TSTP Solution File: KLE109+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE109+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 : n017.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 05:35:55 EDT 2023
% Result : Theorem 264.00s 34.13s
% Output : Proof 267.46s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : KLE109+1 : TPTP v8.1.2. Released v4.0.0.
% 0.06/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n017.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Tue Aug 29 11:08:58 EDT 2023
% 0.12/0.34 % CPUTime :
% 264.00/34.13 Command-line arguments: --ground-connectedness --complete-subsets
% 264.00/34.13
% 264.00/34.13 % SZS status Theorem
% 264.00/34.13
% 266.39/34.51 % SZS output start Proof
% 266.39/34.51 Take the following subset of the input axioms:
% 266.39/34.51 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 266.39/34.51 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 266.39/34.51 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 266.39/34.51 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 266.39/34.51 fof(backward_box, axiom, ![X0, X1]: backward_box(X0, X1)=c(backward_diamond(X0, c(X1)))).
% 266.39/34.51 fof(backward_diamond, axiom, ![X0_2, X1_2]: backward_diamond(X0_2, X1_2)=codomain(multiplication(codomain(X1_2), X0_2))).
% 266.39/34.51 fof(codomain1, axiom, ![X0_2]: multiplication(X0_2, coantidomain(X0_2))=zero).
% 266.39/34.51 fof(codomain2, axiom, ![X0_2, X1_2]: addition(coantidomain(multiplication(X0_2, X1_2)), coantidomain(multiplication(coantidomain(coantidomain(X0_2)), X1_2)))=coantidomain(multiplication(coantidomain(coantidomain(X0_2)), X1_2))).
% 266.39/34.51 fof(codomain3, axiom, ![X0_2]: addition(coantidomain(coantidomain(X0_2)), coantidomain(X0_2))=one).
% 266.39/34.51 fof(codomain4, axiom, ![X0_2]: codomain(X0_2)=coantidomain(coantidomain(X0_2))).
% 266.39/34.51 fof(complement, axiom, ![X0_2]: c(X0_2)=antidomain(domain(X0_2))).
% 266.39/34.51 fof(domain1, axiom, ![X0_2]: multiplication(antidomain(X0_2), X0_2)=zero).
% 266.39/34.51 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))))).
% 266.39/34.51 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 266.39/34.51 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 266.39/34.51 fof(domain_difference, axiom, ![X0_2, X1_2]: domain_difference(X0_2, X1_2)=multiplication(domain(X0_2), antidomain(X1_2))).
% 266.39/34.51 fof(forward_box, axiom, ![X0_2, X1_2]: forward_box(X0_2, X1_2)=c(forward_diamond(X0_2, c(X1_2)))).
% 266.39/34.51 fof(forward_diamond, axiom, ![X0_2, X1_2]: forward_diamond(X0_2, X1_2)=domain(multiplication(X0_2, domain(X1_2)))).
% 266.39/34.51 fof(goals, conjecture, ![X0_2, X1_2]: addition(forward_diamond(X0_2, backward_box(X0_2, domain(X1_2))), domain(X1_2))=domain(X1_2)).
% 266.39/34.51 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 266.39/34.51 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 266.39/34.51 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 266.39/34.51 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 266.39/34.51 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 266.39/34.51 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 266.39/34.51 fof(right_annihilation, axiom, ![A3]: multiplication(A3, zero)=zero).
% 266.39/34.51 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 266.39/34.51
% 266.39/34.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 266.39/34.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 266.39/34.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 266.39/34.51 fresh(y, y, x1...xn) = u
% 266.39/34.51 C => fresh(s, t, x1...xn) = v
% 266.39/34.51 where fresh is a fresh function symbol and x1..xn are the free
% 266.39/34.51 variables of u and v.
% 266.39/34.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 266.39/34.51 input problem has no model of domain size 1).
% 266.39/34.51
% 266.39/34.51 The encoding turns the above axioms into the following unit equations and goals:
% 266.39/34.51
% 266.39/34.51 Axiom 1 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 266.39/34.51 Axiom 2 (complement): c(X) = antidomain(domain(X)).
% 266.39/34.51 Axiom 3 (domain4): domain(X) = antidomain(antidomain(X)).
% 266.39/34.51 Axiom 4 (additive_idempotence): addition(X, X) = X.
% 266.39/34.51 Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 266.39/34.51 Axiom 6 (additive_identity): addition(X, zero) = X.
% 266.39/34.51 Axiom 7 (multiplicative_right_identity): multiplication(X, one) = X.
% 266.39/34.51 Axiom 8 (right_annihilation): multiplication(X, zero) = zero.
% 266.39/34.51 Axiom 9 (multiplicative_left_identity): multiplication(one, X) = X.
% 266.39/34.51 Axiom 10 (left_annihilation): multiplication(zero, X) = zero.
% 266.39/34.51 Axiom 11 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 266.39/34.51 Axiom 12 (domain1): multiplication(antidomain(X), X) = zero.
% 266.39/34.51 Axiom 13 (order): fresh(X, X, Y, Z) = true.
% 266.39/34.51 Axiom 14 (order_1): fresh2(X, X, Y, Z) = Z.
% 266.39/34.51 Axiom 15 (backward_diamond): backward_diamond(X, Y) = codomain(multiplication(codomain(Y), X)).
% 266.39/34.51 Axiom 16 (backward_box): backward_box(X, Y) = c(backward_diamond(X, c(Y))).
% 266.39/34.51 Axiom 17 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
% 266.39/34.51 Axiom 18 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 266.39/34.51 Axiom 19 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 266.39/34.51 Axiom 20 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 266.39/34.51 Axiom 21 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 266.39/34.51 Axiom 22 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 266.39/34.51 Axiom 23 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 266.39/34.51 Axiom 24 (order): fresh(addition(X, Y), Y, X, Y) = leq(X, Y).
% 266.39/34.51 Axiom 25 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 266.39/34.51 Axiom 26 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 266.39/34.51 Axiom 27 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 266.39/34.51 Axiom 28 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
% 266.99/34.51 Axiom 29 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 266.99/34.51
% 266.99/34.51 Lemma 30: addition(domain(X), antidomain(X)) = one.
% 266.99/34.51 Proof:
% 266.99/34.51 addition(domain(X), antidomain(X))
% 266.99/34.51 = { by axiom 3 (domain4) }
% 266.99/34.51 addition(antidomain(antidomain(X)), antidomain(X))
% 266.99/34.51 = { by axiom 23 (domain3) }
% 266.99/34.51 one
% 266.99/34.51
% 266.99/34.51 Lemma 31: multiplication(domain(X), X) = X.
% 266.99/34.51 Proof:
% 266.99/34.51 multiplication(domain(X), X)
% 266.99/34.51 = { by axiom 6 (additive_identity) R->L }
% 266.99/34.51 addition(multiplication(domain(X), X), zero)
% 266.99/34.51 = { by axiom 12 (domain1) R->L }
% 266.99/34.51 addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
% 266.99/34.51 = { by axiom 27 (left_distributivity) R->L }
% 266.99/34.51 multiplication(addition(domain(X), antidomain(X)), X)
% 266.99/34.51 = { by lemma 30 }
% 266.99/34.51 multiplication(one, X)
% 266.99/34.51 = { by axiom 9 (multiplicative_left_identity) }
% 266.99/34.51 X
% 266.99/34.51
% 266.99/34.51 Lemma 32: domain(antidomain(X)) = c(X).
% 266.99/34.51 Proof:
% 266.99/34.51 domain(antidomain(X))
% 266.99/34.51 = { by axiom 3 (domain4) }
% 266.99/34.51 antidomain(antidomain(antidomain(X)))
% 266.99/34.51 = { by axiom 3 (domain4) R->L }
% 266.99/34.51 antidomain(domain(X))
% 266.99/34.51 = { by axiom 2 (complement) R->L }
% 266.99/34.51 c(X)
% 266.99/34.51
% 266.99/34.51 Lemma 33: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 266.99/34.51 Proof:
% 266.99/34.51 multiplication(antidomain(X), addition(X, Y))
% 266.99/34.51 = { by axiom 5 (additive_commutativity) R->L }
% 266.99/34.51 multiplication(antidomain(X), addition(Y, X))
% 266.99/34.51 = { by axiom 26 (right_distributivity) }
% 266.99/34.51 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 266.99/34.51 = { by axiom 12 (domain1) }
% 266.99/34.51 addition(multiplication(antidomain(X), Y), zero)
% 266.99/34.51 = { by axiom 6 (additive_identity) }
% 266.99/34.51 multiplication(antidomain(X), Y)
% 266.99/34.51
% 266.99/34.51 Lemma 34: antidomain(X) = c(X).
% 266.99/34.51 Proof:
% 266.99/34.51 antidomain(X)
% 266.99/34.51 = { by lemma 31 R->L }
% 266.99/34.51 multiplication(domain(antidomain(X)), antidomain(X))
% 266.99/34.51 = { by lemma 32 }
% 266.99/34.51 multiplication(c(X), antidomain(X))
% 266.99/34.51 = { by axiom 2 (complement) }
% 266.99/34.51 multiplication(antidomain(domain(X)), antidomain(X))
% 266.99/34.51 = { by lemma 33 R->L }
% 266.99/34.51 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 266.99/34.51 = { by lemma 30 }
% 266.99/34.51 multiplication(antidomain(domain(X)), one)
% 266.99/34.51 = { by axiom 7 (multiplicative_right_identity) }
% 266.99/34.51 antidomain(domain(X))
% 266.99/34.51 = { by axiom 2 (complement) R->L }
% 266.99/34.51 c(X)
% 266.99/34.51
% 266.99/34.51 Lemma 35: antidomain(one) = zero.
% 266.99/34.51 Proof:
% 266.99/34.51 antidomain(one)
% 266.99/34.51 = { by axiom 7 (multiplicative_right_identity) R->L }
% 266.99/34.51 multiplication(antidomain(one), one)
% 266.99/34.51 = { by axiom 12 (domain1) }
% 266.99/34.51 zero
% 266.99/34.51
% 266.99/34.51 Lemma 36: antidomain(zero) = one.
% 266.99/34.51 Proof:
% 266.99/34.51 antidomain(zero)
% 266.99/34.51 = { by lemma 35 R->L }
% 266.99/34.51 antidomain(antidomain(one))
% 266.99/34.51 = { by axiom 3 (domain4) R->L }
% 266.99/34.51 domain(one)
% 266.99/34.51 = { by axiom 6 (additive_identity) R->L }
% 266.99/34.51 addition(domain(one), zero)
% 266.99/34.51 = { by lemma 35 R->L }
% 266.99/34.51 addition(domain(one), antidomain(one))
% 266.99/34.51 = { by lemma 30 }
% 266.99/34.51 one
% 266.99/34.51
% 266.99/34.51 Lemma 37: coantidomain(one) = zero.
% 266.99/34.51 Proof:
% 266.99/34.51 coantidomain(one)
% 266.99/34.51 = { by axiom 9 (multiplicative_left_identity) R->L }
% 266.99/34.51 multiplication(one, coantidomain(one))
% 266.99/34.51 = { by axiom 11 (codomain1) }
% 266.99/34.51 zero
% 266.99/34.51
% 266.99/34.51 Lemma 38: addition(zero, X) = X.
% 266.99/34.51 Proof:
% 266.99/34.51 addition(zero, X)
% 266.99/34.51 = { by axiom 5 (additive_commutativity) R->L }
% 266.99/34.51 addition(X, zero)
% 266.99/34.51 = { by axiom 6 (additive_identity) }
% 266.99/34.51 X
% 266.99/34.51
% 266.99/34.51 Lemma 39: antidomain(c(X)) = domain(domain(X)).
% 266.99/34.51 Proof:
% 266.99/34.51 antidomain(c(X))
% 266.99/34.51 = { by axiom 2 (complement) }
% 266.99/34.51 antidomain(antidomain(domain(X)))
% 266.99/34.51 = { by axiom 3 (domain4) R->L }
% 266.99/34.51 domain(domain(X))
% 266.99/34.51
% 266.99/34.51 Lemma 40: antidomain(c(X)) = c(antidomain(X)).
% 266.99/34.51 Proof:
% 266.99/34.51 antidomain(c(X))
% 266.99/34.51 = { by lemma 32 R->L }
% 266.99/34.51 antidomain(domain(antidomain(X)))
% 266.99/34.51 = { by axiom 2 (complement) R->L }
% 266.99/34.51 c(antidomain(X))
% 266.99/34.51
% 266.99/34.51 Lemma 41: domain(domain(X)) = c(antidomain(X)).
% 266.99/34.51 Proof:
% 266.99/34.51 domain(domain(X))
% 266.99/34.51 = { by axiom 3 (domain4) }
% 266.99/34.51 domain(antidomain(antidomain(X)))
% 266.99/34.51 = { by lemma 32 }
% 266.99/34.51 c(antidomain(X))
% 266.99/34.51
% 266.99/34.51 Lemma 42: coantidomain(codomain(X)) = codomain(coantidomain(X)).
% 266.99/34.51 Proof:
% 266.99/34.51 coantidomain(codomain(X))
% 266.99/34.51 = { by axiom 1 (codomain4) }
% 266.99/34.51 coantidomain(coantidomain(coantidomain(X)))
% 266.99/34.51 = { by axiom 1 (codomain4) R->L }
% 266.99/34.51 codomain(coantidomain(X))
% 266.99/34.51
% 266.99/34.51 Lemma 43: addition(codomain(X), coantidomain(X)) = one.
% 266.99/34.51 Proof:
% 266.99/34.51 addition(codomain(X), coantidomain(X))
% 266.99/34.51 = { by axiom 1 (codomain4) }
% 266.99/34.51 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 266.99/34.51 = { by axiom 22 (codomain3) }
% 266.99/34.51 one
% 266.99/34.51
% 266.99/34.51 Lemma 44: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
% 266.99/34.51 Proof:
% 266.99/34.51 multiplication(addition(X, Y), coantidomain(X))
% 266.99/34.51 = { by axiom 5 (additive_commutativity) R->L }
% 266.99/34.51 multiplication(addition(Y, X), coantidomain(X))
% 266.99/34.51 = { by axiom 27 (left_distributivity) }
% 266.99/34.51 addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
% 266.99/34.51 = { by axiom 11 (codomain1) }
% 266.99/34.51 addition(multiplication(Y, coantidomain(X)), zero)
% 266.99/34.51 = { by axiom 6 (additive_identity) }
% 266.99/34.51 multiplication(Y, coantidomain(X))
% 266.99/34.51
% 266.99/34.51 Lemma 45: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y).
% 266.99/34.51 Proof:
% 266.99/34.51 multiplication(X, addition(Y, coantidomain(X)))
% 266.99/34.51 = { by axiom 26 (right_distributivity) }
% 266.99/34.51 addition(multiplication(X, Y), multiplication(X, coantidomain(X)))
% 266.99/34.51 = { by axiom 11 (codomain1) }
% 266.99/34.51 addition(multiplication(X, Y), zero)
% 266.99/34.51 = { by axiom 6 (additive_identity) }
% 266.99/34.51 multiplication(X, Y)
% 266.99/34.51
% 266.99/34.51 Lemma 46: multiplication(X, codomain(X)) = X.
% 266.99/34.51 Proof:
% 266.99/34.51 multiplication(X, codomain(X))
% 266.99/34.51 = { by lemma 45 R->L }
% 266.99/34.51 multiplication(X, addition(codomain(X), coantidomain(X)))
% 266.99/34.51 = { by lemma 43 }
% 266.99/34.51 multiplication(X, one)
% 266.99/34.51 = { by axiom 7 (multiplicative_right_identity) }
% 266.99/34.51 X
% 266.99/34.51
% 266.99/34.51 Lemma 47: codomain(coantidomain(X)) = coantidomain(X).
% 266.99/34.51 Proof:
% 266.99/34.51 codomain(coantidomain(X))
% 266.99/34.51 = { by lemma 42 R->L }
% 266.99/34.51 coantidomain(codomain(X))
% 266.99/34.51 = { by axiom 9 (multiplicative_left_identity) R->L }
% 266.99/34.51 multiplication(one, coantidomain(codomain(X)))
% 266.99/34.51 = { by lemma 43 R->L }
% 266.99/34.51 multiplication(addition(codomain(X), coantidomain(X)), coantidomain(codomain(X)))
% 266.99/34.51 = { by lemma 44 }
% 266.99/34.51 multiplication(coantidomain(X), coantidomain(codomain(X)))
% 266.99/34.51 = { by lemma 42 }
% 266.99/34.51 multiplication(coantidomain(X), codomain(coantidomain(X)))
% 266.99/34.51 = { by lemma 46 }
% 266.99/34.51 coantidomain(X)
% 266.99/34.51
% 266.99/34.51 Lemma 48: coantidomain(codomain(X)) = coantidomain(X).
% 266.99/34.51 Proof:
% 266.99/34.51 coantidomain(codomain(X))
% 266.99/34.51 = { by axiom 1 (codomain4) }
% 266.99/34.51 coantidomain(coantidomain(coantidomain(X)))
% 266.99/34.51 = { by axiom 1 (codomain4) R->L }
% 266.99/34.51 codomain(coantidomain(X))
% 266.99/34.51 = { by lemma 47 }
% 266.99/34.51 coantidomain(X)
% 266.99/34.51
% 266.99/34.51 Lemma 49: codomain(codomain(X)) = codomain(X).
% 266.99/34.51 Proof:
% 266.99/34.51 codomain(codomain(X))
% 266.99/34.51 = { by axiom 1 (codomain4) }
% 266.99/34.51 codomain(coantidomain(coantidomain(X)))
% 266.99/34.51 = { by lemma 47 }
% 266.99/34.51 coantidomain(coantidomain(X))
% 266.99/34.51 = { by axiom 1 (codomain4) R->L }
% 266.99/34.51 codomain(X)
% 266.99/34.51
% 266.99/34.51 Lemma 50: c(domain(X)) = c(X).
% 266.99/34.51 Proof:
% 266.99/34.52 c(domain(X))
% 266.99/34.52 = { by lemma 34 R->L }
% 266.99/34.52 antidomain(domain(X))
% 266.99/34.52 = { by axiom 2 (complement) R->L }
% 266.99/34.52 c(X)
% 266.99/34.52
% 266.99/34.52 Lemma 51: multiplication(X, addition(coantidomain(X), Y)) = multiplication(X, Y).
% 266.99/34.52 Proof:
% 266.99/34.52 multiplication(X, addition(coantidomain(X), Y))
% 266.99/34.52 = { by axiom 5 (additive_commutativity) R->L }
% 266.99/34.52 multiplication(X, addition(Y, coantidomain(X)))
% 266.99/34.52 = { by lemma 45 }
% 266.99/34.52 multiplication(X, Y)
% 266.99/34.52
% 266.99/34.52 Lemma 52: addition(X, addition(X, Y)) = addition(X, Y).
% 266.99/34.52 Proof:
% 266.99/34.52 addition(X, addition(X, Y))
% 266.99/34.52 = { by axiom 19 (additive_associativity) }
% 266.99/34.52 addition(addition(X, X), Y)
% 266.99/34.52 = { by axiom 4 (additive_idempotence) }
% 266.99/34.52 addition(X, Y)
% 266.99/34.52
% 266.99/34.52 Lemma 53: addition(one, domain(X)) = one.
% 266.99/34.52 Proof:
% 266.99/34.52 addition(one, domain(X))
% 266.99/34.52 = { by axiom 5 (additive_commutativity) R->L }
% 266.99/34.52 addition(domain(X), one)
% 266.99/34.52 = { by lemma 30 R->L }
% 266.99/34.52 addition(domain(X), addition(domain(X), antidomain(X)))
% 266.99/34.52 = { by lemma 52 }
% 266.99/34.52 addition(domain(X), antidomain(X))
% 266.99/34.52 = { by lemma 30 }
% 266.99/34.52 one
% 266.99/34.52
% 266.99/34.52 Lemma 54: multiplication(antidomain(X), multiplication(X, Y)) = zero.
% 266.99/34.52 Proof:
% 266.99/34.52 multiplication(antidomain(X), multiplication(X, Y))
% 266.99/34.52 = { by axiom 21 (multiplicative_associativity) }
% 266.99/34.52 multiplication(multiplication(antidomain(X), X), Y)
% 266.99/34.52 = { by axiom 12 (domain1) }
% 266.99/34.52 multiplication(zero, Y)
% 266.99/34.52 = { by axiom 10 (left_annihilation) }
% 266.99/34.52 zero
% 266.99/34.52
% 266.99/34.52 Lemma 55: multiplication(c(X), X) = zero.
% 266.99/34.52 Proof:
% 266.99/34.52 multiplication(c(X), X)
% 266.99/34.52 = { by axiom 2 (complement) }
% 266.99/34.52 multiplication(antidomain(domain(X)), X)
% 266.99/34.52 = { by lemma 31 R->L }
% 266.99/34.52 multiplication(antidomain(domain(X)), multiplication(domain(X), X))
% 266.99/34.52 = { by lemma 54 }
% 266.99/34.52 zero
% 266.99/34.52
% 266.99/34.52 Lemma 56: c(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)).
% 266.99/34.52 Proof:
% 266.99/34.52 c(multiplication(X, domain(Y)))
% 266.99/34.52 = { by axiom 2 (complement) }
% 266.99/34.52 antidomain(domain(multiplication(X, domain(Y))))
% 266.99/34.52 = { by axiom 18 (forward_diamond) R->L }
% 266.99/34.52 antidomain(forward_diamond(X, Y))
% 266.99/34.52
% 266.99/34.52 Lemma 57: c(multiplication(X, c(Y))) = forward_box(X, Y).
% 266.99/34.52 Proof:
% 266.99/34.52 c(multiplication(X, c(Y)))
% 266.99/34.52 = { by lemma 32 R->L }
% 266.99/34.52 c(multiplication(X, domain(antidomain(Y))))
% 266.99/34.52 = { by lemma 56 }
% 266.99/34.52 antidomain(forward_diamond(X, antidomain(Y)))
% 266.99/34.52 = { by lemma 34 }
% 266.99/34.52 c(forward_diamond(X, antidomain(Y)))
% 266.99/34.52 = { by lemma 34 }
% 266.99/34.52 c(forward_diamond(X, c(Y)))
% 266.99/34.52 = { by axiom 17 (forward_box) R->L }
% 266.99/34.52 forward_box(X, Y)
% 266.99/34.52
% 266.99/34.52 Lemma 58: addition(c(X), domain(X)) = one.
% 266.99/34.52 Proof:
% 266.99/34.52 addition(c(X), domain(X))
% 266.99/34.52 = { by axiom 3 (domain4) }
% 266.99/34.52 addition(c(X), antidomain(antidomain(X)))
% 266.99/34.52 = { by lemma 32 R->L }
% 266.99/34.52 addition(domain(antidomain(X)), antidomain(antidomain(X)))
% 266.99/34.52 = { by lemma 30 }
% 266.99/34.52 one
% 266.99/34.52
% 266.99/34.52 Lemma 59: c(c(X)) = domain(X).
% 266.99/34.52 Proof:
% 266.99/34.52 c(c(X))
% 266.99/34.52 = { by lemma 34 R->L }
% 266.99/34.52 c(antidomain(X))
% 266.99/34.52 = { by lemma 40 R->L }
% 266.99/34.52 antidomain(c(X))
% 266.99/34.52 = { by axiom 7 (multiplicative_right_identity) R->L }
% 266.99/34.52 multiplication(antidomain(c(X)), one)
% 266.99/34.52 = { by lemma 58 R->L }
% 266.99/34.52 multiplication(antidomain(c(X)), addition(c(X), domain(X)))
% 266.99/34.52 = { by lemma 33 }
% 266.99/34.52 multiplication(antidomain(c(X)), domain(X))
% 266.99/34.52 = { by lemma 40 }
% 266.99/34.52 multiplication(c(antidomain(X)), domain(X))
% 266.99/34.52 = { by lemma 32 R->L }
% 266.99/34.52 multiplication(domain(antidomain(antidomain(X))), domain(X))
% 266.99/34.52 = { by axiom 3 (domain4) }
% 266.99/34.52 multiplication(domain(antidomain(antidomain(X))), antidomain(antidomain(X)))
% 266.99/34.52 = { by lemma 31 }
% 266.99/34.52 antidomain(antidomain(X))
% 266.99/34.52 = { by axiom 3 (domain4) R->L }
% 266.99/34.52 domain(X)
% 266.99/34.52
% 266.99/34.52 Lemma 60: c(forward_diamond(X, Y)) = forward_box(X, c(Y)).
% 266.99/34.52 Proof:
% 266.99/34.52 c(forward_diamond(X, Y))
% 266.99/34.52 = { by lemma 34 R->L }
% 266.99/34.52 antidomain(forward_diamond(X, Y))
% 266.99/34.52 = { by lemma 56 R->L }
% 266.99/34.52 c(multiplication(X, domain(Y)))
% 266.99/34.52 = { by lemma 59 R->L }
% 266.99/34.52 c(multiplication(X, c(c(Y))))
% 266.99/34.52 = { by lemma 57 }
% 266.99/34.52 forward_box(X, c(Y))
% 266.99/34.52
% 266.99/34.52 Lemma 61: multiplication(X, c(coantidomain(X))) = X.
% 266.99/34.52 Proof:
% 266.99/34.52 multiplication(X, c(coantidomain(X)))
% 266.99/34.52 = { by axiom 6 (additive_identity) R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), zero))
% 266.99/34.52 = { by lemma 55 R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(c(multiplication(codomain(X), c(c(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by lemma 57 }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(forward_box(codomain(X), c(coantidomain(X))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by lemma 60 R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(c(forward_diamond(codomain(X), coantidomain(X))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by lemma 34 R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(antidomain(forward_diamond(codomain(X), coantidomain(X))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by lemma 56 R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(c(multiplication(codomain(X), domain(coantidomain(X)))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by lemma 34 R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 3 (domain4) }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(antidomain(multiplication(codomain(X), antidomain(antidomain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 29 (domain2) R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(multiplication(codomain(X), coantidomain(X))), antidomain(multiplication(codomain(X), antidomain(antidomain(coantidomain(X)))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 3 (domain4) R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(multiplication(codomain(X), coantidomain(X))), antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 1 (codomain4) }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(X)), coantidomain(X))), antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by lemma 47 R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(X)), codomain(coantidomain(X)))), antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 1 (codomain4) }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(multiplication(coantidomain(coantidomain(X)), coantidomain(coantidomain(coantidomain(X))))), antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 11 (codomain1) }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(zero), antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by lemma 36 }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(addition(one, antidomain(multiplication(codomain(X), domain(coantidomain(X))))), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 5 (additive_commutativity) R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(addition(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), one), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 25 (order_1) R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(fresh2(leq(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), one), true, antidomain(multiplication(codomain(X), domain(coantidomain(X)))), one), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by lemma 30 R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(fresh2(leq(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), addition(domain(multiplication(codomain(X), domain(coantidomain(X)))), antidomain(multiplication(codomain(X), domain(coantidomain(X)))))), true, antidomain(multiplication(codomain(X), domain(coantidomain(X)))), one), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 5 (additive_commutativity) R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(fresh2(leq(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), addition(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), domain(multiplication(codomain(X), domain(coantidomain(X)))))), true, antidomain(multiplication(codomain(X), domain(coantidomain(X)))), one), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 24 (order) R->L }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(fresh2(fresh(addition(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), addition(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), domain(multiplication(codomain(X), domain(coantidomain(X)))))), addition(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), domain(multiplication(codomain(X), domain(coantidomain(X))))), antidomain(multiplication(codomain(X), domain(coantidomain(X)))), addition(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), domain(multiplication(codomain(X), domain(coantidomain(X)))))), true, antidomain(multiplication(codomain(X), domain(coantidomain(X)))), one), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by lemma 52 }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(fresh2(fresh(addition(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), domain(multiplication(codomain(X), domain(coantidomain(X))))), addition(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), domain(multiplication(codomain(X), domain(coantidomain(X))))), antidomain(multiplication(codomain(X), domain(coantidomain(X)))), addition(antidomain(multiplication(codomain(X), domain(coantidomain(X)))), domain(multiplication(codomain(X), domain(coantidomain(X)))))), true, antidomain(multiplication(codomain(X), domain(coantidomain(X)))), one), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 13 (order) }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(fresh2(true, true, antidomain(multiplication(codomain(X), domain(coantidomain(X)))), one), multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 14 (order_1) }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(one, multiplication(codomain(X), c(c(coantidomain(X)))))))
% 266.99/34.52 = { by axiom 9 (multiplicative_left_identity) }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(codomain(X), c(c(coantidomain(X))))))
% 266.99/34.52 = { by lemma 59 }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), multiplication(codomain(X), domain(coantidomain(X)))))
% 266.99/34.52 = { by axiom 5 (additive_commutativity) R->L }
% 266.99/34.52 multiplication(X, addition(multiplication(codomain(X), domain(coantidomain(X))), c(coantidomain(X))))
% 266.99/34.52 = { by axiom 26 (right_distributivity) }
% 266.99/34.52 addition(multiplication(X, multiplication(codomain(X), domain(coantidomain(X)))), multiplication(X, c(coantidomain(X))))
% 266.99/34.52 = { by axiom 21 (multiplicative_associativity) }
% 266.99/34.52 addition(multiplication(multiplication(X, codomain(X)), domain(coantidomain(X))), multiplication(X, c(coantidomain(X))))
% 266.99/34.52 = { by lemma 46 }
% 266.99/34.52 addition(multiplication(X, domain(coantidomain(X))), multiplication(X, c(coantidomain(X))))
% 266.99/34.52 = { by axiom 26 (right_distributivity) R->L }
% 266.99/34.52 multiplication(X, addition(domain(coantidomain(X)), c(coantidomain(X))))
% 266.99/34.52 = { by axiom 5 (additive_commutativity) }
% 266.99/34.52 multiplication(X, addition(c(coantidomain(X)), domain(coantidomain(X))))
% 266.99/34.52 = { by lemma 58 }
% 266.99/34.52 multiplication(X, one)
% 266.99/34.52 = { by axiom 7 (multiplicative_right_identity) }
% 266.99/34.52 X
% 266.99/34.52
% 266.99/34.52 Lemma 62: c(codomain(X)) = coantidomain(X).
% 266.99/34.52 Proof:
% 266.99/34.52 c(codomain(X))
% 266.99/34.52 = { by lemma 49 R->L }
% 266.99/34.52 c(codomain(codomain(X)))
% 266.99/34.52 = { by axiom 9 (multiplicative_left_identity) R->L }
% 266.99/34.52 multiplication(one, c(codomain(codomain(X))))
% 266.99/34.52 = { by lemma 43 R->L }
% 266.99/34.52 multiplication(addition(codomain(X), coantidomain(X)), c(codomain(codomain(X))))
% 266.99/34.52 = { by axiom 27 (left_distributivity) }
% 266.99/34.52 addition(multiplication(codomain(X), c(codomain(codomain(X)))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by lemma 51 R->L }
% 266.99/34.52 addition(multiplication(codomain(X), addition(coantidomain(codomain(X)), c(codomain(codomain(X))))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by lemma 34 R->L }
% 266.99/34.52 addition(multiplication(codomain(X), addition(coantidomain(codomain(X)), antidomain(codomain(codomain(X))))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by axiom 7 (multiplicative_right_identity) R->L }
% 266.99/34.52 addition(multiplication(codomain(X), addition(coantidomain(codomain(X)), multiplication(antidomain(codomain(codomain(X))), one))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by lemma 43 R->L }
% 266.99/34.52 addition(multiplication(codomain(X), addition(coantidomain(codomain(X)), multiplication(antidomain(codomain(codomain(X))), addition(codomain(codomain(X)), coantidomain(codomain(X)))))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by lemma 33 }
% 266.99/34.52 addition(multiplication(codomain(X), addition(coantidomain(codomain(X)), multiplication(antidomain(codomain(codomain(X))), coantidomain(codomain(X))))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by lemma 34 }
% 266.99/34.52 addition(multiplication(codomain(X), addition(coantidomain(codomain(X)), multiplication(c(codomain(codomain(X))), coantidomain(codomain(X))))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by axiom 9 (multiplicative_left_identity) R->L }
% 266.99/34.52 addition(multiplication(codomain(X), addition(multiplication(one, coantidomain(codomain(X))), multiplication(c(codomain(codomain(X))), coantidomain(codomain(X))))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by axiom 27 (left_distributivity) R->L }
% 266.99/34.52 addition(multiplication(codomain(X), multiplication(addition(one, c(codomain(codomain(X)))), coantidomain(codomain(X)))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by lemma 32 R->L }
% 266.99/34.52 addition(multiplication(codomain(X), multiplication(addition(one, domain(antidomain(codomain(codomain(X))))), coantidomain(codomain(X)))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by lemma 53 }
% 266.99/34.52 addition(multiplication(codomain(X), multiplication(one, coantidomain(codomain(X)))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by axiom 9 (multiplicative_left_identity) }
% 266.99/34.52 addition(multiplication(codomain(X), coantidomain(codomain(X))), multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by axiom 11 (codomain1) }
% 266.99/34.52 addition(zero, multiplication(coantidomain(X), c(codomain(codomain(X)))))
% 266.99/34.52 = { by lemma 38 }
% 266.99/34.52 multiplication(coantidomain(X), c(codomain(codomain(X))))
% 266.99/34.52 = { by lemma 49 }
% 266.99/34.52 multiplication(coantidomain(X), c(codomain(X)))
% 266.99/34.52 = { by axiom 1 (codomain4) }
% 266.99/34.52 multiplication(coantidomain(X), c(coantidomain(coantidomain(X))))
% 266.99/34.52 = { by lemma 61 }
% 266.99/34.52 coantidomain(X)
% 266.99/34.52
% 266.99/34.52 Lemma 63: domain(forward_diamond(X, Y)) = forward_diamond(X, Y).
% 266.99/34.52 Proof:
% 266.99/34.52 domain(forward_diamond(X, Y))
% 266.99/34.52 = { by lemma 59 R->L }
% 266.99/34.52 c(c(forward_diamond(X, Y)))
% 266.99/34.52 = { by lemma 34 R->L }
% 266.99/34.52 c(antidomain(forward_diamond(X, Y)))
% 266.99/34.52 = { by lemma 56 R->L }
% 266.99/34.52 c(c(multiplication(X, domain(Y))))
% 266.99/34.52 = { by lemma 59 }
% 266.99/34.52 domain(multiplication(X, domain(Y)))
% 266.99/34.52 = { by axiom 18 (forward_diamond) R->L }
% 266.99/34.52 forward_diamond(X, Y)
% 266.99/34.52
% 266.99/34.52 Lemma 64: domain(backward_box(X, Y)) = backward_box(X, Y).
% 266.99/34.52 Proof:
% 266.99/34.52 domain(backward_box(X, Y))
% 266.99/34.52 = { by axiom 16 (backward_box) }
% 266.99/34.52 domain(c(backward_diamond(X, c(Y))))
% 266.99/34.52 = { by lemma 34 R->L }
% 266.99/34.52 domain(antidomain(backward_diamond(X, c(Y))))
% 266.99/34.52 = { by lemma 32 }
% 266.99/34.52 c(backward_diamond(X, c(Y)))
% 266.99/34.52 = { by axiom 16 (backward_box) R->L }
% 266.99/34.52 backward_box(X, Y)
% 266.99/34.52
% 266.99/34.52 Lemma 65: domain(multiplication(X, c(Y))) = forward_diamond(X, antidomain(Y)).
% 266.99/34.52 Proof:
% 266.99/34.52 domain(multiplication(X, c(Y)))
% 266.99/34.52 = { by lemma 32 R->L }
% 266.99/34.52 domain(multiplication(X, domain(antidomain(Y))))
% 266.99/34.52 = { by axiom 18 (forward_diamond) R->L }
% 266.99/34.52 forward_diamond(X, antidomain(Y))
% 266.99/34.52
% 266.99/34.52 Lemma 66: forward_diamond(X, c(Y)) = c(forward_box(X, Y)).
% 266.99/34.52 Proof:
% 266.99/34.52 forward_diamond(X, c(Y))
% 266.99/34.52 = { by lemma 34 R->L }
% 266.99/34.52 forward_diamond(X, antidomain(Y))
% 266.99/34.52 = { by lemma 65 R->L }
% 266.99/34.52 domain(multiplication(X, c(Y)))
% 266.99/34.52 = { by lemma 59 R->L }
% 266.99/34.52 c(c(multiplication(X, c(Y))))
% 266.99/34.52 = { by lemma 34 R->L }
% 266.99/34.52 c(antidomain(multiplication(X, c(Y))))
% 266.99/34.52 = { by lemma 41 R->L }
% 266.99/34.52 domain(domain(multiplication(X, c(Y))))
% 266.99/34.52 = { by lemma 39 R->L }
% 266.99/34.52 antidomain(c(multiplication(X, c(Y))))
% 266.99/34.52 = { by lemma 57 }
% 266.99/34.52 antidomain(forward_box(X, Y))
% 266.99/34.52 = { by lemma 34 }
% 266.99/34.52 c(forward_box(X, Y))
% 266.99/34.52
% 266.99/34.52 Lemma 67: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
% 266.99/34.52 Proof:
% 266.99/34.52 multiplication(X, addition(Y, one))
% 266.99/34.52 = { by axiom 5 (additive_commutativity) R->L }
% 266.99/34.52 multiplication(X, addition(one, Y))
% 266.99/34.52 = { by axiom 26 (right_distributivity) }
% 266.99/34.52 addition(multiplication(X, one), multiplication(X, Y))
% 266.99/34.52 = { by axiom 7 (multiplicative_right_identity) }
% 266.99/34.52 addition(X, multiplication(X, Y))
% 266.99/34.52
% 266.99/34.52 Lemma 68: coantidomain(multiplication(codomain(X), Y)) = coantidomain(backward_diamond(Y, X)).
% 266.99/34.52 Proof:
% 266.99/34.52 coantidomain(multiplication(codomain(X), Y))
% 266.99/34.52 = { by lemma 47 R->L }
% 266.99/34.52 codomain(coantidomain(multiplication(codomain(X), Y)))
% 266.99/34.52 = { by lemma 42 R->L }
% 266.99/34.52 coantidomain(codomain(multiplication(codomain(X), Y)))
% 266.99/34.52 = { by axiom 15 (backward_diamond) R->L }
% 266.99/34.52 coantidomain(backward_diamond(Y, X))
% 266.99/34.52
% 266.99/34.52 Lemma 69: multiplication(codomain(X), c(coantidomain(X))) = codomain(X).
% 266.99/34.52 Proof:
% 266.99/34.52 multiplication(codomain(X), c(coantidomain(X)))
% 266.99/34.52 = { by lemma 48 R->L }
% 266.99/34.52 multiplication(codomain(X), c(coantidomain(codomain(X))))
% 266.99/34.52 = { by lemma 61 }
% 266.99/34.53 codomain(X)
% 266.99/34.53
% 266.99/34.53 Goal 1 (goals): addition(forward_diamond(x0, backward_box(x0, domain(x1))), domain(x1)) = domain(x1).
% 266.99/34.53 Proof:
% 266.99/34.53 addition(forward_diamond(x0, backward_box(x0, domain(x1))), domain(x1))
% 266.99/34.53 = { by axiom 5 (additive_commutativity) }
% 266.99/34.53 addition(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))
% 266.99/34.53 = { by axiom 9 (multiplicative_left_identity) R->L }
% 266.99/34.53 addition(domain(x1), multiplication(one, forward_diamond(x0, backward_box(x0, domain(x1)))))
% 266.99/34.53 = { by lemma 58 R->L }
% 266.99/34.53 addition(domain(x1), multiplication(addition(c(x1), domain(x1)), forward_diamond(x0, backward_box(x0, domain(x1)))))
% 266.99/34.53 = { by lemma 50 R->L }
% 266.99/34.53 addition(domain(x1), multiplication(addition(c(domain(x1)), domain(x1)), forward_diamond(x0, backward_box(x0, domain(x1)))))
% 266.99/34.53 = { by axiom 27 (left_distributivity) }
% 266.99/34.53 addition(domain(x1), addition(multiplication(c(domain(x1)), forward_diamond(x0, backward_box(x0, domain(x1)))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.53 = { by lemma 63 R->L }
% 266.99/34.53 addition(domain(x1), addition(multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.53 = { by lemma 31 R->L }
% 266.99/34.53 addition(domain(x1), addition(multiplication(domain(multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.53 = { by lemma 63 }
% 266.99/34.53 addition(domain(x1), addition(multiplication(domain(multiplication(c(domain(x1)), forward_diamond(x0, backward_box(x0, domain(x1))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.53 = { by axiom 18 (forward_diamond) }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain(multiplication(c(domain(x1)), domain(multiplication(x0, domain(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by axiom 18 (forward_diamond) R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(forward_diamond(c(domain(x1)), multiplication(x0, domain(backward_box(x0, domain(x1))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 64 }
% 266.99/34.54 addition(domain(x1), addition(multiplication(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 63 R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by axiom 7 (multiplicative_right_identity) R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(multiplication(domain(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), one), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 36 R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(multiplication(domain(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), antidomain(zero)), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by axiom 20 (domain_difference) R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), zero), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by axiom 11 (codomain1) R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(multiplication(codomain(c(domain(x1))), x0), coantidomain(multiplication(codomain(c(domain(x1))), x0)))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 68 }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(multiplication(codomain(c(domain(x1))), x0), coantidomain(backward_diamond(x0, c(domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by axiom 21 (multiplicative_associativity) R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, coantidomain(backward_diamond(x0, c(domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 68 R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, coantidomain(multiplication(codomain(c(domain(x1))), x0))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 62 R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(codomain(multiplication(codomain(c(domain(x1))), x0)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 69 R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(codomain(multiplication(codomain(c(domain(x1))), x0)), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 34 R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(codomain(multiplication(codomain(c(domain(x1))), x0)), antidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 38 R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(addition(zero, multiplication(codomain(multiplication(codomain(c(domain(x1))), x0)), antidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0))))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by axiom 12 (domain1) R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(addition(multiplication(antidomain(antidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), antidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), multiplication(codomain(multiplication(codomain(c(domain(x1))), x0)), antidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0))))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by axiom 3 (domain4) R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(addition(multiplication(domain(coantidomain(multiplication(codomain(c(domain(x1))), x0))), antidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), multiplication(codomain(multiplication(codomain(c(domain(x1))), x0)), antidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0))))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by axiom 27 (left_distributivity) R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(domain(coantidomain(multiplication(codomain(c(domain(x1))), x0))), codomain(multiplication(codomain(c(domain(x1))), x0))), antidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 34 }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(domain(coantidomain(multiplication(codomain(c(domain(x1))), x0))), codomain(multiplication(codomain(c(domain(x1))), x0))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by axiom 5 (additive_commutativity) }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), domain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 47 R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), domain(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0))))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by axiom 7 (multiplicative_right_identity) R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), multiplication(domain(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), one)), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 43 R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), multiplication(domain(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), addition(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0))), coantidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 52 R->L }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), multiplication(domain(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), addition(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0))), addition(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0))), coantidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0))))))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.54 = { by lemma 43 }
% 266.99/34.54 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), multiplication(domain(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), addition(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0))), one))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 67 }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), addition(domain(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), multiplication(domain(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 31 }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), addition(domain(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0))))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by axiom 5 (additive_commutativity) }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), addition(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0))), domain(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 47 }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), addition(coantidomain(multiplication(codomain(c(domain(x1))), x0)), domain(codomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 47 }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), addition(coantidomain(multiplication(codomain(c(domain(x1))), x0)), domain(coantidomain(multiplication(codomain(c(domain(x1))), x0))))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by axiom 19 (additive_associativity) }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(addition(codomain(multiplication(codomain(c(domain(x1))), x0)), coantidomain(multiplication(codomain(c(domain(x1))), x0))), domain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 43 }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(addition(one, domain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 53 }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(multiplication(one, c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by axiom 9 (multiplicative_left_identity) }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(c(coantidomain(multiplication(codomain(c(domain(x1))), x0))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 62 R->L }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(c(c(codomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 69 R->L }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(c(c(multiplication(codomain(multiplication(codomain(c(domain(x1))), x0)), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 57 }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(c(forward_box(codomain(multiplication(codomain(c(domain(x1))), x0)), coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 66 R->L }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(forward_diamond(codomain(multiplication(codomain(c(domain(x1))), x0)), c(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 34 R->L }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(forward_diamond(codomain(multiplication(codomain(c(domain(x1))), x0)), antidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by axiom 18 (forward_diamond) }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(domain(multiplication(codomain(multiplication(codomain(c(domain(x1))), x0)), domain(antidomain(coantidomain(multiplication(codomain(c(domain(x1))), x0)))))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 32 }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(domain(multiplication(codomain(multiplication(codomain(c(domain(x1))), x0)), c(coantidomain(multiplication(codomain(c(domain(x1))), x0))))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 69 }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(domain(codomain(multiplication(codomain(c(domain(x1))), x0))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by axiom 15 (backward_diamond) R->L }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(domain(backward_diamond(x0, c(domain(x1)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 59 R->L }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(c(c(backward_diamond(x0, c(domain(x1))))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 34 R->L }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(c(antidomain(backward_diamond(x0, c(domain(x1))))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 41 R->L }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(domain(domain(backward_diamond(x0, c(domain(x1))))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by lemma 39 R->L }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(antidomain(c(backward_diamond(x0, c(domain(x1))))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.55 = { by axiom 16 (backward_box) R->L }
% 266.99/34.55 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(antidomain(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.56 = { by lemma 34 }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(domain(x1))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 50 }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(codomain(c(x1)), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 9 (multiplicative_left_identity) R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(multiplication(one, codomain(c(x1))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 58 R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(multiplication(addition(c(x1), domain(x1)), codomain(c(x1))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 27 (left_distributivity) }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(multiplication(c(x1), codomain(c(x1))), multiplication(domain(x1), codomain(c(x1)))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 46 }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(x1), multiplication(domain(x1), codomain(c(x1)))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 50 R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(x1), multiplication(domain(x1), codomain(c(domain(x1))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 50 R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(domain(x1), codomain(c(domain(x1))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 1 (codomain4) }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(domain(x1), coantidomain(coantidomain(c(domain(x1)))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 48 R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(domain(x1), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 44 R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(codomain(coantidomain(c(domain(x1)))), domain(x1)), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 9 (multiplicative_left_identity) R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(codomain(coantidomain(c(domain(x1)))), multiplication(one, domain(x1))), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 43 R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(codomain(coantidomain(c(domain(x1)))), multiplication(addition(codomain(coantidomain(c(domain(x1)))), coantidomain(coantidomain(c(domain(x1))))), domain(x1))), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 5 (additive_commutativity) R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(codomain(coantidomain(c(domain(x1)))), multiplication(addition(coantidomain(coantidomain(c(domain(x1)))), codomain(coantidomain(c(domain(x1))))), domain(x1))), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 5 (additive_commutativity) R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(multiplication(addition(coantidomain(coantidomain(c(domain(x1)))), codomain(coantidomain(c(domain(x1))))), domain(x1)), codomain(coantidomain(c(domain(x1))))), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 27 (left_distributivity) }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(addition(multiplication(coantidomain(coantidomain(c(domain(x1)))), domain(x1)), multiplication(codomain(coantidomain(c(domain(x1)))), domain(x1))), codomain(coantidomain(c(domain(x1))))), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 19 (additive_associativity) R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(multiplication(coantidomain(coantidomain(c(domain(x1)))), domain(x1)), addition(multiplication(codomain(coantidomain(c(domain(x1)))), domain(x1)), codomain(coantidomain(c(domain(x1)))))), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 5 (additive_commutativity) }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(multiplication(coantidomain(coantidomain(c(domain(x1)))), domain(x1)), addition(codomain(coantidomain(c(domain(x1)))), multiplication(codomain(coantidomain(c(domain(x1)))), domain(x1)))), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 67 R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(multiplication(coantidomain(coantidomain(c(domain(x1)))), domain(x1)), multiplication(codomain(coantidomain(c(domain(x1)))), addition(domain(x1), one))), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 30 R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(multiplication(coantidomain(coantidomain(c(domain(x1)))), domain(x1)), multiplication(codomain(coantidomain(c(domain(x1)))), addition(domain(x1), addition(domain(x1), antidomain(x1))))), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 52 }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(multiplication(coantidomain(coantidomain(c(domain(x1)))), domain(x1)), multiplication(codomain(coantidomain(c(domain(x1)))), addition(domain(x1), antidomain(x1)))), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 30 }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(multiplication(coantidomain(coantidomain(c(domain(x1)))), domain(x1)), multiplication(codomain(coantidomain(c(domain(x1)))), one)), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 7 (multiplicative_right_identity) }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(addition(multiplication(coantidomain(coantidomain(c(domain(x1)))), domain(x1)), codomain(coantidomain(c(domain(x1))))), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 27 (left_distributivity) }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), addition(multiplication(multiplication(coantidomain(coantidomain(c(domain(x1)))), domain(x1)), coantidomain(codomain(coantidomain(c(domain(x1)))))), multiplication(codomain(coantidomain(c(domain(x1)))), coantidomain(codomain(coantidomain(c(domain(x1)))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 21 (multiplicative_associativity) R->L }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), addition(multiplication(coantidomain(coantidomain(c(domain(x1)))), multiplication(domain(x1), coantidomain(codomain(coantidomain(c(domain(x1))))))), multiplication(codomain(coantidomain(c(domain(x1)))), coantidomain(codomain(coantidomain(c(domain(x1)))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 11 (codomain1) }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), addition(multiplication(coantidomain(coantidomain(c(domain(x1)))), multiplication(domain(x1), coantidomain(codomain(coantidomain(c(domain(x1))))))), zero)), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 6 (additive_identity) }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(coantidomain(coantidomain(c(domain(x1)))), multiplication(domain(x1), coantidomain(codomain(coantidomain(c(domain(x1)))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by lemma 48 }
% 266.99/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(coantidomain(coantidomain(c(domain(x1)))), multiplication(domain(x1), coantidomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 266.99/34.57 = { by axiom 1 (codomain4) R->L }
% 267.46/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(domain(x1), coantidomain(coantidomain(c(domain(x1))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.57 = { by axiom 1 (codomain4) R->L }
% 267.46/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(domain(x1), codomain(c(domain(x1)))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.57 = { by lemma 46 R->L }
% 267.46/34.57 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(multiplication(codomain(c(domain(x1))), multiplication(domain(x1), codomain(c(domain(x1))))), codomain(multiplication(codomain(c(domain(x1))), multiplication(domain(x1), codomain(c(domain(x1)))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 15 (backward_diamond) R->L }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(multiplication(codomain(c(domain(x1))), multiplication(domain(x1), codomain(c(domain(x1))))), backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 21 (multiplicative_associativity) R->L }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1)))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 7 (multiplicative_right_identity) R->L }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), one)))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 43 R->L }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), addition(codomain(one), coantidomain(one)))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 37 }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), addition(codomain(one), zero))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 6 (additive_identity) }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), codomain(one))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 1 (codomain4) }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), coantidomain(coantidomain(one)))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 37 }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), coantidomain(zero))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 54 R->L }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), coantidomain(multiplication(antidomain(domain(x1)), multiplication(domain(x1), codomain(c(domain(x1)))))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 34 }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), coantidomain(multiplication(c(domain(x1)), multiplication(domain(x1), codomain(c(domain(x1)))))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 51 R->L }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), addition(coantidomain(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1)))), coantidomain(multiplication(c(domain(x1)), multiplication(domain(x1), codomain(c(domain(x1))))))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 5 (additive_commutativity) R->L }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), addition(coantidomain(multiplication(c(domain(x1)), multiplication(domain(x1), codomain(c(domain(x1)))))), coantidomain(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 68 R->L }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), addition(coantidomain(multiplication(c(domain(x1)), multiplication(domain(x1), codomain(c(domain(x1)))))), coantidomain(multiplication(codomain(c(domain(x1))), multiplication(domain(x1), codomain(c(domain(x1))))))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 1 (codomain4) }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), addition(coantidomain(multiplication(c(domain(x1)), multiplication(domain(x1), codomain(c(domain(x1)))))), coantidomain(multiplication(coantidomain(coantidomain(c(domain(x1)))), multiplication(domain(x1), codomain(c(domain(x1))))))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 28 (codomain2) }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), coantidomain(multiplication(coantidomain(coantidomain(c(domain(x1)))), multiplication(domain(x1), codomain(c(domain(x1)))))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 1 (codomain4) R->L }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), coantidomain(multiplication(codomain(c(domain(x1))), multiplication(domain(x1), codomain(c(domain(x1)))))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 68 }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), multiplication(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1))), coantidomain(backward_diamond(multiplication(domain(x1), codomain(c(domain(x1)))), c(domain(x1)))))))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 11 (codomain1) }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(multiplication(domain(x1), codomain(c(domain(x1)))), zero))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 21 (multiplicative_associativity) R->L }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(domain(x1), multiplication(codomain(c(domain(x1))), zero)))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 8 (right_annihilation) }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), multiplication(domain(x1), zero))), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 8 (right_annihilation) }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), multiplication(codomain(c(domain(x1))), zero)), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 8 (right_annihilation) }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(addition(c(domain(x1)), zero), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 6 (additive_identity) }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(c(domain(x1)), multiplication(x0, c(c(backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 59 }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(c(domain(x1)), multiplication(x0, domain(backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 64 }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))), multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by axiom 18 (forward_diamond) }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(domain(multiplication(c(domain(x1)), domain(multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 59 R->L }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(domain(multiplication(c(domain(x1)), c(c(multiplication(x0, backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 65 }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), antidomain(c(multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 40 }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), c(antidomain(multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.58 = { by lemma 34 }
% 267.46/34.58 addition(domain(x1), addition(multiplication(domain_difference(forward_diamond(c(domain(x1)), c(c(multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 66 }
% 267.46/34.59 addition(domain(x1), addition(multiplication(domain_difference(c(forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 34 R->L }
% 267.46/34.59 addition(domain(x1), addition(multiplication(domain_difference(antidomain(forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by axiom 20 (domain_difference) }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(domain(antidomain(forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1))))))), antidomain(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 34 }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(domain(antidomain(forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 32 }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1)))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 60 R->L }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(c(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 34 R->L }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(antidomain(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 56 R->L }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(c(multiplication(c(domain(x1)), domain(multiplication(x0, backward_box(x0, domain(x1))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 34 R->L }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(antidomain(multiplication(c(domain(x1)), domain(multiplication(x0, backward_box(x0, domain(x1))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by axiom 3 (domain4) }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(antidomain(multiplication(c(domain(x1)), antidomain(antidomain(multiplication(x0, backward_box(x0, domain(x1)))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by axiom 29 (domain2) R->L }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(addition(antidomain(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), antidomain(multiplication(c(domain(x1)), antidomain(antidomain(multiplication(x0, backward_box(x0, domain(x1))))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by axiom 3 (domain4) R->L }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(addition(antidomain(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), antidomain(multiplication(c(domain(x1)), domain(multiplication(x0, backward_box(x0, domain(x1)))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 34 }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), antidomain(multiplication(c(domain(x1)), domain(multiplication(x0, backward_box(x0, domain(x1)))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 34 }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), c(multiplication(c(domain(x1)), domain(multiplication(x0, backward_box(x0, domain(x1)))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 56 }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), antidomain(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 34 }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), c(forward_diamond(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 60 }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 34 R->L }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(antidomain(addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1))))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1)))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 33 R->L }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(antidomain(addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1))))))), addition(addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1)))))), c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by axiom 5 (additive_commutativity) }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(antidomain(addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1))))))), addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 34 }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1))))))), addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1)))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 52 }
% 267.46/34.59 addition(domain(x1), addition(multiplication(multiplication(c(addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1))))))), addition(c(multiplication(c(domain(x1)), multiplication(x0, backward_box(x0, domain(x1))))), forward_box(c(domain(x1)), c(multiplication(x0, backward_box(x0, domain(x1))))))), multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 55 }
% 267.46/34.59 addition(domain(x1), addition(multiplication(zero, multiplication(c(domain(x1)), domain(forward_diamond(x0, backward_box(x0, domain(x1)))))), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by axiom 10 (left_annihilation) }
% 267.46/34.59 addition(domain(x1), addition(zero, multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1))))))
% 267.46/34.59 = { by lemma 38 }
% 267.46/34.59 addition(domain(x1), multiplication(domain(x1), forward_diamond(x0, backward_box(x0, domain(x1)))))
% 267.46/34.59 = { by lemma 67 R->L }
% 267.46/34.59 multiplication(domain(x1), addition(forward_diamond(x0, backward_box(x0, domain(x1))), one))
% 267.46/34.59 = { by axiom 5 (additive_commutativity) R->L }
% 267.46/34.59 multiplication(domain(x1), addition(one, forward_diamond(x0, backward_box(x0, domain(x1)))))
% 267.46/34.59 = { by axiom 18 (forward_diamond) }
% 267.46/34.59 multiplication(domain(x1), addition(one, domain(multiplication(x0, domain(backward_box(x0, domain(x1)))))))
% 267.46/34.59 = { by lemma 53 }
% 267.46/34.59 multiplication(domain(x1), one)
% 267.46/34.59 = { by axiom 7 (multiplicative_right_identity) }
% 267.46/34.59 domain(x1)
% 267.46/34.59 % SZS output end Proof
% 267.46/34.59
% 267.46/34.59 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------