0.07/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.13 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.13/0.34 % Computer : n010.cluster.edu 0.13/0.34 % Model : x86_64 x86_64 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.34 % Memory : 8042.1875MB 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 180 0.13/0.34 % DateTime : Thu Aug 29 13:11:36 EDT 2019 0.13/0.35 % CPUTime : 17.85/18.03 % SZS status Unsatisfiable 17.85/18.03 17.85/18.03 % SZS output start Proof 17.85/18.03 Take the following subset of the input axioms: 24.95/25.15 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)). 24.95/25.15 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)). 24.95/25.15 fof(additive_idempotence, axiom, ![A]: A=addition(A, A)). 24.95/25.15 fof(additive_identity, axiom, ![A]: addition(A, zero)=A). 24.95/25.15 fof(backward_box, axiom, ![X0, X1]: backward_box(X0, X1)=c(backward_diamond(X0, c(X1)))). 24.95/25.15 fof(backward_diamond, axiom, ![X0, X1]: codomain(multiplication(codomain(X1), X0))=backward_diamond(X0, X1)). 24.95/25.15 fof(codomain1, axiom, ![X0]: zero=multiplication(X0, coantidomain(X0))). 24.95/25.15 fof(codomain2, axiom, ![X0, X1]: coantidomain(multiplication(coantidomain(coantidomain(X0)), X1))=addition(coantidomain(multiplication(X0, X1)), coantidomain(multiplication(coantidomain(coantidomain(X0)), X1)))). 24.95/25.15 fof(codomain3, axiom, ![X0]: addition(coantidomain(coantidomain(X0)), coantidomain(X0))=one). 24.95/25.15 fof(codomain4, axiom, ![X0]: codomain(X0)=coantidomain(coantidomain(X0))). 24.95/25.15 fof(complement, axiom, ![X0]: antidomain(domain(X0))=c(X0)). 24.95/25.15 fof(domain1, axiom, ![X0]: zero=multiplication(antidomain(X0), X0)). 24.95/25.15 fof(domain2, axiom, ![X0, X1]: antidomain(multiplication(X0, antidomain(antidomain(X1))))=addition(antidomain(multiplication(X0, X1)), antidomain(multiplication(X0, antidomain(antidomain(X1)))))). 24.95/25.15 fof(domain3, axiom, ![X0]: one=addition(antidomain(antidomain(X0)), antidomain(X0))). 24.95/25.15 fof(domain4, axiom, ![X0]: domain(X0)=antidomain(antidomain(X0))). 24.95/25.15 fof(domain_difference, axiom, ![X0, X1]: multiplication(domain(X0), antidomain(X1))=domain_difference(X0, X1)). 24.95/25.15 fof(forward_box, axiom, ![X0, X1]: forward_box(X0, X1)=c(forward_diamond(X0, c(X1)))). 24.95/25.15 fof(forward_diamond, axiom, ![X0, X1]: domain(multiplication(X0, domain(X1)))=forward_diamond(X0, X1)). 24.95/25.15 fof(goals, negated_conjecture, addition(domain(sK2_goals_X1), backward_box(sK3_goals_X0, domain(sK1_goals_X2)))=one). 24.95/25.15 fof(goals_1, negated_conjecture, addition(forward_box(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2))!=one). 24.95/25.15 fof(ifeq_axiom, axiom, ![A, B, C]: ifeq2(A, A, B, C)=B). 24.95/25.15 fof(ifeq_axiom_001, axiom, ![A, B, C]: ifeq(A, A, B, C)=B). 24.95/25.15 fof(left_annihilation, axiom, ![A]: multiplication(zero, A)=zero). 24.95/25.15 fof(left_distributivity, axiom, ![A, B, C]: addition(multiplication(A, C), multiplication(B, C))=multiplication(addition(A, B), C)). 24.95/25.15 fof(multiplicative_associativity, axiom, ![A, B, C]: multiplication(A, multiplication(B, C))=multiplication(multiplication(A, B), C)). 24.95/25.15 fof(multiplicative_left_identity, axiom, ![A]: multiplication(one, A)=A). 24.95/25.15 fof(multiplicative_right_identity, axiom, ![A]: A=multiplication(A, one)). 24.95/25.15 fof(order, axiom, ![A, B]: true=ifeq2(addition(A, B), B, leq(A, B), true)). 24.95/25.15 fof(order_1, axiom, ![A, B]: B=ifeq(leq(A, B), true, addition(A, B), B)). 24.95/25.15 fof(right_annihilation, axiom, ![A]: zero=multiplication(A, zero)). 24.95/25.15 fof(right_distributivity, axiom, ![A, B, C]: multiplication(A, addition(B, C))=addition(multiplication(A, B), multiplication(A, C))). 24.95/25.15 24.95/25.15 Now clausify the problem and encode Horn clauses using encoding 3 of 24.95/25.15 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 24.95/25.15 We repeatedly replace C & s=t => u=v by the two clauses: 24.95/25.15 fresh(y, y, x1...xn) = u 24.95/25.15 C => fresh(s, t, x1...xn) = v 24.95/25.15 where fresh is a fresh function symbol and x1..xn are the free 24.95/25.15 variables of u and v. 24.95/25.15 A predicate p(X) is encoded as p(X)=true (this is sound, because the 24.95/25.15 input problem has no model of domain size 1). 24.95/25.15 24.95/25.15 The encoding turns the above axioms into the following unit equations and goals: 24.95/25.15 24.95/25.15 Axiom 1 (ifeq_axiom_001): ifeq(X, X, Y, Z) = Y. 24.95/25.15 Axiom 2 (codomain2): coantidomain(multiplication(coantidomain(coantidomain(X)), Y)) = addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))). 24.95/25.15 Axiom 3 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))). 24.95/25.15 Axiom 4 (order_1): X = ifeq(leq(Y, X), true, addition(Y, X), X). 24.95/25.15 Axiom 5 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one. 24.95/25.15 Axiom 6 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)). 24.95/25.15 Axiom 7 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y). 24.95/25.15 Axiom 8 (backward_diamond): codomain(multiplication(codomain(X), Y)) = backward_diamond(Y, X). 24.95/25.15 Axiom 9 (additive_identity): addition(X, zero) = X. 24.95/25.15 Axiom 10 (domain2): antidomain(multiplication(X, antidomain(antidomain(Y)))) = addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))). 24.95/25.15 Axiom 11 (complement): antidomain(domain(X)) = c(X). 24.95/25.15 Axiom 12 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y. 24.95/25.15 Axiom 13 (domain4): domain(X) = antidomain(antidomain(X)). 24.95/25.15 Axiom 14 (forward_diamond): domain(multiplication(X, domain(Y))) = forward_diamond(X, Y). 24.95/25.15 Axiom 15 (left_annihilation): multiplication(zero, X) = zero. 24.95/25.15 Axiom 16 (domain1): zero = multiplication(antidomain(X), X). 24.95/25.15 Axiom 17 (additive_commutativity): addition(X, Y) = addition(Y, X). 24.95/25.15 Axiom 18 (order): true = ifeq2(addition(X, Y), Y, leq(X, Y), true). 24.95/25.15 Axiom 19 (goals): addition(domain(sK2_goals_X1), backward_box(sK3_goals_X0, domain(sK1_goals_X2))) = one. 24.95/25.15 Axiom 20 (right_annihilation): zero = multiplication(X, zero). 24.95/25.15 Axiom 21 (multiplicative_left_identity): multiplication(one, X) = X. 24.95/25.15 Axiom 22 (additive_idempotence): X = addition(X, X). 24.95/25.15 Axiom 23 (multiplicative_right_identity): X = multiplication(X, one). 24.95/25.15 Axiom 24 (codomain4): codomain(X) = coantidomain(coantidomain(X)). 24.95/25.15 Axiom 25 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z). 24.95/25.15 Axiom 26 (domain3): one = addition(antidomain(antidomain(X)), antidomain(X)). 24.95/25.15 Axiom 27 (backward_box): backward_box(X, Y) = c(backward_diamond(X, c(Y))). 24.95/25.15 Axiom 28 (codomain1): zero = multiplication(X, coantidomain(X)). 24.95/25.15 Axiom 29 (domain_difference): multiplication(domain(X), antidomain(Y)) = domain_difference(X, Y). 25.12/25.32 Axiom 30 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z). 25.12/25.32 25.12/25.32 Lemma 31: addition(antidomain(X), domain(X)) = one. 25.12/25.32 Proof: 25.12/25.32 addition(antidomain(X), domain(X)) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 addition(domain(X), antidomain(X)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 addition(antidomain(antidomain(X)), antidomain(X)) 25.12/25.32 = { by axiom 26 (domain3) } 25.12/25.32 one 25.12/25.32 25.12/25.32 Lemma 32: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y). 25.12/25.32 Proof: 25.12/25.32 multiplication(antidomain(X), addition(X, Y)) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 multiplication(antidomain(X), addition(Y, X)) 25.12/25.32 = { by axiom 6 (right_distributivity) } 25.12/25.32 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X)) 25.12/25.32 = { by axiom 16 (domain1) } 25.12/25.32 addition(multiplication(antidomain(X), Y), zero) 25.12/25.32 = { by axiom 9 (additive_identity) } 25.12/25.32 multiplication(antidomain(X), Y) 25.12/25.32 25.12/25.32 Lemma 33: multiplication(antidomain(X), addition(Y, X)) = multiplication(antidomain(X), Y). 25.12/25.32 Proof: 25.12/25.32 multiplication(antidomain(X), addition(Y, X)) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 multiplication(antidomain(X), addition(X, Y)) 25.12/25.32 = { by lemma 32 } 25.12/25.32 multiplication(antidomain(X), Y) 25.12/25.32 25.12/25.32 Lemma 34: domain(antidomain(X)) = c(X). 25.12/25.32 Proof: 25.12/25.32 domain(antidomain(X)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 antidomain(antidomain(antidomain(X))) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 antidomain(domain(X)) 25.12/25.32 = { by axiom 11 (complement) } 25.12/25.32 c(X) 25.12/25.32 25.12/25.32 Lemma 35: multiplication(c(X), antidomain(Y)) = domain_difference(antidomain(X), Y). 25.12/25.32 Proof: 25.12/25.32 multiplication(c(X), antidomain(Y)) 25.12/25.32 = { by lemma 34 } 25.12/25.32 multiplication(domain(antidomain(X)), antidomain(Y)) 25.12/25.32 = { by axiom 29 (domain_difference) } 25.12/25.32 domain_difference(antidomain(X), Y) 25.12/25.32 25.12/25.32 Lemma 36: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y). 25.12/25.32 Proof: 25.12/25.32 multiplication(addition(X, antidomain(Y)), Y) 25.12/25.32 = { by axiom 7 (left_distributivity) } 25.12/25.32 addition(multiplication(X, Y), multiplication(antidomain(Y), Y)) 25.12/25.32 = { by axiom 16 (domain1) } 25.12/25.32 addition(multiplication(X, Y), zero) 25.12/25.32 = { by axiom 9 (additive_identity) } 25.12/25.32 multiplication(X, Y) 25.12/25.32 25.12/25.32 Lemma 37: multiplication(addition(antidomain(Y), X), Y) = multiplication(X, Y). 25.12/25.32 Proof: 25.12/25.32 multiplication(addition(antidomain(Y), X), Y) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 multiplication(addition(X, antidomain(Y)), Y) 25.12/25.32 = { by lemma 36 } 25.12/25.32 multiplication(X, Y) 25.12/25.32 25.12/25.32 Lemma 38: multiplication(domain(X), X) = X. 25.12/25.32 Proof: 25.12/25.32 multiplication(domain(X), X) 25.12/25.32 = { by lemma 37 } 25.12/25.32 multiplication(addition(antidomain(X), domain(X)), X) 25.12/25.32 = { by lemma 31 } 25.12/25.32 multiplication(one, X) 25.12/25.32 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.32 X 25.12/25.32 25.12/25.32 Lemma 39: c(X) = antidomain(X). 25.12/25.32 Proof: 25.12/25.32 c(X) 25.12/25.32 = { by axiom 11 (complement) } 25.12/25.32 antidomain(domain(X)) 25.12/25.32 = { by axiom 23 (multiplicative_right_identity) } 25.12/25.32 multiplication(antidomain(domain(X)), one) 25.12/25.32 = { by lemma 31 } 25.12/25.32 multiplication(antidomain(domain(X)), addition(antidomain(X), domain(X))) 25.12/25.32 = { by lemma 33 } 25.12/25.32 multiplication(antidomain(domain(X)), antidomain(X)) 25.12/25.32 = { by axiom 11 (complement) } 25.12/25.32 multiplication(c(X), antidomain(X)) 25.12/25.32 = { by lemma 35 } 25.12/25.32 domain_difference(antidomain(X), X) 25.12/25.32 = { by axiom 29 (domain_difference) } 25.12/25.32 multiplication(domain(antidomain(X)), antidomain(X)) 25.12/25.32 = { by lemma 38 } 25.12/25.32 antidomain(X) 25.12/25.32 25.12/25.32 Lemma 40: antidomain(forward_diamond(X, antidomain(Y))) = forward_box(X, Y). 25.12/25.32 Proof: 25.12/25.32 antidomain(forward_diamond(X, antidomain(Y))) 25.12/25.32 = { by lemma 39 } 25.12/25.32 antidomain(forward_diamond(X, c(Y))) 25.12/25.32 = { by lemma 39 } 25.12/25.32 c(forward_diamond(X, c(Y))) 25.12/25.32 = { by axiom 3 (forward_box) } 25.12/25.32 forward_box(X, Y) 25.12/25.32 25.12/25.32 Lemma 41: domain(domain(X)) = antidomain(c(X)). 25.12/25.32 Proof: 25.12/25.32 domain(domain(X)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 antidomain(antidomain(domain(X))) 25.12/25.32 = { by axiom 11 (complement) } 25.12/25.32 antidomain(c(X)) 25.12/25.32 25.12/25.32 Lemma 42: c(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)). 25.12/25.32 Proof: 25.12/25.32 c(multiplication(X, domain(Y))) 25.12/25.32 = { by axiom 11 (complement) } 25.12/25.32 antidomain(domain(multiplication(X, domain(Y)))) 25.12/25.32 = { by axiom 14 (forward_diamond) } 25.12/25.32 antidomain(forward_diamond(X, Y)) 25.12/25.32 25.12/25.32 Lemma 43: forward_diamond(X, domain(Y)) = forward_diamond(X, Y). 25.12/25.32 Proof: 25.12/25.32 forward_diamond(X, domain(Y)) 25.12/25.32 = { by axiom 14 (forward_diamond) } 25.12/25.32 domain(multiplication(X, domain(domain(Y)))) 25.12/25.32 = { by lemma 41 } 25.12/25.32 domain(multiplication(X, antidomain(c(Y)))) 25.12/25.32 = { by lemma 39 } 25.12/25.32 domain(multiplication(X, antidomain(antidomain(Y)))) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 domain(multiplication(X, domain(Y))) 25.12/25.32 = { by axiom 14 (forward_diamond) } 25.12/25.32 forward_diamond(X, Y) 25.12/25.32 25.12/25.32 Lemma 44: antidomain(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)). 25.12/25.32 Proof: 25.12/25.32 antidomain(multiplication(X, domain(Y))) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 antidomain(multiplication(X, antidomain(antidomain(Y)))) 25.12/25.32 = { by lemma 39 } 25.12/25.32 antidomain(multiplication(X, antidomain(c(Y)))) 25.12/25.32 = { by lemma 39 } 25.12/25.32 c(multiplication(X, antidomain(c(Y)))) 25.12/25.32 = { by lemma 41 } 25.12/25.32 c(multiplication(X, domain(domain(Y)))) 25.12/25.32 = { by lemma 42 } 25.12/25.32 antidomain(forward_diamond(X, domain(Y))) 25.12/25.32 = { by lemma 43 } 25.12/25.32 antidomain(forward_diamond(X, Y)) 25.12/25.32 25.12/25.32 Lemma 45: antidomain(multiplication(X, antidomain(Y))) = forward_box(X, Y). 25.12/25.32 Proof: 25.12/25.32 antidomain(multiplication(X, antidomain(Y))) 25.12/25.32 = { by lemma 39 } 25.12/25.32 antidomain(multiplication(X, c(Y))) 25.12/25.32 = { by lemma 34 } 25.12/25.32 antidomain(multiplication(X, domain(antidomain(Y)))) 25.12/25.32 = { by lemma 44 } 25.12/25.32 antidomain(forward_diamond(X, antidomain(Y))) 25.12/25.32 = { by lemma 40 } 25.12/25.32 forward_box(X, Y) 25.12/25.32 25.12/25.32 Lemma 46: addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))) = one. 25.12/25.32 Proof: 25.12/25.32 addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y))) 25.12/25.32 = { by axiom 14 (forward_diamond) } 25.12/25.32 addition(domain(multiplication(X, domain(Y))), antidomain(forward_diamond(X, Y))) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 addition(antidomain(antidomain(multiplication(X, domain(Y)))), antidomain(forward_diamond(X, Y))) 25.12/25.32 = { by lemma 42 } 25.12/25.32 addition(antidomain(antidomain(multiplication(X, domain(Y)))), c(multiplication(X, domain(Y)))) 25.12/25.32 = { by lemma 34 } 25.12/25.32 addition(antidomain(antidomain(multiplication(X, domain(Y)))), domain(antidomain(multiplication(X, domain(Y))))) 25.12/25.32 = { by lemma 31 } 25.12/25.32 one 25.12/25.32 25.12/25.32 Lemma 47: domain(forward_diamond(X, Y)) = forward_diamond(X, Y). 25.12/25.32 Proof: 25.12/25.32 domain(forward_diamond(X, Y)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 antidomain(antidomain(forward_diamond(X, Y))) 25.12/25.32 = { by axiom 23 (multiplicative_right_identity) } 25.12/25.32 multiplication(antidomain(antidomain(forward_diamond(X, Y))), one) 25.12/25.32 = { by lemma 46 } 25.12/25.32 multiplication(antidomain(antidomain(forward_diamond(X, Y))), addition(forward_diamond(X, Y), antidomain(forward_diamond(X, Y)))) 25.12/25.32 = { by lemma 33 } 25.12/25.32 multiplication(antidomain(antidomain(forward_diamond(X, Y))), forward_diamond(X, Y)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 multiplication(domain(forward_diamond(X, Y)), forward_diamond(X, Y)) 25.12/25.32 = { by lemma 38 } 25.12/25.32 forward_diamond(X, Y) 25.12/25.32 25.12/25.32 Lemma 48: forward_diamond(X, antidomain(Y)) = antidomain(forward_box(X, Y)). 25.12/25.32 Proof: 25.12/25.32 forward_diamond(X, antidomain(Y)) 25.12/25.32 = { by lemma 47 } 25.12/25.32 domain(forward_diamond(X, antidomain(Y))) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 antidomain(antidomain(forward_diamond(X, antidomain(Y)))) 25.12/25.32 = { by lemma 40 } 25.12/25.32 antidomain(forward_box(X, Y)) 25.12/25.32 25.12/25.32 Lemma 49: domain(multiplication(X, c(Y))) = forward_diamond(X, antidomain(Y)). 25.12/25.32 Proof: 25.12/25.32 domain(multiplication(X, c(Y))) 25.12/25.32 = { by lemma 34 } 25.12/25.32 domain(multiplication(X, domain(antidomain(Y)))) 25.12/25.32 = { by axiom 14 (forward_diamond) } 25.12/25.32 forward_diamond(X, antidomain(Y)) 25.12/25.32 25.12/25.32 Lemma 50: domain_difference(domain(X), Y) = domain_difference(X, Y). 25.12/25.32 Proof: 25.12/25.32 domain_difference(domain(X), Y) 25.12/25.32 = { by axiom 29 (domain_difference) } 25.12/25.32 multiplication(domain(domain(X)), antidomain(Y)) 25.12/25.32 = { by lemma 41 } 25.12/25.32 multiplication(antidomain(c(X)), antidomain(Y)) 25.12/25.32 = { by lemma 39 } 25.12/25.32 multiplication(antidomain(antidomain(X)), antidomain(Y)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 multiplication(domain(X), antidomain(Y)) 25.12/25.32 = { by axiom 29 (domain_difference) } 25.12/25.32 domain_difference(X, Y) 25.12/25.32 25.12/25.32 Lemma 51: multiplication(domain(X), domain(Y)) = domain_difference(X, antidomain(Y)). 25.12/25.32 Proof: 25.12/25.32 multiplication(domain(X), domain(Y)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 multiplication(domain(X), antidomain(antidomain(Y))) 25.12/25.32 = { by axiom 29 (domain_difference) } 25.12/25.32 domain_difference(X, antidomain(Y)) 25.12/25.32 25.12/25.32 Lemma 52: domain_difference(antidomain(X), antidomain(Y)) = multiplication(antidomain(X), domain(Y)). 25.12/25.32 Proof: 25.12/25.32 domain_difference(antidomain(X), antidomain(Y)) 25.12/25.32 = { by lemma 51 } 25.12/25.32 multiplication(domain(antidomain(X)), domain(Y)) 25.12/25.32 = { by lemma 34 } 25.12/25.32 multiplication(c(X), domain(Y)) 25.12/25.32 = { by lemma 39 } 25.12/25.32 multiplication(antidomain(X), domain(Y)) 25.12/25.32 25.12/25.32 Lemma 53: multiplication(domain(X), multiplication(antidomain(Y), Z)) = multiplication(domain_difference(X, Y), Z). 25.12/25.32 Proof: 25.12/25.32 multiplication(domain(X), multiplication(antidomain(Y), Z)) 25.12/25.32 = { by axiom 25 (multiplicative_associativity) } 25.12/25.32 multiplication(multiplication(domain(X), antidomain(Y)), Z) 25.12/25.32 = { by axiom 29 (domain_difference) } 25.12/25.32 multiplication(domain_difference(X, Y), Z) 25.12/25.32 25.12/25.32 Lemma 54: multiplication(antidomain(X), multiplication(antidomain(Y), Z)) = multiplication(domain_difference(antidomain(X), Y), Z). 25.12/25.32 Proof: 25.12/25.32 multiplication(antidomain(X), multiplication(antidomain(Y), Z)) 25.12/25.32 = { by lemma 39 } 25.12/25.32 multiplication(c(X), multiplication(antidomain(Y), Z)) 25.12/25.32 = { by lemma 34 } 25.12/25.32 multiplication(domain(antidomain(X)), multiplication(antidomain(Y), Z)) 25.12/25.32 = { by lemma 53 } 25.12/25.32 multiplication(domain_difference(antidomain(X), Y), Z) 25.12/25.32 25.12/25.32 Lemma 55: addition(Y, multiplication(X, Y)) = multiplication(addition(X, one), Y). 25.12/25.32 Proof: 25.12/25.32 addition(Y, multiplication(X, Y)) 25.12/25.32 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.32 addition(multiplication(one, Y), multiplication(X, Y)) 25.12/25.32 = { by axiom 7 (left_distributivity) } 25.12/25.32 multiplication(addition(one, X), Y) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 multiplication(addition(X, one), Y) 25.12/25.32 25.12/25.32 Lemma 56: addition(X, addition(X, Y)) = addition(X, Y). 25.12/25.32 Proof: 25.12/25.32 addition(X, addition(X, Y)) 25.12/25.32 = { by axiom 30 (additive_associativity) } 25.12/25.32 addition(addition(X, X), Y) 25.12/25.32 = { by axiom 22 (additive_idempotence) } 25.12/25.32 addition(X, Y) 25.12/25.32 25.12/25.32 Lemma 57: addition(one, antidomain(X)) = one. 25.12/25.32 Proof: 25.12/25.32 addition(one, antidomain(X)) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 addition(antidomain(X), one) 25.12/25.32 = { by lemma 31 } 25.12/25.32 addition(antidomain(X), addition(antidomain(X), domain(X))) 25.12/25.32 = { by lemma 56 } 25.12/25.32 addition(antidomain(X), domain(X)) 25.12/25.32 = { by lemma 31 } 25.12/25.32 one 25.12/25.32 25.12/25.32 Lemma 58: addition(antidomain(X), domain_difference(Y, X)) = antidomain(X). 25.12/25.32 Proof: 25.12/25.32 addition(antidomain(X), domain_difference(Y, X)) 25.12/25.32 = { by axiom 29 (domain_difference) } 25.12/25.32 addition(antidomain(X), multiplication(domain(Y), antidomain(X))) 25.12/25.32 = { by lemma 55 } 25.12/25.32 multiplication(addition(domain(Y), one), antidomain(X)) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 multiplication(addition(one, domain(Y)), antidomain(X)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 multiplication(addition(one, antidomain(antidomain(Y))), antidomain(X)) 25.12/25.32 = { by lemma 57 } 25.12/25.32 multiplication(one, antidomain(X)) 25.12/25.32 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.32 antidomain(X) 25.12/25.32 25.12/25.32 Lemma 59: multiplication(antidomain(X), antidomain(Y)) = domain_difference(antidomain(X), Y). 25.12/25.32 Proof: 25.12/25.32 multiplication(antidomain(X), antidomain(Y)) 25.12/25.32 = { by lemma 39 } 25.12/25.32 multiplication(c(X), antidomain(Y)) 25.12/25.32 = { by lemma 35 } 25.12/25.32 domain_difference(antidomain(X), Y) 25.12/25.32 25.12/25.32 Lemma 60: multiplication(domain(X), domain_difference(Y, X)) = zero. 25.12/25.32 Proof: 25.12/25.32 multiplication(domain(X), domain_difference(Y, X)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 multiplication(antidomain(antidomain(X)), domain_difference(Y, X)) 25.12/25.32 = { by lemma 32 } 25.12/25.32 multiplication(antidomain(antidomain(X)), addition(antidomain(X), domain_difference(Y, X))) 25.12/25.32 = { by lemma 58 } 25.12/25.32 multiplication(antidomain(antidomain(X)), antidomain(X)) 25.12/25.32 = { by axiom 16 (domain1) } 25.12/25.32 zero 25.12/25.32 25.12/25.32 Lemma 61: addition(zero, X) = X. 25.12/25.32 Proof: 25.12/25.32 addition(zero, X) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 addition(X, zero) 25.12/25.32 = { by axiom 9 (additive_identity) } 25.12/25.32 X 25.12/25.32 25.12/25.32 Lemma 62: multiplication(domain_difference(Y, X), domain(Y)) = multiplication(antidomain(X), domain(Y)). 25.12/25.32 Proof: 25.12/25.32 multiplication(domain_difference(Y, X), domain(Y)) 25.12/25.32 = { by lemma 50 } 25.12/25.32 multiplication(domain_difference(domain(Y), X), domain(Y)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 multiplication(domain_difference(antidomain(antidomain(Y)), X), domain(Y)) 25.12/25.32 = { by lemma 54 } 25.12/25.32 multiplication(antidomain(antidomain(Y)), multiplication(antidomain(X), domain(Y))) 25.12/25.32 = { by lemma 52 } 25.12/25.32 multiplication(antidomain(antidomain(Y)), domain_difference(antidomain(X), antidomain(Y))) 25.12/25.32 = { by lemma 61 } 25.12/25.32 addition(zero, multiplication(antidomain(antidomain(Y)), domain_difference(antidomain(X), antidomain(Y)))) 25.12/25.32 = { by lemma 60 } 25.12/25.32 addition(multiplication(domain(antidomain(Y)), domain_difference(antidomain(X), antidomain(Y))), multiplication(antidomain(antidomain(Y)), domain_difference(antidomain(X), antidomain(Y)))) 25.12/25.32 = { by axiom 7 (left_distributivity) } 25.12/25.32 multiplication(addition(domain(antidomain(Y)), antidomain(antidomain(Y))), domain_difference(antidomain(X), antidomain(Y))) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 multiplication(addition(antidomain(antidomain(Y)), domain(antidomain(Y))), domain_difference(antidomain(X), antidomain(Y))) 25.12/25.32 = { by lemma 31 } 25.12/25.32 multiplication(one, domain_difference(antidomain(X), antidomain(Y))) 25.12/25.32 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.32 domain_difference(antidomain(X), antidomain(Y)) 25.12/25.32 = { by lemma 52 } 25.12/25.32 multiplication(antidomain(X), domain(Y)) 25.12/25.32 25.12/25.32 Lemma 63: antidomain(one) = zero. 25.12/25.32 Proof: 25.12/25.32 antidomain(one) 25.12/25.32 = { by axiom 23 (multiplicative_right_identity) } 25.12/25.32 multiplication(antidomain(one), one) 25.12/25.32 = { by axiom 16 (domain1) } 25.12/25.32 zero 25.12/25.32 25.12/25.32 Lemma 64: domain(one) = antidomain(zero). 25.12/25.32 Proof: 25.12/25.32 domain(one) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 antidomain(antidomain(one)) 25.12/25.32 = { by lemma 63 } 25.12/25.32 antidomain(zero) 25.12/25.32 25.12/25.32 Lemma 65: antidomain(zero) = one. 25.12/25.32 Proof: 25.12/25.32 antidomain(zero) 25.12/25.32 = { by lemma 61 } 25.12/25.32 addition(zero, antidomain(zero)) 25.12/25.32 = { by lemma 63 } 25.12/25.32 addition(antidomain(one), antidomain(zero)) 25.12/25.32 = { by lemma 64 } 25.12/25.32 addition(antidomain(one), domain(one)) 25.12/25.32 = { by lemma 31 } 25.12/25.32 one 25.12/25.32 25.12/25.32 Lemma 66: c(zero) = one. 25.12/25.32 Proof: 25.12/25.32 c(zero) 25.12/25.32 = { by lemma 34 } 25.12/25.32 domain(antidomain(zero)) 25.12/25.32 = { by lemma 65 } 25.12/25.32 domain(one) 25.12/25.32 = { by lemma 64 } 25.12/25.32 antidomain(zero) 25.12/25.32 = { by lemma 65 } 25.12/25.32 one 25.12/25.32 25.12/25.32 Lemma 67: forward_diamond(X, one) = domain(X). 25.12/25.32 Proof: 25.12/25.32 forward_diamond(X, one) 25.12/25.32 = { by lemma 65 } 25.12/25.32 forward_diamond(X, antidomain(zero)) 25.12/25.32 = { by lemma 49 } 25.12/25.32 domain(multiplication(X, c(zero))) 25.12/25.32 = { by lemma 66 } 25.12/25.32 domain(multiplication(X, one)) 25.12/25.32 = { by axiom 23 (multiplicative_right_identity) } 25.12/25.32 domain(X) 25.12/25.32 25.12/25.32 Lemma 68: domain(antidomain(X)) = antidomain(X). 25.12/25.32 Proof: 25.12/25.32 domain(antidomain(X)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 antidomain(antidomain(antidomain(X))) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 antidomain(domain(X)) 25.12/25.32 = { by axiom 11 (complement) } 25.12/25.32 c(X) 25.12/25.32 = { by lemma 39 } 25.12/25.32 antidomain(X) 25.12/25.32 25.12/25.32 Lemma 69: addition(domain_difference(X, Y), multiplication(domain(X), Z)) = multiplication(domain(X), addition(Z, antidomain(Y))). 25.12/25.32 Proof: 25.12/25.32 addition(domain_difference(X, Y), multiplication(domain(X), Z)) 25.12/25.32 = { by axiom 29 (domain_difference) } 25.12/25.32 addition(multiplication(domain(X), antidomain(Y)), multiplication(domain(X), Z)) 25.12/25.32 = { by axiom 6 (right_distributivity) } 25.12/25.32 multiplication(domain(X), addition(antidomain(Y), Z)) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 multiplication(domain(X), addition(Z, antidomain(Y))) 25.12/25.32 25.12/25.32 Lemma 70: addition(multiplication(antidomain(X), Y), domain_difference(antidomain(X), Z)) = multiplication(antidomain(X), addition(Y, antidomain(Z))). 25.12/25.32 Proof: 25.12/25.32 addition(multiplication(antidomain(X), Y), domain_difference(antidomain(X), Z)) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 addition(domain_difference(antidomain(X), Z), multiplication(antidomain(X), Y)) 25.12/25.32 = { by lemma 68 } 25.12/25.32 addition(domain_difference(antidomain(X), Z), multiplication(domain(antidomain(X)), Y)) 25.12/25.32 = { by lemma 69 } 25.12/25.32 multiplication(domain(antidomain(X)), addition(Y, antidomain(Z))) 25.12/25.32 = { by lemma 68 } 25.12/25.32 multiplication(antidomain(X), addition(Y, antidomain(Z))) 25.12/25.32 25.12/25.32 Lemma 71: multiplication(antidomain(Y), domain(X)) = domain_difference(X, Y). 25.12/25.32 Proof: 25.12/25.32 multiplication(antidomain(Y), domain(X)) 25.12/25.32 = { by lemma 62 } 25.12/25.32 multiplication(domain_difference(X, Y), domain(X)) 25.12/25.32 = { by lemma 50 } 25.12/25.32 multiplication(domain_difference(domain(X), Y), domain(X)) 25.12/25.32 = { by lemma 67 } 25.12/25.32 multiplication(domain_difference(forward_diamond(X, one), Y), domain(X)) 25.12/25.32 = { by lemma 67 } 25.12/25.32 multiplication(domain_difference(forward_diamond(X, one), Y), forward_diamond(X, one)) 25.12/25.32 = { by lemma 53 } 25.12/25.32 multiplication(domain(forward_diamond(X, one)), multiplication(antidomain(Y), forward_diamond(X, one))) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 multiplication(antidomain(antidomain(forward_diamond(X, one))), multiplication(antidomain(Y), forward_diamond(X, one))) 25.12/25.32 = { by lemma 33 } 25.12/25.32 multiplication(antidomain(antidomain(forward_diamond(X, one))), addition(multiplication(antidomain(Y), forward_diamond(X, one)), antidomain(forward_diamond(X, one)))) 25.12/25.32 = { by lemma 58 } 25.12/25.32 multiplication(antidomain(antidomain(forward_diamond(X, one))), addition(multiplication(antidomain(Y), forward_diamond(X, one)), addition(antidomain(forward_diamond(X, one)), domain_difference(antidomain(Y), forward_diamond(X, one))))) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 multiplication(antidomain(antidomain(forward_diamond(X, one))), addition(addition(antidomain(forward_diamond(X, one)), domain_difference(antidomain(Y), forward_diamond(X, one))), multiplication(antidomain(Y), forward_diamond(X, one)))) 25.12/25.32 = { by axiom 30 (additive_associativity) } 25.12/25.32 multiplication(antidomain(antidomain(forward_diamond(X, one))), addition(antidomain(forward_diamond(X, one)), addition(domain_difference(antidomain(Y), forward_diamond(X, one)), multiplication(antidomain(Y), forward_diamond(X, one))))) 25.12/25.32 = { by lemma 32 } 25.12/25.32 multiplication(antidomain(antidomain(forward_diamond(X, one))), addition(domain_difference(antidomain(Y), forward_diamond(X, one)), multiplication(antidomain(Y), forward_diamond(X, one)))) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 multiplication(domain(forward_diamond(X, one)), addition(domain_difference(antidomain(Y), forward_diamond(X, one)), multiplication(antidomain(Y), forward_diamond(X, one)))) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 multiplication(domain(forward_diamond(X, one)), addition(multiplication(antidomain(Y), forward_diamond(X, one)), domain_difference(antidomain(Y), forward_diamond(X, one)))) 25.12/25.32 = { by lemma 70 } 25.12/25.32 multiplication(domain(forward_diamond(X, one)), multiplication(antidomain(Y), addition(forward_diamond(X, one), antidomain(forward_diamond(X, one))))) 25.12/25.32 = { by lemma 53 } 25.12/25.32 multiplication(domain_difference(forward_diamond(X, one), Y), addition(forward_diamond(X, one), antidomain(forward_diamond(X, one)))) 25.12/25.32 = { by lemma 46 } 25.12/25.32 multiplication(domain_difference(forward_diamond(X, one), Y), one) 25.12/25.32 = { by axiom 23 (multiplicative_right_identity) } 25.12/25.32 domain_difference(forward_diamond(X, one), Y) 25.12/25.32 = { by lemma 67 } 25.12/25.32 domain_difference(domain(X), Y) 25.12/25.32 = { by lemma 50 } 25.12/25.32 domain_difference(X, Y) 25.12/25.32 25.12/25.32 Lemma 72: addition(multiplication(Z, W), multiplication(addition(X, Z), Y)) = addition(multiplication(X, Y), multiplication(Z, addition(Y, W))). 25.12/25.32 Proof: 25.12/25.32 addition(multiplication(Z, W), multiplication(addition(X, Z), Y)) 25.12/25.32 = { by axiom 7 (left_distributivity) } 25.12/25.32 addition(multiplication(Z, W), addition(multiplication(X, Y), multiplication(Z, Y))) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 addition(multiplication(Z, W), addition(multiplication(Z, Y), multiplication(X, Y))) 25.12/25.32 = { by axiom 30 (additive_associativity) } 25.12/25.32 addition(addition(multiplication(Z, W), multiplication(Z, Y)), multiplication(X, Y)) 25.12/25.32 = { by axiom 6 (right_distributivity) } 25.12/25.32 addition(multiplication(Z, addition(W, Y)), multiplication(X, Y)) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 addition(multiplication(X, Y), multiplication(Z, addition(W, Y))) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 addition(multiplication(X, Y), multiplication(Z, addition(Y, W))) 25.12/25.32 25.12/25.32 Lemma 73: addition(antidomain(X), domain_difference(X, Y)) = addition(antidomain(X), antidomain(Y)). 25.12/25.32 Proof: 25.12/25.32 addition(antidomain(X), domain_difference(X, Y)) 25.12/25.32 = { by lemma 71 } 25.12/25.32 addition(antidomain(X), multiplication(antidomain(Y), domain(X))) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 addition(multiplication(antidomain(Y), domain(X)), antidomain(X)) 25.12/25.32 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.32 addition(multiplication(antidomain(Y), domain(X)), multiplication(one, antidomain(X))) 25.12/25.32 = { by lemma 57 } 25.12/25.32 addition(multiplication(antidomain(Y), domain(X)), multiplication(addition(one, antidomain(Y)), antidomain(X))) 25.12/25.32 = { by lemma 72 } 25.12/25.32 addition(multiplication(one, antidomain(X)), multiplication(antidomain(Y), addition(antidomain(X), domain(X)))) 25.12/25.32 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.32 addition(antidomain(X), multiplication(antidomain(Y), addition(antidomain(X), domain(X)))) 25.12/25.32 = { by lemma 31 } 25.12/25.32 addition(antidomain(X), multiplication(antidomain(Y), one)) 25.12/25.32 = { by axiom 23 (multiplicative_right_identity) } 25.12/25.32 addition(antidomain(X), antidomain(Y)) 25.12/25.32 25.12/25.32 Lemma 74: domain(domain_difference(X, antidomain(Y))) = forward_diamond(domain(X), Y). 25.12/25.32 Proof: 25.12/25.32 domain(domain_difference(X, antidomain(Y))) 25.12/25.32 = { by lemma 51 } 25.12/25.32 domain(multiplication(domain(X), domain(Y))) 25.12/25.32 = { by axiom 14 (forward_diamond) } 25.12/25.32 forward_diamond(domain(X), Y) 25.12/25.32 25.12/25.32 Lemma 75: domain_difference(X, antidomain(Y)) = domain_difference(Y, antidomain(X)). 25.12/25.32 Proof: 25.12/25.32 domain_difference(X, antidomain(Y)) 25.12/25.32 = { by lemma 71 } 25.12/25.32 multiplication(antidomain(antidomain(Y)), domain(X)) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 multiplication(domain(Y), domain(X)) 25.12/25.32 = { by lemma 51 } 25.12/25.32 domain_difference(Y, antidomain(X)) 25.12/25.32 25.12/25.32 Lemma 76: forward_diamond(domain(X), Y) = forward_diamond(domain(Y), X). 25.12/25.32 Proof: 25.12/25.32 forward_diamond(domain(X), Y) 25.12/25.32 = { by lemma 74 } 25.12/25.32 domain(domain_difference(X, antidomain(Y))) 25.12/25.32 = { by lemma 75 } 25.12/25.32 domain(domain_difference(Y, antidomain(X))) 25.12/25.32 = { by lemma 74 } 25.12/25.32 forward_diamond(domain(Y), X) 25.12/25.32 25.12/25.32 Lemma 77: antidomain(domain(X)) = antidomain(X). 25.12/25.32 Proof: 25.12/25.32 antidomain(domain(X)) 25.12/25.32 = { by axiom 11 (complement) } 25.12/25.32 c(X) 25.12/25.32 = { by lemma 39 } 25.12/25.32 antidomain(X) 25.12/25.32 25.12/25.32 Lemma 78: multiplication(domain(X), c(Y)) = domain_difference(X, domain(Y)). 25.12/25.32 Proof: 25.12/25.32 multiplication(domain(X), c(Y)) 25.12/25.32 = { by axiom 11 (complement) } 25.12/25.32 multiplication(domain(X), antidomain(domain(Y))) 25.12/25.32 = { by axiom 29 (domain_difference) } 25.12/25.32 domain_difference(X, domain(Y)) 25.12/25.32 25.12/25.32 Lemma 79: domain_difference(X, domain(Y)) = domain_difference(X, Y). 25.12/25.32 Proof: 25.12/25.32 domain_difference(X, domain(Y)) 25.12/25.32 = { by lemma 78 } 25.12/25.32 multiplication(domain(X), c(Y)) 25.12/25.32 = { by lemma 39 } 25.12/25.32 multiplication(domain(X), antidomain(Y)) 25.12/25.32 = { by axiom 29 (domain_difference) } 25.12/25.32 domain_difference(X, Y) 25.12/25.32 25.12/25.32 Lemma 80: forward_box(domain(X), Y) = antidomain(domain_difference(X, Y)). 25.12/25.32 Proof: 25.12/25.32 forward_box(domain(X), Y) 25.12/25.32 = { by lemma 40 } 25.12/25.32 antidomain(forward_diamond(domain(X), antidomain(Y))) 25.12/25.32 = { by lemma 49 } 25.12/25.32 antidomain(domain(multiplication(domain(X), c(Y)))) 25.12/25.32 = { by lemma 78 } 25.12/25.32 antidomain(domain(domain_difference(X, domain(Y)))) 25.12/25.32 = { by lemma 79 } 25.12/25.32 antidomain(domain(domain_difference(X, Y))) 25.12/25.32 = { by axiom 11 (complement) } 25.12/25.32 c(domain_difference(X, Y)) 25.12/25.32 = { by lemma 39 } 25.12/25.32 antidomain(domain_difference(X, Y)) 25.12/25.32 25.12/25.32 Lemma 81: multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y))) = zero. 25.12/25.32 Proof: 25.12/25.32 multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y))) 25.12/25.32 = { by axiom 13 (domain4) } 25.12/25.32 multiplication(antidomain(multiplication(X, Y)), multiplication(X, antidomain(antidomain(Y)))) 25.12/25.32 = { by lemma 36 } 25.12/25.32 multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))), multiplication(X, antidomain(antidomain(Y)))) 25.12/25.32 = { by axiom 10 (domain2) } 25.12/25.32 multiplication(antidomain(multiplication(X, antidomain(antidomain(Y)))), multiplication(X, antidomain(antidomain(Y)))) 25.12/25.32 = { by axiom 16 (domain1) } 25.12/25.32 zero 25.12/25.32 25.12/25.32 Lemma 82: addition(coantidomain(X), codomain(X)) = one. 25.12/25.32 Proof: 25.12/25.32 addition(coantidomain(X), codomain(X)) 25.12/25.32 = { by axiom 17 (additive_commutativity) } 25.12/25.32 addition(codomain(X), coantidomain(X)) 25.12/25.32 = { by axiom 24 (codomain4) } 25.12/25.32 addition(coantidomain(coantidomain(X)), coantidomain(X)) 25.12/25.32 = { by axiom 5 (codomain3) } 25.12/25.32 one 25.12/25.32 25.12/25.32 Lemma 83: leq(multiplication(X, Y), multiplication(addition(X, Z), Y)) = true. 25.12/25.32 Proof: 25.12/25.32 leq(multiplication(X, Y), multiplication(addition(X, Z), Y)) 25.12/25.32 = { by axiom 7 (left_distributivity) } 25.12/25.32 leq(multiplication(X, Y), addition(multiplication(X, Y), multiplication(Z, Y))) 25.12/25.32 = { by axiom 12 (ifeq_axiom) } 25.12/25.32 ifeq2(addition(multiplication(X, Y), multiplication(Z, Y)), addition(multiplication(X, Y), multiplication(Z, Y)), leq(multiplication(X, Y), addition(multiplication(X, Y), multiplication(Z, Y))), true) 25.12/25.32 = { by lemma 56 } 25.12/25.32 ifeq2(addition(multiplication(X, Y), addition(multiplication(X, Y), multiplication(Z, Y))), addition(multiplication(X, Y), multiplication(Z, Y)), leq(multiplication(X, Y), addition(multiplication(X, Y), multiplication(Z, Y))), true) 25.12/25.32 = { by axiom 18 (order) } 25.12/25.32 true 25.12/25.32 25.12/25.32 Lemma 84: multiplication(antidomain(codomain(X)), coantidomain(X)) = antidomain(codomain(X)). 25.12/25.32 Proof: 25.12/25.32 multiplication(antidomain(codomain(X)), coantidomain(X)) 25.12/25.32 = { by axiom 24 (codomain4) } 25.12/25.32 multiplication(antidomain(coantidomain(coantidomain(X))), coantidomain(X)) 25.12/25.32 = { by lemma 32 } 25.12/25.32 multiplication(antidomain(coantidomain(coantidomain(X))), addition(coantidomain(coantidomain(X)), coantidomain(X))) 25.12/25.32 = { by axiom 5 (codomain3) } 25.12/25.32 multiplication(antidomain(coantidomain(coantidomain(X))), one) 25.12/25.32 = { by axiom 23 (multiplicative_right_identity) } 25.12/25.32 antidomain(coantidomain(coantidomain(X))) 25.12/25.32 = { by axiom 24 (codomain4) } 25.12/25.33 antidomain(codomain(X)) 25.12/25.33 25.12/25.33 Lemma 85: antidomain(codomain(X)) = coantidomain(X). 25.12/25.33 Proof: 25.12/25.33 antidomain(codomain(X)) 25.12/25.33 = { by axiom 4 (order_1) } 25.12/25.33 ifeq(leq(coantidomain(X), antidomain(codomain(X))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by axiom 23 (multiplicative_right_identity) } 25.12/25.33 ifeq(leq(multiplication(coantidomain(X), one), antidomain(codomain(X))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by lemma 31 } 25.12/25.33 ifeq(leq(multiplication(coantidomain(X), addition(antidomain(codomain(X)), domain(codomain(X)))), antidomain(codomain(X))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by axiom 17 (additive_commutativity) } 25.12/25.33 ifeq(leq(multiplication(coantidomain(X), addition(domain(codomain(X)), antidomain(codomain(X)))), antidomain(codomain(X))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by axiom 6 (right_distributivity) } 25.12/25.33 ifeq(leq(addition(multiplication(coantidomain(X), domain(codomain(X))), multiplication(coantidomain(X), antidomain(codomain(X)))), antidomain(codomain(X))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.33 ifeq(leq(addition(multiplication(one, multiplication(coantidomain(X), domain(codomain(X)))), multiplication(coantidomain(X), antidomain(codomain(X)))), antidomain(codomain(X))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by lemma 65 } 25.12/25.33 ifeq(leq(addition(multiplication(antidomain(zero), multiplication(coantidomain(X), domain(codomain(X)))), multiplication(coantidomain(X), antidomain(codomain(X)))), antidomain(codomain(X))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by axiom 28 (codomain1) } 25.12/25.33 ifeq(leq(addition(multiplication(antidomain(multiplication(coantidomain(X), coantidomain(coantidomain(X)))), multiplication(coantidomain(X), domain(codomain(X)))), multiplication(coantidomain(X), antidomain(codomain(X)))), antidomain(codomain(X))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 ifeq(leq(addition(multiplication(antidomain(multiplication(coantidomain(X), codomain(X))), multiplication(coantidomain(X), domain(codomain(X)))), multiplication(coantidomain(X), antidomain(codomain(X)))), antidomain(codomain(X))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by lemma 81 } 25.12/25.33 ifeq(leq(addition(zero, multiplication(coantidomain(X), antidomain(codomain(X)))), antidomain(codomain(X))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by lemma 61 } 25.12/25.33 ifeq(leq(multiplication(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.33 ifeq(leq(multiplication(coantidomain(X), antidomain(codomain(X))), multiplication(one, antidomain(codomain(X)))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by lemma 82 } 25.12/25.33 ifeq(leq(multiplication(coantidomain(X), antidomain(codomain(X))), multiplication(addition(coantidomain(X), codomain(X)), antidomain(codomain(X)))), true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by lemma 83 } 25.12/25.33 ifeq(true, true, addition(coantidomain(X), antidomain(codomain(X))), antidomain(codomain(X))) 25.12/25.33 = { by axiom 1 (ifeq_axiom_001) } 25.12/25.33 addition(coantidomain(X), antidomain(codomain(X))) 25.12/25.33 = { by axiom 17 (additive_commutativity) } 25.12/25.33 addition(antidomain(codomain(X)), coantidomain(X)) 25.12/25.33 = { by axiom 1 (ifeq_axiom_001) } 25.12/25.33 ifeq(true, true, addition(antidomain(codomain(X)), coantidomain(X)), coantidomain(X)) 25.12/25.33 = { by lemma 83 } 25.12/25.33 ifeq(leq(multiplication(antidomain(codomain(X)), coantidomain(X)), multiplication(addition(antidomain(codomain(X)), domain(codomain(X))), coantidomain(X))), true, addition(antidomain(codomain(X)), coantidomain(X)), coantidomain(X)) 25.12/25.33 = { by lemma 84 } 25.12/25.33 ifeq(leq(antidomain(codomain(X)), multiplication(addition(antidomain(codomain(X)), domain(codomain(X))), coantidomain(X))), true, addition(antidomain(codomain(X)), coantidomain(X)), coantidomain(X)) 25.12/25.33 = { by lemma 31 } 25.12/25.33 ifeq(leq(antidomain(codomain(X)), multiplication(one, coantidomain(X))), true, addition(antidomain(codomain(X)), coantidomain(X)), coantidomain(X)) 25.12/25.33 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.33 ifeq(leq(antidomain(codomain(X)), coantidomain(X)), true, addition(antidomain(codomain(X)), coantidomain(X)), coantidomain(X)) 25.12/25.33 = { by axiom 4 (order_1) } 25.12/25.33 coantidomain(X) 25.12/25.33 25.12/25.33 Lemma 86: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y). 25.12/25.33 Proof: 25.12/25.33 multiplication(X, addition(Y, coantidomain(X))) 25.12/25.33 = { by axiom 6 (right_distributivity) } 25.12/25.33 addition(multiplication(X, Y), multiplication(X, coantidomain(X))) 25.12/25.33 = { by axiom 28 (codomain1) } 25.12/25.33 addition(multiplication(X, Y), zero) 25.12/25.33 = { by axiom 9 (additive_identity) } 25.12/25.33 multiplication(X, Y) 25.12/25.33 25.12/25.33 Lemma 87: multiplication(X, codomain(X)) = X. 25.12/25.33 Proof: 25.12/25.33 multiplication(X, codomain(X)) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 multiplication(X, coantidomain(coantidomain(X))) 25.12/25.33 = { by lemma 86 } 25.12/25.33 multiplication(X, addition(coantidomain(coantidomain(X)), coantidomain(X))) 25.12/25.33 = { by axiom 5 (codomain3) } 25.12/25.33 multiplication(X, one) 25.12/25.33 = { by axiom 23 (multiplicative_right_identity) } 25.12/25.33 X 25.12/25.33 25.12/25.33 Lemma 88: multiplication(addition(one, X), codomain(X)) = addition(X, codomain(X)). 25.12/25.33 Proof: 25.12/25.33 multiplication(addition(one, X), codomain(X)) 25.12/25.33 = { by axiom 17 (additive_commutativity) } 25.12/25.33 multiplication(addition(X, one), codomain(X)) 25.12/25.33 = { by lemma 55 } 25.12/25.33 addition(codomain(X), multiplication(X, codomain(X))) 25.12/25.33 = { by lemma 87 } 25.12/25.33 addition(codomain(X), X) 25.12/25.33 = { by axiom 17 (additive_commutativity) } 25.12/25.33 addition(X, codomain(X)) 25.12/25.33 25.12/25.33 Lemma 89: coantidomain(one) = zero. 25.12/25.33 Proof: 25.12/25.33 coantidomain(one) 25.12/25.33 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.33 multiplication(one, coantidomain(one)) 25.12/25.33 = { by axiom 28 (codomain1) } 25.12/25.33 zero 25.12/25.33 25.12/25.33 Lemma 90: coantidomain(zero) = one. 25.12/25.33 Proof: 25.12/25.33 coantidomain(zero) 25.12/25.33 = { by axiom 9 (additive_identity) } 25.12/25.33 addition(coantidomain(zero), zero) 25.12/25.33 = { by lemma 89 } 25.12/25.33 addition(coantidomain(coantidomain(one)), zero) 25.12/25.33 = { by lemma 89 } 25.12/25.33 addition(coantidomain(coantidomain(one)), coantidomain(one)) 25.12/25.33 = { by axiom 5 (codomain3) } 25.12/25.33 one 25.12/25.33 25.12/25.33 Lemma 91: backward_diamond(X, one) = codomain(X). 25.12/25.33 Proof: 25.12/25.33 backward_diamond(X, one) 25.12/25.33 = { by axiom 8 (backward_diamond) } 25.12/25.33 codomain(multiplication(codomain(one), X)) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 codomain(multiplication(coantidomain(coantidomain(one)), X)) 25.12/25.33 = { by lemma 89 } 25.12/25.33 codomain(multiplication(coantidomain(zero), X)) 25.12/25.33 = { by lemma 90 } 25.12/25.33 codomain(multiplication(one, X)) 25.12/25.33 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.33 codomain(X) 25.12/25.33 25.12/25.33 Lemma 92: codomain(coantidomain(X)) = coantidomain(codomain(X)). 25.12/25.33 Proof: 25.12/25.33 codomain(coantidomain(X)) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 coantidomain(coantidomain(coantidomain(X))) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 coantidomain(codomain(X)) 25.12/25.33 25.12/25.33 Lemma 93: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)). 25.12/25.33 Proof: 25.12/25.33 multiplication(addition(X, Y), coantidomain(X)) 25.12/25.33 = { by axiom 17 (additive_commutativity) } 25.12/25.33 multiplication(addition(Y, X), coantidomain(X)) 25.12/25.33 = { by axiom 7 (left_distributivity) } 25.12/25.33 addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X))) 25.12/25.33 = { by axiom 28 (codomain1) } 25.12/25.33 addition(multiplication(Y, coantidomain(X)), zero) 25.12/25.33 = { by axiom 9 (additive_identity) } 25.12/25.33 multiplication(Y, coantidomain(X)) 25.12/25.33 25.12/25.33 Lemma 94: coantidomain(codomain(X)) = coantidomain(X). 25.12/25.33 Proof: 25.12/25.33 coantidomain(codomain(X)) 25.12/25.33 = { by lemma 92 } 25.12/25.33 codomain(coantidomain(X)) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 coantidomain(coantidomain(coantidomain(X))) 25.12/25.33 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.33 multiplication(one, coantidomain(coantidomain(coantidomain(X)))) 25.12/25.33 = { by axiom 5 (codomain3) } 25.12/25.33 multiplication(addition(coantidomain(coantidomain(X)), coantidomain(X)), coantidomain(coantidomain(coantidomain(X)))) 25.12/25.33 = { by lemma 93 } 25.12/25.33 multiplication(coantidomain(X), coantidomain(coantidomain(coantidomain(X)))) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 multiplication(coantidomain(X), codomain(coantidomain(X))) 25.12/25.33 = { by lemma 87 } 25.12/25.33 coantidomain(X) 25.12/25.33 25.12/25.33 Lemma 95: codomain(codomain(X)) = codomain(X). 25.12/25.33 Proof: 25.12/25.33 codomain(codomain(X)) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 coantidomain(coantidomain(codomain(X))) 25.12/25.33 = { by lemma 94 } 25.12/25.33 coantidomain(coantidomain(X)) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 codomain(X) 25.12/25.33 25.12/25.33 Lemma 96: codomain(backward_diamond(X, Y)) = backward_diamond(X, Y). 25.12/25.33 Proof: 25.12/25.33 codomain(backward_diamond(X, Y)) 25.12/25.33 = { by axiom 8 (backward_diamond) } 25.12/25.33 codomain(codomain(multiplication(codomain(Y), X))) 25.12/25.33 = { by lemma 95 } 25.12/25.33 codomain(multiplication(codomain(Y), X)) 25.12/25.33 = { by axiom 8 (backward_diamond) } 25.12/25.33 backward_diamond(X, Y) 25.12/25.33 25.12/25.33 Lemma 97: coantidomain(multiplication(codomain(X), Y)) = coantidomain(backward_diamond(Y, X)). 25.12/25.33 Proof: 25.12/25.33 coantidomain(multiplication(codomain(X), Y)) 25.12/25.33 = { by lemma 94 } 25.12/25.33 coantidomain(codomain(multiplication(codomain(X), Y))) 25.12/25.33 = { by axiom 8 (backward_diamond) } 25.12/25.33 coantidomain(backward_diamond(Y, X)) 25.12/25.33 25.12/25.33 Lemma 98: coantidomain(backward_diamond(X, Y)) = antidomain(backward_diamond(X, Y)). 25.12/25.33 Proof: 25.12/25.33 coantidomain(backward_diamond(X, Y)) 25.12/25.33 = { by lemma 97 } 25.12/25.33 coantidomain(multiplication(codomain(Y), X)) 25.12/25.33 = { by lemma 85 } 25.12/25.33 antidomain(codomain(multiplication(codomain(Y), X))) 25.12/25.33 = { by axiom 8 (backward_diamond) } 25.12/25.33 antidomain(backward_diamond(X, Y)) 25.12/25.33 25.12/25.33 Lemma 99: forward_diamond(one, X) = domain(X). 25.12/25.33 Proof: 25.12/25.33 forward_diamond(one, X) 25.12/25.33 = { by axiom 14 (forward_diamond) } 25.12/25.33 domain(multiplication(one, domain(X))) 25.12/25.33 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.33 domain(domain(X)) 25.12/25.33 = { by lemma 41 } 25.12/25.33 antidomain(c(X)) 25.12/25.33 = { by lemma 39 } 25.12/25.33 antidomain(antidomain(X)) 25.12/25.33 = { by axiom 13 (domain4) } 25.12/25.33 domain(X) 25.12/25.33 25.12/25.33 Lemma 100: codomain(coantidomain(X)) = coantidomain(X). 25.12/25.33 Proof: 25.12/25.33 codomain(coantidomain(X)) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 coantidomain(coantidomain(coantidomain(X))) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 coantidomain(codomain(X)) 25.12/25.33 = { by lemma 94 } 25.12/25.33 coantidomain(X) 25.12/25.33 25.12/25.33 Lemma 101: antidomain(coantidomain(X)) = codomain(X). 25.12/25.33 Proof: 25.12/25.33 antidomain(coantidomain(X)) 25.12/25.33 = { by lemma 100 } 25.12/25.33 antidomain(codomain(coantidomain(X))) 25.12/25.33 = { by lemma 85 } 25.12/25.33 coantidomain(coantidomain(X)) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 codomain(X) 25.12/25.33 25.12/25.33 Lemma 102: domain(codomain(X)) = codomain(X). 25.12/25.33 Proof: 25.12/25.33 domain(codomain(X)) 25.12/25.33 = { by axiom 13 (domain4) } 25.12/25.33 antidomain(antidomain(codomain(X))) 25.12/25.33 = { by lemma 85 } 25.12/25.33 antidomain(coantidomain(X)) 25.12/25.33 = { by lemma 101 } 25.12/25.33 codomain(X) 25.12/25.33 25.12/25.33 Lemma 103: domain(backward_diamond(X, Y)) = backward_diamond(X, Y). 25.12/25.33 Proof: 25.12/25.33 domain(backward_diamond(X, Y)) 25.12/25.33 = { by axiom 8 (backward_diamond) } 25.12/25.33 domain(codomain(multiplication(codomain(Y), X))) 25.12/25.33 = { by lemma 102 } 25.12/25.33 codomain(multiplication(codomain(Y), X)) 25.12/25.33 = { by axiom 8 (backward_diamond) } 25.12/25.33 backward_diamond(X, Y) 25.12/25.33 25.12/25.33 Lemma 104: forward_diamond(backward_diamond(X, Y), backward_diamond(X, Y)) = backward_diamond(X, Y). 25.12/25.33 Proof: 25.12/25.33 forward_diamond(backward_diamond(X, Y), backward_diamond(X, Y)) 25.12/25.33 = { by axiom 14 (forward_diamond) } 25.12/25.33 domain(multiplication(backward_diamond(X, Y), domain(backward_diamond(X, Y)))) 25.12/25.33 = { by lemma 36 } 25.12/25.33 domain(multiplication(addition(backward_diamond(X, Y), antidomain(domain(backward_diamond(X, Y)))), domain(backward_diamond(X, Y)))) 25.12/25.33 = { by axiom 14 (forward_diamond) } 25.12/25.33 forward_diamond(addition(backward_diamond(X, Y), antidomain(domain(backward_diamond(X, Y)))), backward_diamond(X, Y)) 25.12/25.33 = { by axiom 11 (complement) } 25.12/25.33 forward_diamond(addition(backward_diamond(X, Y), c(backward_diamond(X, Y))), backward_diamond(X, Y)) 25.12/25.33 = { by axiom 17 (additive_commutativity) } 25.12/25.33 forward_diamond(addition(c(backward_diamond(X, Y)), backward_diamond(X, Y)), backward_diamond(X, Y)) 25.12/25.33 = { by lemma 39 } 25.12/25.33 forward_diamond(addition(antidomain(backward_diamond(X, Y)), backward_diamond(X, Y)), backward_diamond(X, Y)) 25.12/25.33 = { by axiom 17 (additive_commutativity) } 25.12/25.33 forward_diamond(addition(backward_diamond(X, Y), antidomain(backward_diamond(X, Y))), backward_diamond(X, Y)) 25.12/25.33 = { by lemma 96 } 25.12/25.33 forward_diamond(addition(codomain(backward_diamond(X, Y)), antidomain(backward_diamond(X, Y))), backward_diamond(X, Y)) 25.12/25.33 = { by axiom 24 (codomain4) } 25.12/25.33 forward_diamond(addition(coantidomain(coantidomain(backward_diamond(X, Y))), antidomain(backward_diamond(X, Y))), backward_diamond(X, Y)) 25.12/25.33 = { by lemma 97 } 25.12/25.33 forward_diamond(addition(coantidomain(coantidomain(multiplication(codomain(Y), X))), antidomain(backward_diamond(X, Y))), backward_diamond(X, Y)) 25.12/25.33 = { by lemma 98 } 25.12/25.33 forward_diamond(addition(coantidomain(coantidomain(multiplication(codomain(Y), X))), coantidomain(backward_diamond(X, Y))), backward_diamond(X, Y)) 25.12/25.33 = { by lemma 97 } 25.12/25.33 forward_diamond(addition(coantidomain(coantidomain(multiplication(codomain(Y), X))), coantidomain(multiplication(codomain(Y), X))), backward_diamond(X, Y)) 25.12/25.33 = { by axiom 5 (codomain3) } 25.12/25.33 forward_diamond(one, backward_diamond(X, Y)) 25.12/25.33 = { by lemma 99 } 25.12/25.33 domain(backward_diamond(X, Y)) 25.12/25.33 = { by lemma 103 } 25.12/25.33 backward_diamond(X, Y) 25.12/25.33 25.12/25.33 Lemma 105: domain_difference(multiplication(X, domain(Y)), Z) = multiplication(forward_diamond(X, Y), antidomain(Z)). 25.12/25.33 Proof: 25.12/25.33 domain_difference(multiplication(X, domain(Y)), Z) 25.12/25.33 = { by axiom 29 (domain_difference) } 25.12/25.33 multiplication(domain(multiplication(X, domain(Y))), antidomain(Z)) 25.12/25.33 = { by axiom 14 (forward_diamond) } 25.12/25.33 multiplication(forward_diamond(X, Y), antidomain(Z)) 25.12/25.33 25.12/25.33 Lemma 106: multiplication(forward_diamond(X, Y), antidomain(Z)) = domain_difference(forward_diamond(X, Y), Z). 25.12/25.33 Proof: 25.12/25.33 multiplication(forward_diamond(X, Y), antidomain(Z)) 25.12/25.33 = { by lemma 105 } 25.12/25.33 domain_difference(multiplication(X, domain(Y)), Z) 25.12/25.33 = { by lemma 50 } 25.12/25.33 domain_difference(domain(multiplication(X, domain(Y))), Z) 25.12/25.33 = { by axiom 14 (forward_diamond) } 25.12/25.33 domain_difference(forward_diamond(X, Y), Z) 25.12/25.33 25.12/25.33 Lemma 107: coantidomain(multiplication(codomain(X), Y)) = antidomain(backward_diamond(Y, X)). 25.12/25.33 Proof: 25.12/25.33 coantidomain(multiplication(codomain(X), Y)) 25.12/25.33 = { by lemma 94 } 25.12/25.33 coantidomain(codomain(multiplication(codomain(X), Y))) 25.12/25.33 = { by axiom 8 (backward_diamond) } 25.12/25.33 coantidomain(backward_diamond(Y, X)) 25.12/25.33 = { by lemma 98 } 25.12/25.33 antidomain(backward_diamond(Y, X)) 25.12/25.33 25.12/25.33 Lemma 108: multiplication(domain(X), multiplication(domain(Y), Z)) = multiplication(domain_difference(X, antidomain(Y)), Z). 25.12/25.33 Proof: 25.12/25.33 multiplication(domain(X), multiplication(domain(Y), Z)) 25.12/25.33 = { by axiom 25 (multiplicative_associativity) } 25.12/25.33 multiplication(multiplication(domain(X), domain(Y)), Z) 25.12/25.33 = { by lemma 51 } 25.12/25.33 multiplication(domain_difference(X, antidomain(Y)), Z) 25.12/25.33 25.12/25.33 Lemma 109: multiplication(domain_difference(X, antidomain(Y)), Y) = multiplication(domain(X), Y). 25.12/25.33 Proof: 25.12/25.33 multiplication(domain_difference(X, antidomain(Y)), Y) 25.12/25.33 = { by lemma 108 } 25.12/25.33 multiplication(domain(X), multiplication(domain(Y), Y)) 25.12/25.33 = { by lemma 38 } 25.12/25.33 multiplication(domain(X), Y) 25.12/25.33 25.12/25.33 Lemma 110: domain(multiplication(X, backward_box(Y, Z))) = forward_diamond(X, backward_box(Y, Z)). 25.12/25.33 Proof: 25.12/25.33 domain(multiplication(X, backward_box(Y, Z))) 25.12/25.33 = { by axiom 27 (backward_box) } 25.12/25.33 domain(multiplication(X, c(backward_diamond(Y, c(Z))))) 25.12/25.33 = { by lemma 49 } 25.12/25.33 forward_diamond(X, antidomain(backward_diamond(Y, c(Z)))) 25.12/25.33 = { by lemma 48 } 25.12/25.33 antidomain(forward_box(X, backward_diamond(Y, c(Z)))) 25.12/25.33 = { by axiom 3 (forward_box) } 25.12/25.33 antidomain(c(forward_diamond(X, c(backward_diamond(Y, c(Z)))))) 25.12/25.33 = { by axiom 27 (backward_box) } 25.12/25.33 antidomain(c(forward_diamond(X, backward_box(Y, Z)))) 25.12/25.33 = { by lemma 39 } 25.12/25.33 antidomain(antidomain(forward_diamond(X, backward_box(Y, Z)))) 25.12/25.33 = { by axiom 13 (domain4) } 25.12/25.33 domain(forward_diamond(X, backward_box(Y, Z))) 25.12/25.33 = { by lemma 47 } 25.12/25.33 forward_diamond(X, backward_box(Y, Z)) 25.12/25.33 25.12/25.33 Lemma 111: domain_difference(X, backward_diamond(Y, antidomain(Z))) = multiplication(domain(X), backward_box(Y, Z)). 25.12/25.33 Proof: 25.12/25.33 domain_difference(X, backward_diamond(Y, antidomain(Z))) 25.12/25.33 = { by lemma 39 } 25.12/25.33 domain_difference(X, backward_diamond(Y, c(Z))) 25.12/25.33 = { by lemma 79 } 25.12/25.33 domain_difference(X, domain(backward_diamond(Y, c(Z)))) 25.12/25.33 = { by lemma 78 } 25.12/25.33 multiplication(domain(X), c(backward_diamond(Y, c(Z)))) 25.12/25.33 = { by axiom 27 (backward_box) } 25.12/25.33 multiplication(domain(X), backward_box(Y, Z)) 25.12/25.33 25.12/25.33 Lemma 112: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)). 25.12/25.33 Proof: 25.12/25.33 addition(X, multiplication(X, Y)) 25.12/25.33 = { by axiom 23 (multiplicative_right_identity) } 25.12/25.33 addition(multiplication(X, one), multiplication(X, Y)) 25.12/25.33 = { by axiom 6 (right_distributivity) } 25.12/25.33 multiplication(X, addition(one, Y)) 25.12/25.33 = { by axiom 17 (additive_commutativity) } 25.12/25.33 multiplication(X, addition(Y, one)) 25.12/25.33 25.12/25.33 Lemma 113: addition(domain(X), domain_difference(X, Y)) = domain(X). 25.12/25.33 Proof: 25.12/25.33 addition(domain(X), domain_difference(X, Y)) 25.12/25.33 = { by axiom 29 (domain_difference) } 25.12/25.33 addition(domain(X), multiplication(domain(X), antidomain(Y))) 25.12/25.33 = { by lemma 112 } 25.12/25.33 multiplication(domain(X), addition(antidomain(Y), one)) 25.12/25.33 = { by axiom 17 (additive_commutativity) } 25.12/25.33 multiplication(domain(X), addition(one, antidomain(Y))) 25.12/25.33 = { by lemma 57 } 25.12/25.33 multiplication(domain(X), one) 25.12/25.33 = { by axiom 23 (multiplicative_right_identity) } 25.12/25.34 domain(X) 25.12/25.34 25.12/25.34 Lemma 114: multiplication(codomain(X), backward_box(coantidomain(X), codomain(X))) = codomain(X). 25.12/25.34 Proof: 25.12/25.34 multiplication(codomain(X), backward_box(coantidomain(X), codomain(X))) 25.12/25.34 = { by axiom 24 (codomain4) } 25.12/25.34 multiplication(coantidomain(coantidomain(X)), backward_box(coantidomain(X), codomain(X))) 25.12/25.34 = { by lemma 85 } 25.12/25.34 multiplication(coantidomain(antidomain(codomain(X))), backward_box(coantidomain(X), codomain(X))) 25.12/25.34 = { by lemma 85 } 25.12/25.34 multiplication(antidomain(codomain(antidomain(codomain(X)))), backward_box(coantidomain(X), codomain(X))) 25.12/25.34 = { by lemma 68 } 25.12/25.34 multiplication(domain(antidomain(codomain(antidomain(codomain(X))))), backward_box(coantidomain(X), codomain(X))) 25.12/25.34 = { by lemma 85 } 25.12/25.34 multiplication(domain(coantidomain(antidomain(codomain(X)))), backward_box(coantidomain(X), codomain(X))) 25.12/25.34 = { by lemma 111 } 25.12/25.34 domain_difference(coantidomain(antidomain(codomain(X))), backward_diamond(coantidomain(X), antidomain(codomain(X)))) 25.12/25.34 = { by lemma 84 } 25.12/25.34 domain_difference(coantidomain(multiplication(antidomain(codomain(X)), coantidomain(X))), backward_diamond(coantidomain(X), antidomain(codomain(X)))) 25.12/25.34 = { by lemma 85 } 25.12/25.34 domain_difference(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), backward_diamond(coantidomain(X), antidomain(codomain(X)))) 25.12/25.34 = { by lemma 59 } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), antidomain(backward_diamond(coantidomain(X), antidomain(codomain(X))))) 25.12/25.34 = { by lemma 98 } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X))))) 25.12/25.34 = { by axiom 22 (additive_idempotence) } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), addition(coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))), coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))))) 25.12/25.34 = { by lemma 97 } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), addition(coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))), coantidomain(multiplication(codomain(antidomain(codomain(X))), coantidomain(X))))) 25.12/25.34 = { by axiom 24 (codomain4) } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), addition(coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(codomain(X)))), coantidomain(X))))) 25.12/25.34 = { by axiom 2 (codomain2) } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), addition(coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))), addition(coantidomain(multiplication(antidomain(codomain(X)), coantidomain(X))), coantidomain(multiplication(coantidomain(coantidomain(antidomain(codomain(X)))), coantidomain(X)))))) 25.12/25.34 = { by axiom 24 (codomain4) } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), addition(coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))), addition(coantidomain(multiplication(antidomain(codomain(X)), coantidomain(X))), coantidomain(multiplication(codomain(antidomain(codomain(X))), coantidomain(X)))))) 25.12/25.34 = { by lemma 97 } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), addition(coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))), addition(coantidomain(multiplication(antidomain(codomain(X)), coantidomain(X))), coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X))))))) 25.12/25.34 = { by axiom 17 (additive_commutativity) } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), addition(coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))), addition(coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))), coantidomain(multiplication(antidomain(codomain(X)), coantidomain(X)))))) 25.12/25.34 = { by lemma 56 } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), addition(coantidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))), coantidomain(multiplication(antidomain(codomain(X)), coantidomain(X))))) 25.12/25.34 = { by lemma 98 } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), addition(antidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))), coantidomain(multiplication(antidomain(codomain(X)), coantidomain(X))))) 25.12/25.34 = { by axiom 17 (additive_commutativity) } 25.12/25.34 multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), addition(coantidomain(multiplication(antidomain(codomain(X)), coantidomain(X))), antidomain(backward_diamond(coantidomain(X), antidomain(codomain(X)))))) 25.12/25.34 = { by lemma 70 } 25.12/25.34 addition(multiplication(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), coantidomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), domain_difference(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), backward_diamond(coantidomain(X), antidomain(codomain(X))))) 25.12/25.34 = { by lemma 84 } 25.12/25.34 addition(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), domain_difference(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), backward_diamond(coantidomain(X), antidomain(codomain(X))))) 25.12/25.34 = { by lemma 39 } 25.12/25.34 addition(c(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), domain_difference(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), backward_diamond(coantidomain(X), antidomain(codomain(X))))) 25.12/25.34 = { by lemma 34 } 25.12/25.34 addition(domain(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X))))), domain_difference(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))), backward_diamond(coantidomain(X), antidomain(codomain(X))))) 25.12/25.34 = { by lemma 113 } 25.12/25.34 domain(antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X))))) 25.12/25.34 = { by lemma 34 } 25.12/25.34 c(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))) 25.12/25.34 = { by lemma 39 } 25.12/25.34 antidomain(codomain(multiplication(antidomain(codomain(X)), coantidomain(X)))) 25.12/25.34 = { by lemma 85 } 25.12/25.34 coantidomain(multiplication(antidomain(codomain(X)), coantidomain(X))) 25.12/25.34 = { by lemma 84 } 25.12/25.34 coantidomain(antidomain(codomain(X))) 25.12/25.34 = { by lemma 85 } 25.12/25.34 coantidomain(coantidomain(X)) 25.12/25.34 = { by axiom 24 (codomain4) } 25.12/25.34 codomain(X) 25.12/25.34 25.12/25.34 Lemma 115: forward_diamond(codomain(X), backward_box(coantidomain(X), codomain(X))) = codomain(X). 25.12/25.34 Proof: 25.12/25.34 forward_diamond(codomain(X), backward_box(coantidomain(X), codomain(X))) 25.12/25.34 = { by lemma 110 } 25.12/25.34 domain(multiplication(codomain(X), backward_box(coantidomain(X), codomain(X)))) 25.12/25.34 = { by lemma 114 } 25.12/25.34 domain(codomain(X)) 25.12/25.34 = { by lemma 102 } 25.12/25.34 codomain(X) 25.12/25.34 25.12/25.34 Lemma 116: domain_difference(X, antidomain(forward_diamond(Y, Z))) = multiplication(domain(X), forward_diamond(Y, Z)). 25.12/25.34 Proof: 25.12/25.34 domain_difference(X, antidomain(forward_diamond(Y, Z))) 25.12/25.34 = { by lemma 44 } 25.12/25.34 domain_difference(X, antidomain(multiplication(Y, domain(Z)))) 25.12/25.34 = { by lemma 51 } 25.12/25.34 multiplication(domain(X), domain(multiplication(Y, domain(Z)))) 25.12/25.34 = { by axiom 14 (forward_diamond) } 25.12/25.34 multiplication(domain(X), forward_diamond(Y, Z)) 25.12/25.34 25.12/25.34 Lemma 117: multiplication(domain(X), codomain(Y)) = domain_difference(X, coantidomain(Y)). 25.12/25.34 Proof: 25.12/25.34 multiplication(domain(X), codomain(Y)) 25.12/25.34 = { by lemma 115 } 25.12/25.34 multiplication(domain(X), forward_diamond(codomain(Y), backward_box(coantidomain(Y), codomain(Y)))) 25.12/25.34 = { by lemma 116 } 25.12/25.34 domain_difference(X, antidomain(forward_diamond(codomain(Y), backward_box(coantidomain(Y), codomain(Y))))) 25.12/25.34 = { by lemma 115 } 25.12/25.34 domain_difference(X, antidomain(codomain(Y))) 25.12/25.34 = { by lemma 85 } 25.12/25.34 domain_difference(X, coantidomain(Y)) 25.12/25.34 25.12/25.34 Lemma 118: addition(one, codomain(X)) = one. 25.12/25.34 Proof: 25.12/25.34 addition(one, codomain(X)) 25.12/25.34 = { by axiom 17 (additive_commutativity) } 25.12/25.34 addition(codomain(X), one) 25.12/25.34 = { by axiom 24 (codomain4) } 25.12/25.34 addition(coantidomain(coantidomain(X)), one) 25.12/25.34 = { by axiom 5 (codomain3) } 25.12/25.34 addition(coantidomain(coantidomain(X)), addition(coantidomain(coantidomain(X)), coantidomain(X))) 25.12/25.34 = { by lemma 56 } 25.12/25.34 addition(coantidomain(coantidomain(X)), coantidomain(X)) 25.12/25.34 = { by axiom 5 (codomain3) } 25.12/25.34 one 25.12/25.34 25.12/25.34 Lemma 119: addition(X, multiplication(codomain(Y), addition(X, Z))) = addition(X, multiplication(codomain(Y), Z)). 25.12/25.34 Proof: 25.12/25.34 addition(X, multiplication(codomain(Y), addition(X, Z))) 25.12/25.34 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.34 addition(multiplication(one, X), multiplication(codomain(Y), addition(X, Z))) 25.12/25.34 = { by lemma 72 } 25.12/25.34 addition(multiplication(codomain(Y), Z), multiplication(addition(one, codomain(Y)), X)) 25.12/25.34 = { by lemma 118 } 25.12/25.34 addition(multiplication(codomain(Y), Z), multiplication(one, X)) 25.12/25.34 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.34 addition(multiplication(codomain(Y), Z), X) 25.12/25.34 = { by axiom 17 (additive_commutativity) } 25.12/25.34 addition(X, multiplication(codomain(Y), Z)) 25.12/25.34 25.12/25.34 Lemma 120: domain_difference(one, X) = antidomain(X). 25.12/25.34 Proof: 25.12/25.34 domain_difference(one, X) 25.12/25.34 = { by lemma 65 } 25.12/25.34 domain_difference(antidomain(zero), X) 25.12/25.34 = { by lemma 35 } 25.12/25.34 multiplication(c(zero), antidomain(X)) 25.12/25.34 = { by lemma 66 } 25.12/25.34 multiplication(one, antidomain(X)) 25.12/25.34 = { by axiom 21 (multiplicative_left_identity) } 25.12/25.34 antidomain(X) 25.12/25.34 25.12/25.34 Lemma 121: domain(zero) = zero. 25.12/25.34 Proof: 25.12/25.34 domain(zero) 25.12/25.34 = { by axiom 13 (domain4) } 25.12/25.34 antidomain(antidomain(zero)) 25.12/25.34 = { by lemma 65 } 25.12/25.34 antidomain(one) 25.12/25.34 = { by lemma 63 } 25.12/25.34 zero 25.12/25.34 25.12/25.34 Lemma 122: forward_diamond(zero, X) = domain(zero). 25.12/25.34 Proof: 25.12/25.34 forward_diamond(zero, X) 25.12/25.34 = { by axiom 14 (forward_diamond) } 25.12/25.34 domain(multiplication(zero, domain(X))) 25.12/25.34 = { by axiom 15 (left_annihilation) } 25.12/25.34 domain(zero) 25.12/25.34 25.12/25.34 Lemma 123: forward_box(X, forward_diamond(Y, c(Z))) = c(forward_diamond(X, forward_box(Y, Z))). 25.12/25.34 Proof: 25.12/25.34 forward_box(X, forward_diamond(Y, c(Z))) 25.12/25.34 = { by axiom 3 (forward_box) } 25.12/25.34 c(forward_diamond(X, c(forward_diamond(Y, c(Z))))) 25.12/25.34 = { by axiom 3 (forward_box) } 25.12/25.34 c(forward_diamond(X, forward_box(Y, Z))) 25.12/25.34 25.12/25.34 Lemma 124: c(domain(X)) = domain(c(X)). 25.12/25.34 Proof: 25.12/25.34 c(domain(X)) 25.12/25.34 = { by lemma 34 } 25.12/25.34 domain(antidomain(domain(X))) 25.12/25.34 = { by axiom 11 (complement) } 25.12/25.34 domain(c(X)) 25.12/25.34 25.12/25.34 Lemma 125: forward_box(zero, X) = one. 25.12/25.34 Proof: 25.12/25.34 forward_box(zero, X) 25.12/25.34 = { by axiom 3 (forward_box) } 25.12/25.34 c(forward_diamond(zero, c(X))) 25.12/25.34 = { by lemma 122 } 25.12/25.34 c(domain(zero)) 25.12/25.34 = { by lemma 124 } 25.12/25.34 domain(c(zero)) 25.12/25.34 = { by lemma 66 } 25.12/25.34 domain(one) 25.12/25.34 = { by lemma 64 } 25.12/25.34 antidomain(zero) 25.12/25.34 = { by lemma 65 } 25.12/25.34 one 25.12/25.34 25.12/25.34 Lemma 126: backward_diamond(X, domain_difference(Y, X)) = zero. 25.12/25.34 Proof: 25.12/25.34 backward_diamond(X, domain_difference(Y, X)) 25.12/25.34 = { by lemma 96 } 25.12/25.34 codomain(backward_diamond(X, domain_difference(Y, X))) 25.12/25.34 = { by axiom 24 (codomain4) } 25.12/25.34 coantidomain(coantidomain(backward_diamond(X, domain_difference(Y, X)))) 25.12/25.34 = { by lemma 97 } 25.12/25.34 coantidomain(coantidomain(multiplication(codomain(domain_difference(Y, X)), X))) 25.12/25.34 = { by axiom 24 (codomain4) } 25.12/25.34 coantidomain(coantidomain(multiplication(coantidomain(coantidomain(domain_difference(Y, X))), X))) 25.12/25.34 = { by axiom 2 (codomain2) } 25.12/25.34 coantidomain(addition(coantidomain(multiplication(domain_difference(Y, X), X)), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(Y, X))), X)))) 25.12/25.34 = { by lemma 53 } 25.12/25.34 coantidomain(addition(coantidomain(multiplication(domain(Y), multiplication(antidomain(X), X))), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(Y, X))), X)))) 25.12/25.34 = { by axiom 16 (domain1) } 25.12/25.34 coantidomain(addition(coantidomain(multiplication(domain(Y), zero)), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(Y, X))), X)))) 25.12/25.34 = { by axiom 20 (right_annihilation) } 25.12/25.34 coantidomain(addition(coantidomain(zero), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(Y, X))), X)))) 25.12/25.34 = { by lemma 90 } 25.12/25.34 coantidomain(addition(one, coantidomain(multiplication(coantidomain(coantidomain(domain_difference(Y, X))), X)))) 25.12/25.34 = { by axiom 24 (codomain4) } 25.12/25.34 coantidomain(addition(one, coantidomain(multiplication(codomain(domain_difference(Y, X)), X)))) 25.12/25.34 = { by axiom 17 (additive_commutativity) } 25.12/25.34 coantidomain(addition(coantidomain(multiplication(codomain(domain_difference(Y, X)), X)), one)) 25.12/25.34 = { by lemma 82 } 25.12/25.34 coantidomain(addition(coantidomain(multiplication(codomain(domain_difference(Y, X)), X)), addition(coantidomain(multiplication(codomain(domain_difference(Y, X)), X)), codomain(multiplication(codomain(domain_difference(Y, X)), X))))) 25.12/25.34 = { by lemma 56 } 25.12/25.34 coantidomain(addition(coantidomain(multiplication(codomain(domain_difference(Y, X)), X)), codomain(multiplication(codomain(domain_difference(Y, X)), X)))) 25.12/25.34 = { by lemma 82 } 25.12/25.34 coantidomain(one) 25.12/25.34 = { by lemma 89 } 25.12/25.34 zero 25.12/25.34 25.12/25.34 Lemma 127: multiplication(codomain(X), antidomain(Y)) = domain_difference(codomain(X), Y). 25.12/25.34 Proof: 25.12/25.34 multiplication(codomain(X), antidomain(Y)) 25.12/25.34 = { by lemma 115 } 25.12/25.34 multiplication(forward_diamond(codomain(X), backward_box(coantidomain(X), codomain(X))), antidomain(Y)) 25.12/25.34 = { by lemma 110 } 25.12/25.34 multiplication(domain(multiplication(codomain(X), backward_box(coantidomain(X), codomain(X)))), antidomain(Y)) 25.12/25.34 = { by axiom 29 (domain_difference) } 25.12/25.34 domain_difference(multiplication(codomain(X), backward_box(coantidomain(X), codomain(X))), Y) 25.12/25.34 = { by lemma 114 } 25.12/25.34 domain_difference(codomain(X), Y) 25.12/25.34 25.12/25.34 Lemma 128: domain(multiplication(X, antidomain(Y))) = antidomain(forward_box(X, Y)). 25.12/25.34 Proof: 25.12/25.34 domain(multiplication(X, antidomain(Y))) 25.12/25.34 = { by lemma 39 } 25.12/25.34 domain(multiplication(X, c(Y))) 25.12/25.34 = { by lemma 49 } 25.12/25.34 forward_diamond(X, antidomain(Y)) 25.12/25.34 = { by lemma 48 } 25.12/25.34 antidomain(forward_box(X, Y)) 25.12/25.34 25.12/25.34 Lemma 129: multiplication(X, multiplication(Y, coantidomain(multiplication(X, Y)))) = zero. 25.12/25.34 Proof: 25.12/25.34 multiplication(X, multiplication(Y, coantidomain(multiplication(X, Y)))) 25.12/25.34 = { by axiom 25 (multiplicative_associativity) } 25.12/25.34 multiplication(multiplication(X, Y), coantidomain(multiplication(X, Y))) 25.12/25.34 = { by axiom 28 (codomain1) } 25.18/25.40 zero 25.18/25.40 25.18/25.40 Lemma 130: codomain(antidomain(X)) = antidomain(X). 25.18/25.40 Proof: 25.18/25.40 codomain(antidomain(X)) 25.18/25.40 = { by axiom 21 (multiplicative_left_identity) } 25.18/25.40 multiplication(one, codomain(antidomain(X))) 25.18/25.40 = { by lemma 57 } 25.18/25.40 multiplication(addition(one, antidomain(X)), codomain(antidomain(X))) 25.18/25.40 = { by lemma 88 } 25.18/25.40 addition(antidomain(X), codomain(antidomain(X))) 25.18/25.40 = { by axiom 23 (multiplicative_right_identity) } 25.18/25.40 addition(antidomain(X), multiplication(codomain(antidomain(X)), one)) 25.18/25.40 = { by lemma 31 } 25.18/25.40 addition(antidomain(X), multiplication(codomain(antidomain(X)), addition(antidomain(X), domain(X)))) 25.18/25.40 = { by lemma 119 } 25.18/25.40 addition(antidomain(X), multiplication(codomain(antidomain(X)), domain(X))) 25.18/25.40 = { by lemma 91 } 25.18/25.40 addition(antidomain(X), multiplication(backward_diamond(antidomain(X), one), domain(X))) 25.18/25.40 = { by lemma 104 } 25.18/25.40 addition(antidomain(X), multiplication(forward_diamond(backward_diamond(antidomain(X), one), backward_diamond(antidomain(X), one)), domain(X))) 25.18/25.40 = { by axiom 14 (forward_diamond) } 25.18/25.40 addition(antidomain(X), multiplication(domain(multiplication(backward_diamond(antidomain(X), one), domain(backward_diamond(antidomain(X), one)))), domain(X))) 25.18/25.40 = { by lemma 51 } 25.18/25.40 addition(antidomain(X), domain_difference(multiplication(backward_diamond(antidomain(X), one), domain(backward_diamond(antidomain(X), one))), antidomain(X))) 25.18/25.40 = { by lemma 105 } 25.18/25.40 addition(antidomain(X), multiplication(forward_diamond(backward_diamond(antidomain(X), one), backward_diamond(antidomain(X), one)), antidomain(antidomain(X)))) 25.18/25.40 = { by lemma 106 } 25.18/25.40 addition(antidomain(X), domain_difference(forward_diamond(backward_diamond(antidomain(X), one), backward_diamond(antidomain(X), one)), antidomain(X))) 25.18/25.40 = { by lemma 104 } 25.18/25.40 addition(antidomain(X), domain_difference(backward_diamond(antidomain(X), one), antidomain(X))) 25.18/25.40 = { by lemma 75 } 25.18/25.40 addition(antidomain(X), domain_difference(X, antidomain(backward_diamond(antidomain(X), one)))) 25.18/25.40 = { by lemma 107 } 25.18/25.40 addition(antidomain(X), domain_difference(X, coantidomain(multiplication(codomain(one), antidomain(X))))) 25.18/25.40 = { by axiom 29 (domain_difference) } 25.18/25.40 addition(antidomain(X), multiplication(domain(X), antidomain(coantidomain(multiplication(codomain(one), antidomain(X)))))) 25.18/25.40 = { by axiom 23 (multiplicative_right_identity) } 25.18/25.40 addition(antidomain(X), multiplication(domain(X), multiplication(antidomain(coantidomain(multiplication(codomain(one), antidomain(X)))), one))) 25.18/25.40 = { by lemma 82 } 25.18/25.40 addition(antidomain(X), multiplication(domain(X), multiplication(antidomain(coantidomain(multiplication(codomain(one), antidomain(X)))), addition(coantidomain(multiplication(codomain(one), antidomain(X))), codomain(multiplication(codomain(one), antidomain(X))))))) 25.18/25.40 = { by lemma 32 } 25.18/25.40 addition(antidomain(X), multiplication(domain(X), multiplication(antidomain(coantidomain(multiplication(codomain(one), antidomain(X)))), codomain(multiplication(codomain(one), antidomain(X)))))) 25.18/25.40 = { by lemma 53 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(X, coantidomain(multiplication(codomain(one), antidomain(X)))), codomain(multiplication(codomain(one), antidomain(X))))) 25.18/25.40 = { by lemma 107 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(X, antidomain(backward_diamond(antidomain(X), one))), codomain(multiplication(codomain(one), antidomain(X))))) 25.18/25.40 = { by axiom 8 (backward_diamond) } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(X, antidomain(backward_diamond(antidomain(X), one))), backward_diamond(antidomain(X), one))) 25.18/25.40 = { by lemma 109 } 25.18/25.40 addition(antidomain(X), multiplication(domain(X), backward_diamond(antidomain(X), one))) 25.18/25.40 = { by lemma 91 } 25.18/25.40 addition(antidomain(X), multiplication(domain(X), codomain(antidomain(X)))) 25.18/25.40 = { by lemma 117 } 25.18/25.40 addition(antidomain(X), domain_difference(X, coantidomain(antidomain(X)))) 25.18/25.40 = { by lemma 120 } 25.18/25.40 addition(antidomain(X), domain_difference(X, coantidomain(domain_difference(one, X)))) 25.18/25.40 = { by lemma 117 } 25.18/25.40 addition(antidomain(X), multiplication(domain(X), codomain(domain_difference(one, X)))) 25.18/25.40 = { by axiom 13 (domain4) } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(antidomain(X)), codomain(domain_difference(one, X)))) 25.18/25.40 = { by lemma 102 } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(antidomain(X)), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 62 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(codomain(domain_difference(one, X)), antidomain(X)), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 68 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(codomain(domain_difference(one, X)), domain(antidomain(X))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 39 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(codomain(domain_difference(one, X)), domain(c(X))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 124 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(codomain(domain_difference(one, X)), c(domain(X))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 67 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(codomain(domain_difference(one, X)), c(forward_diamond(X, one))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 125 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(codomain(domain_difference(one, X)), c(forward_diamond(X, forward_box(zero, ?)))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 123 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(codomain(domain_difference(one, X)), forward_box(X, forward_diamond(zero, c(?)))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 122 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(codomain(domain_difference(one, X)), forward_box(X, domain(zero))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 121 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(codomain(domain_difference(one, X)), forward_box(X, zero)), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 126 } 25.18/25.40 addition(antidomain(X), multiplication(domain_difference(codomain(domain_difference(one, X)), forward_box(X, backward_diamond(X, domain_difference(one, X)))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 127 } 25.18/25.40 addition(antidomain(X), multiplication(multiplication(codomain(domain_difference(one, X)), antidomain(forward_box(X, backward_diamond(X, domain_difference(one, X))))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 128 } 25.18/25.40 addition(antidomain(X), multiplication(multiplication(codomain(domain_difference(one, X)), domain(multiplication(X, antidomain(backward_diamond(X, domain_difference(one, X)))))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 107 } 25.18/25.40 addition(antidomain(X), multiplication(multiplication(codomain(domain_difference(one, X)), domain(multiplication(X, coantidomain(multiplication(codomain(domain_difference(one, X)), X))))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by axiom 21 (multiplicative_left_identity) } 25.18/25.40 addition(antidomain(X), multiplication(multiplication(one, multiplication(codomain(domain_difference(one, X)), domain(multiplication(X, coantidomain(multiplication(codomain(domain_difference(one, X)), X)))))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 65 } 25.18/25.40 addition(antidomain(X), multiplication(multiplication(antidomain(zero), multiplication(codomain(domain_difference(one, X)), domain(multiplication(X, coantidomain(multiplication(codomain(domain_difference(one, X)), X)))))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 129 } 25.18/25.40 addition(antidomain(X), multiplication(multiplication(antidomain(multiplication(codomain(domain_difference(one, X)), multiplication(X, coantidomain(multiplication(codomain(domain_difference(one, X)), X))))), multiplication(codomain(domain_difference(one, X)), domain(multiplication(X, coantidomain(multiplication(codomain(domain_difference(one, X)), X)))))), domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by lemma 81 } 25.18/25.40 addition(antidomain(X), multiplication(zero, domain(codomain(domain_difference(one, X))))) 25.18/25.40 = { by axiom 15 (left_annihilation) } 25.18/25.40 addition(antidomain(X), zero) 25.18/25.40 = { by axiom 9 (additive_identity) } 25.18/25.40 antidomain(X) 25.18/25.40 25.18/25.40 Lemma 131: coantidomain(antidomain(X)) = domain(X). 25.18/25.40 Proof: 25.18/25.40 coantidomain(antidomain(X)) 25.18/25.40 = { by lemma 85 } 25.18/25.40 antidomain(codomain(antidomain(X))) 25.18/25.40 = { by lemma 130 } 25.18/25.40 antidomain(antidomain(X)) 25.18/25.40 = { by axiom 13 (domain4) } 25.18/25.40 domain(X) 25.18/25.40 25.18/25.40 Lemma 132: multiplication(domain(X), coantidomain(antidomain(X))) = coantidomain(antidomain(X)). 25.18/25.40 Proof: 25.18/25.40 multiplication(domain(X), coantidomain(antidomain(X))) 25.18/25.40 = { by lemma 93 } 25.18/25.40 multiplication(addition(antidomain(X), domain(X)), coantidomain(antidomain(X))) 25.18/25.40 = { by lemma 31 } 25.18/25.40 multiplication(one, coantidomain(antidomain(X))) 25.18/25.40 = { by axiom 21 (multiplicative_left_identity) } 25.18/25.40 coantidomain(antidomain(X)) 25.18/25.40 25.18/25.40 Lemma 133: multiplication(antidomain(X), multiplication(X, Y)) = zero. 25.18/25.40 Proof: 25.18/25.40 multiplication(antidomain(X), multiplication(X, Y)) 25.18/25.40 = { by axiom 25 (multiplicative_associativity) } 25.18/25.40 multiplication(multiplication(antidomain(X), X), Y) 25.18/25.40 = { by axiom 16 (domain1) } 25.18/25.40 multiplication(zero, Y) 25.18/25.40 = { by axiom 15 (left_annihilation) } 25.18/25.40 zero 25.18/25.40 25.18/25.40 Lemma 134: multiplication(antidomain(addition(X, Y)), addition(X, Z)) = multiplication(antidomain(addition(X, Y)), Z). 25.18/25.40 Proof: 25.18/25.40 multiplication(antidomain(addition(X, Y)), addition(X, Z)) 25.18/25.40 = { by axiom 17 (additive_commutativity) } 25.18/25.40 multiplication(antidomain(addition(X, Y)), addition(Z, X)) 25.18/25.40 = { by lemma 32 } 25.18/25.40 multiplication(antidomain(addition(X, Y)), addition(addition(X, Y), addition(Z, X))) 25.18/25.40 = { by axiom 17 (additive_commutativity) } 25.18/25.40 multiplication(antidomain(addition(X, Y)), addition(addition(Z, X), addition(X, Y))) 25.18/25.40 = { by axiom 30 (additive_associativity) } 25.18/25.40 multiplication(antidomain(addition(X, Y)), addition(Z, addition(X, addition(X, Y)))) 25.18/25.40 = { by lemma 56 } 25.18/25.40 multiplication(antidomain(addition(X, Y)), addition(Z, addition(X, Y))) 25.18/25.40 = { by lemma 33 } 25.18/25.40 multiplication(antidomain(addition(X, Y)), Z) 25.18/25.40 25.18/25.40 Lemma 135: addition(antidomain(X), antidomain(addition(Y, domain(X)))) = antidomain(X). 25.18/25.40 Proof: 25.18/25.40 addition(antidomain(X), antidomain(addition(Y, domain(X)))) 25.18/25.40 = { by axiom 23 (multiplicative_right_identity) } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(addition(Y, domain(X))), one)) 25.18/25.40 = { by lemma 134 } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(addition(Y, domain(X))), addition(Y, one))) 25.18/25.40 = { by lemma 31 } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(addition(Y, domain(X))), addition(Y, addition(antidomain(X), domain(X))))) 25.18/25.40 = { by axiom 30 (additive_associativity) } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(addition(Y, domain(X))), addition(addition(Y, antidomain(X)), domain(X)))) 25.18/25.40 = { by axiom 17 (additive_commutativity) } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(addition(Y, domain(X))), addition(addition(antidomain(X), Y), domain(X)))) 25.18/25.40 = { by axiom 30 (additive_associativity) } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(addition(Y, domain(X))), addition(antidomain(X), addition(Y, domain(X))))) 25.18/25.40 = { by lemma 33 } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(addition(Y, domain(X))), antidomain(X))) 25.18/25.40 = { by lemma 59 } 25.18/25.40 addition(antidomain(X), domain_difference(antidomain(addition(Y, domain(X))), X)) 25.18/25.40 = { by lemma 58 } 25.18/25.40 antidomain(X) 25.18/25.40 25.18/25.40 Lemma 136: addition(antidomain(X), domain(domain_difference(Y, X))) = antidomain(X). 25.18/25.40 Proof: 25.18/25.40 addition(antidomain(X), domain(domain_difference(Y, X))) 25.18/25.40 = { by axiom 13 (domain4) } 25.18/25.40 addition(antidomain(X), antidomain(antidomain(domain_difference(Y, X)))) 25.18/25.40 = { by lemma 73 } 25.18/25.40 addition(antidomain(X), domain_difference(X, antidomain(domain_difference(Y, X)))) 25.18/25.40 = { by axiom 21 (multiplicative_left_identity) } 25.18/25.40 addition(antidomain(X), multiplication(one, domain_difference(X, antidomain(domain_difference(Y, X))))) 25.18/25.40 = { by lemma 65 } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(zero), domain_difference(X, antidomain(domain_difference(Y, X))))) 25.18/25.40 = { by lemma 60 } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(multiplication(domain(X), domain_difference(Y, X))), domain_difference(X, antidomain(domain_difference(Y, X))))) 25.18/25.40 = { by lemma 51 } 25.18/25.40 addition(antidomain(X), multiplication(antidomain(multiplication(domain(X), domain_difference(Y, X))), multiplication(domain(X), domain(domain_difference(Y, X))))) 25.18/25.40 = { by lemma 81 } 25.18/25.40 addition(antidomain(X), zero) 25.18/25.40 = { by axiom 9 (additive_identity) } 25.18/25.40 antidomain(X) 25.18/25.40 25.18/25.40 Lemma 137: forward_box(X, antidomain(Y)) = antidomain(forward_diamond(X, Y)). 25.18/25.40 Proof: 25.18/25.40 forward_box(X, antidomain(Y)) 25.18/25.40 = { by lemma 40 } 25.18/25.40 antidomain(forward_diamond(X, antidomain(antidomain(Y)))) 25.18/25.40 = { by axiom 13 (domain4) } 25.18/25.40 antidomain(forward_diamond(X, domain(Y))) 25.18/25.40 = { by lemma 43 } 25.18/25.40 antidomain(forward_diamond(X, Y)) 25.18/25.40 25.18/25.40 Lemma 138: domain_difference(X, zero) = domain(X). 25.18/25.40 Proof: 25.18/25.40 domain_difference(X, zero) 25.18/25.40 = { by axiom 29 (domain_difference) } 25.18/25.40 multiplication(domain(X), antidomain(zero)) 25.18/25.40 = { by lemma 65 } 25.18/25.40 multiplication(domain(X), one) 25.18/25.40 = { by axiom 23 (multiplicative_right_identity) } 25.18/25.40 domain(X) 25.18/25.40 25.18/25.40 Lemma 139: domain_difference(antidomain(X), Y) = domain_difference(antidomain(Y), X). 25.18/25.40 Proof: 25.18/25.40 domain_difference(antidomain(X), Y) 25.18/25.40 = { by lemma 71 } 25.18/25.40 multiplication(antidomain(Y), domain(antidomain(X))) 25.18/25.40 = { by lemma 68 } 25.18/25.40 multiplication(antidomain(Y), antidomain(X)) 25.18/25.40 = { by lemma 59 } 25.18/25.40 domain_difference(antidomain(Y), X) 25.18/25.40 25.18/25.40 Lemma 140: multiplication(domain_difference(antidomain(X), multiplication(antidomain(X), Y)), domain(Y)) = zero. 25.18/25.40 Proof: 25.18/25.40 multiplication(domain_difference(antidomain(X), multiplication(antidomain(X), Y)), domain(Y)) 25.18/25.40 = { by lemma 139 } 25.18/25.40 multiplication(domain_difference(antidomain(multiplication(antidomain(X), Y)), X), domain(Y)) 25.18/25.40 = { by lemma 54 } 25.18/25.40 multiplication(antidomain(multiplication(antidomain(X), Y)), multiplication(antidomain(X), domain(Y))) 25.18/25.40 = { by lemma 81 } 25.18/25.40 zero 25.18/25.40 25.18/25.40 Lemma 141: antidomain(forward_diamond(X, Y)) = antidomain(multiplication(X, Y)). 25.18/25.40 Proof: 25.18/25.40 antidomain(forward_diamond(X, Y)) 25.18/25.40 = { by lemma 137 } 25.18/25.40 forward_box(X, antidomain(Y)) 25.18/25.40 = { by lemma 45 } 25.18/25.40 antidomain(multiplication(X, antidomain(antidomain(Y)))) 25.18/25.40 = { by axiom 10 (domain2) } 25.18/25.40 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) 25.18/25.40 = { by lemma 45 } 25.18/25.40 addition(antidomain(multiplication(X, Y)), forward_box(X, antidomain(Y))) 25.18/25.40 = { by lemma 137 } 25.18/25.40 addition(antidomain(multiplication(X, Y)), antidomain(forward_diamond(X, Y))) 25.18/25.40 = { by lemma 73 } 25.18/25.40 addition(antidomain(multiplication(X, Y)), domain_difference(multiplication(X, Y), forward_diamond(X, Y))) 25.18/25.40 = { by lemma 71 } 25.18/25.40 addition(antidomain(multiplication(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), domain(multiplication(X, Y)))) 25.18/25.40 = { by lemma 52 } 25.18/25.40 addition(antidomain(multiplication(X, Y)), domain_difference(antidomain(forward_diamond(X, Y)), antidomain(multiplication(X, Y)))) 25.18/25.40 = { by lemma 51 } 25.18/25.40 addition(antidomain(multiplication(X, Y)), multiplication(domain(antidomain(forward_diamond(X, Y))), domain(multiplication(X, Y)))) 25.18/25.40 = { by lemma 138 } 25.18/25.40 addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), zero), domain(multiplication(X, Y)))) 25.18/25.40 = { by lemma 133 } 25.18/25.40 addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(multiplication(X, domain(Y))), multiplication(multiplication(X, domain(Y)), Y))), domain(multiplication(X, Y)))) 25.18/25.40 = { by lemma 44 } 25.18/25.40 addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), multiplication(multiplication(X, domain(Y)), Y))), domain(multiplication(X, Y)))) 25.18/25.40 = { by axiom 25 (multiplicative_associativity) } 25.18/25.40 addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), multiplication(X, multiplication(domain(Y), Y)))), domain(multiplication(X, Y)))) 25.18/25.40 = { by lemma 38 } 25.18/25.40 addition(antidomain(multiplication(X, Y)), multiplication(domain_difference(antidomain(forward_diamond(X, Y)), multiplication(antidomain(forward_diamond(X, Y)), multiplication(X, Y))), domain(multiplication(X, Y)))) 25.18/25.40 = { by lemma 140 } 25.18/25.40 addition(antidomain(multiplication(X, Y)), zero) 25.18/25.40 = { by axiom 9 (additive_identity) } 25.18/25.40 antidomain(multiplication(X, Y)) 25.18/25.40 25.18/25.40 Lemma 142: domain(multiplication(X, Y)) = forward_diamond(X, Y). 25.18/25.40 Proof: 25.18/25.40 domain(multiplication(X, Y)) 25.18/25.40 = { by axiom 13 (domain4) } 25.18/25.40 antidomain(antidomain(multiplication(X, Y))) 25.18/25.40 = { by lemma 141 } 25.18/25.40 antidomain(antidomain(forward_diamond(X, Y))) 25.18/25.40 = { by axiom 13 (domain4) } 25.18/25.40 domain(forward_diamond(X, Y)) 25.18/25.40 = { by lemma 47 } 25.18/25.40 forward_diamond(X, Y) 25.18/25.40 25.18/25.40 Lemma 143: domain_difference(forward_diamond(X, Y), Z) = domain_difference(multiplication(X, Y), Z). 25.18/25.40 Proof: 25.18/25.40 domain_difference(forward_diamond(X, Y), Z) 25.18/25.40 = { by lemma 106 } 25.18/25.40 multiplication(forward_diamond(X, Y), antidomain(Z)) 25.18/25.40 = { by lemma 142 } 25.18/25.40 multiplication(domain(multiplication(X, Y)), antidomain(Z)) 25.18/25.40 = { by axiom 29 (domain_difference) } 25.18/25.40 domain_difference(multiplication(X, Y), Z) 25.18/25.40 25.18/25.40 Lemma 144: multiplication(domain(X), forward_diamond(Z, Y)) = multiplication(forward_diamond(Z, Y), domain(X)). 25.18/25.40 Proof: 25.18/25.40 multiplication(domain(X), forward_diamond(Z, Y)) 25.18/25.40 = { by lemma 116 } 25.18/25.40 domain_difference(X, antidomain(forward_diamond(Z, Y))) 25.18/25.40 = { by lemma 75 } 25.18/25.40 domain_difference(forward_diamond(Z, Y), antidomain(X)) 25.18/25.40 = { by lemma 106 } 25.18/25.40 multiplication(forward_diamond(Z, Y), antidomain(antidomain(X))) 25.18/25.40 = { by lemma 105 } 25.18/25.40 domain_difference(multiplication(Z, domain(Y)), antidomain(X)) 25.18/25.40 = { by lemma 51 } 25.18/25.40 multiplication(domain(multiplication(Z, domain(Y))), domain(X)) 25.18/25.40 = { by axiom 14 (forward_diamond) } 25.18/25.40 multiplication(forward_diamond(Z, Y), domain(X)) 25.18/25.40 25.18/25.40 Lemma 145: multiplication(domain(Z), domain_difference(multiplication(X, Y), W)) = multiplication(forward_diamond(X, Y), domain_difference(Z, W)). 25.18/25.40 Proof: 25.18/25.40 multiplication(domain(Z), domain_difference(multiplication(X, Y), W)) 25.18/25.40 = { by lemma 143 } 25.18/25.40 multiplication(domain(Z), domain_difference(forward_diamond(X, Y), W)) 25.18/25.40 = { by lemma 106 } 25.18/25.40 multiplication(domain(Z), multiplication(forward_diamond(X, Y), antidomain(W))) 25.18/25.40 = { by axiom 25 (multiplicative_associativity) } 25.18/25.40 multiplication(multiplication(domain(Z), forward_diamond(X, Y)), antidomain(W)) 25.18/25.40 = { by lemma 144 } 25.18/25.40 multiplication(multiplication(forward_diamond(X, Y), domain(Z)), antidomain(W)) 25.18/25.40 = { by axiom 25 (multiplicative_associativity) } 25.18/25.40 multiplication(forward_diamond(X, Y), multiplication(domain(Z), antidomain(W))) 25.18/25.40 = { by axiom 29 (domain_difference) } 25.18/25.40 multiplication(forward_diamond(X, Y), domain_difference(Z, W)) 25.18/25.40 25.18/25.40 Lemma 146: multiplication(domain(X), domain_difference(Y, Z)) = multiplication(domain(Y), domain_difference(X, Z)). 25.18/25.40 Proof: 25.18/25.40 multiplication(domain(X), domain_difference(Y, Z)) 25.18/25.40 = { by lemma 99 } 25.18/25.40 multiplication(forward_diamond(one, X), domain_difference(Y, Z)) 25.18/25.40 = { by lemma 145 } 25.18/25.40 multiplication(domain(Y), domain_difference(multiplication(one, X), Z)) 25.18/25.40 = { by axiom 21 (multiplicative_left_identity) } 25.23/25.43 multiplication(domain(Y), domain_difference(X, Z)) 25.23/25.43 25.23/25.43 Lemma 147: domain(domain_difference(X, Y)) = domain_difference(X, Y). 25.23/25.43 Proof: 25.23/25.43 domain(domain_difference(X, Y)) 25.23/25.43 = { by axiom 13 (domain4) } 25.23/25.43 antidomain(antidomain(domain_difference(X, Y))) 25.23/25.43 = { by lemma 135 } 25.23/25.43 antidomain(addition(antidomain(domain_difference(X, Y)), antidomain(addition(domain(X), domain(domain_difference(X, Y)))))) 25.23/25.43 = { by axiom 17 (additive_commutativity) } 25.23/25.43 antidomain(addition(antidomain(addition(domain(X), domain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y)))) 25.23/25.43 = { by axiom 23 (multiplicative_right_identity) } 25.23/25.43 multiplication(antidomain(addition(antidomain(addition(domain(X), domain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y)))), one) 25.23/25.43 = { by lemma 31 } 25.23/25.43 multiplication(antidomain(addition(antidomain(addition(domain(X), domain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y)))), addition(antidomain(addition(domain(X), domain(domain_difference(X, Y)))), domain(addition(domain(X), domain(domain_difference(X, Y)))))) 25.23/25.43 = { by lemma 134 } 25.23/25.43 multiplication(antidomain(addition(antidomain(addition(domain(X), domain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y)))), domain(addition(domain(X), domain(domain_difference(X, Y))))) 25.23/25.43 = { by lemma 71 } 25.23/25.43 domain_difference(addition(domain(X), domain(domain_difference(X, Y))), addition(antidomain(addition(domain(X), domain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y)))) 25.23/25.43 = { by axiom 17 (additive_commutativity) } 25.23/25.43 domain_difference(addition(domain(X), domain(domain_difference(X, Y))), addition(antidomain(domain_difference(X, Y)), antidomain(addition(domain(X), domain(domain_difference(X, Y)))))) 25.23/25.43 = { by lemma 135 } 25.23/25.43 domain_difference(addition(domain(X), domain(domain_difference(X, Y))), antidomain(domain_difference(X, Y))) 25.23/25.43 = { by lemma 75 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(domain(X), domain(domain_difference(X, Y))))) 25.23/25.43 = { by axiom 13 (domain4) } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), domain(domain_difference(X, Y))))) 25.23/25.43 = { by axiom 13 (domain4) } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), antidomain(antidomain(domain_difference(X, Y)))))) 25.23/25.43 = { by lemma 80 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), antidomain(forward_box(domain(X), Y))))) 25.23/25.43 = { by lemma 131 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), antidomain(forward_box(coantidomain(antidomain(X)), Y))))) 25.23/25.43 = { by lemma 73 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), domain_difference(antidomain(X), forward_box(coantidomain(antidomain(X)), Y))))) 25.23/25.43 = { by lemma 39 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), domain_difference(c(X), forward_box(coantidomain(antidomain(X)), Y))))) 25.23/25.43 = { by axiom 11 (complement) } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), domain_difference(antidomain(domain(X)), forward_box(coantidomain(antidomain(X)), Y))))) 25.23/25.43 = { by lemma 132 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), domain_difference(antidomain(domain(X)), forward_box(multiplication(domain(X), coantidomain(antidomain(X))), Y))))) 25.23/25.43 = { by lemma 59 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), multiplication(antidomain(domain(X)), antidomain(forward_box(multiplication(domain(X), coantidomain(antidomain(X))), Y)))))) 25.23/25.43 = { by lemma 120 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), multiplication(domain_difference(one, domain(X)), antidomain(forward_box(multiplication(domain(X), coantidomain(antidomain(X))), Y)))))) 25.23/25.43 = { by lemma 65 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), multiplication(domain_difference(antidomain(zero), domain(X)), antidomain(forward_box(multiplication(domain(X), coantidomain(antidomain(X))), Y)))))) 25.23/25.43 = { by lemma 48 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), multiplication(domain_difference(antidomain(zero), domain(X)), forward_diamond(multiplication(domain(X), coantidomain(antidomain(X))), antidomain(Y)))))) 25.23/25.43 = { by axiom 14 (forward_diamond) } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), multiplication(domain_difference(antidomain(zero), domain(X)), domain(multiplication(multiplication(domain(X), coantidomain(antidomain(X))), domain(antidomain(Y)))))))) 25.23/25.43 = { by axiom 25 (multiplicative_associativity) } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), multiplication(domain_difference(antidomain(zero), domain(X)), domain(multiplication(domain(X), multiplication(coantidomain(antidomain(X)), domain(antidomain(Y))))))))) 25.23/25.43 = { by lemma 34 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), multiplication(domain_difference(antidomain(zero), domain(X)), domain(multiplication(domain(X), multiplication(coantidomain(antidomain(X)), c(Y)))))))) 25.23/25.43 = { by lemma 39 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), multiplication(domain_difference(antidomain(zero), domain(X)), domain(multiplication(domain(X), multiplication(coantidomain(antidomain(X)), antidomain(Y)))))))) 25.23/25.43 = { by lemma 54 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), multiplication(antidomain(zero), multiplication(antidomain(domain(X)), domain(multiplication(domain(X), multiplication(coantidomain(antidomain(X)), antidomain(Y))))))))) 25.23/25.43 = { by lemma 133 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), multiplication(antidomain(multiplication(antidomain(domain(X)), multiplication(domain(X), multiplication(coantidomain(antidomain(X)), antidomain(Y))))), multiplication(antidomain(domain(X)), domain(multiplication(domain(X), multiplication(coantidomain(antidomain(X)), antidomain(Y))))))))) 25.23/25.43 = { by lemma 81 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(addition(antidomain(antidomain(X)), zero))) 25.23/25.43 = { by axiom 9 (additive_identity) } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(antidomain(antidomain(X)))) 25.23/25.43 = { by axiom 13 (domain4) } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(domain(X))) 25.23/25.43 = { by lemma 77 } 25.23/25.43 domain_difference(domain_difference(X, Y), antidomain(X)) 25.23/25.43 = { by lemma 75 } 25.23/25.43 domain_difference(X, antidomain(domain_difference(X, Y))) 25.23/25.43 = { by lemma 51 } 25.23/25.43 multiplication(domain(X), domain(domain_difference(X, Y))) 25.23/25.43 = { by lemma 113 } 25.23/25.43 multiplication(domain(X), addition(domain(domain_difference(X, Y)), domain_difference(domain_difference(X, Y), Y))) 25.23/25.43 = { by lemma 50 } 25.23/25.43 multiplication(domain(X), addition(domain(domain_difference(X, Y)), domain_difference(domain(domain_difference(X, Y)), Y))) 25.23/25.43 = { by axiom 17 (additive_commutativity) } 25.23/25.43 multiplication(domain(X), addition(domain_difference(domain(domain_difference(X, Y)), Y), domain(domain_difference(X, Y)))) 25.23/25.43 = { by lemma 38 } 25.23/25.43 multiplication(domain(X), addition(domain_difference(domain(domain_difference(X, Y)), Y), multiplication(domain(domain(domain_difference(X, Y))), domain(domain_difference(X, Y))))) 25.23/25.43 = { by lemma 69 } 25.23/25.43 multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), addition(domain(domain_difference(X, Y)), antidomain(Y)))) 25.23/25.43 = { by axiom 17 (additive_commutativity) } 25.23/25.43 multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), addition(antidomain(Y), domain(domain_difference(X, Y))))) 25.23/25.43 = { by lemma 136 } 25.23/25.43 multiplication(domain(X), multiplication(domain(domain(domain_difference(X, Y))), antidomain(Y))) 25.23/25.43 = { by axiom 29 (domain_difference) } 25.23/25.43 multiplication(domain(X), domain_difference(domain(domain_difference(X, Y)), Y)) 25.23/25.43 = { by lemma 50 } 25.23/25.43 multiplication(domain(X), domain_difference(domain_difference(X, Y), Y)) 25.23/25.43 = { by lemma 146 } 25.23/25.43 multiplication(domain(domain_difference(X, Y)), domain_difference(X, Y)) 25.23/25.43 = { by lemma 38 } 25.23/25.43 domain_difference(X, Y) 25.23/25.43 25.23/25.43 Lemma 148: domain_difference(X, antidomain(Y)) = forward_diamond(domain(X), Y). 25.23/25.43 Proof: 25.23/25.43 domain_difference(X, antidomain(Y)) 25.23/25.43 = { by lemma 147 } 25.23/25.43 domain(domain_difference(X, antidomain(Y))) 25.23/25.43 = { by lemma 74 } 25.23/25.43 forward_diamond(domain(X), Y) 25.23/25.43 25.23/25.43 Lemma 149: addition(domain(Y), domain_difference(antidomain(X), Y)) = addition(antidomain(X), domain(Y)). 25.23/25.43 Proof: 25.23/25.43 addition(domain(Y), domain_difference(antidomain(X), Y)) 25.23/25.43 = { by axiom 13 (domain4) } 25.23/25.43 addition(antidomain(antidomain(Y)), domain_difference(antidomain(X), Y)) 25.23/25.43 = { by lemma 139 } 25.23/25.43 addition(antidomain(antidomain(Y)), domain_difference(antidomain(Y), X)) 25.23/25.43 = { by lemma 73 } 25.23/25.43 addition(antidomain(antidomain(Y)), antidomain(X)) 25.23/25.43 = { by axiom 13 (domain4) } 25.23/25.43 addition(domain(Y), antidomain(X)) 25.23/25.43 = { by axiom 17 (additive_commutativity) } 25.23/25.43 addition(antidomain(X), domain(Y)) 25.23/25.43 25.23/25.43 Lemma 150: forward_diamond(antidomain(X), Y) = domain(domain_difference(Y, X)). 25.23/25.43 Proof: 25.23/25.43 forward_diamond(antidomain(X), Y) 25.23/25.43 = { by axiom 14 (forward_diamond) } 25.23/25.43 domain(multiplication(antidomain(X), domain(Y))) 25.23/25.43 = { by lemma 71 } 25.23/25.43 domain(domain_difference(Y, X)) 25.23/25.43 25.23/25.43 Lemma 151: forward_diamond(addition(Y, addition(antidomain(X), Z)), X) = forward_diamond(addition(Y, Z), X). 25.23/25.43 Proof: 25.23/25.43 forward_diamond(addition(Y, addition(antidomain(X), Z)), X) 25.23/25.43 = { by axiom 30 (additive_associativity) } 25.23/25.43 forward_diamond(addition(addition(Y, antidomain(X)), Z), X) 25.23/25.43 = { by axiom 17 (additive_commutativity) } 25.23/25.43 forward_diamond(addition(Z, addition(Y, antidomain(X))), X) 25.23/25.43 = { by lemma 77 } 25.23/25.43 forward_diamond(addition(Z, addition(Y, antidomain(domain(X)))), X) 25.23/25.43 = { by axiom 14 (forward_diamond) } 25.23/25.43 domain(multiplication(addition(Z, addition(Y, antidomain(domain(X)))), domain(X))) 25.23/25.43 = { by axiom 17 (additive_commutativity) } 25.23/25.43 domain(multiplication(addition(Z, addition(antidomain(domain(X)), Y)), domain(X))) 25.23/25.43 = { by axiom 30 (additive_associativity) } 25.23/25.43 domain(multiplication(addition(addition(Z, antidomain(domain(X))), Y), domain(X))) 25.23/25.43 = { by axiom 7 (left_distributivity) } 25.23/25.43 domain(addition(multiplication(addition(Z, antidomain(domain(X))), domain(X)), multiplication(Y, domain(X)))) 25.23/25.43 = { by lemma 36 } 25.23/25.43 domain(addition(multiplication(Z, domain(X)), multiplication(Y, domain(X)))) 25.23/25.43 = { by axiom 7 (left_distributivity) } 25.23/25.43 domain(multiplication(addition(Z, Y), domain(X))) 25.23/25.43 = { by axiom 14 (forward_diamond) } 25.23/25.43 forward_diamond(addition(Z, Y), X) 25.23/25.43 = { by axiom 17 (additive_commutativity) } 25.23/25.43 forward_diamond(addition(Y, Z), X) 25.23/25.43 25.23/25.43 Lemma 152: domain_difference(X, antidomain(forward_box(Y, Z))) = multiplication(domain(X), forward_box(Y, Z)). 25.23/25.43 Proof: 25.23/25.43 domain_difference(X, antidomain(forward_box(Y, Z))) 25.23/25.43 = { by lemma 48 } 25.23/25.43 domain_difference(X, forward_diamond(Y, antidomain(Z))) 25.23/25.43 = { by lemma 39 } 25.23/25.43 domain_difference(X, forward_diamond(Y, c(Z))) 25.23/25.43 = { by lemma 79 } 25.23/25.43 domain_difference(X, domain(forward_diamond(Y, c(Z)))) 25.23/25.43 = { by lemma 78 } 25.23/25.43 multiplication(domain(X), c(forward_diamond(Y, c(Z)))) 25.23/25.43 = { by axiom 3 (forward_box) } 25.23/25.43 multiplication(domain(X), forward_box(Y, Z)) 25.23/25.43 25.23/25.43 Lemma 153: domain(multiplication(X, forward_box(Y, Z))) = forward_diamond(X, forward_box(Y, Z)). 25.23/25.43 Proof: 25.23/25.43 domain(multiplication(X, forward_box(Y, Z))) 25.23/25.43 = { by axiom 3 (forward_box) } 25.23/25.43 domain(multiplication(X, c(forward_diamond(Y, c(Z))))) 25.23/25.43 = { by lemma 49 } 25.23/25.43 forward_diamond(X, antidomain(forward_diamond(Y, c(Z)))) 25.23/25.43 = { by lemma 48 } 25.23/25.43 antidomain(forward_box(X, forward_diamond(Y, c(Z)))) 25.23/25.43 = { by lemma 123 } 25.23/25.43 antidomain(c(forward_diamond(X, forward_box(Y, Z)))) 25.23/25.43 = { by lemma 39 } 25.23/25.43 antidomain(antidomain(forward_diamond(X, forward_box(Y, Z)))) 25.23/25.43 = { by axiom 13 (domain4) } 25.23/25.43 domain(forward_diamond(X, forward_box(Y, Z))) 25.23/25.43 = { by lemma 47 } 25.23/25.43 forward_diamond(X, forward_box(Y, Z)) 25.23/25.43 25.23/25.43 Lemma 154: addition(one, domain_difference(X, Y)) = one. 25.23/25.43 Proof: 25.23/25.43 addition(one, domain_difference(X, Y)) 25.23/25.43 = { by axiom 17 (additive_commutativity) } 25.23/25.43 addition(domain_difference(X, Y), one) 25.23/25.43 = { by axiom 17 (additive_commutativity) } 25.23/25.43 addition(one, domain_difference(X, Y)) 25.23/25.43 = { by lemma 57 } 25.23/25.43 addition(addition(one, antidomain(Y)), domain_difference(X, Y)) 25.23/25.43 = { by axiom 30 (additive_associativity) } 25.23/25.43 addition(one, addition(antidomain(Y), domain_difference(X, Y))) 25.23/25.43 = { by lemma 58 } 25.23/25.43 addition(one, antidomain(Y)) 25.23/25.43 = { by lemma 57 } 25.23/25.43 one 25.23/25.43 25.23/25.43 Lemma 155: backward_diamond(one, X) = codomain(codomain(X)). 25.23/25.43 Proof: 25.23/25.43 backward_diamond(one, X) 25.23/25.43 = { by axiom 8 (backward_diamond) } 25.23/25.43 codomain(multiplication(codomain(X), one)) 25.23/25.43 = { by axiom 23 (multiplicative_right_identity) } 25.23/25.43 codomain(codomain(X)) 25.23/25.43 25.23/25.43 Lemma 156: domain_difference(X, codomain(antidomain(X))) = coantidomain(antidomain(X)). 25.23/25.43 Proof: 25.23/25.43 domain_difference(X, codomain(antidomain(X))) 25.23/25.43 = { by lemma 95 } 25.23/25.43 domain_difference(X, codomain(codomain(antidomain(X)))) 25.23/25.43 = { by lemma 155 } 25.23/25.43 domain_difference(X, backward_diamond(one, antidomain(X))) 25.23/25.43 = { by lemma 111 } 25.23/25.43 multiplication(domain(X), backward_box(one, X)) 25.23/25.43 = { by axiom 27 (backward_box) } 25.23/25.43 multiplication(domain(X), c(backward_diamond(one, c(X)))) 25.23/25.43 = { by lemma 155 } 25.23/25.43 multiplication(domain(X), c(codomain(codomain(c(X))))) 25.23/25.43 = { by lemma 39 } 25.23/25.43 multiplication(domain(X), antidomain(codomain(codomain(c(X))))) 25.23/25.43 = { by lemma 95 } 25.23/25.43 multiplication(domain(X), antidomain(codomain(c(X)))) 25.23/25.43 = { by lemma 39 } 25.23/25.43 multiplication(domain(X), antidomain(codomain(antidomain(X)))) 25.23/25.43 = { by lemma 85 } 25.23/25.43 multiplication(domain(X), coantidomain(antidomain(X))) 25.23/25.43 = { by lemma 132 } 25.23/25.43 coantidomain(antidomain(X)) 25.23/25.43 25.23/25.43 Lemma 157: addition(Z, multiplication(X, multiplication(Y, Z))) = multiplication(addition(one, multiplication(X, Y)), Z). 25.23/25.43 Proof: 25.23/25.43 addition(Z, multiplication(X, multiplication(Y, Z))) 25.23/25.43 = { by axiom 25 (multiplicative_associativity) } 25.23/25.43 addition(Z, multiplication(multiplication(X, Y), Z)) 25.23/25.43 = { by lemma 55 } 25.23/25.43 multiplication(addition(multiplication(X, Y), one), Z) 25.23/25.43 = { by axiom 17 (additive_commutativity) } 25.23/25.43 multiplication(addition(one, multiplication(X, Y)), Z) 25.23/25.43 25.23/25.43 Lemma 158: codomain(domain(X)) = domain(X). 25.23/25.43 Proof: 25.23/25.43 codomain(domain(X)) 25.23/25.43 = { by axiom 13 (domain4) } 25.23/25.43 codomain(antidomain(antidomain(X))) 25.23/25.43 = { by lemma 130 } 25.23/25.43 antidomain(antidomain(X)) 25.23/25.43 = { by axiom 13 (domain4) } 25.23/25.44 domain(X) 25.23/25.44 25.23/25.44 Lemma 159: codomain(domain_difference(X, Y)) = domain_difference(X, Y). 25.23/25.44 Proof: 25.23/25.44 codomain(domain_difference(X, Y)) 25.23/25.44 = { by axiom 21 (multiplicative_left_identity) } 25.23/25.44 multiplication(one, codomain(domain_difference(X, Y))) 25.23/25.44 = { by lemma 154 } 25.23/25.44 multiplication(addition(one, domain_difference(X, Y)), codomain(domain_difference(X, Y))) 25.23/25.44 = { by lemma 88 } 25.23/25.44 addition(domain_difference(X, Y), codomain(domain_difference(X, Y))) 25.23/25.44 = { by lemma 147 } 25.23/25.44 addition(domain(domain_difference(X, Y)), codomain(domain_difference(X, Y))) 25.23/25.44 = { by lemma 131 } 25.23/25.44 addition(coantidomain(antidomain(domain_difference(X, Y))), codomain(domain_difference(X, Y))) 25.23/25.44 = { by lemma 147 } 25.23/25.44 addition(coantidomain(antidomain(domain_difference(X, Y))), codomain(domain(domain_difference(X, Y)))) 25.23/25.44 = { by axiom 17 (additive_commutativity) } 25.23/25.44 addition(codomain(domain(domain_difference(X, Y))), coantidomain(antidomain(domain_difference(X, Y)))) 25.23/25.44 = { by lemma 156 } 25.23/25.44 addition(codomain(domain(domain_difference(X, Y))), domain_difference(domain_difference(X, Y), codomain(antidomain(domain_difference(X, Y))))) 25.23/25.44 = { by lemma 79 } 25.23/25.44 addition(codomain(domain(domain_difference(X, Y))), domain_difference(domain_difference(X, Y), domain(codomain(antidomain(domain_difference(X, Y)))))) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 addition(codomain(domain(domain_difference(X, Y))), domain_difference(domain_difference(X, Y), antidomain(antidomain(codomain(antidomain(domain_difference(X, Y))))))) 25.23/25.44 = { by lemma 75 } 25.23/25.44 addition(codomain(domain(domain_difference(X, Y))), domain_difference(antidomain(codomain(antidomain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y)))) 25.23/25.44 = { by lemma 51 } 25.23/25.44 addition(codomain(domain(domain_difference(X, Y))), multiplication(domain(antidomain(codomain(antidomain(domain_difference(X, Y))))), domain(domain_difference(X, Y)))) 25.23/25.44 = { by lemma 87 } 25.23/25.44 addition(codomain(domain(domain_difference(X, Y))), multiplication(domain(antidomain(codomain(antidomain(domain_difference(X, Y))))), multiplication(domain(domain_difference(X, Y)), codomain(domain(domain_difference(X, Y)))))) 25.23/25.44 = { by lemma 157 } 25.23/25.44 multiplication(addition(one, multiplication(domain(antidomain(codomain(antidomain(domain_difference(X, Y))))), domain(domain_difference(X, Y)))), codomain(domain(domain_difference(X, Y)))) 25.23/25.44 = { by lemma 51 } 25.23/25.44 multiplication(addition(one, domain_difference(antidomain(codomain(antidomain(domain_difference(X, Y)))), antidomain(domain_difference(X, Y)))), codomain(domain(domain_difference(X, Y)))) 25.23/25.44 = { by lemma 154 } 25.23/25.44 multiplication(one, codomain(domain(domain_difference(X, Y)))) 25.23/25.44 = { by axiom 21 (multiplicative_left_identity) } 25.23/25.44 codomain(domain(domain_difference(X, Y))) 25.23/25.44 = { by lemma 158 } 25.23/25.44 domain(domain_difference(X, Y)) 25.23/25.44 = { by lemma 147 } 25.23/25.44 domain_difference(X, Y) 25.23/25.44 25.23/25.44 Lemma 160: domain(domain(X)) = domain(X). 25.23/25.44 Proof: 25.23/25.44 domain(domain(X)) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 antidomain(antidomain(domain(X))) 25.23/25.44 = { by axiom 11 (complement) } 25.23/25.44 antidomain(c(X)) 25.23/25.44 = { by lemma 39 } 25.23/25.44 antidomain(antidomain(X)) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 domain(X) 25.23/25.44 25.23/25.44 Lemma 161: forward_diamond(X, addition(Y, one)) = domain(X). 25.23/25.44 Proof: 25.23/25.44 forward_diamond(X, addition(Y, one)) 25.23/25.44 = { by axiom 14 (forward_diamond) } 25.23/25.44 domain(multiplication(X, domain(addition(Y, one)))) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 domain(multiplication(X, antidomain(antidomain(addition(Y, one))))) 25.23/25.44 = { by axiom 17 (additive_commutativity) } 25.23/25.44 domain(multiplication(X, antidomain(antidomain(addition(one, Y))))) 25.23/25.44 = { by axiom 23 (multiplicative_right_identity) } 25.23/25.44 domain(multiplication(X, antidomain(multiplication(antidomain(addition(one, Y)), one)))) 25.23/25.44 = { by lemma 33 } 25.23/25.44 domain(multiplication(X, antidomain(multiplication(antidomain(addition(one, Y)), addition(one, addition(one, Y)))))) 25.23/25.44 = { by lemma 56 } 25.23/25.44 domain(multiplication(X, antidomain(multiplication(antidomain(addition(one, Y)), addition(one, Y))))) 25.23/25.44 = { by axiom 16 (domain1) } 25.23/25.44 domain(multiplication(X, antidomain(zero))) 25.23/25.44 = { by lemma 65 } 25.23/25.44 domain(multiplication(X, one)) 25.23/25.44 = { by axiom 23 (multiplicative_right_identity) } 25.23/25.44 domain(X) 25.23/25.44 25.23/25.44 Lemma 162: forward_diamond(X, addition(one, Y)) = domain(X). 25.23/25.44 Proof: 25.23/25.44 forward_diamond(X, addition(one, Y)) 25.23/25.44 = { by axiom 17 (additive_commutativity) } 25.23/25.44 forward_diamond(X, addition(Y, one)) 25.23/25.44 = { by lemma 161 } 25.23/25.44 domain(X) 25.23/25.44 25.23/25.44 Lemma 163: codomain(multiplication(coantidomain(Y), X)) = backward_diamond(X, coantidomain(Y)). 25.23/25.44 Proof: 25.23/25.44 codomain(multiplication(coantidomain(Y), X)) 25.23/25.44 = { by lemma 94 } 25.23/25.44 codomain(multiplication(coantidomain(codomain(Y)), X)) 25.23/25.44 = { by lemma 92 } 25.23/25.44 codomain(multiplication(codomain(coantidomain(Y)), X)) 25.23/25.44 = { by axiom 8 (backward_diamond) } 25.23/25.44 backward_diamond(X, coantidomain(Y)) 25.23/25.44 25.23/25.44 Lemma 164: backward_box(X, forward_diamond(Y, c(Z))) = c(backward_diamond(X, forward_box(Y, Z))). 25.23/25.44 Proof: 25.23/25.44 backward_box(X, forward_diamond(Y, c(Z))) 25.23/25.44 = { by axiom 27 (backward_box) } 25.23/25.44 c(backward_diamond(X, c(forward_diamond(Y, c(Z))))) 25.23/25.44 = { by axiom 3 (forward_box) } 25.23/25.44 c(backward_diamond(X, forward_box(Y, Z))) 25.23/25.44 25.23/25.44 Lemma 165: backward_box(X, zero) = coantidomain(X). 25.23/25.44 Proof: 25.23/25.44 backward_box(X, zero) 25.23/25.44 = { by lemma 121 } 25.23/25.44 backward_box(X, domain(zero)) 25.23/25.44 = { by lemma 122 } 25.23/25.44 backward_box(X, forward_diamond(zero, c(?))) 25.23/25.44 = { by lemma 164 } 25.23/25.44 c(backward_diamond(X, forward_box(zero, ?))) 25.23/25.44 = { by lemma 125 } 25.23/25.44 c(backward_diamond(X, one)) 25.23/25.44 = { by lemma 91 } 25.23/25.44 c(codomain(X)) 25.23/25.44 = { by lemma 39 } 25.23/25.44 antidomain(codomain(X)) 25.23/25.44 = { by lemma 85 } 25.23/25.44 coantidomain(X) 25.23/25.44 25.23/25.44 Lemma 166: antidomain(backward_diamond(X, antidomain(Y))) = backward_box(X, Y). 25.23/25.44 Proof: 25.23/25.44 antidomain(backward_diamond(X, antidomain(Y))) 25.23/25.44 = { by lemma 39 } 25.23/25.44 antidomain(backward_diamond(X, c(Y))) 25.23/25.44 = { by lemma 39 } 25.23/25.44 c(backward_diamond(X, c(Y))) 25.23/25.44 = { by axiom 27 (backward_box) } 25.23/25.44 backward_box(X, Y) 25.23/25.44 25.23/25.44 Lemma 167: multiplication(backward_box(X, Y), antidomain(Z)) = domain_difference(backward_box(X, Y), Z). 25.23/25.44 Proof: 25.23/25.44 multiplication(backward_box(X, Y), antidomain(Z)) 25.23/25.44 = { by axiom 27 (backward_box) } 25.23/25.44 multiplication(c(backward_diamond(X, c(Y))), antidomain(Z)) 25.23/25.44 = { by lemma 35 } 25.23/25.44 domain_difference(antidomain(backward_diamond(X, c(Y))), Z) 25.23/25.44 = { by lemma 39 } 25.23/25.44 domain_difference(antidomain(backward_diamond(X, antidomain(Y))), Z) 25.23/25.44 = { by lemma 166 } 25.23/25.44 domain_difference(backward_box(X, Y), Z) 25.23/25.44 25.23/25.44 Lemma 168: multiplication(coantidomain(X), antidomain(Y)) = domain_difference(coantidomain(X), Y). 25.23/25.44 Proof: 25.23/25.44 multiplication(coantidomain(X), antidomain(Y)) 25.23/25.44 = { by lemma 165 } 25.23/25.44 multiplication(backward_box(X, zero), antidomain(Y)) 25.23/25.44 = { by lemma 167 } 25.23/25.44 domain_difference(backward_box(X, zero), Y) 25.23/25.44 = { by lemma 165 } 25.23/25.44 domain_difference(coantidomain(X), Y) 25.23/25.44 25.23/25.44 Lemma 169: backward_diamond(X, codomain(Y)) = backward_diamond(X, Y). 25.23/25.44 Proof: 25.23/25.44 backward_diamond(X, codomain(Y)) 25.23/25.44 = { by axiom 8 (backward_diamond) } 25.23/25.44 codomain(multiplication(codomain(codomain(Y)), X)) 25.23/25.44 = { by lemma 95 } 25.23/25.44 codomain(multiplication(codomain(Y), X)) 25.23/25.44 = { by axiom 8 (backward_diamond) } 25.23/25.44 backward_diamond(X, Y) 25.23/25.44 25.23/25.44 Lemma 170: codomain(domain_difference(codomain(X), Y)) = backward_diamond(antidomain(Y), X). 25.23/25.44 Proof: 25.23/25.44 codomain(domain_difference(codomain(X), Y)) 25.23/25.44 = { by axiom 24 (codomain4) } 25.23/25.44 codomain(domain_difference(coantidomain(coantidomain(X)), Y)) 25.23/25.44 = { by lemma 168 } 25.23/25.44 codomain(multiplication(coantidomain(coantidomain(X)), antidomain(Y))) 25.23/25.44 = { by lemma 163 } 25.23/25.44 backward_diamond(antidomain(Y), coantidomain(coantidomain(X))) 25.23/25.44 = { by axiom 24 (codomain4) } 25.23/25.44 backward_diamond(antidomain(Y), codomain(X)) 25.23/25.44 = { by lemma 169 } 25.23/25.44 backward_diamond(antidomain(Y), X) 25.23/25.44 25.23/25.44 Lemma 171: domain_difference(codomain(X), Y) = backward_diamond(antidomain(Y), X). 25.23/25.44 Proof: 25.23/25.44 domain_difference(codomain(X), Y) 25.23/25.44 = { by lemma 159 } 25.23/25.44 codomain(domain_difference(codomain(X), Y)) 25.23/25.44 = { by lemma 170 } 25.23/25.44 backward_diamond(antidomain(Y), X) 25.23/25.44 25.23/25.44 Lemma 172: antidomain(domain_difference(coantidomain(X), Y)) = forward_box(coantidomain(X), Y). 25.23/25.44 Proof: 25.23/25.44 antidomain(domain_difference(coantidomain(X), Y)) 25.23/25.44 = { by lemma 165 } 25.23/25.44 antidomain(domain_difference(backward_box(X, zero), Y)) 25.23/25.44 = { by lemma 167 } 25.23/25.44 antidomain(multiplication(backward_box(X, zero), antidomain(Y))) 25.23/25.44 = { by lemma 45 } 25.23/25.44 forward_box(backward_box(X, zero), Y) 25.23/25.44 = { by lemma 165 } 25.23/25.44 forward_box(coantidomain(X), Y) 25.23/25.44 25.23/25.44 Lemma 173: domain(multiplication(X, coantidomain(Y))) = forward_diamond(X, coantidomain(Y)). 25.23/25.44 Proof: 25.23/25.44 domain(multiplication(X, coantidomain(Y))) 25.23/25.44 = { by lemma 165 } 25.23/25.44 domain(multiplication(X, backward_box(Y, zero))) 25.23/25.44 = { by lemma 110 } 25.23/25.44 forward_diamond(X, backward_box(Y, zero)) 25.23/25.44 = { by lemma 165 } 25.23/25.44 forward_diamond(X, coantidomain(Y)) 25.23/25.44 25.23/25.44 Lemma 174: forward_diamond(one, X) = antidomain(c(X)). 25.23/25.44 Proof: 25.23/25.44 forward_diamond(one, X) 25.23/25.44 = { by axiom 14 (forward_diamond) } 25.23/25.44 domain(multiplication(one, domain(X))) 25.23/25.44 = { by axiom 21 (multiplicative_left_identity) } 25.23/25.44 domain(domain(X)) 25.23/25.44 = { by lemma 41 } 25.23/25.44 antidomain(c(X)) 25.23/25.44 25.23/25.44 Lemma 175: backward_box(X, domain(Y)) = backward_box(X, Y). 25.23/25.44 Proof: 25.23/25.44 backward_box(X, domain(Y)) 25.23/25.44 = { by lemma 166 } 25.23/25.44 antidomain(backward_diamond(X, antidomain(domain(Y)))) 25.23/25.44 = { by axiom 11 (complement) } 25.23/25.44 antidomain(backward_diamond(X, c(Y))) 25.23/25.44 = { by lemma 39 } 25.23/25.44 antidomain(backward_diamond(X, antidomain(Y))) 25.23/25.44 = { by lemma 166 } 25.23/25.44 backward_box(X, Y) 25.23/25.44 25.23/25.44 Lemma 176: antidomain(backward_diamond(X, domain(Y))) = backward_box(X, antidomain(Y)). 25.23/25.44 Proof: 25.23/25.44 antidomain(backward_diamond(X, domain(Y))) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 antidomain(backward_diamond(X, antidomain(antidomain(Y)))) 25.23/25.44 = { by lemma 39 } 25.23/25.44 antidomain(backward_diamond(X, antidomain(c(Y)))) 25.23/25.44 = { by lemma 39 } 25.23/25.44 antidomain(backward_diamond(X, c(c(Y)))) 25.23/25.44 = { by lemma 34 } 25.23/25.44 antidomain(backward_diamond(X, domain(antidomain(c(Y))))) 25.23/25.44 = { by lemma 39 } 25.23/25.44 antidomain(backward_diamond(X, domain(c(c(Y))))) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 antidomain(backward_diamond(X, antidomain(antidomain(c(c(Y)))))) 25.23/25.44 = { by lemma 39 } 25.23/25.44 antidomain(backward_diamond(X, antidomain(c(c(c(Y)))))) 25.23/25.44 = { by lemma 34 } 25.23/25.44 antidomain(backward_diamond(X, antidomain(domain(antidomain(c(c(Y))))))) 25.23/25.44 = { by axiom 11 (complement) } 25.23/25.44 antidomain(backward_diamond(X, c(antidomain(c(c(Y)))))) 25.23/25.44 = { by lemma 174 } 25.23/25.44 antidomain(backward_diamond(X, c(forward_diamond(one, c(Y))))) 25.23/25.44 = { by axiom 3 (forward_box) } 25.23/25.44 antidomain(backward_diamond(X, forward_box(one, Y))) 25.23/25.44 = { by lemma 39 } 25.23/25.44 c(backward_diamond(X, forward_box(one, Y))) 25.23/25.44 = { by lemma 164 } 25.23/25.44 backward_box(X, forward_diamond(one, c(Y))) 25.23/25.44 = { by lemma 174 } 25.23/25.44 backward_box(X, antidomain(c(c(Y)))) 25.23/25.44 = { by lemma 39 } 25.23/25.44 backward_box(X, antidomain(antidomain(c(Y)))) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 backward_box(X, domain(c(Y))) 25.23/25.44 = { by lemma 175 } 25.23/25.44 backward_box(X, c(Y)) 25.23/25.44 = { by lemma 39 } 25.23/25.44 backward_box(X, antidomain(Y)) 25.23/25.44 25.23/25.44 Lemma 177: multiplication(codomain(X), codomain(X)) = codomain(X). 25.23/25.44 Proof: 25.23/25.44 multiplication(codomain(X), codomain(X)) 25.23/25.44 = { by lemma 86 } 25.23/25.44 multiplication(codomain(X), addition(codomain(X), coantidomain(codomain(X)))) 25.23/25.44 = { by axiom 17 (additive_commutativity) } 25.23/25.44 multiplication(codomain(X), addition(coantidomain(codomain(X)), codomain(X))) 25.23/25.44 = { by axiom 24 (codomain4) } 25.23/25.44 multiplication(codomain(X), addition(coantidomain(coantidomain(coantidomain(X))), codomain(X))) 25.23/25.44 = { by axiom 24 (codomain4) } 25.23/25.44 multiplication(codomain(X), addition(coantidomain(coantidomain(coantidomain(X))), coantidomain(coantidomain(X)))) 25.23/25.44 = { by axiom 5 (codomain3) } 25.23/25.44 multiplication(codomain(X), one) 25.23/25.44 = { by axiom 23 (multiplicative_right_identity) } 25.23/25.44 codomain(X) 25.23/25.44 25.23/25.44 Lemma 178: addition(c(X), antidomain(c(X))) = one. 25.23/25.44 Proof: 25.23/25.44 addition(c(X), antidomain(c(X))) 25.23/25.44 = { by axiom 11 (complement) } 25.23/25.44 addition(antidomain(domain(X)), antidomain(c(X))) 25.23/25.44 = { by lemma 41 } 25.23/25.44 addition(antidomain(domain(X)), domain(domain(X))) 25.23/25.44 = { by lemma 31 } 25.23/25.44 one 25.23/25.44 25.23/25.44 Lemma 179: addition(backward_box(X, Y), antidomain(backward_box(X, Y))) = one. 25.23/25.44 Proof: 25.23/25.44 addition(backward_box(X, Y), antidomain(backward_box(X, Y))) 25.23/25.44 = { by axiom 27 (backward_box) } 25.23/25.44 addition(c(backward_diamond(X, c(Y))), antidomain(backward_box(X, Y))) 25.23/25.44 = { by axiom 27 (backward_box) } 25.23/25.44 addition(c(backward_diamond(X, c(Y))), antidomain(c(backward_diamond(X, c(Y))))) 25.23/25.44 = { by lemma 178 } 25.23/25.44 one 25.23/25.44 25.23/25.44 Lemma 180: domain(backward_box(X, Y)) = backward_box(X, Y). 25.23/25.44 Proof: 25.23/25.44 domain(backward_box(X, Y)) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 antidomain(antidomain(backward_box(X, Y))) 25.23/25.44 = { by axiom 23 (multiplicative_right_identity) } 25.23/25.44 multiplication(antidomain(antidomain(backward_box(X, Y))), one) 25.23/25.44 = { by lemma 179 } 25.23/25.44 multiplication(antidomain(antidomain(backward_box(X, Y))), addition(backward_box(X, Y), antidomain(backward_box(X, Y)))) 25.23/25.44 = { by lemma 33 } 25.23/25.44 multiplication(antidomain(antidomain(backward_box(X, Y))), backward_box(X, Y)) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 multiplication(domain(backward_box(X, Y)), backward_box(X, Y)) 25.23/25.44 = { by lemma 38 } 25.23/25.44 backward_box(X, Y) 25.23/25.44 25.23/25.44 Lemma 181: domain_difference(multiplication(X, domain(Y)), Z) = domain_difference(forward_diamond(X, Y), Z). 25.23/25.44 Proof: 25.23/25.44 domain_difference(multiplication(X, domain(Y)), Z) 25.23/25.44 = { by axiom 29 (domain_difference) } 25.23/25.44 multiplication(domain(multiplication(X, domain(Y))), antidomain(Z)) 25.23/25.44 = { by axiom 14 (forward_diamond) } 25.23/25.44 multiplication(forward_diamond(X, Y), antidomain(Z)) 25.23/25.44 = { by lemma 106 } 25.23/25.44 domain_difference(forward_diamond(X, Y), Z) 25.23/25.44 25.23/25.44 Lemma 182: domain(multiplication(X, forward_diamond(Y, Z))) = forward_diamond(X, forward_diamond(Y, Z)). 25.23/25.44 Proof: 25.23/25.44 domain(multiplication(X, forward_diamond(Y, Z))) 25.23/25.44 = { by axiom 14 (forward_diamond) } 25.23/25.44 domain(multiplication(X, domain(multiplication(Y, domain(Z))))) 25.23/25.44 = { by axiom 14 (forward_diamond) } 25.23/25.44 forward_diamond(X, multiplication(Y, domain(Z))) 25.23/25.44 = { by lemma 43 } 25.23/25.44 forward_diamond(X, domain(multiplication(Y, domain(Z)))) 25.23/25.44 = { by axiom 14 (forward_diamond) } 25.23/25.44 forward_diamond(X, forward_diamond(Y, Z)) 25.23/25.44 25.23/25.44 Lemma 183: addition(antidomain(X), codomain(domain(X))) = one. 25.23/25.44 Proof: 25.23/25.44 addition(antidomain(X), codomain(domain(X))) 25.23/25.44 = { by axiom 21 (multiplicative_left_identity) } 25.23/25.44 addition(antidomain(X), multiplication(one, codomain(domain(X)))) 25.23/25.44 = { by lemma 31 } 25.23/25.44 addition(antidomain(X), multiplication(addition(antidomain(X), domain(X)), codomain(domain(X)))) 25.23/25.44 = { by axiom 17 (additive_commutativity) } 25.23/25.44 addition(antidomain(X), multiplication(addition(domain(X), antidomain(X)), codomain(domain(X)))) 25.23/25.44 = { by axiom 7 (left_distributivity) } 25.23/25.44 addition(antidomain(X), addition(multiplication(domain(X), codomain(domain(X))), multiplication(antidomain(X), codomain(domain(X))))) 25.23/25.44 = { by lemma 87 } 25.23/25.44 addition(antidomain(X), addition(domain(X), multiplication(antidomain(X), codomain(domain(X))))) 25.23/25.44 = { by axiom 17 (additive_commutativity) } 25.23/25.44 addition(antidomain(X), addition(multiplication(antidomain(X), codomain(domain(X))), domain(X))) 25.23/25.44 = { by axiom 30 (additive_associativity) } 25.23/25.44 addition(addition(antidomain(X), multiplication(antidomain(X), codomain(domain(X)))), domain(X)) 25.23/25.44 = { by lemma 112 } 25.23/25.44 addition(multiplication(antidomain(X), addition(codomain(domain(X)), one)), domain(X)) 25.23/25.44 = { by axiom 17 (additive_commutativity) } 25.23/25.44 addition(multiplication(antidomain(X), addition(one, codomain(domain(X)))), domain(X)) 25.23/25.44 = { by lemma 118 } 25.23/25.44 addition(multiplication(antidomain(X), one), domain(X)) 25.23/25.44 = { by axiom 23 (multiplicative_right_identity) } 25.23/25.44 addition(antidomain(X), domain(X)) 25.23/25.44 = { by lemma 31 } 25.23/25.44 one 25.23/25.44 25.23/25.44 Lemma 184: forward_diamond(domain(X), coantidomain(Y)) = domain(domain_difference(X, codomain(Y))). 25.23/25.44 Proof: 25.23/25.44 forward_diamond(domain(X), coantidomain(Y)) 25.23/25.44 = { by lemma 173 } 25.23/25.44 domain(multiplication(domain(X), coantidomain(Y))) 25.23/25.44 = { by lemma 165 } 25.23/25.44 domain(multiplication(domain(X), backward_box(Y, zero))) 25.23/25.44 = { by lemma 109 } 25.23/25.44 domain(multiplication(domain_difference(X, antidomain(backward_box(Y, zero))), backward_box(Y, zero))) 25.23/25.44 = { by lemma 53 } 25.23/25.44 domain(multiplication(domain(X), multiplication(antidomain(antidomain(backward_box(Y, zero))), backward_box(Y, zero)))) 25.23/25.44 = { by lemma 32 } 25.23/25.44 domain(multiplication(domain(X), multiplication(antidomain(antidomain(backward_box(Y, zero))), addition(antidomain(backward_box(Y, zero)), backward_box(Y, zero))))) 25.23/25.44 = { by lemma 53 } 25.23/25.44 domain(multiplication(domain_difference(X, antidomain(backward_box(Y, zero))), addition(antidomain(backward_box(Y, zero)), backward_box(Y, zero)))) 25.23/25.44 = { by axiom 17 (additive_commutativity) } 25.23/25.44 domain(multiplication(domain_difference(X, antidomain(backward_box(Y, zero))), addition(backward_box(Y, zero), antidomain(backward_box(Y, zero))))) 25.23/25.44 = { by lemma 179 } 25.23/25.44 domain(multiplication(domain_difference(X, antidomain(backward_box(Y, zero))), one)) 25.23/25.44 = { by axiom 23 (multiplicative_right_identity) } 25.23/25.44 domain(domain_difference(X, antidomain(backward_box(Y, zero)))) 25.23/25.44 = { by lemma 165 } 25.23/25.44 domain(domain_difference(X, antidomain(coantidomain(Y)))) 25.23/25.44 = { by lemma 101 } 25.23/25.44 domain(domain_difference(X, codomain(Y))) 25.23/25.44 25.23/25.44 Lemma 185: backward_diamond(X, antidomain(Y)) = antidomain(backward_box(X, Y)). 25.23/25.44 Proof: 25.23/25.44 backward_diamond(X, antidomain(Y)) 25.23/25.44 = { by lemma 103 } 25.23/25.44 domain(backward_diamond(X, antidomain(Y))) 25.23/25.44 = { by lemma 39 } 25.23/25.44 domain(backward_diamond(X, c(Y))) 25.23/25.44 = { by lemma 34 } 25.23/25.44 domain(backward_diamond(X, domain(antidomain(Y)))) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 antidomain(antidomain(backward_diamond(X, domain(antidomain(Y))))) 25.23/25.44 = { by lemma 176 } 25.23/25.44 antidomain(backward_box(X, antidomain(antidomain(Y)))) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 antidomain(backward_box(X, domain(Y))) 25.23/25.44 = { by lemma 175 } 25.23/25.44 antidomain(backward_box(X, Y)) 25.23/25.44 25.23/25.44 Lemma 186: backward_box(addition(X, Y), X) = backward_box(Y, X). 25.23/25.44 Proof: 25.23/25.44 backward_box(addition(X, Y), X) 25.23/25.44 = { by axiom 17 (additive_commutativity) } 25.23/25.44 backward_box(addition(Y, X), X) 25.23/25.44 = { by lemma 180 } 25.23/25.44 domain(backward_box(addition(Y, X), X)) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 antidomain(antidomain(backward_box(addition(Y, X), X))) 25.23/25.44 = { by lemma 185 } 25.23/25.44 antidomain(backward_diamond(addition(Y, X), antidomain(X))) 25.23/25.44 = { by axiom 23 (multiplicative_right_identity) } 25.23/25.44 antidomain(backward_diamond(addition(Y, multiplication(X, one)), antidomain(X))) 25.23/25.44 = { by lemma 65 } 25.23/25.44 antidomain(backward_diamond(addition(Y, multiplication(X, antidomain(zero))), antidomain(X))) 25.23/25.44 = { by lemma 126 } 25.23/25.44 antidomain(backward_diamond(addition(Y, multiplication(X, antidomain(backward_diamond(X, domain_difference(one, X))))), antidomain(X))) 25.23/25.44 = { by lemma 107 } 25.23/25.44 antidomain(backward_diamond(addition(Y, multiplication(X, coantidomain(multiplication(codomain(domain_difference(one, X)), X)))), antidomain(X))) 25.23/25.44 = { by lemma 120 } 25.23/25.44 antidomain(backward_diamond(addition(Y, multiplication(X, coantidomain(multiplication(codomain(domain_difference(one, X)), X)))), domain_difference(one, X))) 25.23/25.44 = { by axiom 8 (backward_diamond) } 25.23/25.44 antidomain(codomain(multiplication(codomain(domain_difference(one, X)), addition(Y, multiplication(X, coantidomain(multiplication(codomain(domain_difference(one, X)), X))))))) 25.23/25.44 = { by axiom 17 (additive_commutativity) } 25.23/25.44 antidomain(codomain(multiplication(codomain(domain_difference(one, X)), addition(multiplication(X, coantidomain(multiplication(codomain(domain_difference(one, X)), X))), Y)))) 25.23/25.44 = { by axiom 6 (right_distributivity) } 25.23/25.44 antidomain(codomain(addition(multiplication(codomain(domain_difference(one, X)), multiplication(X, coantidomain(multiplication(codomain(domain_difference(one, X)), X)))), multiplication(codomain(domain_difference(one, X)), Y)))) 25.23/25.44 = { by lemma 129 } 25.23/25.44 antidomain(codomain(addition(zero, multiplication(codomain(domain_difference(one, X)), Y)))) 25.23/25.44 = { by lemma 61 } 25.23/25.44 antidomain(codomain(multiplication(codomain(domain_difference(one, X)), Y))) 25.23/25.44 = { by axiom 8 (backward_diamond) } 25.23/25.44 antidomain(backward_diamond(Y, domain_difference(one, X))) 25.23/25.44 = { by lemma 120 } 25.23/25.44 antidomain(backward_diamond(Y, antidomain(X))) 25.23/25.44 = { by lemma 185 } 25.23/25.44 antidomain(antidomain(backward_box(Y, X))) 25.23/25.44 = { by axiom 13 (domain4) } 25.23/25.44 domain(backward_box(Y, X)) 25.23/25.44 = { by lemma 180 } 25.28/25.49 backward_box(Y, X) 25.28/25.49 25.28/25.49 Lemma 187: backward_diamond(domain(Y), domain_difference(X, Z)) = multiplication(domain(X), domain_difference(Y, Z)). 25.28/25.49 Proof: 25.28/25.49 backward_diamond(domain(Y), domain_difference(X, Z)) 25.28/25.49 = { by axiom 13 (domain4) } 25.28/25.49 backward_diamond(antidomain(antidomain(Y)), domain_difference(X, Z)) 25.28/25.49 = { by lemma 170 } 25.28/25.49 codomain(domain_difference(codomain(domain_difference(X, Z)), antidomain(Y))) 25.28/25.49 = { by lemma 148 } 25.28/25.49 codomain(forward_diamond(domain(codomain(domain_difference(X, Z))), Y)) 25.28/25.49 = { by axiom 21 (multiplicative_left_identity) } 25.28/25.49 codomain(multiplication(one, forward_diamond(domain(codomain(domain_difference(X, Z))), Y))) 25.28/25.49 = { by lemma 183 } 25.28/25.49 codomain(multiplication(addition(antidomain(forward_diamond(domain(codomain(domain_difference(X, Z))), Y)), codomain(domain(forward_diamond(domain(codomain(domain_difference(X, Z))), Y)))), forward_diamond(domain(codomain(domain_difference(X, Z))), Y))) 25.28/25.49 = { by lemma 37 } 25.28/25.49 codomain(multiplication(codomain(domain(forward_diamond(domain(codomain(domain_difference(X, Z))), Y))), forward_diamond(domain(codomain(domain_difference(X, Z))), Y))) 25.28/25.49 = { by axiom 8 (backward_diamond) } 25.28/25.49 backward_diamond(forward_diamond(domain(codomain(domain_difference(X, Z))), Y), domain(forward_diamond(domain(codomain(domain_difference(X, Z))), Y))) 25.28/25.49 = { by lemma 142 } 25.28/25.49 backward_diamond(domain(multiplication(domain(codomain(domain_difference(X, Z))), Y)), domain(forward_diamond(domain(codomain(domain_difference(X, Z))), Y))) 25.28/25.49 = { by lemma 47 } 25.28/25.49 backward_diamond(domain(multiplication(domain(codomain(domain_difference(X, Z))), Y)), forward_diamond(domain(codomain(domain_difference(X, Z))), Y)) 25.28/25.49 = { by lemma 142 } 25.28/25.49 backward_diamond(domain(multiplication(domain(codomain(domain_difference(X, Z))), Y)), domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))) 25.28/25.49 = { by axiom 8 (backward_diamond) } 25.28/25.49 codomain(multiplication(codomain(domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))), domain(multiplication(domain(codomain(domain_difference(X, Z))), Y)))) 25.28/25.49 = { by lemma 61 } 25.28/25.49 codomain(addition(zero, multiplication(codomain(domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))), domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))))) 25.28/25.49 = { by axiom 16 (domain1) } 25.28/25.49 codomain(addition(multiplication(antidomain(domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))), domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))), multiplication(codomain(domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))), domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))))) 25.28/25.49 = { by axiom 11 (complement) } 25.28/25.49 codomain(addition(multiplication(c(multiplication(domain(codomain(domain_difference(X, Z))), Y)), domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))), multiplication(codomain(domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))), domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))))) 25.28/25.49 = { by axiom 7 (left_distributivity) } 25.28/25.49 codomain(multiplication(addition(c(multiplication(domain(codomain(domain_difference(X, Z))), Y)), codomain(domain(multiplication(domain(codomain(domain_difference(X, Z))), Y)))), domain(multiplication(domain(codomain(domain_difference(X, Z))), Y)))) 25.28/25.49 = { by lemma 39 } 25.28/25.49 codomain(multiplication(addition(antidomain(multiplication(domain(codomain(domain_difference(X, Z))), Y)), codomain(domain(multiplication(domain(codomain(domain_difference(X, Z))), Y)))), domain(multiplication(domain(codomain(domain_difference(X, Z))), Y)))) 25.28/25.49 = { by lemma 183 } 25.28/25.49 codomain(multiplication(one, domain(multiplication(domain(codomain(domain_difference(X, Z))), Y)))) 25.28/25.49 = { by axiom 21 (multiplicative_left_identity) } 25.28/25.49 codomain(domain(multiplication(domain(codomain(domain_difference(X, Z))), Y))) 25.28/25.49 = { by lemma 158 } 25.28/25.49 domain(multiplication(domain(codomain(domain_difference(X, Z))), Y)) 25.28/25.49 = { by lemma 142 } 25.28/25.49 forward_diamond(domain(codomain(domain_difference(X, Z))), Y) 25.28/25.49 = { by lemma 102 } 25.28/25.49 forward_diamond(codomain(domain_difference(X, Z)), Y) 25.28/25.49 = { by lemma 47 } 25.28/25.49 domain(forward_diamond(codomain(domain_difference(X, Z)), Y)) 25.28/25.49 = { by axiom 13 (domain4) } 25.28/25.49 antidomain(antidomain(forward_diamond(codomain(domain_difference(X, Z)), Y))) 25.28/25.49 = { by lemma 137 } 25.28/25.49 antidomain(forward_box(codomain(domain_difference(X, Z)), antidomain(Y))) 25.28/25.49 = { by axiom 24 (codomain4) } 25.28/25.49 antidomain(forward_box(coantidomain(coantidomain(domain_difference(X, Z))), antidomain(Y))) 25.28/25.49 = { by lemma 128 } 25.28/25.49 domain(multiplication(coantidomain(coantidomain(domain_difference(X, Z))), antidomain(antidomain(Y)))) 25.28/25.49 = { by lemma 168 } 25.28/25.49 domain(domain_difference(coantidomain(coantidomain(domain_difference(X, Z))), antidomain(Y))) 25.28/25.49 = { by axiom 24 (codomain4) } 25.28/25.49 domain(domain_difference(codomain(domain_difference(X, Z)), antidomain(Y))) 25.28/25.49 = { by lemma 75 } 25.28/25.49 domain(domain_difference(Y, antidomain(codomain(domain_difference(X, Z))))) 25.28/25.49 = { by lemma 74 } 25.28/25.49 forward_diamond(domain(Y), codomain(domain_difference(X, Z))) 25.28/25.49 = { by lemma 148 } 25.28/25.49 domain_difference(Y, antidomain(codomain(domain_difference(X, Z)))) 25.28/25.49 = { by lemma 85 } 25.28/25.49 domain_difference(Y, coantidomain(domain_difference(X, Z))) 25.28/25.49 = { by axiom 21 (multiplicative_left_identity) } 25.28/25.49 domain_difference(Y, coantidomain(multiplication(one, domain_difference(X, Z)))) 25.28/25.49 = { by lemma 183 } 25.28/25.49 domain_difference(Y, coantidomain(multiplication(addition(antidomain(X), codomain(domain(X))), domain_difference(X, Z)))) 25.28/25.49 = { by axiom 17 (additive_commutativity) } 25.28/25.49 domain_difference(Y, coantidomain(multiplication(addition(codomain(domain(X)), antidomain(X)), domain_difference(X, Z)))) 25.28/25.49 = { by lemma 77 } 25.28/25.49 domain_difference(Y, coantidomain(multiplication(addition(codomain(domain(X)), antidomain(domain(X))), domain_difference(X, Z)))) 25.28/25.49 = { by axiom 17 (additive_commutativity) } 25.28/25.49 domain_difference(Y, coantidomain(multiplication(addition(antidomain(domain(X)), codomain(domain(X))), domain_difference(X, Z)))) 25.28/25.49 = { by axiom 29 (domain_difference) } 25.28/25.49 domain_difference(Y, coantidomain(multiplication(addition(antidomain(domain(X)), codomain(domain(X))), multiplication(domain(X), antidomain(Z))))) 25.28/25.49 = { by axiom 7 (left_distributivity) } 25.28/25.49 domain_difference(Y, coantidomain(addition(multiplication(antidomain(domain(X)), multiplication(domain(X), antidomain(Z))), multiplication(codomain(domain(X)), multiplication(domain(X), antidomain(Z)))))) 25.28/25.49 = { by lemma 133 } 25.28/25.49 domain_difference(Y, coantidomain(addition(zero, multiplication(codomain(domain(X)), multiplication(domain(X), antidomain(Z)))))) 25.28/25.49 = { by lemma 61 } 25.28/25.49 domain_difference(Y, coantidomain(multiplication(codomain(domain(X)), multiplication(domain(X), antidomain(Z))))) 25.28/25.49 = { by axiom 29 (domain_difference) } 25.28/25.49 domain_difference(Y, coantidomain(multiplication(codomain(domain(X)), domain_difference(X, Z)))) 25.28/25.49 = { by lemma 107 } 25.28/25.49 domain_difference(Y, antidomain(backward_diamond(domain_difference(X, Z), domain(X)))) 25.28/25.49 = { by lemma 176 } 25.28/25.49 domain_difference(Y, backward_box(domain_difference(X, Z), antidomain(X))) 25.28/25.49 = { by lemma 186 } 25.28/25.49 domain_difference(Y, backward_box(addition(antidomain(X), domain_difference(X, Z)), antidomain(X))) 25.28/25.49 = { by lemma 73 } 25.28/25.49 domain_difference(Y, backward_box(addition(antidomain(X), antidomain(Z)), antidomain(X))) 25.28/25.49 = { by lemma 186 } 25.28/25.49 domain_difference(Y, backward_box(antidomain(Z), antidomain(X))) 25.28/25.49 = { by axiom 29 (domain_difference) } 25.28/25.49 multiplication(domain(Y), antidomain(backward_box(antidomain(Z), antidomain(X)))) 25.28/25.49 = { by lemma 176 } 25.28/25.49 multiplication(domain(Y), antidomain(antidomain(backward_diamond(antidomain(Z), domain(X))))) 25.28/25.49 = { by axiom 13 (domain4) } 25.28/25.49 multiplication(domain(Y), domain(backward_diamond(antidomain(Z), domain(X)))) 25.28/25.49 = { by lemma 103 } 25.28/25.49 multiplication(domain(Y), backward_diamond(antidomain(Z), domain(X))) 25.28/25.49 = { by lemma 171 } 25.28/25.49 multiplication(domain(Y), domain_difference(codomain(domain(X)), Z)) 25.28/25.49 = { by lemma 177 } 25.28/25.49 multiplication(domain(Y), domain_difference(multiplication(codomain(domain(X)), codomain(domain(X))), Z)) 25.28/25.49 = { by lemma 145 } 25.28/25.49 multiplication(forward_diamond(codomain(domain(X)), codomain(domain(X))), domain_difference(Y, Z)) 25.28/25.49 = { by lemma 102 } 25.28/25.49 multiplication(forward_diamond(domain(codomain(domain(X))), codomain(domain(X))), domain_difference(Y, Z)) 25.28/25.49 = { by lemma 74 } 25.28/25.49 multiplication(domain(domain_difference(codomain(domain(X)), antidomain(codomain(domain(X))))), domain_difference(Y, Z)) 25.28/25.49 = { by lemma 51 } 25.28/25.49 multiplication(domain(multiplication(domain(codomain(domain(X))), domain(codomain(domain(X))))), domain_difference(Y, Z)) 25.28/25.49 = { by axiom 13 (domain4) } 25.28/25.49 multiplication(domain(multiplication(antidomain(antidomain(codomain(domain(X)))), domain(codomain(domain(X))))), domain_difference(Y, Z)) 25.28/25.49 = { by lemma 32 } 25.28/25.49 multiplication(domain(multiplication(antidomain(antidomain(codomain(domain(X)))), addition(antidomain(codomain(domain(X))), domain(codomain(domain(X)))))), domain_difference(Y, Z)) 25.28/25.49 = { by lemma 31 } 25.28/25.49 multiplication(domain(multiplication(antidomain(antidomain(codomain(domain(X)))), one)), domain_difference(Y, Z)) 25.28/25.49 = { by axiom 23 (multiplicative_right_identity) } 25.28/25.49 multiplication(domain(antidomain(antidomain(codomain(domain(X))))), domain_difference(Y, Z)) 25.28/25.49 = { by axiom 13 (domain4) } 25.28/25.49 multiplication(domain(domain(codomain(domain(X)))), domain_difference(Y, Z)) 25.28/25.49 = { by lemma 41 } 25.28/25.49 multiplication(antidomain(c(codomain(domain(X)))), domain_difference(Y, Z)) 25.28/25.49 = { by lemma 39 } 25.28/25.49 multiplication(antidomain(antidomain(codomain(domain(X)))), domain_difference(Y, Z)) 25.28/25.49 = { by axiom 13 (domain4) } 25.28/25.49 multiplication(domain(codomain(domain(X))), domain_difference(Y, Z)) 25.28/25.49 = { by lemma 102 } 25.28/25.49 multiplication(codomain(domain(X)), domain_difference(Y, Z)) 25.28/25.49 = { by lemma 158 } 25.28/25.49 multiplication(domain(X), domain_difference(Y, Z)) 25.28/25.49 25.28/25.49 Lemma 188: multiplication(domain_difference(X, antidomain(Y)), antidomain(Z)) = multiplication(domain(X), domain_difference(Y, Z)). 25.28/25.49 Proof: 25.28/25.49 multiplication(domain_difference(X, antidomain(Y)), antidomain(Z)) 25.28/25.49 = { by lemma 108 } 25.28/25.49 multiplication(domain(X), multiplication(domain(Y), antidomain(Z))) 25.28/25.49 = { by axiom 29 (domain_difference) } 25.28/25.49 multiplication(domain(X), domain_difference(Y, Z)) 25.28/25.49 25.28/25.49 Lemma 189: antidomain(multiplication(domain(X), domain_difference(Y, Z))) = forward_box(domain_difference(X, antidomain(Y)), Z). 25.28/25.49 Proof: 25.28/25.49 antidomain(multiplication(domain(X), domain_difference(Y, Z))) 25.28/25.49 = { by lemma 188 } 25.28/25.49 antidomain(multiplication(domain_difference(X, antidomain(Y)), antidomain(Z))) 25.28/25.49 = { by lemma 45 } 25.28/25.49 forward_box(domain_difference(X, antidomain(Y)), Z) 25.28/25.49 25.28/25.49 Lemma 190: forward_box(domain_difference(X, Y), Z) = forward_box(domain_difference(X, Z), Y). 25.28/25.49 Proof: 25.28/25.49 forward_box(domain_difference(X, Y), Z) 25.28/25.49 = { by lemma 79 } 25.28/25.49 forward_box(domain_difference(X, domain(Y)), Z) 25.28/25.49 = { by axiom 13 (domain4) } 25.28/25.49 forward_box(domain_difference(X, antidomain(antidomain(Y))), Z) 25.28/25.49 = { by lemma 189 } 25.28/25.49 antidomain(multiplication(domain(X), domain_difference(antidomain(Y), Z))) 25.28/25.49 = { by lemma 139 } 25.28/25.49 antidomain(multiplication(domain(X), domain_difference(antidomain(Z), Y))) 25.28/25.49 = { by lemma 189 } 25.28/25.49 forward_box(domain_difference(X, antidomain(antidomain(Z))), Y) 25.28/25.49 = { by axiom 13 (domain4) } 25.28/25.49 forward_box(domain_difference(X, domain(Z)), Y) 25.28/25.49 = { by lemma 79 } 25.28/25.49 forward_box(domain_difference(X, Z), Y) 25.28/25.49 25.28/25.49 Lemma 191: antidomain(multiplication(domain(X), domain_difference(Y, Z))) = antidomain(forward_diamond(domain_difference(X, Z), Y)). 25.28/25.49 Proof: 25.28/25.49 antidomain(multiplication(domain(X), domain_difference(Y, Z))) 25.28/25.49 = { by lemma 188 } 25.28/25.49 antidomain(multiplication(domain_difference(X, antidomain(Y)), antidomain(Z))) 25.28/25.49 = { by lemma 45 } 25.28/25.49 forward_box(domain_difference(X, antidomain(Y)), Z) 25.28/25.49 = { by lemma 190 } 25.28/25.49 forward_box(domain_difference(X, Z), antidomain(Y)) 25.28/25.49 = { by lemma 137 } 25.28/25.52 antidomain(forward_diamond(domain_difference(X, Z), Y)) 25.28/25.52 25.28/25.52 Lemma 192: forward_diamond(domain_difference(X, W), multiplication(Y, Z)) = multiplication(forward_diamond(Y, Z), domain_difference(X, W)). 25.28/25.52 Proof: 25.28/25.52 forward_diamond(domain_difference(X, W), multiplication(Y, Z)) 25.28/25.52 = { by axiom 14 (forward_diamond) } 25.28/25.52 domain(multiplication(domain_difference(X, W), domain(multiplication(Y, Z)))) 25.28/25.52 = { by lemma 142 } 25.28/25.52 domain(multiplication(domain_difference(X, W), forward_diamond(Y, Z))) 25.28/25.52 = { by lemma 182 } 25.28/25.52 forward_diamond(domain_difference(X, W), forward_diamond(Y, Z)) 25.28/25.52 = { by lemma 47 } 25.28/25.52 domain(forward_diamond(domain_difference(X, W), forward_diamond(Y, Z))) 25.28/25.52 = { by axiom 13 (domain4) } 25.28/25.52 antidomain(antidomain(forward_diamond(domain_difference(X, W), forward_diamond(Y, Z)))) 25.28/25.52 = { by lemma 191 } 25.28/25.52 antidomain(antidomain(multiplication(domain(X), domain_difference(forward_diamond(Y, Z), W)))) 25.28/25.52 = { by axiom 13 (domain4) } 25.28/25.52 antidomain(antidomain(multiplication(antidomain(antidomain(X)), domain_difference(forward_diamond(Y, Z), W)))) 25.28/25.52 = { by lemma 37 } 25.28/25.52 antidomain(antidomain(multiplication(addition(antidomain(domain_difference(forward_diamond(Y, Z), W)), antidomain(antidomain(X))), domain_difference(forward_diamond(Y, Z), W)))) 25.28/25.52 = { by lemma 80 } 25.28/25.52 antidomain(antidomain(multiplication(addition(forward_box(domain(forward_diamond(Y, Z)), W), antidomain(antidomain(X))), domain_difference(forward_diamond(Y, Z), W)))) 25.28/25.52 = { by axiom 29 (domain_difference) } 25.28/25.52 antidomain(antidomain(multiplication(addition(forward_box(domain(forward_diamond(Y, Z)), W), antidomain(antidomain(X))), multiplication(domain(forward_diamond(Y, Z)), antidomain(W))))) 25.28/25.52 = { by lemma 39 } 25.28/25.52 antidomain(antidomain(multiplication(addition(forward_box(domain(forward_diamond(Y, Z)), W), antidomain(antidomain(X))), multiplication(domain(forward_diamond(Y, Z)), c(W))))) 25.28/25.52 = { by lemma 34 } 25.28/25.52 antidomain(antidomain(multiplication(addition(forward_box(domain(forward_diamond(Y, Z)), W), antidomain(antidomain(X))), multiplication(domain(forward_diamond(Y, Z)), domain(antidomain(W)))))) 25.28/25.52 = { by lemma 39 } 25.28/25.52 antidomain(c(multiplication(addition(forward_box(domain(forward_diamond(Y, Z)), W), antidomain(antidomain(X))), multiplication(domain(forward_diamond(Y, Z)), domain(antidomain(W)))))) 25.28/25.52 = { by axiom 25 (multiplicative_associativity) } 25.28/25.52 antidomain(c(multiplication(multiplication(addition(forward_box(domain(forward_diamond(Y, Z)), W), antidomain(antidomain(X))), domain(forward_diamond(Y, Z))), domain(antidomain(W))))) 25.28/25.52 = { by lemma 42 } 25.28/25.52 antidomain(antidomain(forward_diamond(multiplication(addition(forward_box(domain(forward_diamond(Y, Z)), W), antidomain(antidomain(X))), domain(forward_diamond(Y, Z))), antidomain(W)))) 25.28/25.52 = { by lemma 40 } 25.28/25.52 antidomain(forward_box(multiplication(addition(forward_box(domain(forward_diamond(Y, Z)), W), antidomain(antidomain(X))), domain(forward_diamond(Y, Z))), W)) 25.28/25.52 = { by lemma 68 } 25.28/25.52 antidomain(forward_box(multiplication(addition(forward_box(domain(forward_diamond(Y, Z)), W), domain(antidomain(antidomain(X)))), domain(forward_diamond(Y, Z))), W)) 25.28/25.52 = { by axiom 17 (additive_commutativity) } 25.28/25.52 antidomain(forward_box(multiplication(addition(domain(antidomain(antidomain(X))), forward_box(domain(forward_diamond(Y, Z)), W)), domain(forward_diamond(Y, Z))), W)) 25.28/25.52 = { by axiom 7 (left_distributivity) } 25.28/25.52 antidomain(forward_box(addition(multiplication(domain(antidomain(antidomain(X))), domain(forward_diamond(Y, Z))), multiplication(forward_box(domain(forward_diamond(Y, Z)), W), domain(forward_diamond(Y, Z)))), W)) 25.28/25.52 = { by lemma 51 } 25.28/25.52 antidomain(forward_box(addition(domain_difference(antidomain(antidomain(X)), antidomain(forward_diamond(Y, Z))), multiplication(forward_box(domain(forward_diamond(Y, Z)), W), domain(forward_diamond(Y, Z)))), W)) 25.28/25.52 = { by axiom 17 (additive_commutativity) } 25.28/25.52 antidomain(forward_box(addition(multiplication(forward_box(domain(forward_diamond(Y, Z)), W), domain(forward_diamond(Y, Z))), domain_difference(antidomain(antidomain(X)), antidomain(forward_diamond(Y, Z)))), W)) 25.28/25.52 = { by lemma 75 } 25.28/25.52 antidomain(forward_box(addition(multiplication(forward_box(domain(forward_diamond(Y, Z)), W), domain(forward_diamond(Y, Z))), domain_difference(forward_diamond(Y, Z), antidomain(antidomain(antidomain(X))))), W)) 25.28/25.52 = { by axiom 13 (domain4) } 25.28/25.52 antidomain(forward_box(addition(multiplication(forward_box(domain(forward_diamond(Y, Z)), W), domain(forward_diamond(Y, Z))), domain_difference(forward_diamond(Y, Z), domain(antidomain(X)))), W)) 25.28/25.52 = { by lemma 79 } 25.28/25.52 antidomain(forward_box(addition(multiplication(forward_box(domain(forward_diamond(Y, Z)), W), domain(forward_diamond(Y, Z))), domain_difference(forward_diamond(Y, Z), antidomain(X))), W)) 25.28/25.52 = { by axiom 17 (additive_commutativity) } 25.28/25.52 antidomain(forward_box(addition(domain_difference(forward_diamond(Y, Z), antidomain(X)), multiplication(forward_box(domain(forward_diamond(Y, Z)), W), domain(forward_diamond(Y, Z)))), W)) 25.28/25.52 = { by lemma 45 } 25.28/25.52 antidomain(forward_box(addition(domain_difference(forward_diamond(Y, Z), antidomain(X)), multiplication(antidomain(multiplication(domain(forward_diamond(Y, Z)), antidomain(W))), domain(forward_diamond(Y, Z)))), W)) 25.28/25.52 = { by lemma 45 } 25.28/25.52 antidomain(antidomain(multiplication(addition(domain_difference(forward_diamond(Y, Z), antidomain(X)), multiplication(antidomain(multiplication(domain(forward_diamond(Y, Z)), antidomain(W))), domain(forward_diamond(Y, Z)))), antidomain(W)))) 25.28/25.52 = { by axiom 17 (additive_commutativity) } 25.28/25.52 antidomain(antidomain(multiplication(addition(multiplication(antidomain(multiplication(domain(forward_diamond(Y, Z)), antidomain(W))), domain(forward_diamond(Y, Z))), domain_difference(forward_diamond(Y, Z), antidomain(X))), antidomain(W)))) 25.28/25.52 = { by axiom 7 (left_distributivity) } 25.28/25.52 antidomain(antidomain(addition(multiplication(multiplication(antidomain(multiplication(domain(forward_diamond(Y, Z)), antidomain(W))), domain(forward_diamond(Y, Z))), antidomain(W)), multiplication(domain_difference(forward_diamond(Y, Z), antidomain(X)), antidomain(W))))) 25.28/25.52 = { by axiom 25 (multiplicative_associativity) } 25.28/25.52 antidomain(antidomain(addition(multiplication(antidomain(multiplication(domain(forward_diamond(Y, Z)), antidomain(W))), multiplication(domain(forward_diamond(Y, Z)), antidomain(W))), multiplication(domain_difference(forward_diamond(Y, Z), antidomain(X)), antidomain(W))))) 25.28/25.52 = { by axiom 16 (domain1) } 25.28/25.52 antidomain(antidomain(addition(zero, multiplication(domain_difference(forward_diamond(Y, Z), antidomain(X)), antidomain(W))))) 25.28/25.52 = { by lemma 61 } 25.28/25.52 antidomain(antidomain(multiplication(domain_difference(forward_diamond(Y, Z), antidomain(X)), antidomain(W)))) 25.28/25.52 = { by lemma 45 } 25.28/25.52 antidomain(forward_box(domain_difference(forward_diamond(Y, Z), antidomain(X)), W)) 25.28/25.52 = { by lemma 190 } 25.28/25.52 antidomain(forward_box(domain_difference(forward_diamond(Y, Z), W), antidomain(X))) 25.28/25.52 = { by lemma 137 } 25.28/25.52 antidomain(antidomain(forward_diamond(domain_difference(forward_diamond(Y, Z), W), X))) 25.28/25.52 = { by lemma 191 } 25.28/25.52 antidomain(antidomain(multiplication(domain(forward_diamond(Y, Z)), domain_difference(X, W)))) 25.28/25.52 = { by axiom 13 (domain4) } 25.28/25.52 domain(multiplication(domain(forward_diamond(Y, Z)), domain_difference(X, W))) 25.28/25.52 = { by lemma 131 } 25.28/25.52 coantidomain(antidomain(multiplication(domain(forward_diamond(Y, Z)), domain_difference(X, W)))) 25.28/25.52 = { by lemma 187 } 25.28/25.52 coantidomain(antidomain(backward_diamond(domain(X), domain_difference(forward_diamond(Y, Z), W)))) 25.28/25.52 = { by lemma 107 } 25.28/25.52 coantidomain(coantidomain(multiplication(codomain(domain_difference(forward_diamond(Y, Z), W)), domain(X)))) 25.28/25.52 = { by axiom 24 (codomain4) } 25.28/25.52 codomain(multiplication(codomain(domain_difference(forward_diamond(Y, Z), W)), domain(X))) 25.28/25.52 = { by axiom 8 (backward_diamond) } 25.28/25.52 backward_diamond(domain(X), domain_difference(forward_diamond(Y, Z), W)) 25.28/25.52 = { by lemma 187 } 25.28/25.52 multiplication(domain(forward_diamond(Y, Z)), domain_difference(X, W)) 25.28/25.52 = { by lemma 146 } 25.28/25.52 multiplication(domain(X), domain_difference(forward_diamond(Y, Z), W)) 25.28/25.52 = { by lemma 106 } 25.28/25.52 multiplication(domain(X), multiplication(forward_diamond(Y, Z), antidomain(W))) 25.28/25.52 = { by axiom 25 (multiplicative_associativity) } 25.28/25.52 multiplication(multiplication(domain(X), forward_diamond(Y, Z)), antidomain(W)) 25.28/25.52 = { by lemma 144 } 25.28/25.52 multiplication(multiplication(forward_diamond(Y, Z), domain(X)), antidomain(W)) 25.28/25.52 = { by axiom 25 (multiplicative_associativity) } 25.28/25.52 multiplication(forward_diamond(Y, Z), multiplication(domain(X), antidomain(W))) 25.28/25.52 = { by axiom 29 (domain_difference) } 32.39/32.60 multiplication(forward_diamond(Y, Z), domain_difference(X, W)) 32.39/32.60 32.39/32.60 Goal 1 (goals_1): addition(forward_box(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)) = one. 32.39/32.60 Proof: 32.39/32.60 addition(forward_box(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)) 32.39/32.60 = { by lemma 40 } 32.39/32.60 addition(antidomain(forward_diamond(sK3_goals_X0, antidomain(domain(sK2_goals_X1)))), domain(sK1_goals_X2)) 32.39/32.60 = { by axiom 11 (complement) } 32.39/32.60 addition(antidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), domain(sK1_goals_X2)) 32.39/32.60 = { by lemma 39 } 32.39/32.60 addition(antidomain(forward_diamond(sK3_goals_X0, antidomain(sK2_goals_X1))), domain(sK1_goals_X2)) 32.39/32.60 = { by lemma 40 } 32.39/32.60 addition(forward_box(sK3_goals_X0, sK2_goals_X1), domain(sK1_goals_X2)) 32.39/32.60 = { by lemma 45 } 32.39/32.60 addition(antidomain(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1))), domain(sK1_goals_X2)) 32.39/32.60 = { by axiom 13 (domain4) } 32.39/32.60 addition(antidomain(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1))), antidomain(antidomain(sK1_goals_X2))) 32.39/32.60 = { by lemma 73 } 32.39/32.60 addition(antidomain(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.60 = { by lemma 45 } 32.39/32.60 addition(forward_box(sK3_goals_X0, sK2_goals_X1), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.60 = { by lemma 38 } 32.39/32.60 addition(multiplication(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.60 = { by axiom 13 (domain4) } 32.39/32.60 addition(multiplication(antidomain(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))), forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.60 = { by lemma 33 } 32.39/32.60 addition(multiplication(antidomain(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))), addition(forward_box(sK3_goals_X0, sK2_goals_X1), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.60 = { by axiom 3 (forward_box) } 32.39/32.60 addition(multiplication(antidomain(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))), addition(c(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.60 = { by axiom 3 (forward_box) } 32.39/32.60 addition(multiplication(antidomain(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))), addition(c(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), antidomain(c(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.60 = { by lemma 178 } 32.39/32.60 addition(multiplication(antidomain(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))), one), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.60 = { by axiom 23 (multiplicative_right_identity) } 32.39/32.61 addition(antidomain(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 13 (domain4) } 32.39/32.61 addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 9 (additive_identity) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), zero), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 121 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(zero)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 140 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(multiplication(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain(multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 14 (forward_diamond) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 87 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(domain_difference(antidomain(sK1_goals_X2), multiplication(multiplication(antidomain(sK1_goals_X2), codomain(antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 107 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(domain_difference(antidomain(sK1_goals_X2), multiplication(multiplication(antidomain(sK1_goals_X2), codomain(antidomain(sK1_goals_X2))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(codomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), coantidomain(domain(sK2_goals_X1)))))), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 25 (multiplicative_associativity) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(codomain(antidomain(sK1_goals_X2)), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(codomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), coantidomain(domain(sK2_goals_X1))))))), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 25 (multiplicative_associativity) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(multiplication(codomain(antidomain(sK1_goals_X2)), sK3_goals_X0), multiplication(coantidomain(multiplication(codomain(antidomain(sK1_goals_X2)), sK3_goals_X0)), coantidomain(domain(sK2_goals_X1)))))), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 25 (multiplicative_associativity) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(multiplication(multiplication(codomain(antidomain(sK1_goals_X2)), sK3_goals_X0), coantidomain(multiplication(codomain(antidomain(sK1_goals_X2)), sK3_goals_X0))), coantidomain(domain(sK2_goals_X1))))), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 28 (codomain1) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(zero, coantidomain(domain(sK2_goals_X1))))), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 15 (left_annihilation) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), zero)), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 20 (right_annihilation) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(domain_difference(antidomain(sK1_goals_X2), zero), multiplication(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 192 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), multiplication(forward_diamond(sK3_goals_X0, multiplication(antidomain(backward_diamond(sK3_goals_X0, antidomain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), domain_difference(antidomain(sK1_goals_X2), zero))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 185 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), multiplication(forward_diamond(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), domain_difference(antidomain(sK1_goals_X2), zero))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 192 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(domain_difference(antidomain(sK1_goals_X2), zero), multiplication(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 138 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(domain(antidomain(sK1_goals_X2)), multiplication(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 74 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(domain_difference(antidomain(sK1_goals_X2), antidomain(multiplication(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 141 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(domain_difference(antidomain(sK1_goals_X2), antidomain(forward_diamond(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 75 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(domain_difference(forward_diamond(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), antidomain(antidomain(sK1_goals_X2))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 106 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(multiplication(forward_diamond(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), antidomain(antidomain(antidomain(sK1_goals_X2)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 105 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(domain_difference(multiplication(sK3_goals_X0, domain(multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1))))), antidomain(antidomain(sK1_goals_X2))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 51 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(multiplication(domain(multiplication(sK3_goals_X0, domain(multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))))), domain(antidomain(sK1_goals_X2))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 14 (forward_diamond) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(multiplication(forward_diamond(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), domain(antidomain(sK1_goals_X2))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 14 (forward_diamond) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(forward_diamond(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), antidomain(sK1_goals_X2))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 48 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), antidomain(forward_box(forward_diamond(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), sK1_goals_X2))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 128 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(multiplication(forward_diamond(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), antidomain(sK1_goals_X2)))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 106 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(domain_difference(forward_diamond(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), sK1_goals_X2))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 181 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(domain_difference(multiplication(sK3_goals_X0, domain(multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1))))), sK1_goals_X2))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 147 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, domain(multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 181 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(forward_diamond(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 143 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, multiplication(antidomain(antidomain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 13 (domain4) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, multiplication(domain(backward_box(sK3_goals_X0, sK1_goals_X2)), coantidomain(domain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 180 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), coantidomain(domain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 175 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, domain(sK1_goals_X2)), coantidomain(domain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 93 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, multiplication(addition(domain(sK2_goals_X1), backward_box(sK3_goals_X0, domain(sK1_goals_X2))), coantidomain(domain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 19 (goals) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, multiplication(one, coantidomain(domain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 21 (multiplicative_left_identity) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(domain(sK2_goals_X1))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 85 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(codomain(domain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 95 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(codomain(codomain(domain(sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 177 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(codomain(multiplication(codomain(domain(sK2_goals_X1)), codomain(domain(sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 8 (backward_diamond) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(backward_diamond(codomain(domain(sK2_goals_X1)), domain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 176 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, backward_box(codomain(domain(sK2_goals_X1)), antidomain(sK2_goals_X1))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 130 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, backward_box(codomain(domain(sK2_goals_X1)), codomain(antidomain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 101 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, backward_box(codomain(domain(sK2_goals_X1)), antidomain(coantidomain(antidomain(sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 165 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, backward_box(codomain(domain(sK2_goals_X1)), antidomain(backward_box(antidomain(sK2_goals_X1), zero)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 176 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(backward_diamond(codomain(domain(sK2_goals_X1)), domain(backward_box(antidomain(sK2_goals_X1), zero))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 180 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(backward_diamond(codomain(domain(sK2_goals_X1)), backward_box(antidomain(sK2_goals_X1), zero)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 165 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(backward_diamond(codomain(domain(sK2_goals_X1)), coantidomain(antidomain(sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 98 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(backward_diamond(codomain(domain(sK2_goals_X1)), coantidomain(antidomain(sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 163 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(codomain(multiplication(coantidomain(antidomain(sK2_goals_X1)), codomain(domain(sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 94 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(multiplication(coantidomain(antidomain(sK2_goals_X1)), codomain(domain(sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 156 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(multiplication(domain_difference(sK2_goals_X1, codomain(antidomain(sK2_goals_X1))), codomain(domain(sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 79 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(multiplication(domain_difference(sK2_goals_X1, domain(codomain(antidomain(sK2_goals_X1)))), codomain(domain(sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 13 (domain4) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(multiplication(domain_difference(sK2_goals_X1, antidomain(antidomain(codomain(antidomain(sK2_goals_X1))))), codomain(domain(sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 75 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(multiplication(domain_difference(antidomain(codomain(antidomain(sK2_goals_X1))), antidomain(sK2_goals_X1)), codomain(domain(sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 108 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(multiplication(domain(antidomain(codomain(antidomain(sK2_goals_X1)))), multiplication(domain(sK2_goals_X1), codomain(domain(sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 87 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(multiplication(domain(antidomain(codomain(antidomain(sK2_goals_X1)))), domain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 51 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(domain_difference(antidomain(codomain(antidomain(sK2_goals_X1))), antidomain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 75 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(domain_difference(sK2_goals_X1, antidomain(antidomain(codomain(antidomain(sK2_goals_X1))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 13 (domain4) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(domain_difference(sK2_goals_X1, domain(codomain(antidomain(sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 79 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(domain_difference(sK2_goals_X1, codomain(antidomain(sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 156 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, coantidomain(coantidomain(antidomain(sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 24 (codomain4) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, codomain(antidomain(sK2_goals_X1))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 130 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 39 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, c(sK2_goals_X1)), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 50 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(domain(multiplication(sK3_goals_X0, c(sK2_goals_X1))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 49 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(forward_diamond(sK3_goals_X0, antidomain(sK2_goals_X1)), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 48 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 139 } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(antidomain(sK1_goals_X2), forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 149 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), domain(forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 68 } 32.39/32.61 addition(addition(domain(antidomain(sK1_goals_X2)), domain(forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 17 (additive_commutativity) } 32.39/32.61 addition(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(antidomain(sK1_goals_X2))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 13 (domain4) } 32.39/32.61 addition(addition(antidomain(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))), domain(antidomain(sK1_goals_X2))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 149 } 32.39/32.61 addition(addition(domain(antidomain(sK1_goals_X2)), domain_difference(antidomain(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))), antidomain(sK1_goals_X2))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 68 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), domain_difference(antidomain(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))), antidomain(sK1_goals_X2))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 13 (domain4) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), domain_difference(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), antidomain(sK1_goals_X2))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 50 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), domain_difference(forward_box(sK3_goals_X0, sK2_goals_X1), antidomain(sK1_goals_X2))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 148 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 9 (additive_identity) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), zero), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 160 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(domain(forward_box(sK3_goals_X0, sK2_goals_X1))), zero), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 162 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), zero), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 28 (codomain1) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 2 (codomain2) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(addition(coantidomain(multiplication(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 17 (additive_commutativity) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(addition(coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), coantidomain(multiplication(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 93 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(coantidomain(multiplication(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 54 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(coantidomain(multiplication(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 16 (domain1) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(coantidomain(zero), coantidomain(coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 90 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(one, coantidomain(coantidomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 24 (codomain4) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(one, codomain(multiplication(coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 163 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(one, backward_diamond(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), coantidomain(coantidomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 24 (codomain4) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(one, backward_diamond(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), codomain(domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 169 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(one, backward_diamond(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 21 (multiplicative_left_identity) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 77 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(domain(forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 161 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(forward_diamond(forward_box(sK3_goals_X0, sK2_goals_X1), addition(forward_box(sK3_goals_X0, sK2_goals_X1), one))), domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 141 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), addition(forward_box(sK3_goals_X0, sK2_goals_X1), one))), domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 112 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(addition(forward_box(sK3_goals_X0, sK2_goals_X1), multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), forward_box(sK3_goals_X0, sK2_goals_X1)))), domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 55 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(multiplication(addition(forward_box(sK3_goals_X0, sK2_goals_X1), one), forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 141 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(forward_diamond(addition(forward_box(sK3_goals_X0, sK2_goals_X1), one), forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 44 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(multiplication(addition(forward_box(sK3_goals_X0, sK2_goals_X1), one), domain(forward_box(sK3_goals_X0, sK2_goals_X1)))), domain_difference(antidomain(multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 139 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(multiplication(addition(forward_box(sK3_goals_X0, sK2_goals_X1), one), domain(forward_box(sK3_goals_X0, sK2_goals_X1)))), domain_difference(antidomain(antidomain(sK1_goals_X2)), multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 13 (domain4) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(multiplication(addition(forward_box(sK3_goals_X0, sK2_goals_X1), one), domain(forward_box(sK3_goals_X0, sK2_goals_X1)))), domain_difference(domain(sK1_goals_X2), multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 50 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(multiplication(addition(forward_box(sK3_goals_X0, sK2_goals_X1), one), domain(forward_box(sK3_goals_X0, sK2_goals_X1)))), domain_difference(sK1_goals_X2, multiplication(antidomain(antidomain(sK1_goals_X2)), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 13 (domain4) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), backward_diamond(antidomain(multiplication(addition(forward_box(sK3_goals_X0, sK2_goals_X1), one), domain(forward_box(sK3_goals_X0, sK2_goals_X1)))), domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 171 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(addition(forward_box(sK3_goals_X0, sK2_goals_X1), one), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 17 (additive_commutativity) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(addition(one, forward_box(sK3_goals_X0, sK2_goals_X1)), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 38 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(addition(one, multiplication(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_box(sK3_goals_X0, sK2_goals_X1))), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 157 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), multiplication(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 112 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))), one)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 17 (additive_commutativity) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 29 (domain_difference) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(domain(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))))), antidomain(multiplication(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1)))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 141 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(domain(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))))), antidomain(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1)))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 29 (domain_difference) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 127 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), antidomain(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1)))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 119 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), antidomain(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 46 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), multiplication(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), one)), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 23 (multiplicative_right_identity) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1))))), codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 17 (additive_commutativity) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), forward_diamond(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(one, multiplication(forward_box(sK3_goals_X0, sK2_goals_X1), domain(forward_box(sK3_goals_X0, sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 162 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), domain(domain(forward_box(sK3_goals_X0, sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 160 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), domain(forward_box(sK3_goals_X0, sK2_goals_X1))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 17 (additive_commutativity) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), codomain(domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 159 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(sK1_goals_X2, multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 29 (domain_difference) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), multiplication(domain(sK1_goals_X2), antidomain(multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 45 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), multiplication(domain(sK1_goals_X2), forward_box(domain(sK1_goals_X2), forward_box(sK3_goals_X0, sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 152 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(sK1_goals_X2, antidomain(forward_box(domain(sK1_goals_X2), forward_box(sK3_goals_X0, sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 147 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(domain_difference(sK1_goals_X2, antidomain(forward_box(domain(sK1_goals_X2), forward_box(sK3_goals_X0, sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 75 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(domain_difference(forward_box(domain(sK1_goals_X2), forward_box(sK3_goals_X0, sK2_goals_X1)), antidomain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 45 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(domain_difference(antidomain(multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), antidomain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 51 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(multiplication(domain(antidomain(multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))))), domain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 34 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(multiplication(c(multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), domain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 39 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(multiplication(antidomain(multiplication(domain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), domain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 45 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(multiplication(forward_box(domain(sK1_goals_X2), forward_box(sK3_goals_X0, sK2_goals_X1)), domain(sK1_goals_X2)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 14 (forward_diamond) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(forward_box(domain(sK1_goals_X2), forward_box(sK3_goals_X0, sK2_goals_X1)), sK1_goals_X2)), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 80 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), forward_diamond(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), sK1_goals_X2)), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 150 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(domain_difference(sK1_goals_X2, domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 147 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(sK1_goals_X2, domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 151 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(antidomain(sK1_goals_X2), domain_difference(sK1_goals_X2, domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 73 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), addition(antidomain(sK1_goals_X2), antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 151 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(domain(forward_box(sK3_goals_X0, sK2_goals_X1)), antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 17 (additive_commutativity) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), domain(forward_box(sK3_goals_X0, sK2_goals_X1))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 13 (domain4) } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), antidomain(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 136 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), antidomain(addition(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1)))))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 135 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), forward_diamond(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 150 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), domain(domain_difference(sK1_goals_X2, domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 147 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), domain_difference(sK1_goals_X2, domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1)))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 73 } 32.39/32.61 addition(addition(antidomain(sK1_goals_X2), antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1)))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 17 (additive_commutativity) } 32.39/32.61 addition(addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), antidomain(sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 80 } 32.39/32.61 addition(addition(forward_box(domain(sK1_goals_X2), forward_box(sK3_goals_X0, sK2_goals_X1)), antidomain(sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 131 } 32.39/32.61 addition(addition(forward_box(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1)), antidomain(sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 172 } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), antidomain(sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 73 } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1)), sK1_goals_X2)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 71 } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(antidomain(sK1_goals_X2), domain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 68 } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain(antidomain(sK1_goals_X2)), domain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 138 } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain_difference(antidomain(sK1_goals_X2), zero), domain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 28 (codomain1) } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), coantidomain(antidomain(sK1_goals_X2)))), domain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 21 (multiplicative_left_identity) } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(one, coantidomain(antidomain(sK1_goals_X2))))), domain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 57 } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(addition(one, antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))), coantidomain(antidomain(sK1_goals_X2))))), domain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 17 (additive_commutativity) } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(addition(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), one), coantidomain(antidomain(sK1_goals_X2))))), domain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 25 (multiplicative_associativity) } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain_difference(antidomain(sK1_goals_X2), multiplication(multiplication(antidomain(sK1_goals_X2), addition(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), one)), coantidomain(antidomain(sK1_goals_X2)))), domain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 112 } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain_difference(antidomain(sK1_goals_X2), multiplication(addition(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)))), coantidomain(antidomain(sK1_goals_X2)))), domain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 93 } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain_difference(antidomain(sK1_goals_X2), multiplication(multiplication(antidomain(sK1_goals_X2), antidomain(forward_box(sK3_goals_X0, sK2_goals_X1))), coantidomain(antidomain(sK1_goals_X2)))), domain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 25 (multiplicative_associativity) } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), coantidomain(antidomain(sK1_goals_X2))))), domain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 150 } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), coantidomain(antidomain(sK1_goals_X2))))), forward_diamond(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), coantidomain(antidomain(sK1_goals_X2))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 173 } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(domain_difference(antidomain(sK1_goals_X2), multiplication(antidomain(sK1_goals_X2), multiplication(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), coantidomain(antidomain(sK1_goals_X2))))), domain(multiplication(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), coantidomain(antidomain(sK1_goals_X2)))))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 140 } 32.39/32.61 addition(addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), zero), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by axiom 9 (additive_identity) } 32.39/32.61 addition(antidomain(domain_difference(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 172 } 32.39/32.61 addition(forward_box(coantidomain(antidomain(sK1_goals_X2)), forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 131 } 32.39/32.61 addition(forward_box(domain(sK1_goals_X2), forward_box(sK3_goals_X0, sK2_goals_X1)), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 80 } 32.39/32.61 addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(multiplication(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 39 } 32.39/32.61 addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(multiplication(sK3_goals_X0, c(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 50 } 32.39/32.61 addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(domain(multiplication(sK3_goals_X0, c(sK2_goals_X1))), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 49 } 32.39/32.61 addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(forward_diamond(sK3_goals_X0, antidomain(sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 48 } 32.39/32.61 addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), antidomain(sK1_goals_X2))) 32.39/32.61 = { by lemma 52 } 32.39/32.61 addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), multiplication(antidomain(forward_box(sK3_goals_X0, sK2_goals_X1)), domain(sK1_goals_X2))) 32.39/32.61 = { by lemma 71 } 32.39/32.61 addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))) 32.39/32.61 = { by lemma 147 } 32.39/32.61 addition(antidomain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1))), domain(domain_difference(sK1_goals_X2, forward_box(sK3_goals_X0, sK2_goals_X1)))) 32.39/32.61 = { by lemma 31 } 32.39/32.61 one 32.39/32.61 % SZS output end Proof 32.39/32.61 32.39/32.61 RESULT: Unsatisfiable (the axioms are contradictory). 32.39/32.62 EOF