemma 110: backward_box(domain(X), zero) = backward_box(one, c(X)).
% 142.23/18.83 Proof:
% 142.23/18.83 backward_box(domain(X), zero)
% 142.23/18.83 = { by lemma 73 R->L }
% 142.23/18.83 backward_box(forward_diamond(one, X), zero)
% 142.23/18.83 = { by lemma 68 R->L }
% 142.23/18.83 backward_box(c(antidomain(X)), zero)
% 142.23/18.83 = { by lemma 66 }
% 142.23/18.83 backward_box(one, antidomain(X))
% 142.23/18.83 = { by lemma 35 }
% 142.23/18.83 backward_box(one, c(X))
% 142.23/18.83
% 142.23/18.83 Lemma 111: addition(c(X), codomain(domain(X))) = one.
% 142.23/18.83 Proof:
% 142.23/18.83 addition(c(X), codomain(domain(X)))
% 142.23/18.83 = { by axiom 5 (additive_commutativity) R->L }
% 142.23/18.83 addition(codomain(domain(X)), c(X))
% 142.23/18.83 = { by axiom 7 (multiplicative_right_identity) R->L }
% 142.23/18.83 addition(codomain(domain(X)), multiplication(c(X), one))
% 142.23/18.83 = { by axiom 14 (order_1) R->L }
% 142.23/18.83 addition(codomain(domain(X)), multiplication(c(X), fresh2(true, true, coantidomain(domain(X)), one)))
% 142.23/18.83 = { by lemma 47 R->L }
% 142.23/18.83 addition(codomain(domain(X)), multiplication(c(X), fresh2(leq(coantidomain(domain(X)), addition(codomain(domain(X)), coantidomain(domain(X)))), true, coantidomain(domain(X)), one)))
% 142.23/18.83 = { by lemma 39 }
% 142.23/18.83 addition(codomain(domain(X)), multiplication(c(X), fresh2(leq(coantidomain(domain(X)), one), true, coantidomain(domain(X)), one)))
% 142.23/18.83 = { by axiom 25 (order_1) }
% 142.23/18.83 addition(codomain(domain(X)), multiplication(c(X), addition(coantidomain(domain(X)), one)))
% 142.23/18.83 = { by axiom 5 (additive_commutativity) R->L }
% 142.23/18.83 addition(codomain(domain(X)), multiplication(c(X), addition(one, coantidomain(domain(X)))))
% 142.23/18.83 = { by axiom 26 (right_distributivity) }
% 142.23/18.83 addition(codomain(domain(X)), addition(multiplication(c(X), one), multiplication(c(X), coantidomain(domain(X)))))
% 142.23/18.83 = { by axiom 7 (multiplicative_right_identity) }
% 142.23/18.83 addition(codomain(domain(X)), addition(c(X), multiplication(c(X), coantidomain(domain(X)))))
% 142.23/18.83 = { by lemma 35 R->L }
% 142.23/18.83 addition(codomain(domain(X)), addition(c(X), multiplication(antidomain(X), coantidomain(domain(X)))))
% 142.23/18.83 = { by lemma 53 R->L }
% 142.23/18.83 addition(codomain(domain(X)), addition(c(X), multiplication(addition(domain(X), antidomain(X)), coantidomain(domain(X)))))
% 142.23/18.83 = { by lemma 31 }
% 142.23/18.83 addition(codomain(domain(X)), addition(c(X), multiplication(one, coantidomain(domain(X)))))
% 142.23/18.83 = { by axiom 9 (multiplicative_left_identity) }
% 142.23/18.83 addition(codomain(domain(X)), addition(c(X), coantidomain(domain(X))))
% 142.23/18.83 = { by axiom 5 (additive_commutativity) R->L }
% 142.23/18.83 addition(codomain(domain(X)), addition(coantidomain(domain(X)), c(X)))
% 142.23/18.83 = { by axiom 19 (additive_associativity) }
% 142.23/18.83 addition(addition(codomain(domain(X)), coantidomain(domain(X))), c(X))
% 142.23/18.83 = { by lemma 39 }
% 142.23/18.83 addition(one, c(X))
% 142.23/18.83 = { by lemma 60 }
% 142.23/18.83 one
% 142.23/18.83
% 142.23/18.83 Lemma 112: forward_diamond(X, domain(Y)) = forward_diamond(X, Y).
% 142.23/18.83 Proof:
% 142.23/18.83 forward_diamond(X, domain(Y))
% 142.23/18.83 = { by lemma 73 R->L }
% 142.23/18.83 forward_diamond(X, forward_diamond(one, Y))
% 142.23/18.83 = { by lemma 68 R->L }
% 142.23/18.83 forward_diamond(X, c(antidomain(Y)))
% 142.23/18.83 = { by lemma 44 R->L }
% 142.23/18.83 forward_diamond(X, antidomain(c(Y)))
% 142.23/18.83 = { by lemma 74 R->L }
% 142.23/18.83 domain(multiplication(X, c(c(Y))))
% 142.23/18.83 = { by lemma 70 }
% 142.23/18.83 domain(multiplication(X, domain(Y)))
% 142.23/18.83 = { by axiom 18 (forward_diamond) R->L }
% 142.23/18.83 forward_diamond(X, Y)
% 142.23/18.83
% 142.23/18.83 Lemma 113: c(backward_box(X, Y)) = backward_diamond(X, c(Y)).
% 142.23/18.83 Proof:
% 142.23/18.83 c(backward_box(X, Y))
% 142.23/18.83 = { by lemma 105 R->L }
% 142.23/18.83 c(backward_box(X, domain(Y)))
% 142.23/18.83 = { by lemma 106 R->L }
% 142.23/18.83 c(backward_box(X, forward_box(c(Y), Y)))
% 142.23/18.83 = { by lemma 107 R->L }
% 142.23/18.83 c(backward_box(X, forward_box(c(Y), domain(Y))))
% 142.23/18.83 = { by lemma 70 R->L }
% 142.23/18.83 c(backward_box(X, forward_box(c(Y), c(c(Y)))))
% 142.23/18.83 = { by lemma 108 R->L }
% 142.23/18.83 c(backward_box(X, c(multiplication(c(Y), domain(c(Y))))))
% 142.23/18.83 = { by lemma 109 R->L }
% 142.23/18.83 c(c(backward_diamond(X, domain(multiplication(c(Y), domain(c(Y)))))))
% 142.23/18.83 = { by axiom 18 (forward_diamond) R->L }
% 142.23/18.83 c(c(backward_diamond(X, forward_diamond(c(Y), c(Y)))))
% 142.23/18.83 = { by lemma 103 }
% 142.23/18.83 c(backward_box(X, coantidomain(forward_diamond(c(Y), c(Y)))))
% 142.23/18.83 = { by lemma 81 R->L }
% 142.23/18.83 c(backward_box(X, coantidomain(c(forward_box(c(Y), Y)))))
% 142.23/18.83 = { by lemma 106 }
% 142.23/18.83 c(backward_box(X, coantidomain(c(domain(Y)))))
% 142.23/18.83 = { by lemma 69 }
% 142.23/18.83 c(backward_box(X, coantidomain(c(Y))))
% 142.23/18.83 = { by lemma 67 }
% 142.23/18.83 c(backward_box(X, backward_box(one, Y)))
% 142.23/18.83 = { by lemma 105 R->L }
% 142.23/18.83 c(backward_box(X, backward_box(one, domain(Y))))
% 142.23/18.83 = { by lemma 73 R->L }
% 142.23/18.83 c(backward_box(X, backward_box(one, forward_diamond(one, Y))))
% 142.23/18.83 = { by lemma 68 R->L }
% 142.23/18.83 c(backward_box(X, backward_box(one, c(antidomain(Y)))))
% 142.23/18.83 = { by lemma 99 R->L }
% 142.23/18.83 c(backward_box(X, domain(backward_box(one, c(antidomain(Y))))))
% 142.23/18.83 = { by lemma 110 R->L }
% 142.23/18.83 c(backward_box(X, domain(backward_box(domain(antidomain(Y)), zero))))
% 142.23/18.83 = { by lemma 58 R->L }
% 142.23/18.83 c(backward_box(X, domain(c(codomain(domain(antidomain(Y)))))))
% 142.23/18.83 = { by axiom 7 (multiplicative_right_identity) R->L }
% 142.23/18.83 c(backward_box(X, domain(multiplication(c(codomain(domain(antidomain(Y)))), one))))
% 142.23/18.83 = { by lemma 111 R->L }
% 142.23/18.83 c(backward_box(X, domain(multiplication(c(codomain(domain(antidomain(Y)))), addition(c(antidomain(Y)), codomain(domain(antidomain(Y))))))))
% 142.23/18.83 = { by lemma 35 R->L }
% 142.23/18.83 c(backward_box(X, domain(multiplication(antidomain(codomain(domain(antidomain(Y)))), addition(c(antidomain(Y)), codomain(domain(antidomain(Y))))))))
% 142.23/18.83 = { by axiom 5 (additive_commutativity) R->L }
% 142.23/18.83 c(backward_box(X, domain(multiplication(antidomain(codomain(domain(antidomain(Y)))), addition(codomain(domain(antidomain(Y))), c(antidomain(Y)))))))
% 142.23/18.83 = { by lemma 34 }
% 142.23/18.83 c(backward_box(X, domain(multiplication(antidomain(codomain(domain(antidomain(Y)))), c(antidomain(Y))))))
% 142.23/18.83 = { by lemma 35 }
% 142.23/18.83 c(backward_box(X, domain(multiplication(c(codomain(domain(antidomain(Y)))), c(antidomain(Y))))))
% 142.23/18.83 = { by lemma 33 R->L }
% 142.23/18.83 c(backward_box(X, domain(multiplication(domain(antidomain(codomain(domain(antidomain(Y))))), c(antidomain(Y))))))
% 142.23/18.83 = { by lemma 75 }
% 142.23/18.83 c(backward_box(X, domain(domain_difference(antidomain(codomain(domain(antidomain(Y)))), domain(antidomain(Y))))))
% 142.23/18.83 = { by lemma 76 }
% 142.23/18.83 c(backward_box(X, domain(domain_difference(antidomain(codomain(domain(antidomain(Y)))), antidomain(Y)))))
% 142.23/18.83 = { by lemma 35 }
% 142.23/18.83 c(backward_box(X, domain(domain_difference(c(codomain(domain(antidomain(Y)))), antidomain(Y)))))
% 142.23/18.83 = { by lemma 58 }
% 142.23/18.83 c(backward_box(X, domain(domain_difference(backward_box(domain(antidomain(Y)), zero), antidomain(Y)))))
% 142.23/18.83 = { by lemma 110 }
% 142.23/18.83 c(backward_box(X, domain(domain_difference(backward_box(one, c(antidomain(Y))), antidomain(Y)))))
% 142.23/18.83 = { by lemma 78 }
% 142.23/18.83 c(backward_box(X, forward_diamond(domain(backward_box(one, c(antidomain(Y)))), Y)))
% 142.23/18.83 = { by lemma 99 }
% 142.23/18.83 c(backward_box(X, forward_diamond(backward_box(one, c(antidomain(Y))), Y)))
% 142.23/18.83 = { by lemma 68 }
% 142.23/18.83 c(backward_box(X, forward_diamond(backward_box(one, forward_diamond(one, Y)), Y)))
% 142.23/18.83 = { by lemma 73 }
% 142.23/18.83 c(backward_box(X, forward_diamond(backward_box(one, domain(Y)), Y)))
% 142.23/18.83 = { by lemma 105 }
% 142.23/18.83 c(backward_box(X, forward_diamond(backward_box(one, Y), Y)))
% 142.23/18.83 = { by lemma 112 R->L }
% 142.23/18.83 c(backward_box(X, forward_diamond(backward_box(one, Y), domain(Y))))
% 142.23/18.83 = { by lemma 73 R->L }
% 142.23/18.83 c(backward_box(X, forward_diamond(backward_box(one, Y), forward_diamond(one, Y))))
% 142.23/18.83 = { by lemma 68 R->L }
% 142.23/18.83 c(backward_box(X, forward_diamond(backward_box(one, Y), c(antidomain(Y)))))
% 142.23/18.83 = { by lemma 35 R->L }
% 142.23/18.83 antidomain(backward_box(X, forward_diamond(backward_box(one, Y), c(antidomain(Y)))))
% 142.23/18.83 = { by axiom 17 (backward_box) }
% 142.23/18.83 antidomain(c(backward_diamond(X, c(forward_diamond(backward_box(one, Y), c(antidomain(Y)))))))
% 142.23/18.83 = { by axiom 16 (forward_box) R->L }
% 142.23/18.83 antidomain(c(backward_diamond(X, forward_box(backward_box(one, Y), antidomain(Y)))))
% 142.23/18.83 = { by lemma 43 }
% 142.23/18.83 domain(domain(backward_diamond(X, forward_box(backward_box(one, Y), antidomain(Y)))))
% 142.23/18.83 = { by lemma 45 }
% 142.23/18.83 forward_diamond(one, backward_diamond(X, forward_box(backward_box(one, Y), antidomain(Y))))
% 142.23/18.83 = { by lemma 73 }
% 142.23/18.83 domain(backward_diamond(X, forward_box(backward_box(one, Y), antidomain(Y))))
% 142.23/18.83 = { by lemma 35 }
% 142.23/18.83 domain(backward_diamond(X, forward_box(backward_box(one, Y), c(Y))))
% 142.23/18.83 = { by lemma 67 R->L }
% 142.23/18.83 domain(backward_diamond(X, forward_box(coantidomain(c(Y)), c(Y))))
% 142.23/18.83 = { by lemma 62 R->L }
% 142.23/18.83 domain(backward_diamond(X, forward_box(coantidomain(domain(c(Y))), c(Y))))
% 142.23/18.84 = { by lemma 80 R->L }
% 142.23/18.84 domain(backward_diamond(X, c(multiplication(coantidomain(domain(c(Y))), c(c(Y))))))
% 142.23/18.84 = { by lemma 42 R->L }
% 142.23/18.84 domain(backward_diamond(X, c(addition(zero, multiplication(coantidomain(domain(c(Y))), c(c(Y)))))))
% 142.23/18.84 = { by axiom 11 (codomain1) R->L }
% 142.23/18.84 domain(backward_diamond(X, c(addition(multiplication(coantidomain(domain(c(Y))), coantidomain(coantidomain(domain(c(Y))))), multiplication(coantidomain(domain(c(Y))), c(c(Y)))))))
% 142.23/18.84 = { by axiom 1 (codomain4) R->L }
% 142.23/18.84 domain(backward_diamond(X, c(addition(multiplication(coantidomain(domain(c(Y))), codomain(domain(c(Y)))), multiplication(coantidomain(domain(c(Y))), c(c(Y)))))))
% 142.23/18.84 = { by axiom 26 (right_distributivity) R->L }
% 142.23/18.84 domain(backward_diamond(X, c(multiplication(coantidomain(domain(c(Y))), addition(codomain(domain(c(Y))), c(c(Y)))))))
% 142.23/18.84 = { by axiom 5 (additive_commutativity) }
% 142.23/18.84 domain(backward_diamond(X, c(multiplication(coantidomain(domain(c(Y))), addition(c(c(Y)), codomain(domain(c(Y))))))))
% 142.23/18.84 = { by lemma 111 }
% 142.23/18.84 domain(backward_diamond(X, c(multiplication(coantidomain(domain(c(Y))), one))))
% 142.23/18.84 = { by axiom 7 (multiplicative_right_identity) }
% 142.23/18.84 domain(backward_diamond(X, c(coantidomain(domain(c(Y))))))
% 142.23/18.84 = { by lemma 61 }
% 142.23/18.84 domain(backward_diamond(X, codomain(domain(c(Y)))))
% 142.23/18.84 = { by lemma 62 }
% 142.23/18.84 domain(backward_diamond(X, codomain(c(Y))))
% 142.23/18.84 = { by axiom 15 (backward_diamond) }
% 142.23/18.84 domain(codomain(multiplication(codomain(codomain(c(Y))), X)))
% 142.23/18.84 = { by lemma 63 }
% 142.23/18.84 codomain(multiplication(codomain(codomain(c(Y))), X))
% 142.23/18.84 = { by axiom 15 (backward_diamond) R->L }
% 142.23/18.84 backward_diamond(X, codomain(c(Y)))
% 142.23/18.84 = { by lemma 102 }
% 142.23/18.84 backward_diamond(X, c(Y))
% 142.23/18.84
% 142.23/18.84 Lemma 114: multiplication(forward_diamond(X, Y), c(Z)) = domain_difference(multiplication(X, domain(Y)), Z).
% 142.23/18.84 Proof:
% 142.23/18.84 multiplication(forward_diamond(X, Y), c(Z))
% 142.23/18.84 = { by lemma 35 R->L }
% 142.23/18.84 multiplication(forward_diamond(X, Y), antidomain(Z))
% 142.23/18.84 = { by axiom 18 (forward_diamond) }
% 142.23/18.84 multiplication(domain(multiplication(X, domain(Y))), antidomain(Z))
% 142.23/18.84 = { by axiom 20 (domain_difference) R->L }
% 142.23/18.84 domain_difference(multiplication(X, domain(Y)), Z)
% 142.23/18.84
% 142.23/18.84 Lemma 115: domain_difference(multiplication(X, domain(Y)), Z) = domain_difference(forward_diamond(X, Y), Z).
% 142.23/18.84 Proof:
% 142.23/18.84 domain_difference(multiplication(X, domain(Y)), Z)
% 142.23/18.84 = { by lemma 114 R->L }
% 142.23/18.84 multiplication(forward_diamond(X, Y), c(Z))
% 142.23/18.84 = { by lemma 35 R->L }
% 142.23/18.84 multiplication(forward_diamond(X, Y), antidomain(Z))
% 142.23/18.84 = { by lemma 98 R->L }
% 142.23/18.84 multiplication(domain(forward_diamond(X, Y)), antidomain(Z))
% 142.23/18.84 = { by axiom 20 (domain_difference) R->L }
% 142.23/18.84 domain_difference(forward_diamond(X, Y), Z)
% 142.23/18.84
% 142.23/18.84 Lemma 116: multiplication(c(X), domain(Y)) = domain_difference(c(X), c(Y)).
% 142.23/18.84 Proof:
% 142.23/18.84 multiplication(c(X), domain(Y))
% 142.23/18.84 = { by lemma 33 R->L }
% 142.23/18.84 multiplication(domain(antidomain(X)), domain(Y))
% 142.23/18.84 = { by lemma 71 }
% 142.23/18.84 domain_difference(antidomain(X), antidomain(Y))
% 142.23/18.84 = { by lemma 35 }
% 142.23/18.84 domain_difference(c(X), antidomain(Y))
% 142.23/18.84 = { by lemma 35 }
% 142.23/18.84 domain_difference(c(X), c(Y))
% 142.23/18.84
% 142.23/18.84 Lemma 117: multiplication(c(X), domain(X)) = zero.
% 142.23/18.84 Proof:
% 142.23/18.84 multiplication(c(X), domain(X))
% 142.23/18.84 = { by axiom 2 (complement) }
% 142.23/18.84 multiplication(antidomain(domain(X)), domain(X))
% 142.23/18.84 = { by axiom 12 (domain1) }
% 142.23/18.84 zero
% 142.23/18.84
% 142.23/18.84 Lemma 118: multiplication(c(X), addition(domain(X), Y)) = multiplication(c(X), Y).
% 142.23/18.84 Proof:
% 142.23/18.84 multiplication(c(X), addition(domain(X), Y))
% 142.23/18.84 = { by axiom 26 (right_distributivity) }
% 142.23/18.84 addition(multiplication(c(X), domain(X)), multiplication(c(X), Y))
% 142.23/18.84 = { by lemma 117 }
% 142.23/18.84 addition(zero, multiplication(c(X), Y))
% 142.23/18.84 = { by lemma 42 }
% 142.23/18.84 multiplication(c(X), Y)
% 142.23/18.84
% 142.23/18.84 Lemma 119: forward_diamond(c(X), Y) = domain_difference(Y, X).
% 142.23/18.84 Proof:
% 142.23/18.84 forward_diamond(c(X), Y)
% 142.23/18.84 = { by lemma 98 R->L }
% 142.23/18.84 domain(forward_diamond(c(X), Y))
% 142.23/18.84 = { by lemma 85 R->L }
% 142.23/18.84 addition(domain_difference(forward_diamond(c(X), Y), c(X)), domain_difference(forward_diamond(c(X), Y), c(c(X))))
% 142.23/18.84 = { by lemma 115 R->L }
% 142.23/18.84 addition(domain_difference(multiplication(c(X), domain(Y)), c(X)), domain_difference(forward_diamond(c(X), Y), c(c(X))))
% 142.23/18.84 = { by lemma 93 }
% 142.23/18.84 addition(zero, domain_difference(forward_diamond(c(X), Y), c(c(X))))
% 142.23/18.84 = { by lemma 42 }
% 142.23/18.84 domain_difference(forward_diamond(c(X), Y), c(c(X)))
% 142.23/18.84 = { by lemma 62 R->L }
% 142.23/18.84 domain_difference(forward_diamond(domain(c(X)), Y), c(c(X)))
% 142.23/18.84 = { by lemma 115 R->L }
% 142.23/18.84 domain_difference(multiplication(domain(c(X)), domain(Y)), c(c(X)))
% 142.23/18.84 = { by lemma 114 R->L }
% 142.23/18.84 multiplication(forward_diamond(domain(c(X)), Y), c(c(c(X))))
% 142.23/18.84 = { by lemma 35 R->L }
% 142.23/18.84 multiplication(forward_diamond(domain(c(X)), Y), antidomain(c(c(X))))
% 142.23/18.84 = { by lemma 78 R->L }
% 142.23/18.84 multiplication(domain(domain_difference(c(X), antidomain(Y))), antidomain(c(c(X))))
% 142.23/18.84 = { by axiom 20 (domain_difference) R->L }
% 142.23/18.84 domain_difference(domain_difference(c(X), antidomain(Y)), c(c(X)))
% 142.23/18.84 = { by lemma 35 }
% 142.23/18.84 domain_difference(domain_difference(c(X), c(Y)), c(c(X)))
% 142.23/18.84 = { by lemma 116 R->L }
% 142.23/18.84 domain_difference(multiplication(c(X), domain(Y)), c(c(X)))
% 142.23/18.84 = { by lemma 118 R->L }
% 142.23/18.84 domain_difference(multiplication(c(X), addition(domain(X), domain(Y))), c(c(X)))
% 142.23/18.84 = { by lemma 70 R->L }
% 142.23/18.84 domain_difference(multiplication(c(X), addition(c(c(X)), domain(Y))), c(c(X)))
% 142.23/18.84 = { by lemma 85 R->L }
% 142.23/18.84 domain_difference(multiplication(c(X), addition(c(c(X)), addition(domain_difference(Y, X), domain_difference(Y, c(X))))), c(c(X)))
% 142.23/18.84 = { by axiom 5 (additive_commutativity) R->L }
% 142.23/18.84 domain_difference(multiplication(c(X), addition(c(c(X)), addition(domain_difference(Y, c(X)), domain_difference(Y, X)))), c(c(X)))
% 142.23/18.84 = { by axiom 19 (additive_associativity) }
% 142.23/18.84 domain_difference(multiplication(c(X), addition(addition(c(c(X)), domain_difference(Y, c(X))), domain_difference(Y, X))), c(c(X)))
% 142.23/18.84 = { by lemma 82 }
% 142.23/18.84 domain_difference(multiplication(c(X), addition(c(c(X)), domain_difference(Y, X))), c(c(X)))
% 142.23/18.84 = { by axiom 5 (additive_commutativity) }
% 142.23/18.84 domain_difference(multiplication(c(X), addition(domain_difference(Y, X), c(c(X)))), c(c(X)))
% 142.23/18.84 = { by lemma 70 }
% 142.23/18.84 domain_difference(multiplication(c(X), addition(domain_difference(Y, X), domain(X))), c(c(X)))
% 142.23/18.84 = { by axiom 5 (additive_commutativity) }
% 142.23/18.84 domain_difference(multiplication(c(X), addition(domain(X), domain_difference(Y, X))), c(c(X)))
% 142.23/18.84 = { by lemma 118 }
% 142.23/18.84 domain_difference(multiplication(c(X), domain_difference(Y, X)), c(c(X)))
% 142.23/18.84 = { by lemma 42 R->L }
% 142.23/18.84 addition(zero, domain_difference(multiplication(c(X), domain_difference(Y, X)), c(c(X))))
% 142.23/18.84 = { by lemma 93 R->L }
% 142.23/18.84 addition(domain_difference(multiplication(c(X), domain_difference(Y, X)), c(X)), domain_difference(multiplication(c(X), domain_difference(Y, X)), c(c(X))))
% 142.23/18.84 = { by lemma 85 }
% 142.23/18.84 domain(multiplication(c(X), domain_difference(Y, X)))
% 142.23/18.84 = { by lemma 92 }
% 142.23/18.84 forward_diamond(c(X), domain_difference(Y, X))
% 142.23/18.84 = { by lemma 69 R->L }
% 142.23/18.84 forward_diamond(c(domain(X)), domain_difference(Y, X))
% 142.23/18.84 = { by lemma 86 R->L }
% 142.23/18.84 forward_diamond(c(domain_difference(X, domain_difference(Y, X))), domain_difference(Y, X))
% 142.23/18.84 = { by lemma 84 }
% 142.23/18.84 forward_diamond(forward_box(domain(X), domain_difference(Y, X)), domain_difference(Y, X))
% 142.23/18.84 = { by lemma 70 R->L }
% 142.23/18.84 forward_diamond(forward_box(c(c(X)), domain_difference(Y, X)), domain_difference(Y, X))
% 142.23/18.84 = { by axiom 18 (forward_diamond) }
% 142.23/18.84 domain(multiplication(forward_box(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X))))
% 142.23/18.84 = { by lemma 42 R->L }
% 142.23/18.84 domain(addition(zero, multiplication(forward_box(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X)))))
% 142.23/18.84 = { by axiom 8 (right_annihilation) R->L }
% 142.23/18.84 domain(addition(multiplication(domain(c(c(X))), zero), multiplication(forward_box(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X)))))
% 142.23/18.84 = { by lemma 117 R->L }
% 142.23/18.84 domain(addition(multiplication(domain(c(c(X))), multiplication(c(domain_difference(Y, X)), domain(domain_difference(Y, X)))), multiplication(forward_box(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X)))))
% 142.23/18.84 = { by lemma 94 }
% 142.23/18.84 domain(addition(multiplication(domain_difference(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X))), multiplication(forward_box(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X)))))
% 142.23/18.84 = { by axiom 27 (left_distributivity) R->L }
% 142.23/18.84 domain(multiplication(addition(domain_difference(c(c(X)), domain_difference(Y, X)), forward_box(c(c(X)), domain_difference(Y, X))), domain(domain_difference(Y, X))))
% 142.23/18.84 = { by axiom 5 (additive_commutativity) }
% 142.23/18.84 domain(multiplication(addition(forward_box(c(c(X)), domain_difference(Y, X)), domain_difference(c(c(X)), domain_difference(Y, X))), domain(domain_difference(Y, X))))
% 142.23/18.84 = { by axiom 18 (forward_diamond) R->L }
% 142.23/18.84 forward_diamond(addition(forward_box(c(c(X)), domain_difference(Y, X)), domain_difference(c(c(X)), domain_difference(Y, X))), domain_difference(Y, X))
% 142.23/18.84 = { by axiom 5 (additive_commutativity) }
% 142.23/18.84 forward_diamond(addition(domain_difference(c(c(X)), domain_difference(Y, X)), forward_box(c(c(X)), domain_difference(Y, X))), domain_difference(Y, X))
% 142.23/18.84 = { by lemma 107 R->L }
% 142.23/18.84 forward_diamond(addition(domain_difference(c(c(X)), domain_difference(Y, X)), forward_box(c(c(X)), domain(domain_difference(Y, X)))), domain_difference(Y, X))
% 142.23/18.84 = { by lemma 70 R->L }
% 142.23/18.84 forward_diamond(addition(domain_difference(c(c(X)), domain_difference(Y, X)), forward_box(c(c(X)), c(c(domain_difference(Y, X))))), domain_difference(Y, X))
% 142.23/18.84 = { by lemma 95 R->L }
% 142.23/18.84 forward_diamond(addition(forward_diamond(c(c(X)), c(domain_difference(Y, X))), forward_box(c(c(X)), c(c(domain_difference(Y, X))))), domain_difference(Y, X))
% 142.23/18.84 = { by axiom 5 (additive_commutativity) R->L }
% 142.23/18.84 forward_diamond(addition(forward_box(c(c(X)), c(c(domain_difference(Y, X)))), forward_diamond(c(c(X)), c(domain_difference(Y, X)))), domain_difference(Y, X))
% 142.23/18.84 = { by axiom 18 (forward_diamond) }
% 142.23/18.84 forward_diamond(addition(forward_box(c(c(X)), c(c(domain_difference(Y, X)))), domain(multiplication(c(c(X)), domain(c(domain_difference(Y, X)))))), domain_difference(Y, X))
% 142.23/18.84 = { by lemma 90 R->L }
% 142.23/18.84 forward_diamond(addition(c(forward_diamond(c(c(X)), c(domain_difference(Y, X)))), domain(multiplication(c(c(X)), domain(c(domain_difference(Y, X)))))), domain_difference(Y, X))
% 142.23/18.84 = { by lemma 35 R->L }
% 142.23/18.84 forward_diamond(addition(antidomain(forward_diamond(c(c(X)), c(domain_difference(Y, X)))), domain(multiplication(c(c(X)), domain(c(domain_difference(Y, X)))))), domain_difference(Y, X))
% 142.23/18.84 = { by lemma 79 R->L }
% 142.23/18.84 forward_diamond(addition(c(multiplication(c(c(X)), domain(c(domain_difference(Y, X))))), domain(multiplication(c(c(X)), domain(c(domain_difference(Y, X)))))), domain_difference(Y, X))
% 142.23/18.84 = { by lemma 50 }
% 142.23/18.84 forward_diamond(one, domain_difference(Y, X))
% 142.23/18.84 = { by lemma 73 }
% 142.23/18.84 domain(domain_difference(Y, X))
% 142.23/18.84 = { by lemma 97 }
% 142.23/18.84 domain_difference(Y, X)
% 142.23/18.84
% 142.23/18.84 Lemma 120: forward_diamond(domain(X), Y) = domain_difference(X, c(Y)).
% 142.23/18.84 Proof:
% 142.23/18.84 forward_diamond(domain(X), Y)
% 142.23/18.84 = { by lemma 91 R->L }
% 142.23/18.84 domain(domain_difference(X, c(Y)))
% 142.23/18.84 = { by lemma 97 }
% 142.23/18.84 domain_difference(X, c(Y))
% 142.23/18.84
% 142.23/18.84 Lemma 121: domain_difference(X, c(Y)) = domain_difference(Y, c(X)).
% 142.23/18.84 Proof:
% 142.23/18.84 domain_difference(X, c(Y))
% 142.23/18.84 = { by lemma 119 R->L }
% 142.23/18.84 forward_diamond(c(c(Y)), X)
% 142.23/18.84 = { by lemma 70 }
% 142.23/18.84 forward_diamond(domain(Y), X)
% 142.23/18.84 = { by lemma 120 }
% 142.23/18.84 domain_difference(Y, c(X))
% 142.23/18.84
% 142.23/18.84 Lemma 122: coantidomain(multiplication(codomain(X), Y)) = backward_box(Y, coantidomain(X)).
% 142.23/18.84 Proof:
% 142.23/18.84 coantidomain(multiplication(codomain(X), Y))
% 142.23/18.84 = { by lemma 55 R->L }
% 142.23/18.84 codomain(coantidomain(multiplication(codomain(X), Y)))
% 142.23/18.84 = { by lemma 52 R->L }
% 142.23/18.84 coantidomain(codomain(multiplication(codomain(X), Y)))
% 142.23/18.84 = { by axiom 15 (backward_diamond) R->L }
% 142.23/18.84 coantidomain(backward_diamond(Y, X))
% 142.23/18.84 = { by lemma 104 }
% 142.23/18.84 backward_box(Y, coantidomain(X))
% 142.23/18.84
% 142.23/18.84 Lemma 123: backward_box(codomain(X), coantidomain(X)) = coantidomain(X).
% 142.23/18.84 Proof:
% 142.23/18.84 backward_box(codomain(X), coantidomain(X))
% 142.23/18.84 = { by lemma 65 R->L }
% 142.23/18.84 backward_box(codomain(codomain(X)), coantidomain(X))
% 142.23/18.84 = { by lemma 61 R->L }
% 142.23/18.84 backward_box(c(coantidomain(codomain(X))), coantidomain(X))
% 142.23/18.84 = { by lemma 104 R->L }
% 142.23/18.84 coantidomain(backward_diamond(c(coantidomain(codomain(X))), X))
% 142.23/18.84 = { by lemma 100 R->L }
% 142.23/18.84 coantidomain(multiplication(codomain(X), c(coantidomain(codomain(X)))))
% 142.23/18.84 = { by lemma 51 }
% 142.23/18.84 coantidomain(codomain(X))
% 142.23/18.84 = { by lemma 56 }
% 142.23/18.84 coantidomain(X)
% 142.23/18.84
% 142.23/18.84 Lemma 124: domain_difference(X, backward_diamond(Y, c(Z))) = forward_diamond(backward_box(Y, Z), X).
% 142.23/18.84 Proof:
% 142.23/18.84 domain_difference(X, backward_diamond(Y, c(Z)))
% 142.23/18.84 = { by lemma 113 R->L }
% 142.23/18.84 domain_difference(X, c(backward_box(Y, Z)))
% 142.23/18.84 = { by lemma 121 R->L }
% 142.23/18.84 domain_difference(backward_box(Y, Z), c(X))
% 142.23/18.84 = { by lemma 120 R->L }
% 142.23/18.84 forward_diamond(domain(backward_box(Y, Z)), X)
% 142.23/18.84 = { by lemma 99 }
% 142.23/18.84 forward_diamond(backward_box(Y, Z), X)
% 142.23/18.84
% 142.23/18.84 Lemma 125: domain_difference(X, forward_box(Y, c(Z))) = forward_diamond(forward_diamond(Y, Z), X).
% 142.23/18.84 Proof:
% 142.23/18.84 domain_difference(X, forward_box(Y, c(Z)))
% 142.23/18.84 = { by lemma 90 R->L }
% 142.23/18.84 domain_difference(X, c(forward_diamond(Y, Z)))
% 142.23/18.84 = { by lemma 121 R->L }
% 142.23/18.84 domain_difference(forward_diamond(Y, Z), c(X))
% 142.23/18.84 = { by lemma 115 R->L }
% 142.23/18.84 domain_difference(multiplication(Y, domain(Z)), c(X))
% 142.23/18.84 = { by lemma 120 R->L }
% 142.23/18.84 forward_diamond(domain(multiplication(Y, domain(Z))), X)
% 142.23/18.84 = { by axiom 18 (forward_diamond) R->L }
% 142.23/18.84 forward_diamond(forward_diamond(Y, Z), X)
% 142.23/18.84
% 142.23/18.84 Goal 1 (goals_1): multiplication(forward_diamond(x0, domain(x1)), domain(x2)) = zero.
% 142.23/18.84 Proof:
% 142.23/18.84 multiplication(forward_diamond(x0, domain(x1)), domain(x2))
% 142.23/18.84 = { by lemma 112 }
% 142.23/18.84 multiplication(forward_diamond(x0, x1), domain(x2))
% 142.23/18.84 = { by lemma 73 R->L }
% 142.23/18.84 multiplication(forward_diamond(x0, x1), forward_diamond(one, x2))
% 142.23/18.84 = { by lemma 68 R->L }
% 142.23/18.84 multiplication(forward_diamond(x0, x1), c(antidomain(x2)))
% 142.23/18.84 = { by lemma 114 }
% 142.23/18.84 domain_difference(multiplication(x0, domain(x1)), antidomain(x2))
% 142.23/18.84 = { by lemma 115 }
% 142.23/18.84 domain_difference(forward_diamond(x0, x1), antidomain(x2))
% 142.23/18.84 = { by lemma 35 }
% 142.23/18.84 domain_difference(forward_diamond(x0, x1), c(x2))
% 142.23/18.84 = { by lemma 121 }
% 142.23/18.84 domain_difference(x2, c(forward_diamond(x0, x1)))
% 142.23/18.84 = { by lemma 90 }
% 142.23/18.84 domain_difference(x2, forward_box(x0, c(x1)))
% 142.23/18.84 = { by lemma 125 }
% 142.23/18.84 forward_diamond(forward_diamond(x0, x1), x2)
% 142.23/18.84 = { by lemma 112 R->L }
% 142.23/18.84 forward_diamond(forward_diamond(x0, domain(x1)), x2)
% 142.23/18.84 = { by lemma 125 R->L }
% 142.23/18.84 domain_difference(x2, forward_box(x0, c(domain(x1))))
% 142.23/18.84 = { by lemma 69 R->L }
% 142.23/18.84 domain_difference(x2, forward_box(x0, c(domain(domain(x1)))))
% 142.23/18.84 = { by axiom 7 (multiplicative_right_identity) R->L }
% 142.23/18.84 domain_difference(x2, forward_box(x0, c(domain(multiplication(domain(x1), one)))))
% 142.23/18.84 = { by lemma 37 R->L }
% 142.23/18.84 domain_difference(x2, forward_box(x0, c(domain(multiplication(domain(x1), domain(one))))))
% 142.23/18.84 = { by axiom 18 (forward_diamond) R->L }
% 142.23/18.84 domain_difference(x2, forward_box(x0, c(forward_diamond(domain(x1), one))))
% 142.23/18.84 = { by lemma 41 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(x0, c(forward_diamond(domain(x1), c(zero)))))
% 142.23/18.85 = { by axiom 16 (forward_box) R->L }
% 142.23/18.85 domain_difference(x2, forward_box(x0, forward_box(domain(x1), zero)))
% 142.23/18.85 = { by lemma 84 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(x0, c(domain_difference(x1, zero))))
% 142.23/18.85 = { by lemma 108 R->L }
% 142.23/18.85 domain_difference(x2, c(multiplication(x0, domain(domain_difference(x1, zero)))))
% 142.23/18.85 = { by lemma 97 }
% 142.23/18.85 domain_difference(x2, c(multiplication(x0, domain_difference(x1, zero))))
% 142.23/18.85 = { by lemma 76 R->L }
% 142.23/18.85 domain_difference(x2, c(multiplication(x0, domain_difference(x1, domain(zero)))))
% 142.23/18.85 = { by lemma 75 R->L }
% 142.23/18.85 domain_difference(x2, c(multiplication(x0, multiplication(domain(x1), c(zero)))))
% 142.23/18.85 = { by lemma 33 R->L }
% 142.23/18.85 domain_difference(x2, c(multiplication(x0, multiplication(domain(x1), domain(antidomain(zero))))))
% 142.23/18.85 = { by axiom 21 (multiplicative_associativity) }
% 142.23/18.85 domain_difference(x2, c(multiplication(multiplication(x0, domain(x1)), domain(antidomain(zero)))))
% 142.23/18.85 = { by lemma 79 }
% 142.23/18.85 domain_difference(x2, antidomain(forward_diamond(multiplication(x0, domain(x1)), antidomain(zero))))
% 142.23/18.85 = { by lemma 35 }
% 142.23/18.85 domain_difference(x2, c(forward_diamond(multiplication(x0, domain(x1)), antidomain(zero))))
% 142.23/18.85 = { by lemma 90 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), c(antidomain(zero))))
% 142.23/18.85 = { by lemma 68 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), forward_diamond(one, zero)))
% 142.23/18.85 = { by lemma 73 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), domain(zero)))
% 142.23/18.85 = { by lemma 107 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), zero))
% 142.23/18.85 = { by lemma 38 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), coantidomain(one)))
% 142.23/18.85 = { by lemma 40 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), coantidomain(codomain(one))))
% 142.23/18.85 = { by axiom 1 (codomain4) }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), coantidomain(coantidomain(coantidomain(one)))))
% 142.23/18.85 = { by lemma 38 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), coantidomain(coantidomain(zero))))
% 142.23/18.85 = { by axiom 1 (codomain4) R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), codomain(zero)))
% 142.23/18.85 = { by axiom 10 (left_annihilation) R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), codomain(multiplication(zero, domain(x1)))))
% 142.23/18.85 = { by axiom 11 (codomain1) R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), codomain(multiplication(multiplication(multiplication(codomain(domain(x2)), x0), coantidomain(multiplication(codomain(domain(x2)), x0))), domain(x1)))))
% 142.23/18.85 = { by axiom 21 (multiplicative_associativity) R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), codomain(multiplication(multiplication(codomain(domain(x2)), x0), multiplication(coantidomain(multiplication(codomain(domain(x2)), x0)), domain(x1))))))
% 142.23/18.85 = { by axiom 21 (multiplicative_associativity) R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), codomain(multiplication(codomain(domain(x2)), multiplication(x0, multiplication(coantidomain(multiplication(codomain(domain(x2)), x0)), domain(x1)))))))
% 142.23/18.85 = { by axiom 15 (backward_diamond) R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, multiplication(coantidomain(multiplication(codomain(domain(x2)), x0)), domain(x1))), domain(x2))))
% 142.23/18.85 = { by lemma 122 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, multiplication(backward_box(x0, coantidomain(domain(x2))), domain(x1))), domain(x2))))
% 142.23/18.85 = { by axiom 17 (backward_box) }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, multiplication(c(backward_diamond(x0, c(coantidomain(domain(x2))))), domain(x1))), domain(x2))))
% 142.23/18.85 = { by lemma 116 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(c(backward_diamond(x0, c(coantidomain(domain(x2))))), c(x1))), domain(x2))))
% 142.23/18.85 = { by axiom 17 (backward_box) R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(backward_box(x0, coantidomain(domain(x2))), c(x1))), domain(x2))))
% 142.23/18.85 = { by lemma 121 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(x1, c(backward_box(x0, coantidomain(domain(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 113 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(x1, backward_diamond(x0, c(coantidomain(domain(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 124 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, forward_diamond(backward_box(x0, coantidomain(domain(x2))), x1)), domain(x2))))
% 142.23/18.85 = { by lemma 64 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, forward_diamond(backward_box(x0, backward_box(domain(x2), zero)), x1)), domain(x2))))
% 142.23/18.85 = { by axiom 17 (backward_box) }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, forward_diamond(backward_box(x0, c(backward_diamond(domain(x2), c(zero)))), x1)), domain(x2))))
% 142.23/18.85 = { by lemma 41 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, forward_diamond(backward_box(x0, c(backward_diamond(domain(x2), one))), x1)), domain(x2))))
% 142.23/18.85 = { by lemma 57 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, forward_diamond(backward_box(x0, c(codomain(domain(x2)))), x1)), domain(x2))))
% 142.23/18.85 = { by lemma 124 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(x1, backward_diamond(x0, c(c(codomain(domain(x2))))))), domain(x2))))
% 142.23/18.85 = { by lemma 113 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(x1, c(backward_box(x0, c(codomain(domain(x2))))))), domain(x2))))
% 142.23/18.85 = { by lemma 121 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(backward_box(x0, c(codomain(domain(x2)))), c(x1))), domain(x2))))
% 142.23/18.85 = { by lemma 119 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, forward_diamond(c(c(x1)), backward_box(x0, c(codomain(domain(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 92 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain(multiplication(c(c(x1)), backward_box(x0, c(codomain(domain(x2))))))), domain(x2))))
% 142.23/18.85 = { by lemma 33 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain(multiplication(domain(antidomain(c(x1))), backward_box(x0, c(codomain(domain(x2))))))), domain(x2))))
% 142.23/18.85 = { by axiom 17 (backward_box) }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain(multiplication(domain(antidomain(c(x1))), c(backward_diamond(x0, c(c(codomain(domain(x2))))))))), domain(x2))))
% 142.23/18.85 = { by lemma 75 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain(domain_difference(antidomain(c(x1)), domain(backward_diamond(x0, c(c(codomain(domain(x2))))))))), domain(x2))))
% 142.23/18.85 = { by lemma 76 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain(domain_difference(antidomain(c(x1)), backward_diamond(x0, c(c(codomain(domain(x2)))))))), domain(x2))))
% 142.23/18.85 = { by lemma 35 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain(domain_difference(c(c(x1)), backward_diamond(x0, c(c(codomain(domain(x2)))))))), domain(x2))))
% 142.23/18.85 = { by lemma 97 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(c(c(x1)), backward_diamond(x0, c(c(codomain(domain(x2))))))), domain(x2))))
% 142.23/18.85 = { by lemma 70 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(c(c(x1)), backward_diamond(x0, domain(codomain(domain(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 70 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(domain(x1), backward_diamond(x0, domain(codomain(domain(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 96 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(x1, backward_diamond(x0, domain(codomain(domain(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 63 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(x1, backward_diamond(x0, codomain(domain(x2))))), domain(x2))))
% 142.23/18.85 = { by lemma 102 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(x1, backward_diamond(x0, domain(x2)))), domain(x2))))
% 142.23/18.85 = { by lemma 70 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(x1, backward_diamond(x0, c(c(x2))))), domain(x2))))
% 142.23/18.85 = { by lemma 113 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain_difference(x1, c(backward_box(x0, c(x2))))), domain(x2))))
% 142.23/18.85 = { by lemma 42 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, addition(zero, domain_difference(x1, c(backward_box(x0, c(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 49 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, addition(multiplication(c(multiplication(domain(x1), backward_diamond(x0, domain(x2)))), multiplication(domain(x1), domain(backward_diamond(x0, domain(x2))))), domain_difference(x1, c(backward_box(x0, c(x2)))))), domain(x2))))
% 142.23/18.85 = { by axiom 28 (goals) }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, addition(multiplication(c(zero), multiplication(domain(x1), domain(backward_diamond(x0, domain(x2))))), domain_difference(x1, c(backward_box(x0, c(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 41 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, addition(multiplication(one, multiplication(domain(x1), domain(backward_diamond(x0, domain(x2))))), domain_difference(x1, c(backward_box(x0, c(x2)))))), domain(x2))))
% 142.23/18.85 = { by axiom 9 (multiplicative_left_identity) }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, addition(multiplication(domain(x1), domain(backward_diamond(x0, domain(x2)))), domain_difference(x1, c(backward_box(x0, c(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 71 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, addition(domain_difference(x1, antidomain(backward_diamond(x0, domain(x2)))), domain_difference(x1, c(backward_box(x0, c(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 35 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, addition(domain_difference(x1, c(backward_diamond(x0, domain(x2)))), domain_difference(x1, c(backward_box(x0, c(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 109 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, addition(domain_difference(x1, backward_box(x0, c(x2))), domain_difference(x1, c(backward_box(x0, c(x2)))))), domain(x2))))
% 142.23/18.85 = { by lemma 85 }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain(x1)), domain(x2))))
% 142.23/18.85 = { by lemma 102 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain(x1)), codomain(domain(x2)))))
% 142.23/18.85 = { by lemma 61 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), backward_diamond(multiplication(x0, domain(x1)), c(coantidomain(domain(x2))))))
% 142.23/18.85 = { by lemma 113 R->L }
% 142.23/18.85 domain_difference(x2, forward_box(multiplication(x0, domain(x1)), c(backward_box(multiplication(x0, domain(x1)), coantidomain(domain(x2))))))
% 142.23/18.85 = { by lemma 90 R->L }
% 142.23/18.85 domain_difference(x2, c(forward_diamond(multiplication(x0, domain(x1)), backward_box(multiplication(x0, domain(x1)), coantidomain(domain(x2))))))
% 142.23/18.85 = { by lemma 35 R->L }
% 142.23/18.85 domain_difference(x2, antidomain(forward_diamond(multiplication(x0, domain(x1)), backward_box(multiplication(x0, domain(x1)), coantidomain(domain(x2))))))
% 142.23/18.85 = { by lemma 79 R->L }
% 142.23/18.85 domain_difference(x2, c(multiplication(multiplication(x0, domain(x1)), domain(backward_box(multiplication(x0, domain(x1)), coantidomain(domain(x2)))))))
% 142.23/18.85 = { by lemma 35 R->L }
% 142.23/18.85 domain_difference(x2, antidomain(multiplication(multiplication(x0, domain(x1)), domain(backward_box(multiplication(x0, domain(x1)), coantidomain(domain(x2)))))))
% 142.23/18.85 = { by lemma 71 R->L }
% 142.23/18.85 multiplication(domain(x2), domain(multiplication(multiplication(x0, domain(x1)), domain(backward_box(multiplication(x0, domain(x1)), coantidomain(domain(x2)))))))
% 142.23/18.85 = { by axiom 18 (forward_diamond) R->L }
% 142.23/18.85 multiplication(domain(x2), forward_diamond(multiplication(x0, domain(x1)), backward_box(multiplication(x0, domain(x1)), coantidomain(domain(x2)))))
% 142.23/18.85 = { by lemma 122 R->L }
% 142.23/18.85 multiplication(domain(x2), forward_diamond(multiplication(x0, domain(x1)), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1))))))
% 142.23/18.85 = { by lemma 54 R->L }
% 142.23/18.85 multiplication(multiplication(domain(x2), codomain(domain(x2))), forward_diamond(multiplication(x0, domain(x1)), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1))))))
% 142.23/18.85 = { by axiom 21 (multiplicative_associativity) R->L }
% 142.23/18.85 multiplication(domain(x2), multiplication(codomain(domain(x2)), forward_diamond(multiplication(x0, domain(x1)), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1)))))))
% 142.23/18.85 = { by lemma 123 R->L }
% 142.23/18.85 multiplication(domain(x2), multiplication(codomain(domain(x2)), forward_diamond(multiplication(x0, domain(x1)), backward_box(codomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1)))), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1))))))))
% 142.23/18.85 = { by axiom 18 (forward_diamond) }
% 142.23/18.85 multiplication(domain(x2), multiplication(codomain(domain(x2)), domain(multiplication(multiplication(x0, domain(x1)), domain(backward_box(codomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1)))), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1))))))))))
% 142.23/18.86 = { by lemma 99 }
% 142.23/18.86 multiplication(domain(x2), multiplication(codomain(domain(x2)), domain(multiplication(multiplication(x0, domain(x1)), backward_box(codomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1)))), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1)))))))))
% 142.23/18.86 = { by lemma 123 }
% 142.23/18.86 multiplication(domain(x2), multiplication(codomain(domain(x2)), domain(multiplication(multiplication(x0, domain(x1)), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1))))))))
% 142.23/18.86 = { by axiom 9 (multiplicative_left_identity) R->L }
% 142.23/18.86 multiplication(domain(x2), multiplication(one, multiplication(codomain(domain(x2)), domain(multiplication(multiplication(x0, domain(x1)), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1)))))))))
% 142.23/18.86 = { by lemma 41 R->L }
% 142.23/18.86 multiplication(domain(x2), multiplication(c(zero), multiplication(codomain(domain(x2)), domain(multiplication(multiplication(x0, domain(x1)), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1)))))))))
% 142.23/18.86 = { by axiom 11 (codomain1) R->L }
% 142.23/18.86 multiplication(domain(x2), multiplication(c(multiplication(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1))), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1)))))), multiplication(codomain(domain(x2)), domain(multiplication(multiplication(x0, domain(x1)), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1)))))))))
% 142.23/18.86 = { by axiom 21 (multiplicative_associativity) R->L }
% 142.23/18.86 multiplication(domain(x2), multiplication(c(multiplication(codomain(domain(x2)), multiplication(multiplication(x0, domain(x1)), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1))))))), multiplication(codomain(domain(x2)), domain(multiplication(multiplication(x0, domain(x1)), coantidomain(multiplication(codomain(domain(x2)), multiplication(x0, domain(x1)))))))))
% 142.23/18.86 = { by lemma 49 }
% 142.23/18.86 multiplication(domain(x2), zero)
% 142.23/18.86 = { by axiom 8 (right_annihilation) }
% 142.23/18.86 zero
% 142.23/18.86 % SZS output end Proof
% 142.23/18.86
% 142.23/18.86 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------