0.06/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.06/0.12 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.12/0.33 % Computer : n014.cluster.edu 0.12/0.33 % Model : x86_64 x86_64 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.33 % Memory : 8042.1875MB 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.33 % CPULimit : 180 0.12/0.33 % DateTime : Thu Aug 29 13:20:54 EDT 2019 0.12/0.33 % CPUTime : 40.60/40.79 % SZS status Unsatisfiable 40.60/40.79 40.60/40.79 % SZS output start Proof 40.60/40.79 Take the following subset of the input axioms: 43.70/43.89 fof(goals, negated_conjecture, product(eta(x0), product(x0, x1))!=product(i(i(x0)), x1)). 43.70/43.89 fof(sos01, axiom, ![A]: A=product(A, one)). 43.70/43.89 fof(sos02, axiom, ![A]: product(one, A)=A). 43.70/43.89 fof(sos03, axiom, ![A, B]: product(A, difference(A, B))=B). 43.70/43.89 fof(sos04, axiom, ![A, B]: B=difference(A, product(A, B))). 43.70/43.89 fof(sos05, axiom, ![A, B]: A=quotient(product(A, B), B)). 43.70/43.89 fof(sos06, axiom, ![A, B]: A=product(quotient(A, B), B)). 43.70/43.89 fof(sos07, axiom, ![A, B, C]: difference(A, product(product(A, B), C))=quotient(product(B, product(C, A)), A)). 43.70/43.89 fof(sos08, axiom, ![A, B, C]: quotient(quotient(product(C, product(A, B)), B), A)=difference(product(A, B), product(A, product(B, C)))). 43.70/43.89 fof(sos09, axiom, ![A]: i(A)=difference(A, one)). 43.70/43.89 fof(sos10, axiom, ![A]: j(A)=quotient(one, A)). 43.70/43.89 fof(sos11, axiom, ![A]: product(A, j(A))=product(i(A), A)). 43.70/43.89 fof(sos12, axiom, ![A]: eta(A)=product(i(A), A)). 43.70/43.89 43.70/43.89 Now clausify the problem and encode Horn clauses using encoding 3 of 43.70/43.89 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 43.70/43.89 We repeatedly replace C & s=t => u=v by the two clauses: 43.70/43.89 fresh(y, y, x1...xn) = u 43.70/43.89 C => fresh(s, t, x1...xn) = v 43.70/43.89 where fresh is a fresh function symbol and x1..xn are the free 43.70/43.89 variables of u and v. 43.70/43.89 A predicate p(X) is encoded as p(X)=true (this is sound, because the 43.70/43.89 input problem has no model of domain size 1). 43.70/43.89 43.70/43.89 The encoding turns the above axioms into the following unit equations and goals: 43.70/43.89 43.70/43.89 Axiom 1 (sos06): X = product(quotient(X, Y), Y). 43.70/43.89 Axiom 2 (sos10): j(X) = quotient(one, X). 43.70/43.89 Axiom 3 (sos08): quotient(quotient(product(X, product(Y, Z)), Z), Y) = difference(product(Y, Z), product(Y, product(Z, X))). 43.70/43.89 Axiom 4 (sos09): i(X) = difference(X, one). 43.70/43.89 Axiom 5 (sos02): product(one, X) = X. 43.70/43.89 Axiom 6 (sos11): product(X, j(X)) = product(i(X), X). 43.70/43.89 Axiom 7 (sos07): difference(X, product(product(X, Y), Z)) = quotient(product(Y, product(Z, X)), X). 43.70/43.89 Axiom 8 (sos03): product(X, difference(X, Y)) = Y. 43.70/43.89 Axiom 9 (sos04): X = difference(Y, product(Y, X)). 43.70/43.89 Axiom 10 (sos12): eta(X) = product(i(X), X). 43.70/43.89 Axiom 11 (sos05): X = quotient(product(X, Y), Y). 43.70/43.91 Axiom 12 (sos01): X = product(X, one). 43.70/43.91 43.70/43.91 Lemma 13: product(X, j(X)) = eta(X). 43.70/43.91 Proof: 43.70/43.91 product(X, j(X)) 43.70/43.91 = { by axiom 6 (sos11) } 43.70/43.91 product(i(X), X) 43.70/43.91 = { by axiom 10 (sos12) } 43.70/43.91 eta(X) 43.70/43.91 43.70/43.91 Lemma 14: product(X, i(X)) = one. 43.70/43.91 Proof: 43.70/43.91 product(X, i(X)) 43.70/43.91 = { by axiom 4 (sos09) } 43.70/43.91 product(X, difference(X, one)) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 one 43.70/43.91 43.70/43.91 Lemma 15: j(i(X)) = X. 43.70/43.91 Proof: 43.70/43.91 j(i(X)) 43.70/43.91 = { by axiom 2 (sos10) } 43.70/43.91 quotient(one, i(X)) 43.70/43.91 = { by lemma 14 } 43.70/43.91 quotient(product(X, i(X)), i(X)) 43.70/43.91 = { by axiom 11 (sos05) } 43.70/43.91 X 43.70/43.91 43.70/43.91 Lemma 16: eta(i(X)) = eta(X). 43.70/43.91 Proof: 43.70/43.91 eta(i(X)) 43.70/43.91 = { by lemma 13 } 43.70/43.91 product(i(X), j(i(X))) 43.70/43.91 = { by lemma 15 } 43.70/43.91 product(i(X), X) 43.70/43.91 = { by axiom 6 (sos11) } 43.70/43.91 product(X, j(X)) 43.70/43.91 = { by lemma 13 } 43.70/43.91 eta(X) 43.70/43.91 43.70/43.91 Lemma 17: difference(one, X) = X. 43.70/43.91 Proof: 43.70/43.91 difference(one, X) 43.70/43.91 = { by axiom 5 (sos02) } 43.70/43.91 product(one, difference(one, X)) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 X 43.70/43.91 43.70/43.91 Lemma 18: quotient(quotient(X, i(Y)), Y) = product(Y, product(i(Y), X)). 43.70/43.91 Proof: 43.70/43.91 quotient(quotient(X, i(Y)), Y) 43.70/43.91 = { by axiom 12 (sos01) } 43.70/43.91 quotient(quotient(product(X, one), i(Y)), Y) 43.70/43.91 = { by lemma 14 } 43.70/43.91 quotient(quotient(product(X, product(Y, i(Y))), i(Y)), Y) 43.70/43.91 = { by axiom 3 (sos08) } 43.70/43.91 difference(product(Y, i(Y)), product(Y, product(i(Y), X))) 43.70/43.91 = { by lemma 14 } 43.70/43.91 difference(one, product(Y, product(i(Y), X))) 43.70/43.91 = { by lemma 17 } 43.70/43.91 product(Y, product(i(Y), X)) 43.70/43.91 43.70/43.91 Lemma 19: product(product(Y, product(i(Y), X)), Y) = quotient(X, i(Y)). 43.70/43.91 Proof: 43.70/43.91 product(product(Y, product(i(Y), X)), Y) 43.70/43.91 = { by lemma 18 } 43.70/43.91 product(quotient(quotient(X, i(Y)), Y), Y) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 quotient(X, i(Y)) 43.70/43.91 43.70/43.91 Lemma 20: product(X, quotient(product(Y, product(Z, X)), X)) = product(product(X, Y), Z). 43.70/43.91 Proof: 43.70/43.91 product(X, quotient(product(Y, product(Z, X)), X)) 43.70/43.91 = { by axiom 7 (sos07) } 43.70/43.91 product(X, difference(X, product(product(X, Y), Z))) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 product(product(X, Y), Z) 43.70/43.91 43.70/43.91 Lemma 21: product(j(X), X) = one. 43.70/43.91 Proof: 43.70/43.91 product(j(X), X) 43.70/43.91 = { by axiom 2 (sos10) } 43.70/43.91 product(quotient(one, X), X) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 one 43.70/43.91 43.70/43.91 Lemma 22: product(product(X, j(product(Y, X))), Y) = eta(X). 43.70/43.91 Proof: 43.70/43.91 product(product(X, j(product(Y, X))), Y) 43.70/43.91 = { by lemma 20 } 43.70/43.91 product(X, quotient(product(j(product(Y, X)), product(Y, X)), X)) 43.70/43.91 = { by lemma 21 } 43.70/43.91 product(X, quotient(one, X)) 43.70/43.91 = { by axiom 2 (sos10) } 43.70/43.91 product(X, j(X)) 43.70/43.91 = { by lemma 13 } 43.70/43.91 eta(X) 43.70/43.91 43.70/43.91 Lemma 23: product(product(difference(X, Y), j(Y)), X) = eta(difference(X, Y)). 43.70/43.91 Proof: 43.70/43.91 product(product(difference(X, Y), j(Y)), X) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 product(product(difference(X, Y), j(product(X, difference(X, Y)))), X) 43.70/43.91 = { by lemma 22 } 43.70/43.91 eta(difference(X, Y)) 43.70/43.91 43.70/43.91 Lemma 24: quotient(product(j(X), product(Y, X)), X) = difference(X, product(eta(X), Y)). 43.70/43.91 Proof: 43.70/43.91 quotient(product(j(X), product(Y, X)), X) 43.70/43.91 = { by axiom 7 (sos07) } 43.70/43.91 difference(X, product(product(X, j(X)), Y)) 43.70/43.91 = { by lemma 13 } 43.70/43.91 difference(X, product(eta(X), Y)) 43.70/43.91 43.70/43.91 Lemma 25: difference(X, product(eta(X), j(X))) = quotient(j(X), X). 43.70/43.91 Proof: 43.70/43.91 difference(X, product(eta(X), j(X))) 43.70/43.91 = { by lemma 24 } 43.70/43.91 quotient(product(j(X), product(j(X), X)), X) 43.70/43.91 = { by lemma 21 } 43.70/43.91 quotient(product(j(X), one), X) 43.70/43.91 = { by axiom 12 (sos01) } 43.70/43.91 quotient(j(X), X) 43.70/43.91 43.70/43.91 Lemma 26: product(X, quotient(j(X), X)) = product(eta(X), j(X)). 43.70/43.91 Proof: 43.70/43.91 product(X, quotient(j(X), X)) 43.70/43.91 = { by lemma 25 } 43.70/43.91 product(X, difference(X, product(eta(X), j(X)))) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 product(eta(X), j(X)) 43.70/43.91 43.70/43.91 Lemma 27: quotient(eta(X), X) = i(X). 43.70/43.91 Proof: 43.70/43.91 quotient(eta(X), X) 43.70/43.91 = { by lemma 13 } 43.70/43.91 quotient(product(X, j(X)), X) 43.70/43.91 = { by axiom 6 (sos11) } 43.70/43.91 quotient(product(i(X), X), X) 43.70/43.91 = { by axiom 11 (sos05) } 43.70/43.91 i(X) 43.70/43.91 43.70/43.91 Lemma 28: quotient(product(i(X), product(Y, X)), X) = difference(X, Y). 43.70/43.91 Proof: 43.70/43.91 quotient(product(i(X), product(Y, X)), X) 43.70/43.91 = { by axiom 7 (sos07) } 43.70/43.91 difference(X, product(product(X, i(X)), Y)) 43.70/43.91 = { by lemma 14 } 43.70/43.91 difference(X, product(one, Y)) 43.70/43.91 = { by axiom 5 (sos02) } 43.70/43.91 difference(X, Y) 43.70/43.91 43.70/43.91 Lemma 29: quotient(product(i(X), Y), X) = difference(X, quotient(Y, X)). 43.70/43.91 Proof: 43.70/43.91 quotient(product(i(X), Y), X) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 quotient(product(i(X), product(quotient(Y, X), X)), X) 43.70/43.91 = { by lemma 28 } 43.70/43.91 difference(X, quotient(Y, X)) 43.70/43.91 43.70/43.91 Lemma 30: quotient(quotient(X, Y), j(Y)) = product(j(Y), product(Y, X)). 43.70/43.91 Proof: 43.70/43.91 quotient(quotient(X, Y), j(Y)) 43.70/43.91 = { by axiom 12 (sos01) } 43.70/43.91 quotient(quotient(product(X, one), Y), j(Y)) 43.70/43.91 = { by lemma 21 } 43.70/43.91 quotient(quotient(product(X, product(j(Y), Y)), Y), j(Y)) 43.70/43.91 = { by axiom 3 (sos08) } 43.70/43.91 difference(product(j(Y), Y), product(j(Y), product(Y, X))) 43.70/43.91 = { by lemma 21 } 43.70/43.91 difference(one, product(j(Y), product(Y, X))) 43.70/43.91 = { by lemma 17 } 43.70/43.91 product(j(Y), product(Y, X)) 43.70/43.91 43.70/43.91 Lemma 31: i(j(X)) = X. 43.70/43.91 Proof: 43.70/43.91 i(j(X)) 43.70/43.91 = { by axiom 4 (sos09) } 43.70/43.91 difference(j(X), one) 43.70/43.91 = { by lemma 21 } 43.70/43.91 difference(j(X), product(j(X), X)) 43.70/43.91 = { by axiom 9 (sos04) } 43.70/43.91 X 43.70/43.91 43.70/43.91 Lemma 32: quotient(i(i(X)), X) = eta(X). 43.70/43.91 Proof: 43.70/43.91 quotient(i(i(X)), X) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 product(X, difference(X, quotient(i(i(X)), X))) 43.70/43.91 = { by lemma 29 } 43.70/43.91 product(X, quotient(product(i(X), i(i(X))), X)) 43.70/43.91 = { by lemma 14 } 43.70/43.91 product(X, quotient(one, X)) 43.70/43.91 = { by axiom 2 (sos10) } 43.70/43.91 product(X, j(X)) 43.70/43.91 = { by lemma 13 } 43.70/43.91 eta(X) 43.70/43.91 43.70/43.91 Lemma 33: eta(j(X)) = eta(X). 43.70/43.91 Proof: 43.70/43.91 eta(j(X)) 43.70/43.91 = { by lemma 13 } 43.70/43.91 product(j(X), j(j(X))) 43.70/43.91 = { by axiom 6 (sos11) } 43.70/43.91 product(i(j(X)), j(X)) 43.70/43.91 = { by lemma 31 } 43.70/43.91 product(X, j(X)) 43.70/43.91 = { by lemma 13 } 43.70/43.91 eta(X) 43.70/43.91 43.70/43.91 Lemma 34: product(j(X), product(X, eta(X))) = eta(X). 43.70/43.91 Proof: 43.70/43.91 product(j(X), product(X, eta(X))) 43.70/43.91 = { by lemma 30 } 43.70/43.91 quotient(quotient(eta(X), X), j(X)) 43.70/43.91 = { by lemma 27 } 43.70/43.91 quotient(i(X), j(X)) 43.70/43.91 = { by lemma 31 } 43.70/43.91 quotient(i(i(j(X))), j(X)) 43.70/43.91 = { by lemma 32 } 43.70/43.91 eta(j(X)) 43.70/43.91 = { by lemma 33 } 43.70/43.91 eta(X) 43.70/43.91 43.70/43.91 Lemma 35: difference(X, eta(X)) = j(X). 43.70/43.91 Proof: 43.70/43.91 difference(X, eta(X)) 43.70/43.91 = { by lemma 13 } 43.70/43.91 difference(X, product(X, j(X))) 43.70/43.91 = { by axiom 9 (sos04) } 43.70/43.91 j(X) 43.70/43.91 43.70/43.91 Lemma 36: product(X, eta(X)) = j(j(X)). 43.70/43.91 Proof: 43.70/43.91 product(X, eta(X)) 43.70/43.91 = { by axiom 9 (sos04) } 43.70/43.91 difference(j(X), product(j(X), product(X, eta(X)))) 43.70/43.91 = { by lemma 34 } 43.70/43.91 difference(j(X), eta(X)) 43.70/43.91 = { by lemma 33 } 43.70/43.91 difference(j(X), eta(j(X))) 43.70/43.91 = { by lemma 35 } 43.70/43.91 j(j(X)) 43.70/43.91 43.70/43.91 Lemma 37: difference(X, i(X)) = quotient(j(X), X). 43.70/43.91 Proof: 43.70/43.91 difference(X, i(X)) 43.70/43.91 = { by lemma 27 } 43.70/43.91 difference(X, quotient(eta(X), X)) 43.70/43.91 = { by lemma 16 } 43.70/43.91 difference(X, quotient(eta(i(X)), X)) 43.70/43.91 = { by lemma 29 } 43.70/43.91 quotient(product(i(X), eta(i(X))), X) 43.70/43.91 = { by lemma 36 } 43.70/43.91 quotient(j(j(i(X))), X) 43.70/43.91 = { by lemma 15 } 43.70/43.91 quotient(j(X), X) 43.70/43.91 43.70/43.91 Lemma 38: product(eta(X), j(X)) = i(X). 43.70/43.91 Proof: 43.70/43.91 product(eta(X), j(X)) 43.70/43.91 = { by lemma 26 } 43.70/43.91 product(X, quotient(j(X), X)) 43.70/43.91 = { by lemma 37 } 43.70/43.91 product(X, difference(X, i(X))) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 i(X) 43.70/43.91 43.70/43.91 Lemma 39: difference(Y, product(difference(X, Z), X)) = product(difference(product(X, Y), Z), X). 43.70/43.91 Proof: 43.70/43.91 difference(Y, product(difference(X, Z), X)) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 difference(Y, product(difference(X, product(product(X, Y), difference(product(X, Y), Z))), X)) 43.70/43.91 = { by axiom 7 (sos07) } 43.70/43.91 difference(Y, product(quotient(product(Y, product(difference(product(X, Y), Z), X)), X), X)) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 difference(Y, product(Y, product(difference(product(X, Y), Z), X))) 43.70/43.91 = { by axiom 9 (sos04) } 43.70/43.91 product(difference(product(X, Y), Z), X) 43.70/43.91 43.70/43.91 Lemma 40: product(difference(product(X, Y), i(X)), X) = difference(Y, j(X)). 43.70/43.91 Proof: 43.70/43.91 product(difference(product(X, Y), i(X)), X) 43.70/43.91 = { by lemma 38 } 43.70/43.91 product(difference(product(X, Y), product(eta(X), j(X))), X) 43.70/43.91 = { by lemma 39 } 43.70/43.91 difference(Y, product(difference(X, product(eta(X), j(X))), X)) 43.70/43.91 = { by lemma 25 } 43.70/43.91 difference(Y, product(quotient(j(X), X), X)) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 difference(Y, j(X)) 43.70/43.91 43.70/43.91 Lemma 41: difference(product(X, Y), i(X)) = quotient(difference(Y, j(X)), X). 43.70/43.91 Proof: 43.70/43.91 difference(product(X, Y), i(X)) 43.70/43.91 = { by axiom 11 (sos05) } 43.70/43.91 quotient(product(difference(product(X, Y), i(X)), X), X) 43.70/43.91 = { by lemma 40 } 43.70/43.91 quotient(difference(Y, j(X)), X) 43.70/43.91 43.70/43.91 Lemma 42: quotient(product(difference(X, Y), product(Z, X)), X) = difference(X, product(Y, Z)). 43.70/43.91 Proof: 43.70/43.91 quotient(product(difference(X, Y), product(Z, X)), X) 43.70/43.91 = { by axiom 7 (sos07) } 43.70/43.91 difference(X, product(product(X, difference(X, Y)), Z)) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 difference(X, product(Y, Z)) 43.70/43.91 43.70/43.91 Lemma 43: product(difference(X, product(Y, Z)), X) = product(difference(X, Y), product(Z, X)). 43.70/43.91 Proof: 43.70/43.91 product(difference(X, product(Y, Z)), X) 43.70/43.91 = { by lemma 42 } 43.70/43.91 product(quotient(product(difference(X, Y), product(Z, X)), X), X) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 product(difference(X, Y), product(Z, X)) 43.70/43.91 43.70/43.91 Lemma 44: product(difference(X, j(Y)), product(Y, X)) = eta(X). 43.70/43.91 Proof: 43.70/43.91 product(difference(X, j(Y)), product(Y, X)) 43.70/43.91 = { by lemma 43 } 43.70/43.91 product(difference(X, product(j(Y), Y)), X) 43.70/43.91 = { by lemma 21 } 43.70/43.91 product(difference(X, one), X) 43.70/43.91 = { by axiom 4 (sos09) } 43.70/43.91 product(i(X), X) 43.70/43.91 = { by axiom 6 (sos11) } 43.70/43.91 product(X, j(X)) 43.70/43.91 = { by lemma 13 } 43.70/43.91 eta(X) 43.70/43.91 43.70/43.91 Lemma 45: eta(difference(Y, i(X))) = eta(difference(X, Y)). 43.70/43.91 Proof: 43.70/43.91 eta(difference(Y, i(X))) 43.70/43.91 = { by axiom 11 (sos05) } 43.70/43.91 eta(quotient(product(difference(Y, i(X)), X), X)) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 eta(quotient(product(difference(product(X, difference(X, Y)), i(X)), X), X)) 43.70/43.91 = { by lemma 40 } 43.70/43.91 eta(quotient(difference(difference(X, Y), j(X)), X)) 43.70/43.91 = { by lemma 41 } 43.70/43.91 eta(difference(product(X, difference(X, Y)), i(X))) 43.70/43.91 = { by lemma 23 } 43.70/43.91 product(product(difference(product(X, difference(X, Y)), i(X)), j(i(X))), product(X, difference(X, Y))) 43.70/43.91 = { by lemma 41 } 43.70/43.91 product(product(quotient(difference(difference(X, Y), j(X)), X), j(i(X))), product(X, difference(X, Y))) 43.70/43.91 = { by lemma 15 } 43.70/43.91 product(product(quotient(difference(difference(X, Y), j(X)), X), X), product(X, difference(X, Y))) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 product(difference(difference(X, Y), j(X)), product(X, difference(X, Y))) 43.70/43.91 = { by lemma 44 } 43.70/43.91 eta(difference(X, Y)) 43.70/43.91 43.70/43.91 Lemma 46: eta(difference(j(X), Y)) = eta(difference(Y, X)). 43.70/43.91 Proof: 43.70/43.91 eta(difference(j(X), Y)) 43.70/43.91 = { by lemma 45 } 43.70/43.91 eta(difference(Y, i(j(X)))) 43.70/43.91 = { by lemma 23 } 43.70/43.91 product(product(difference(Y, i(j(X))), j(i(j(X)))), Y) 43.70/43.91 = { by lemma 31 } 43.70/43.91 product(product(difference(Y, X), j(i(j(X)))), Y) 43.70/43.91 = { by lemma 15 } 43.70/43.91 product(product(difference(Y, X), j(X)), Y) 43.70/43.91 = { by lemma 23 } 43.70/43.91 eta(difference(Y, X)) 43.70/43.91 43.70/43.91 Lemma 47: eta(difference(i(X), Y)) = eta(difference(X, j(Y))). 43.70/43.91 Proof: 43.70/43.91 eta(difference(i(X), Y)) 43.70/43.91 = { by lemma 46 } 43.70/43.91 eta(difference(j(Y), i(X))) 43.70/43.91 = { by lemma 45 } 43.70/43.91 eta(difference(X, j(Y))) 43.70/43.91 43.70/43.91 Lemma 48: quotient(difference(i(X), Y), i(X)) = difference(i(X), product(Y, X)). 43.70/43.91 Proof: 43.70/43.91 quotient(difference(i(X), Y), i(X)) 43.70/43.91 = { by axiom 12 (sos01) } 43.70/43.91 quotient(product(difference(i(X), Y), one), i(X)) 43.70/43.91 = { by lemma 14 } 43.70/43.91 quotient(product(difference(i(X), Y), product(X, i(X))), i(X)) 43.70/43.91 = { by lemma 42 } 43.70/43.91 difference(i(X), product(Y, X)) 43.70/43.91 43.70/43.91 Lemma 49: quotient(difference(i(X), product(Y, X)), X) = product(X, Y). 43.70/43.91 Proof: 43.70/43.91 quotient(difference(i(X), product(Y, X)), X) 43.70/43.91 = { by lemma 48 } 43.70/43.91 quotient(quotient(difference(i(X), Y), i(X)), X) 43.70/43.91 = { by lemma 15 } 43.70/43.91 quotient(quotient(difference(i(X), Y), i(X)), j(i(X))) 43.70/43.91 = { by lemma 30 } 43.70/43.91 product(j(i(X)), product(i(X), difference(i(X), Y))) 43.70/43.91 = { by lemma 15 } 43.70/43.91 product(X, product(i(X), difference(i(X), Y))) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 product(X, Y) 43.70/43.91 43.70/43.91 Lemma 50: difference(i(X), product(Y, X)) = product(product(X, Y), X). 43.70/43.91 Proof: 43.70/43.91 difference(i(X), product(Y, X)) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 product(quotient(difference(i(X), product(Y, X)), X), X) 43.70/43.91 = { by lemma 49 } 43.70/43.91 product(product(X, Y), X) 43.70/43.91 43.70/43.91 Lemma 51: difference(quotient(Y, X), Y) = X. 43.70/43.91 Proof: 43.70/43.91 difference(quotient(Y, X), Y) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 difference(quotient(Y, X), product(quotient(Y, X), X)) 43.70/43.91 = { by axiom 9 (sos04) } 43.70/43.91 X 43.70/43.91 43.70/43.91 Lemma 52: product(product(product(X, Y), X), i(X)) = difference(i(X), Y). 43.70/43.91 Proof: 43.70/43.91 product(product(product(X, Y), X), i(X)) 43.70/43.91 = { by lemma 51 } 43.70/43.91 product(product(product(X, Y), X), difference(quotient(difference(i(X), Y), i(X)), difference(i(X), Y))) 43.70/43.91 = { by lemma 48 } 43.70/43.91 product(product(product(X, Y), X), difference(difference(i(X), product(Y, X)), difference(i(X), Y))) 43.70/43.91 = { by lemma 50 } 43.70/43.91 product(product(product(X, Y), X), difference(product(product(X, Y), X), difference(i(X), Y))) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 difference(i(X), Y) 43.70/43.91 43.70/43.91 Lemma 53: product(product(X, quotient(Y, X)), X) = difference(i(X), Y). 43.70/43.91 Proof: 43.70/43.91 product(product(X, quotient(Y, X)), X) 43.70/43.91 = { by lemma 50 } 43.70/43.91 difference(i(X), product(quotient(Y, X), X)) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 difference(i(X), Y) 43.70/43.91 43.70/43.91 Lemma 54: product(product(X, Y), j(X)) = product(X, quotient(Y, X)). 43.70/43.91 Proof: 43.70/43.91 product(product(X, Y), j(X)) 43.70/43.91 = { by lemma 20 } 43.70/43.91 product(X, quotient(product(Y, product(j(X), X)), X)) 43.70/43.91 = { by lemma 21 } 43.70/43.91 product(X, quotient(product(Y, one), X)) 43.70/43.91 = { by axiom 12 (sos01) } 43.70/43.91 product(X, quotient(Y, X)) 43.70/43.91 43.70/43.91 Lemma 55: product(X, quotient(difference(X, Y), X)) = product(Y, j(X)). 43.70/43.91 Proof: 43.70/43.91 product(X, quotient(difference(X, Y), X)) 43.70/43.91 = { by lemma 54 } 43.70/43.91 product(product(X, difference(X, Y)), j(X)) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 product(Y, j(X)) 43.70/43.91 43.70/43.91 Lemma 56: product(product(X, j(Y)), Y) = product(product(X, Y), i(Y)). 43.70/43.91 Proof: 43.70/43.91 product(product(X, j(Y)), Y) 43.70/43.91 = { by lemma 55 } 43.70/43.91 product(product(Y, quotient(difference(Y, X), Y)), Y) 43.70/43.91 = { by lemma 53 } 43.70/43.91 difference(i(Y), difference(Y, X)) 43.70/43.91 = { by lemma 52 } 43.70/43.91 product(product(product(Y, difference(Y, X)), Y), i(Y)) 43.70/43.91 = { by axiom 8 (sos03) } 43.70/43.91 product(product(X, Y), i(Y)) 43.70/43.91 43.70/43.91 Lemma 57: quotient(product(X, quotient(Y, X)), Z) = product(X, quotient(Y, product(Z, X))). 43.70/43.91 Proof: 43.70/43.91 quotient(product(X, quotient(Y, X)), Z) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 quotient(product(X, quotient(product(quotient(Y, product(Z, X)), product(Z, X)), X)), Z) 43.70/43.91 = { by lemma 20 } 43.70/43.91 quotient(product(product(X, quotient(Y, product(Z, X))), Z), Z) 43.70/43.91 = { by axiom 11 (sos05) } 43.70/43.91 product(X, quotient(Y, product(Z, X))) 43.70/43.91 43.70/43.91 Lemma 58: product(X, quotient(j(X), product(Y, X))) = quotient(i(X), Y). 43.70/43.91 Proof: 43.70/43.91 product(X, quotient(j(X), product(Y, X))) 43.70/43.91 = { by lemma 57 } 43.70/43.91 quotient(product(X, quotient(j(X), X)), Y) 43.70/43.91 = { by lemma 26 } 43.70/43.91 quotient(product(eta(X), j(X)), Y) 43.70/43.91 = { by lemma 38 } 43.70/43.91 quotient(i(X), Y) 43.70/43.91 43.70/43.91 Lemma 59: product(product(X, Y), quotient(Z, X)) = product(X, quotient(product(Y, Z), X)). 43.70/43.91 Proof: 43.70/43.91 product(product(X, Y), quotient(Z, X)) 43.70/43.91 = { by lemma 20 } 43.70/43.91 product(X, quotient(product(Y, product(quotient(Z, X), X)), X)) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 product(X, quotient(product(Y, Z), X)) 43.70/43.91 43.70/43.91 Lemma 60: eta(quotient(j(X), product(Y, X))) = eta(Y). 43.70/43.91 Proof: 43.70/43.91 eta(quotient(j(X), product(Y, X))) 43.70/43.91 = { by lemma 44 } 43.70/43.91 product(difference(quotient(j(X), product(Y, X)), j(X)), product(X, quotient(j(X), product(Y, X)))) 43.70/43.91 = { by lemma 51 } 43.70/43.91 product(product(Y, X), product(X, quotient(j(X), product(Y, X)))) 43.70/43.91 = { by lemma 58 } 43.70/43.91 product(product(Y, X), quotient(i(X), Y)) 43.70/43.91 = { by lemma 59 } 43.70/43.91 product(Y, quotient(product(X, i(X)), Y)) 43.70/43.91 = { by lemma 14 } 43.70/43.91 product(Y, quotient(one, Y)) 43.70/43.91 = { by axiom 2 (sos10) } 43.70/43.91 product(Y, j(Y)) 43.70/43.91 = { by lemma 13 } 43.70/43.91 eta(Y) 43.70/43.91 43.70/43.91 Lemma 61: eta(quotient(j(Y), X)) = eta(quotient(X, Y)). 43.70/43.91 Proof: 43.70/43.91 eta(quotient(j(Y), X)) 43.70/43.91 = { by axiom 1 (sos06) } 43.70/43.91 eta(quotient(j(Y), product(quotient(X, Y), Y))) 43.70/43.91 = { by lemma 60 } 43.70/43.91 eta(quotient(X, Y)) 43.70/43.91 43.70/43.91 Lemma 62: eta(quotient(X, i(Y))) = eta(quotient(Y, X)). 43.70/43.91 Proof: 43.70/43.91 eta(quotient(X, i(Y))) 43.70/43.91 = { by lemma 61 } 43.70/43.91 eta(quotient(j(i(Y)), X)) 43.70/43.91 = { by lemma 15 } 43.70/43.91 eta(quotient(Y, X)) 43.70/43.91 43.70/43.91 Lemma 63: quotient(product(X, product(Y, j(X))), j(X)) = difference(j(X), Y). 43.70/43.91 Proof: 43.70/43.91 quotient(product(X, product(Y, j(X))), j(X)) 43.70/43.91 = { by axiom 7 (sos07) } 43.70/43.91 difference(j(X), product(product(j(X), X), Y)) 43.70/43.91 = { by lemma 21 } 43.70/43.91 difference(j(X), product(one, Y)) 43.70/43.91 = { by axiom 5 (sos02) } 43.70/43.91 difference(j(X), Y) 43.70/43.91 43.70/43.91 Lemma 64: quotient(product(X, eta(X)), j(X)) = difference(j(X), X). 43.70/43.91 Proof: 43.70/43.91 quotient(product(X, eta(X)), j(X)) 43.70/43.91 = { by lemma 33 } 43.70/43.91 quotient(product(X, eta(j(X))), j(X)) 43.70/43.91 = { by lemma 13 } 43.70/43.91 quotient(product(X, product(j(X), j(j(X)))), j(X)) 43.70/43.91 = { by axiom 6 (sos11) } 43.70/43.91 quotient(product(X, product(i(j(X)), j(X))), j(X)) 43.70/43.91 = { by lemma 63 } 43.70/43.91 difference(j(X), i(j(X))) 43.70/43.91 = { by lemma 31 } 43.70/43.92 difference(j(X), X) 43.70/43.92 43.70/43.92 Lemma 65: eta(eta(X)) = one. 43.70/43.92 Proof: 43.70/43.92 eta(eta(X)) 43.70/43.92 = { by lemma 22 } 43.70/43.92 product(product(eta(X), j(product(i(X), eta(X)))), i(X)) 43.70/43.92 = { by lemma 34 } 43.70/43.92 product(product(product(j(X), product(X, eta(X))), j(product(i(X), eta(X)))), i(X)) 43.70/43.92 = { by lemma 16 } 43.70/43.92 product(product(product(j(X), product(X, eta(X))), j(product(i(X), eta(i(X))))), i(X)) 43.70/43.92 = { by lemma 36 } 43.70/43.92 product(product(product(j(X), product(X, eta(X))), j(j(j(i(X))))), i(X)) 43.70/43.92 = { by lemma 15 } 43.70/43.92 product(product(product(j(X), product(X, eta(X))), j(j(X))), i(X)) 43.70/43.92 = { by lemma 54 } 43.70/43.92 product(product(j(X), quotient(product(X, eta(X)), j(X))), i(X)) 43.70/43.92 = { by lemma 64 } 43.70/43.92 product(product(j(X), difference(j(X), X)), i(X)) 43.70/43.92 = { by axiom 8 (sos03) } 43.70/43.92 product(X, i(X)) 43.70/43.92 = { by lemma 14 } 43.70/43.92 one 43.70/43.92 43.70/43.92 Lemma 66: j(eta(X)) = i(eta(X)). 43.70/43.92 Proof: 43.70/43.92 j(eta(X)) 43.70/43.92 = { by lemma 35 } 43.70/43.92 difference(eta(X), eta(eta(X))) 43.70/43.92 = { by lemma 65 } 43.70/43.92 difference(eta(X), one) 43.70/43.92 = { by axiom 4 (sos09) } 43.70/43.92 i(eta(X)) 43.70/43.92 43.70/43.92 Lemma 67: product(X, j(product(Y, X))) = quotient(eta(X), Y). 43.70/43.92 Proof: 43.70/43.92 product(X, j(product(Y, X))) 43.70/43.92 = { by axiom 11 (sos05) } 43.70/43.92 quotient(product(product(X, j(product(Y, X))), Y), Y) 43.70/43.92 = { by lemma 22 } 43.70/43.92 quotient(eta(X), Y) 43.70/43.92 43.70/43.92 Lemma 68: quotient(eta(X), quotient(Y, X)) = product(X, j(Y)). 43.70/43.92 Proof: 43.70/43.92 quotient(eta(X), quotient(Y, X)) 43.70/43.92 = { by lemma 67 } 43.70/43.92 product(X, j(product(quotient(Y, X), X))) 43.70/43.92 = { by axiom 1 (sos06) } 43.70/43.92 product(X, j(Y)) 43.70/43.92 43.70/43.92 Lemma 69: quotient(difference(i(X), Y), X) = product(X, quotient(Y, X)). 43.70/43.92 Proof: 43.70/43.92 quotient(difference(i(X), Y), X) 43.70/43.92 = { by axiom 1 (sos06) } 43.70/43.92 quotient(difference(i(X), product(quotient(Y, X), X)), X) 43.70/43.92 = { by lemma 49 } 43.70/43.92 product(X, quotient(Y, X)) 43.70/43.92 43.70/43.92 Lemma 70: difference(i(X), eta(X)) = X. 43.70/43.92 Proof: 43.70/43.92 difference(i(X), eta(X)) 43.70/43.92 = { by lemma 27 } 43.70/43.92 difference(quotient(eta(X), X), eta(X)) 43.70/43.92 = { by lemma 51 } 43.70/43.92 X 43.70/43.92 43.70/43.92 Lemma 71: product(difference(Y, eta(X)), i(X)) = i(difference(i(X), Y)). 43.70/43.92 Proof: 43.70/43.92 product(difference(Y, eta(X)), i(X)) 43.70/43.92 = { by axiom 8 (sos03) } 43.70/43.92 product(difference(product(i(X), difference(i(X), Y)), eta(X)), i(X)) 43.70/43.92 = { by lemma 39 } 43.70/43.92 difference(difference(i(X), Y), product(difference(i(X), eta(X)), i(X))) 43.70/43.92 = { by lemma 70 } 43.70/43.92 difference(difference(i(X), Y), product(X, i(X))) 43.70/43.92 = { by lemma 14 } 43.70/43.92 difference(difference(i(X), Y), one) 43.70/43.92 = { by axiom 4 (sos09) } 43.70/43.92 i(difference(i(X), Y)) 43.70/43.92 43.70/43.92 Lemma 72: quotient(product(X, quotient(Y, X)), j(X)) = product(X, Y). 43.70/43.92 Proof: 43.70/43.92 quotient(product(X, quotient(Y, X)), j(X)) 43.70/43.92 = { by lemma 54 } 43.70/43.92 quotient(product(product(X, Y), j(X)), j(X)) 43.70/43.92 = { by axiom 11 (sos05) } 43.70/43.92 product(X, Y) 43.70/43.92 43.70/43.92 Lemma 73: quotient(product(X, Y), j(X)) = product(X, product(Y, X)). 43.70/43.92 Proof: 43.70/43.92 quotient(product(X, Y), j(X)) 43.70/43.92 = { by axiom 11 (sos05) } 43.70/43.92 quotient(product(X, quotient(product(Y, X), X)), j(X)) 43.70/43.92 = { by lemma 72 } 43.76/43.92 product(X, product(Y, X)) 43.76/43.92 43.76/43.92 Lemma 74: difference(difference(Y, X), eta(Y)) = product(i(X), Y). 43.76/43.92 Proof: 43.76/43.92 difference(difference(Y, X), eta(Y)) 43.76/43.92 = { by lemma 13 } 43.76/43.92 difference(difference(Y, X), product(Y, j(Y))) 43.76/43.92 = { by axiom 6 (sos11) } 43.76/43.92 difference(difference(Y, X), product(i(Y), Y)) 43.76/43.92 = { by axiom 4 (sos09) } 43.76/43.92 difference(difference(Y, X), product(difference(Y, one), Y)) 43.76/43.92 = { by lemma 14 } 43.76/43.92 difference(difference(Y, X), product(difference(Y, product(X, i(X))), Y)) 43.76/43.92 = { by lemma 43 } 43.76/43.92 difference(difference(Y, X), product(difference(Y, X), product(i(X), Y))) 43.76/43.92 = { by axiom 9 (sos04) } 43.76/43.92 product(i(X), Y) 43.76/43.92 43.76/43.92 Lemma 75: product(i(product(X, Y)), X) = difference(Y, eta(X)). 43.76/43.92 Proof: 43.76/43.92 product(i(product(X, Y)), X) 43.76/43.92 = { by lemma 74 } 43.76/43.92 difference(difference(X, product(X, Y)), eta(X)) 43.76/43.92 = { by axiom 9 (sos04) } 43.76/43.92 difference(Y, eta(X)) 43.76/43.92 43.76/43.92 Lemma 76: quotient(difference(Y, eta(X)), X) = i(product(X, Y)). 43.76/43.92 Proof: 43.76/43.92 quotient(difference(Y, eta(X)), X) 43.76/43.92 = { by lemma 75 } 43.76/43.92 quotient(product(i(product(X, Y)), X), X) 43.76/43.92 = { by axiom 11 (sos05) } 43.76/43.92 i(product(X, Y)) 43.76/43.92 43.76/43.92 Lemma 77: eta(quotient(eta(X), Y)) = eta(product(eta(X), Y)). 43.76/43.92 Proof: 43.76/43.92 eta(quotient(eta(X), Y)) 43.76/43.92 = { by lemma 62 } 43.76/43.92 eta(quotient(Y, i(eta(X)))) 43.76/43.92 = { by axiom 11 (sos05) } 43.76/43.92 eta(quotient(quotient(product(Y, eta(X)), eta(X)), i(eta(X)))) 43.76/43.92 = { by lemma 66 } 43.76/43.92 eta(quotient(quotient(product(Y, eta(X)), eta(X)), j(eta(X)))) 43.76/43.92 = { by lemma 30 } 43.76/43.92 eta(product(j(eta(X)), product(eta(X), product(Y, eta(X))))) 43.76/43.92 = { by lemma 73 } 43.76/43.92 eta(product(j(eta(X)), quotient(product(eta(X), Y), j(eta(X))))) 43.76/43.92 = { by lemma 69 } 43.76/43.92 eta(quotient(difference(i(j(eta(X))), product(eta(X), Y)), j(eta(X)))) 43.76/43.92 = { by lemma 62 } 43.76/43.92 eta(quotient(j(eta(X)), i(difference(i(j(eta(X))), product(eta(X), Y))))) 43.76/43.92 = { by lemma 62 } 43.76/43.92 eta(quotient(i(difference(i(j(eta(X))), product(eta(X), Y))), i(j(eta(X))))) 43.76/43.92 = { by lemma 61 } 43.76/43.92 eta(quotient(j(i(j(eta(X)))), i(difference(i(j(eta(X))), product(eta(X), Y))))) 43.76/43.92 = { by lemma 71 } 43.76/43.92 eta(quotient(j(i(j(eta(X)))), product(difference(product(eta(X), Y), eta(j(eta(X)))), i(j(eta(X)))))) 43.76/43.92 = { by lemma 60 } 43.76/43.92 eta(difference(product(eta(X), Y), eta(j(eta(X))))) 43.76/43.92 = { by lemma 15 } 43.76/43.92 eta(difference(product(eta(X), j(i(Y))), eta(j(eta(X))))) 43.76/43.92 = { by lemma 68 } 43.76/43.92 eta(difference(quotient(eta(eta(X)), quotient(i(Y), eta(X))), eta(j(eta(X))))) 43.76/43.92 = { by lemma 33 } 43.76/43.92 eta(difference(quotient(eta(eta(X)), quotient(i(Y), eta(X))), eta(eta(X)))) 43.76/43.92 = { by lemma 51 } 43.76/43.92 eta(quotient(i(Y), eta(X))) 43.76/43.92 = { by axiom 4 (sos09) } 43.76/43.92 eta(quotient(difference(Y, one), eta(X))) 43.76/43.92 = { by lemma 65 } 43.76/43.92 eta(quotient(difference(Y, eta(eta(X))), eta(X))) 43.76/43.92 = { by lemma 76 } 43.76/43.92 eta(i(product(eta(X), Y))) 43.76/43.92 = { by lemma 16 } 43.76/43.92 eta(product(eta(X), Y)) 43.76/43.92 43.76/43.92 Lemma 78: product(X, product(difference(X, Y), X)) = quotient(Y, j(X)). 43.76/43.92 Proof: 43.76/43.92 product(X, product(difference(X, Y), X)) 43.76/43.92 = { by lemma 73 } 43.76/43.92 quotient(product(X, difference(X, Y)), j(X)) 43.76/43.92 = { by axiom 8 (sos03) } 43.76/43.92 quotient(Y, j(X)) 43.76/43.92 43.76/43.92 Lemma 79: quotient(eta(X), product(Y, X)) = difference(X, j(Y)). 43.76/43.92 Proof: 43.76/43.92 quotient(eta(X), product(Y, X)) 43.76/43.92 = { by lemma 44 } 43.76/43.92 quotient(product(difference(X, j(Y)), product(Y, X)), product(Y, X)) 43.76/43.92 = { by axiom 11 (sos05) } 43.76/43.92 difference(X, j(Y)) 43.76/43.92 43.76/43.92 Lemma 80: difference(X, j(quotient(Y, X))) = quotient(eta(X), Y). 43.76/43.92 Proof: 43.76/43.92 difference(X, j(quotient(Y, X))) 43.76/43.92 = { by lemma 79 } 43.76/43.92 quotient(eta(X), product(quotient(Y, X), X)) 43.76/43.92 = { by axiom 1 (sos06) } 43.76/43.92 quotient(eta(X), Y) 43.76/43.92 43.76/43.92 Lemma 81: eta(product(difference(X, Y), X)) = eta(quotient(eta(X), Y)). 43.76/43.92 Proof: 43.76/43.92 eta(product(difference(X, Y), X)) 43.76/43.92 = { by axiom 9 (sos04) } 43.76/43.92 eta(difference(X, product(X, product(difference(X, Y), X)))) 43.76/43.92 = { by lemma 78 } 43.76/43.92 eta(difference(X, quotient(Y, j(X)))) 43.76/43.92 = { by lemma 46 } 43.76/43.92 eta(difference(j(quotient(Y, j(X))), X)) 43.76/43.92 = { by lemma 46 } 43.76/43.92 eta(difference(j(X), j(quotient(Y, j(X))))) 43.76/43.92 = { by lemma 80 } 43.76/43.92 eta(quotient(eta(j(X)), Y)) 43.76/43.92 = { by lemma 33 } 43.76/43.92 eta(quotient(eta(X), Y)) 43.76/43.92 43.76/43.92 Lemma 82: eta(product(eta(X), product(X, Y))) = eta(product(Y, X)). 43.76/43.92 Proof: 43.76/43.92 eta(product(eta(X), product(X, Y))) 43.76/43.92 = { by lemma 77 } 43.76/43.92 eta(quotient(eta(X), product(X, Y))) 43.76/43.92 = { by lemma 81 } 43.76/43.92 eta(product(difference(X, product(X, Y)), X)) 43.76/43.92 = { by axiom 9 (sos04) } 43.76/43.92 eta(product(Y, X)) 43.76/43.92 43.76/43.92 Lemma 83: product(difference(j(X), Y), j(X)) = product(X, product(Y, j(X))). 43.76/43.92 Proof: 43.76/43.92 product(difference(j(X), Y), j(X)) 43.76/43.92 = { by lemma 63 } 43.76/43.92 product(quotient(product(X, product(Y, j(X))), j(X)), j(X)) 43.76/43.92 = { by axiom 1 (sos06) } 43.76/43.92 product(X, product(Y, j(X))) 43.76/43.92 43.76/43.92 Lemma 84: i(i(eta(X))) = eta(X). 43.76/43.92 Proof: 43.76/43.92 i(i(eta(X))) 43.76/43.92 = { by axiom 4 (sos09) } 43.76/43.92 difference(i(eta(X)), one) 43.76/43.92 = { by lemma 65 } 43.76/43.92 difference(i(eta(X)), eta(eta(X))) 43.76/43.92 = { by lemma 70 } 43.76/43.92 eta(X) 43.76/43.92 43.76/43.92 Lemma 85: difference(Y, quotient(eta(Y), X)) = j(product(X, Y)). 43.76/43.92 Proof: 43.76/43.92 difference(Y, quotient(eta(Y), X)) 43.76/43.92 = { by lemma 67 } 43.76/43.92 difference(Y, product(Y, j(product(X, Y)))) 43.76/43.92 = { by axiom 9 (sos04) } 43.76/43.92 j(product(X, Y)) 43.76/43.92 43.76/43.92 Lemma 86: eta(product(X, i(eta(Y)))) = eta(difference(eta(Y), X)). 43.76/43.92 Proof: 43.76/43.92 eta(product(X, i(eta(Y)))) 43.76/43.92 = { by lemma 66 } 43.76/43.92 eta(product(X, j(eta(Y)))) 43.76/43.92 = { by lemma 33 } 43.76/43.92 eta(j(product(X, j(eta(Y))))) 43.76/43.92 = { by lemma 85 } 43.76/43.92 eta(difference(j(eta(Y)), quotient(eta(j(eta(Y))), X))) 43.76/43.92 = { by lemma 46 } 43.76/43.92 eta(difference(quotient(eta(j(eta(Y))), X), eta(Y))) 43.76/43.92 = { by lemma 33 } 43.76/43.92 eta(difference(quotient(eta(eta(Y)), X), eta(Y))) 43.76/43.92 = { by lemma 65 } 43.76/43.92 eta(difference(quotient(one, X), eta(Y))) 43.76/43.92 = { by axiom 2 (sos10) } 43.76/43.92 eta(difference(j(X), eta(Y))) 43.76/43.92 = { by lemma 46 } 43.76/43.92 eta(difference(eta(Y), X)) 43.76/43.92 43.76/43.92 Lemma 87: product(j(X), product(X, Y)) = product(X, product(i(X), Y)). 43.76/43.92 Proof: 43.76/43.92 product(j(X), product(X, Y)) 43.76/43.92 = { by lemma 30 } 43.76/43.92 quotient(quotient(Y, X), j(X)) 43.76/43.92 = { by lemma 78 } 43.76/43.92 product(X, product(difference(X, quotient(Y, X)), X)) 43.76/43.92 = { by lemma 73 } 43.76/43.92 quotient(product(X, difference(X, quotient(Y, X))), j(X)) 43.76/43.92 = { by lemma 29 } 43.76/43.92 quotient(product(X, quotient(product(i(X), Y), X)), j(X)) 43.76/43.92 = { by lemma 72 } 43.76/43.92 product(X, product(i(X), Y)) 43.76/43.92 43.76/43.92 Lemma 88: product(X, product(i(X), product(Y, j(X)))) = quotient(Y, j(j(X))). 43.76/43.92 Proof: 43.76/43.92 product(X, product(i(X), product(Y, j(X)))) 43.76/43.92 = { by lemma 87 } 43.76/43.92 product(j(X), product(X, product(Y, j(X)))) 43.76/43.92 = { by lemma 72 } 43.76/43.92 quotient(product(j(X), quotient(product(X, product(Y, j(X))), j(X))), j(j(X))) 43.76/43.92 = { by lemma 63 } 43.76/43.92 quotient(product(j(X), difference(j(X), Y)), j(j(X))) 43.76/43.92 = { by lemma 73 } 43.76/43.92 product(j(X), product(difference(j(X), Y), j(X))) 43.76/43.92 = { by lemma 78 } 43.76/43.92 quotient(Y, j(j(X))) 43.76/43.92 43.76/43.92 Lemma 89: quotient(difference(X, Y), i(X)) = product(product(j(X), Y), X). 43.76/43.92 Proof: 43.76/43.92 quotient(difference(X, Y), i(X)) 43.76/43.92 = { by lemma 31 } 43.76/43.92 quotient(difference(i(j(X)), Y), i(X)) 43.76/43.92 = { by lemma 53 } 43.76/43.92 quotient(product(product(j(X), quotient(Y, j(X))), j(X)), i(X)) 43.76/43.92 = { by lemma 19 } 43.76/43.92 product(product(X, product(i(X), product(product(j(X), quotient(Y, j(X))), j(X)))), X) 43.76/43.92 = { by lemma 88 } 43.76/43.92 product(quotient(product(j(X), quotient(Y, j(X))), j(j(X))), X) 43.76/43.92 = { by lemma 72 } 43.76/43.92 product(product(j(X), Y), X) 43.76/43.92 43.76/43.92 Lemma 90: quotient(product(difference(j(X), Y), eta(X)), j(X)) = difference(j(X), product(Y, X)). 43.76/43.92 Proof: 43.76/43.92 quotient(product(difference(j(X), Y), eta(X)), j(X)) 43.76/43.92 = { by lemma 13 } 43.76/43.92 quotient(product(difference(j(X), Y), product(X, j(X))), j(X)) 43.76/43.92 = { by lemma 42 } 43.76/43.92 difference(j(X), product(Y, X)) 43.76/43.92 43.76/43.92 Lemma 91: quotient(difference(j(X), product(Y, X)), X) = difference(eta(X), product(X, Y)). 43.76/43.92 Proof: 43.76/43.92 quotient(difference(j(X), product(Y, X)), X) 43.76/43.92 = { by lemma 90 } 43.76/43.92 quotient(quotient(product(difference(j(X), Y), eta(X)), j(X)), X) 43.76/43.92 = { by lemma 13 } 43.76/43.92 quotient(quotient(product(difference(j(X), Y), product(X, j(X))), j(X)), X) 43.76/43.92 = { by axiom 3 (sos08) } 43.76/43.92 difference(product(X, j(X)), product(X, product(j(X), difference(j(X), Y)))) 43.76/43.92 = { by lemma 13 } 43.76/43.92 difference(eta(X), product(X, product(j(X), difference(j(X), Y)))) 43.76/43.92 = { by axiom 8 (sos03) } 43.76/43.92 difference(eta(X), product(X, Y)) 43.76/43.92 43.76/43.92 Lemma 92: product(product(j(X), product(Y, i(X))), X) = difference(eta(X), product(i(X), Y)). 43.76/43.92 Proof: 43.76/43.92 product(product(j(X), product(Y, i(X))), X) 43.76/43.92 = { by lemma 89 } 43.76/43.92 quotient(difference(X, product(Y, i(X))), i(X)) 43.76/43.92 = { by lemma 15 } 43.76/43.92 quotient(difference(j(i(X)), product(Y, i(X))), i(X)) 43.76/43.92 = { by lemma 91 } 43.76/43.92 difference(eta(i(X)), product(i(X), Y)) 43.76/43.92 = { by lemma 16 } 43.76/43.92 difference(eta(X), product(i(X), Y)) 43.76/43.92 43.76/43.92 Lemma 93: eta(product(X, eta(Y))) = eta(product(eta(Y), X)). 43.76/43.92 Proof: 43.76/43.92 eta(product(X, eta(Y))) 43.76/43.92 = { by axiom 9 (sos04) } 43.76/43.92 eta(difference(eta(Y), product(eta(Y), product(X, eta(Y))))) 43.76/43.92 = { by lemma 15 } 43.76/43.92 eta(difference(eta(Y), product(j(i(eta(Y))), product(X, eta(Y))))) 43.76/43.92 = { by lemma 84 } 43.76/43.92 eta(difference(eta(Y), product(j(i(eta(Y))), product(X, i(i(eta(Y))))))) 43.76/43.92 = { by lemma 86 } 43.76/43.92 eta(product(product(j(i(eta(Y))), product(X, i(i(eta(Y))))), i(eta(Y)))) 43.76/43.92 = { by lemma 92 } 43.76/43.92 eta(difference(eta(i(eta(Y))), product(i(i(eta(Y))), X))) 43.76/43.92 = { by lemma 16 } 43.76/43.92 eta(difference(eta(eta(Y)), product(i(i(eta(Y))), X))) 43.76/43.92 = { by lemma 65 } 43.76/43.92 eta(difference(one, product(i(i(eta(Y))), X))) 43.76/43.92 = { by lemma 84 } 43.76/43.92 eta(difference(one, product(eta(Y), X))) 43.76/43.92 = { by lemma 17 } 43.76/43.92 eta(product(eta(Y), X)) 43.76/43.92 43.76/43.92 Lemma 94: product(i(X), product(Y, X)) = product(difference(X, Y), X). 43.76/43.92 Proof: 43.76/43.92 product(i(X), product(Y, X)) 43.76/43.92 = { by axiom 1 (sos06) } 43.76/43.92 product(quotient(product(i(X), product(Y, X)), X), X) 43.76/43.92 = { by lemma 28 } 43.76/43.92 product(difference(X, Y), X) 43.76/43.92 43.76/43.92 Lemma 95: eta(product(i(X), Y)) = eta(product(X, j(Y))). 43.76/43.92 Proof: 43.76/43.92 eta(product(i(X), Y)) 43.76/43.92 = { by axiom 1 (sos06) } 43.76/43.92 eta(product(i(X), product(quotient(Y, X), X))) 43.76/43.92 = { by lemma 94 } 43.76/43.92 eta(product(difference(X, quotient(Y, X)), X)) 43.76/43.92 = { by lemma 81 } 43.76/43.92 eta(quotient(eta(X), quotient(Y, X))) 43.76/43.92 = { by lemma 68 } 43.76/43.92 eta(product(X, j(Y))) 43.76/43.92 43.76/43.92 Lemma 96: eta(product(X, quotient(Y, X))) = eta(product(Y, eta(X))). 43.76/43.92 Proof: 43.76/43.92 eta(product(X, quotient(Y, X))) 43.76/43.92 = { by lemma 69 } 43.76/43.92 eta(quotient(difference(i(X), Y), X)) 43.76/43.92 = { by lemma 62 } 43.76/43.92 eta(quotient(X, i(difference(i(X), Y)))) 43.76/43.92 = { by lemma 62 } 43.76/43.92 eta(quotient(i(difference(i(X), Y)), i(X))) 43.76/43.92 = { by lemma 61 } 43.76/43.92 eta(quotient(j(i(X)), i(difference(i(X), Y)))) 43.76/43.92 = { by lemma 71 } 43.76/43.92 eta(quotient(j(i(X)), product(difference(Y, eta(X)), i(X)))) 43.76/43.92 = { by lemma 60 } 43.76/43.92 eta(difference(Y, eta(X))) 43.76/43.92 = { by lemma 45 } 43.76/43.92 eta(difference(eta(X), i(Y))) 43.76/43.92 = { by lemma 86 } 43.76/43.92 eta(product(i(Y), i(eta(X)))) 43.76/43.92 = { by lemma 95 } 43.76/43.92 eta(product(Y, j(i(eta(X))))) 43.76/43.92 = { by lemma 15 } 43.76/43.92 eta(product(Y, eta(X))) 43.76/43.92 43.76/43.92 Lemma 97: eta(product(j(X), Y)) = eta(product(X, i(Y))). 43.76/43.92 Proof: 43.76/43.92 eta(product(j(X), Y)) 43.76/43.92 = { by lemma 15 } 43.76/43.92 eta(product(j(X), j(i(Y)))) 43.76/43.92 = { by lemma 68 } 43.76/43.92 eta(quotient(eta(j(X)), quotient(i(Y), j(X)))) 43.76/43.92 = { by lemma 81 } 43.76/43.92 eta(product(difference(j(X), quotient(i(Y), j(X))), j(X))) 43.76/43.92 = { by lemma 63 } 43.76/43.92 eta(product(quotient(product(X, product(quotient(i(Y), j(X)), j(X))), j(X)), j(X))) 43.76/43.92 = { by axiom 1 (sos06) } 43.76/43.92 eta(product(quotient(product(X, i(Y)), j(X)), j(X))) 43.76/43.92 = { by axiom 1 (sos06) } 43.76/43.92 eta(product(X, i(Y))) 43.76/43.92 43.76/43.92 Lemma 98: quotient(quotient(X, Y), j(Y)) = product(Y, product(i(Y), X)). 43.76/43.92 Proof: 43.76/43.92 quotient(quotient(X, Y), j(Y)) 43.76/43.92 = { by axiom 12 (sos01) } 43.76/43.92 quotient(quotient(product(X, one), Y), j(Y)) 43.76/43.92 = { by lemma 21 } 43.76/43.92 quotient(quotient(product(X, product(j(Y), Y)), Y), j(Y)) 43.76/43.92 = { by axiom 3 (sos08) } 43.76/43.92 difference(product(j(Y), Y), product(j(Y), product(Y, X))) 43.76/43.92 = { by lemma 21 } 43.76/43.92 difference(one, product(j(Y), product(Y, X))) 43.76/43.92 = { by lemma 17 } 43.76/43.92 product(j(Y), product(Y, X)) 43.76/43.92 = { by lemma 87 } 43.76/43.92 product(Y, product(i(Y), X)) 43.76/43.92 43.76/43.92 Lemma 99: quotient(difference(difference(X, Y), product(Z, X)), X) = difference(Y, product(X, Z)). 43.76/43.92 Proof: 43.76/43.92 quotient(difference(difference(X, Y), product(Z, X)), X) 43.76/43.92 = { by lemma 42 } 43.76/43.92 quotient(quotient(product(difference(difference(X, Y), Z), product(X, difference(X, Y))), difference(X, Y)), X) 43.76/43.92 = { by axiom 3 (sos08) } 43.76/43.92 difference(product(X, difference(X, Y)), product(X, product(difference(X, Y), difference(difference(X, Y), Z)))) 43.76/43.92 = { by axiom 8 (sos03) } 43.76/43.92 difference(Y, product(X, product(difference(X, Y), difference(difference(X, Y), Z)))) 43.76/43.92 = { by axiom 8 (sos03) } 43.76/43.92 difference(Y, product(X, Z)) 43.76/43.92 43.76/43.92 Lemma 100: difference(X, product(Y, quotient(Z, Y))) = quotient(difference(difference(Y, X), Z), Y). 43.76/43.92 Proof: 43.76/43.92 difference(X, product(Y, quotient(Z, Y))) 43.76/43.92 = { by lemma 99 } 43.76/43.92 quotient(difference(difference(Y, X), product(quotient(Z, Y), Y)), Y) 43.76/43.92 = { by axiom 1 (sos06) } 43.76/43.92 quotient(difference(difference(Y, X), Z), Y) 43.76/43.92 43.76/43.92 Lemma 101: difference(product(Z, X), product(Z, Y)) = quotient(difference(X, product(Y, Z)), Z). 43.76/43.92 Proof: 43.76/43.92 difference(product(Z, X), product(Z, Y)) 43.76/43.92 = { by axiom 8 (sos03) } 43.76/43.92 difference(product(Z, X), product(Z, product(X, difference(X, Y)))) 43.76/43.92 = { by axiom 3 (sos08) } 43.76/43.92 quotient(quotient(product(difference(X, Y), product(Z, X)), X), Z) 43.76/43.92 = { by lemma 42 } 43.76/43.92 quotient(difference(X, product(Y, Z)), Z) 43.76/43.92 43.76/43.92 Lemma 102: product(X, product(i(X), i(difference(X, Y)))) = i(product(j(X), Y)). 43.76/43.92 Proof: 43.76/43.92 product(X, product(i(X), i(difference(X, Y)))) 43.76/43.92 = { by axiom 4 (sos09) } 43.76/43.92 product(X, product(i(X), difference(difference(X, Y), one))) 43.76/43.92 = { by lemma 98 } 43.76/43.92 quotient(quotient(difference(difference(X, Y), one), X), j(X)) 43.76/43.92 = { by lemma 100 } 43.76/43.92 quotient(difference(Y, product(X, quotient(one, X))), j(X)) 43.76/43.92 = { by axiom 2 (sos10) } 43.76/43.92 quotient(difference(Y, product(X, quotient(one, X))), quotient(one, X)) 43.76/43.92 = { by lemma 101 } 43.76/43.92 difference(product(quotient(one, X), Y), product(quotient(one, X), X)) 43.76/43.92 = { by axiom 1 (sos06) } 43.76/43.92 difference(product(quotient(one, X), Y), one) 43.76/43.92 = { by axiom 4 (sos09) } 43.76/43.92 i(product(quotient(one, X), Y)) 43.76/43.92 = { by axiom 2 (sos10) } 43.76/43.92 i(product(j(X), Y)) 43.76/43.92 43.76/43.92 Lemma 103: difference(X, i(product(j(X), Y))) = product(i(X), i(difference(X, Y))). 43.76/43.92 Proof: 43.76/43.92 difference(X, i(product(j(X), Y))) 43.76/43.92 = { by lemma 102 } 43.76/43.92 difference(X, product(X, product(i(X), i(difference(X, Y))))) 43.76/43.92 = { by axiom 9 (sos04) } 43.76/43.93 product(i(X), i(difference(X, Y))) 43.76/43.93 43.76/43.93 Lemma 104: eta(product(product(X, Y), i(Y))) = eta(X). 43.76/43.93 Proof: 43.76/43.93 eta(product(product(X, Y), i(Y))) 43.76/43.93 = { by lemma 56 } 43.76/43.93 eta(product(product(X, j(Y)), Y)) 43.76/43.93 = { by lemma 82 } 43.76/43.93 eta(product(eta(Y), product(Y, product(X, j(Y))))) 43.76/43.93 = { by lemma 33 } 43.76/43.93 eta(product(eta(j(Y)), product(Y, product(X, j(Y))))) 43.76/43.93 = { by lemma 83 } 43.76/43.93 eta(product(eta(j(Y)), product(difference(j(Y), X), j(Y)))) 43.76/43.93 = { by lemma 93 } 43.76/43.93 eta(product(product(difference(j(Y), X), j(Y)), eta(j(Y)))) 43.76/43.93 = { by lemma 96 } 43.76/43.93 eta(product(j(Y), quotient(product(difference(j(Y), X), j(Y)), j(Y)))) 43.76/43.93 = { by axiom 11 (sos05) } 43.76/43.93 eta(product(j(Y), difference(j(Y), X))) 43.76/43.93 = { by lemma 97 } 43.76/43.93 eta(product(Y, i(difference(j(Y), X)))) 43.76/43.93 = { by lemma 31 } 43.76/43.93 eta(product(Y, i(j(i(difference(j(Y), X)))))) 43.76/43.93 = { by lemma 97 } 43.76/43.93 eta(product(j(Y), j(i(difference(j(Y), X))))) 43.76/43.93 = { by lemma 95 } 43.76/43.93 eta(product(i(j(Y)), i(difference(j(Y), X)))) 43.76/43.93 = { by lemma 103 } 43.76/43.93 eta(difference(j(Y), i(product(j(j(Y)), X)))) 43.76/43.93 = { by lemma 46 } 43.76/43.93 eta(difference(i(product(j(j(Y)), X)), Y)) 43.76/43.93 = { by lemma 47 } 43.76/43.93 eta(difference(product(j(j(Y)), X), j(Y))) 43.76/43.93 = { by lemma 46 } 43.76/43.93 eta(difference(j(j(Y)), product(j(j(Y)), X))) 43.76/43.93 = { by axiom 9 (sos04) } 43.76/43.93 eta(X) 43.76/43.93 43.76/43.93 Lemma 105: eta(difference(X, j(Y))) = eta(product(X, Y)). 43.76/43.93 Proof: 43.76/43.93 eta(difference(X, j(Y))) 43.76/43.93 = { by lemma 47 } 43.76/43.93 eta(difference(i(X), Y)) 43.76/43.93 = { by lemma 52 } 43.76/43.93 eta(product(product(product(X, Y), X), i(X))) 43.76/43.93 = { by lemma 104 } 43.76/43.93 eta(product(X, Y)) 43.76/43.93 43.76/43.93 Lemma 106: product(X, quotient(product(difference(X, Y), Z), X)) = product(Y, quotient(Z, X)). 43.76/43.93 Proof: 43.76/43.93 product(X, quotient(product(difference(X, Y), Z), X)) 43.76/43.93 = { by lemma 59 } 43.76/43.93 product(product(X, difference(X, Y)), quotient(Z, X)) 43.76/43.93 = { by axiom 8 (sos03) } 43.76/43.93 product(Y, quotient(Z, X)) 43.76/43.93 43.76/43.93 Lemma 107: quotient(product(X, product(j(product(Y, X)), Y)), Y) = difference(Y, eta(product(Y, X))). 43.76/43.93 Proof: 43.76/43.93 quotient(product(X, product(j(product(Y, X)), Y)), Y) 43.76/43.93 = { by axiom 7 (sos07) } 43.76/43.93 difference(Y, product(product(Y, X), j(product(Y, X)))) 43.76/43.93 = { by lemma 13 } 43.76/43.93 difference(Y, eta(product(Y, X))) 43.76/43.93 43.76/43.93 Lemma 108: product(i(X), eta(Y)) = difference(X, eta(Y)). 43.76/43.93 Proof: 43.76/43.93 product(i(X), eta(Y)) 43.76/43.93 = { by axiom 8 (sos03) } 43.76/43.93 product(i(X), eta(product(X, difference(X, Y)))) 43.76/43.93 = { by lemma 105 } 43.76/43.93 product(i(X), eta(difference(X, j(difference(X, Y))))) 43.76/43.93 = { by lemma 47 } 43.76/43.93 product(i(X), eta(difference(i(X), difference(X, Y)))) 43.76/43.93 = { by lemma 13 } 43.76/43.93 product(i(X), product(difference(i(X), difference(X, Y)), j(difference(i(X), difference(X, Y))))) 43.76/43.93 = { by axiom 9 (sos04) } 43.76/43.93 difference(X, product(X, product(i(X), product(difference(i(X), difference(X, Y)), j(difference(i(X), difference(X, Y))))))) 43.76/43.93 = { by lemma 18 } 43.76/43.93 difference(X, quotient(quotient(product(difference(i(X), difference(X, Y)), j(difference(i(X), difference(X, Y)))), i(X)), X)) 43.76/43.93 = { by lemma 29 } 43.76/43.93 quotient(product(i(X), quotient(product(difference(i(X), difference(X, Y)), j(difference(i(X), difference(X, Y)))), i(X))), X) 43.76/43.93 = { by lemma 106 } 43.76/43.93 quotient(product(difference(X, Y), quotient(j(difference(i(X), difference(X, Y))), i(X))), X) 43.76/43.93 = { by lemma 52 } 43.76/43.93 quotient(product(difference(X, Y), quotient(j(product(product(product(X, difference(X, Y)), X), i(X))), i(X))), X) 43.76/43.93 = { by lemma 19 } 43.76/43.93 quotient(product(difference(X, Y), product(product(X, product(i(X), j(product(product(product(X, difference(X, Y)), X), i(X))))), X)), X) 43.76/43.93 = { by lemma 67 } 43.76/43.93 quotient(product(difference(X, Y), product(product(X, quotient(eta(i(X)), product(product(X, difference(X, Y)), X))), X)), X) 43.76/43.93 = { by lemma 16 } 43.76/43.93 quotient(product(difference(X, Y), product(product(X, quotient(eta(X), product(product(X, difference(X, Y)), X))), X)), X) 43.76/43.93 = { by lemma 80 } 43.76/43.93 quotient(product(difference(X, Y), product(product(X, difference(X, j(quotient(product(product(X, difference(X, Y)), X), X)))), X)), X) 43.76/43.93 = { by axiom 8 (sos03) } 43.76/43.93 quotient(product(difference(X, Y), product(j(quotient(product(product(X, difference(X, Y)), X), X)), X)), X) 43.76/43.93 = { by axiom 11 (sos05) } 43.76/43.93 quotient(product(difference(X, Y), product(j(product(X, difference(X, Y))), X)), X) 43.76/43.93 = { by lemma 107 } 43.76/43.93 difference(X, eta(product(X, difference(X, Y)))) 43.76/43.93 = { by axiom 8 (sos03) } 43.76/43.93 difference(X, eta(Y)) 43.76/43.93 43.76/43.93 Lemma 109: quotient(eta(X), i(Y)) = product(eta(X), Y). 43.76/43.93 Proof: 43.76/43.93 quotient(eta(X), i(Y)) 43.76/43.93 = { by lemma 19 } 43.76/43.93 product(product(Y, product(i(Y), eta(X))), Y) 43.76/43.93 = { by lemma 108 } 43.76/43.93 product(product(Y, difference(Y, eta(X))), Y) 43.76/43.93 = { by axiom 8 (sos03) } 43.76/43.93 product(eta(X), Y) 43.76/43.93 43.76/43.93 Lemma 110: quotient(i(difference(X, Y)), X) = difference(Y, eta(X)). 43.76/43.93 Proof: 43.76/43.93 quotient(i(difference(X, Y)), X) 43.76/43.93 = { by axiom 9 (sos04) } 43.76/43.93 difference(Y, product(Y, quotient(i(difference(X, Y)), X))) 43.76/43.93 = { by lemma 106 } 43.76/43.93 difference(Y, product(X, quotient(product(difference(X, Y), i(difference(X, Y))), X))) 43.76/43.93 = { by lemma 14 } 43.76/43.93 difference(Y, product(X, quotient(one, X))) 43.76/43.93 = { by axiom 2 (sos10) } 43.76/43.93 difference(Y, product(X, j(X))) 43.76/43.93 = { by lemma 13 } 43.76/43.93 difference(Y, eta(X)) 43.76/43.93 43.76/43.93 Lemma 111: quotient(Y, difference(X, Y)) = X. 43.76/43.93 Proof: 43.76/43.93 quotient(Y, difference(X, Y)) 43.76/43.93 = { by axiom 8 (sos03) } 43.76/43.93 quotient(product(X, difference(X, Y)), difference(X, Y)) 43.76/43.93 = { by axiom 11 (sos05) } 43.76/43.93 X 43.76/43.93 43.76/43.93 Lemma 112: eta(product(X, i(Y))) = eta(quotient(X, Y)). 43.76/43.93 Proof: 43.76/43.93 eta(product(X, i(Y))) 43.76/43.93 = { by axiom 1 (sos06) } 43.76/43.93 eta(product(product(quotient(X, Y), Y), i(Y))) 43.76/43.93 = { by lemma 104 } 43.76/43.93 eta(quotient(X, Y)) 43.76/43.93 43.76/43.93 Lemma 113: eta(difference(X, Y)) = eta(quotient(X, Y)). 43.76/43.93 Proof: 43.76/43.93 eta(difference(X, Y)) 43.76/43.93 = { by lemma 15 } 43.76/43.93 eta(difference(X, j(i(Y)))) 43.76/43.93 = { by lemma 105 } 43.76/43.93 eta(product(X, i(Y))) 43.76/43.93 = { by lemma 112 } 43.76/43.93 eta(quotient(X, Y)) 43.76/43.93 43.76/43.93 Lemma 114: difference(i(X), X) = product(X, X). 43.76/43.93 Proof: 43.76/43.93 difference(i(X), X) 43.76/43.93 = { by axiom 1 (sos06) } 43.76/43.93 product(quotient(difference(i(X), X), X), X) 43.76/43.93 = { by lemma 28 } 43.76/43.93 product(quotient(quotient(product(i(i(X)), product(X, i(X))), i(X)), X), X) 43.76/43.93 = { by lemma 14 } 43.76/43.93 product(quotient(quotient(product(i(i(X)), one), i(X)), X), X) 43.76/43.93 = { by axiom 12 (sos01) } 43.76/43.93 product(quotient(quotient(i(i(X)), i(X)), X), X) 43.76/43.93 = { by lemma 15 } 43.76/43.93 product(quotient(quotient(i(i(X)), i(X)), j(i(X))), X) 43.76/43.93 = { by lemma 30 } 43.76/43.93 product(product(j(i(X)), product(i(X), i(i(X)))), X) 43.76/43.93 = { by lemma 15 } 43.76/43.93 product(product(X, product(i(X), i(i(X)))), X) 43.76/43.93 = { by lemma 14 } 43.76/43.93 product(product(X, one), X) 43.76/43.93 = { by axiom 12 (sos01) } 43.76/43.93 product(X, X) 43.76/43.93 43.76/43.93 Lemma 115: difference(X, j(X)) = quotient(i(X), X). 43.76/43.93 Proof: 43.76/43.93 difference(X, j(X)) 43.76/43.93 = { by lemma 28 } 43.76/43.93 quotient(product(i(X), product(j(X), X)), X) 43.76/43.93 = { by lemma 21 } 43.76/43.93 quotient(product(i(X), one), X) 43.76/43.93 = { by axiom 12 (sos01) } 43.76/43.93 quotient(i(X), X) 43.76/43.93 43.76/43.93 Lemma 116: product(j(X), j(X)) = quotient(i(X), X). 43.76/43.93 Proof: 43.76/43.93 product(j(X), j(X)) 43.76/43.93 = { by lemma 114 } 43.76/43.93 difference(i(j(X)), j(X)) 43.76/43.93 = { by lemma 31 } 43.76/43.93 difference(X, j(X)) 43.76/43.93 = { by lemma 115 } 43.76/43.93 quotient(i(X), X) 43.76/43.93 43.76/43.93 Lemma 117: difference(j(X), quotient(Y, X)) = product(X, product(Y, i(X))). 43.76/43.93 Proof: 43.76/43.93 difference(j(X), quotient(Y, X)) 43.76/43.93 = { by axiom 1 (sos06) } 43.76/43.93 difference(j(X), product(quotient(quotient(Y, X), j(X)), j(X))) 43.76/43.93 = { by lemma 30 } 43.76/43.93 difference(j(X), product(product(j(X), product(X, Y)), j(X))) 43.76/43.93 = { by axiom 7 (sos07) } 43.76/43.93 quotient(product(product(X, Y), product(j(X), j(X))), j(X)) 43.76/43.93 = { by lemma 116 } 43.76/43.93 quotient(product(product(X, Y), quotient(i(X), X)), j(X)) 43.76/43.93 = { by lemma 59 } 43.76/43.93 quotient(product(X, quotient(product(Y, i(X)), X)), j(X)) 43.76/43.93 = { by lemma 72 } 43.76/43.93 product(X, product(Y, i(X))) 43.76/43.93 43.76/43.93 Lemma 118: eta(quotient(X, Y)) = eta(product(X, Y)). 43.76/43.93 Proof: 43.76/43.93 eta(quotient(X, Y)) 43.76/43.93 = { by axiom 11 (sos05) } 43.76/43.93 eta(quotient(quotient(product(X, Y), Y), Y)) 43.76/43.93 = { by lemma 113 } 43.76/43.93 eta(difference(quotient(product(X, Y), Y), Y)) 43.76/43.93 = { by lemma 46 } 43.76/43.93 eta(difference(j(Y), quotient(product(X, Y), Y))) 43.76/43.93 = { by lemma 117 } 43.76/43.93 eta(product(Y, product(product(X, Y), i(Y)))) 43.76/43.93 = { by lemma 104 } 43.76/43.93 eta(product(product(product(Y, product(product(X, Y), i(Y))), eta(Y)), i(eta(Y)))) 43.76/43.93 = { by lemma 117 } 43.76/43.93 eta(product(product(difference(j(Y), quotient(product(X, Y), Y)), eta(Y)), i(eta(Y)))) 43.76/43.93 = { by axiom 1 (sos06) } 43.76/43.93 eta(product(product(quotient(product(difference(j(Y), quotient(product(X, Y), Y)), eta(Y)), j(Y)), j(Y)), i(eta(Y)))) 43.76/43.93 = { by lemma 90 } 43.76/43.93 eta(product(product(difference(j(Y), product(quotient(product(X, Y), Y), Y)), j(Y)), i(eta(Y)))) 43.76/43.93 = { by lemma 83 } 43.76/43.93 eta(product(product(Y, product(product(quotient(product(X, Y), Y), Y), j(Y))), i(eta(Y)))) 43.76/43.93 = { by axiom 1 (sos06) } 43.76/43.93 eta(product(product(Y, product(product(X, Y), j(Y))), i(eta(Y)))) 43.76/43.93 = { by lemma 86 } 43.76/43.93 eta(difference(eta(Y), product(Y, product(product(X, Y), j(Y))))) 43.76/43.93 = { by lemma 113 } 43.76/43.93 eta(quotient(eta(Y), product(Y, product(product(X, Y), j(Y))))) 43.76/43.93 = { by lemma 77 } 43.76/43.93 eta(product(eta(Y), product(Y, product(product(X, Y), j(Y))))) 43.76/43.93 = { by lemma 82 } 43.76/43.93 eta(product(product(product(X, Y), j(Y)), Y)) 43.76/43.93 = { by lemma 56 } 43.76/43.93 eta(product(product(product(X, Y), Y), i(Y))) 43.76/43.93 = { by lemma 104 } 43.76/43.93 eta(product(X, Y)) 43.76/43.93 43.76/43.93 Lemma 119: eta(product(X, Y)) = eta(product(Y, X)). 43.76/43.93 Proof: 43.76/43.93 eta(product(X, Y)) 43.76/43.93 = { by axiom 1 (sos06) } 43.76/43.93 product(quotient(eta(product(X, Y)), Y), Y) 43.76/43.93 = { by lemma 118 } 43.76/43.93 product(quotient(eta(quotient(X, Y)), Y), Y) 43.76/43.93 = { by lemma 112 } 43.76/43.93 product(quotient(eta(product(X, i(Y))), Y), Y) 43.76/43.93 = { by lemma 118 } 43.76/43.93 product(quotient(eta(quotient(X, i(Y))), Y), Y) 43.76/43.93 = { by lemma 113 } 43.76/43.93 product(quotient(eta(difference(X, i(Y))), Y), Y) 43.76/43.93 = { by lemma 46 } 43.76/43.93 product(quotient(eta(difference(j(i(Y)), X)), Y), Y) 43.76/43.93 = { by lemma 15 } 43.76/43.93 product(quotient(eta(difference(j(i(Y)), X)), j(i(Y))), Y) 43.76/43.93 = { by lemma 67 } 43.76/43.93 product(product(difference(j(i(Y)), X), j(product(j(i(Y)), difference(j(i(Y)), X)))), Y) 43.76/43.93 = { by lemma 15 } 43.76/43.93 product(product(difference(Y, X), j(product(j(i(Y)), difference(j(i(Y)), X)))), Y) 43.76/43.93 = { by axiom 8 (sos03) } 43.76/43.93 product(product(difference(Y, X), j(X)), Y) 43.76/43.93 = { by lemma 23 } 43.76/43.93 eta(difference(Y, X)) 43.76/43.93 = { by lemma 113 } 43.76/43.93 eta(quotient(Y, X)) 43.76/43.93 = { by lemma 118 } 43.76/43.93 eta(product(Y, X)) 43.76/43.93 43.76/43.93 Lemma 120: eta(product(eta(Y), X)) = eta(X). 43.76/43.93 Proof: 43.76/43.93 eta(product(eta(Y), X)) 43.76/43.93 = { by lemma 93 } 43.76/43.93 eta(product(X, eta(Y))) 43.76/43.93 = { by lemma 96 } 43.76/43.93 eta(product(Y, quotient(X, Y))) 43.76/43.93 = { by lemma 119 } 43.76/43.93 eta(product(quotient(X, Y), Y)) 43.76/43.93 = { by axiom 1 (sos06) } 43.76/43.93 eta(X) 43.76/43.93 43.76/43.93 Lemma 121: difference(i(product(X, Y)), difference(Y, eta(X))) = X. 43.76/43.93 Proof: 43.76/43.93 difference(i(product(X, Y)), difference(Y, eta(X))) 43.76/43.93 = { by lemma 15 } 43.76/43.93 difference(i(product(j(i(X)), Y)), difference(Y, eta(X))) 43.76/43.93 = { by axiom 4 (sos09) } 43.76/43.93 difference(difference(product(j(i(X)), Y), one), difference(Y, eta(X))) 43.76/43.93 = { by lemma 21 } 43.76/43.93 difference(difference(product(j(i(X)), Y), product(j(i(X)), i(X))), difference(Y, eta(X))) 43.76/43.93 = { by lemma 101 } 43.76/43.93 difference(quotient(difference(Y, product(i(X), j(i(X)))), j(i(X))), difference(Y, eta(X))) 43.76/43.93 = { by lemma 13 } 43.76/43.93 difference(quotient(difference(Y, eta(i(X))), j(i(X))), difference(Y, eta(X))) 43.76/43.93 = { by lemma 16 } 43.76/43.93 difference(quotient(difference(Y, eta(i(X))), j(i(X))), difference(Y, eta(i(X)))) 43.76/43.93 = { by lemma 51 } 43.76/43.93 j(i(X)) 43.76/43.93 = { by lemma 15 } 43.76/43.93 X 43.76/43.93 43.76/43.93 Lemma 122: product(difference(product(X, Y), X), product(Y, product(X, Y))) = product(X, Y). 43.76/43.93 Proof: 43.76/43.93 product(difference(product(X, Y), X), product(Y, product(X, Y))) 43.76/43.93 = { by lemma 43 } 43.76/43.93 product(difference(product(X, Y), product(X, Y)), product(X, Y)) 43.76/43.93 = { by axiom 12 (sos01) } 43.76/43.93 product(difference(product(X, Y), product(product(X, Y), one)), product(X, Y)) 43.76/43.93 = { by axiom 9 (sos04) } 43.76/43.93 product(one, product(X, Y)) 43.76/43.93 = { by axiom 5 (sos02) } 43.76/43.93 product(X, Y) 43.76/43.93 43.76/43.93 Lemma 123: quotient(product(X, Y), product(Y, product(X, Y))) = difference(product(X, Y), X). 43.76/43.93 Proof: 43.76/43.93 quotient(product(X, Y), product(Y, product(X, Y))) 43.76/43.93 = { by lemma 122 } 43.76/43.93 quotient(product(difference(product(X, Y), X), product(Y, product(X, Y))), product(Y, product(X, Y))) 43.76/43.93 = { by axiom 11 (sos05) } 43.76/43.93 difference(product(X, Y), X) 43.76/43.93 43.76/43.93 Lemma 124: difference(X, quotient(X, Y)) = quotient(X, product(Y, X)). 43.76/43.93 Proof: 43.76/43.93 difference(X, quotient(X, Y)) 43.76/43.93 = { by axiom 12 (sos01) } 43.76/43.93 difference(X, quotient(product(X, one), Y)) 43.76/43.93 = { by axiom 11 (sos05) } 43.76/43.93 difference(X, quotient(product(X, quotient(product(one, X), X)), Y)) 43.76/43.93 = { by axiom 5 (sos02) } 43.76/43.93 difference(X, quotient(product(X, quotient(X, X)), Y)) 43.76/43.93 = { by lemma 57 } 43.76/43.93 difference(X, product(X, quotient(X, product(Y, X)))) 43.76/43.93 = { by axiom 9 (sos04) } 43.76/43.93 quotient(X, product(Y, X)) 43.76/43.93 43.76/43.93 Lemma 125: difference(product(X, Y), X) = quotient(difference(Y, X), X). 43.76/43.93 Proof: 43.76/43.93 difference(product(X, Y), X) 43.76/43.93 = { by lemma 123 } 43.76/43.93 quotient(product(X, Y), product(Y, product(X, Y))) 43.76/43.93 = { by lemma 124 } 43.76/43.93 difference(product(X, Y), quotient(product(X, Y), Y)) 43.76/43.93 = { by axiom 1 (sos06) } 43.76/43.93 difference(product(quotient(product(X, Y), Y), Y), quotient(product(X, Y), Y)) 43.76/43.93 = { by axiom 12 (sos01) } 43.76/43.93 difference(product(quotient(product(X, Y), Y), Y), product(quotient(product(X, Y), Y), one)) 43.76/43.93 = { by lemma 14 } 43.76/43.93 difference(product(quotient(product(X, Y), Y), Y), product(quotient(product(X, Y), Y), product(Y, i(Y)))) 43.76/43.93 = { by axiom 3 (sos08) } 43.76/43.93 quotient(quotient(product(i(Y), product(quotient(product(X, Y), Y), Y)), Y), quotient(product(X, Y), Y)) 43.76/43.93 = { by axiom 1 (sos06) } 43.76/43.93 quotient(quotient(product(i(Y), product(X, Y)), Y), quotient(product(X, Y), Y)) 43.76/43.93 = { by lemma 29 } 43.76/43.93 quotient(difference(Y, quotient(product(X, Y), Y)), quotient(product(X, Y), Y)) 43.76/43.93 = { by axiom 11 (sos05) } 43.76/43.93 quotient(difference(Y, X), quotient(product(X, Y), Y)) 43.76/43.93 = { by axiom 11 (sos05) } 43.76/43.93 quotient(difference(Y, X), X) 43.76/43.93 43.76/43.94 Lemma 126: product(i(X), quotient(Y, i(X))) = product(product(i(X), Y), X). 43.76/43.94 Proof: 43.76/43.94 product(i(X), quotient(Y, i(X))) 43.76/43.94 = { by axiom 12 (sos01) } 43.76/43.94 product(i(X), quotient(product(Y, one), i(X))) 43.76/43.94 = { by lemma 14 } 43.76/43.94 product(i(X), quotient(product(Y, product(X, i(X))), i(X))) 43.76/43.94 = { by lemma 20 } 43.76/43.94 product(product(i(X), Y), X) 43.76/43.94 43.76/43.94 Lemma 127: quotient(product(difference(difference(X, Y), Z), Y), difference(X, Y)) = product(difference(Y, product(X, Z)), X). 43.76/43.94 Proof: 43.76/43.94 quotient(product(difference(difference(X, Y), Z), Y), difference(X, Y)) 43.76/43.94 = { by axiom 8 (sos03) } 43.76/43.94 quotient(product(difference(difference(X, Y), Z), product(X, difference(X, Y))), difference(X, Y)) 43.76/43.94 = { by lemma 42 } 43.76/43.94 difference(difference(X, Y), product(Z, X)) 43.76/43.94 = { by axiom 1 (sos06) } 43.76/43.94 product(quotient(difference(difference(X, Y), product(Z, X)), X), X) 43.76/43.94 = { by lemma 99 } 43.76/43.94 product(difference(Y, product(X, Z)), X) 43.76/43.94 43.76/43.94 Lemma 128: quotient(product(difference(X, Y), eta(X)), X) = difference(X, product(Y, i(X))). 43.76/43.94 Proof: 43.76/43.94 quotient(product(difference(X, Y), eta(X)), X) 43.76/43.94 = { by lemma 13 } 43.76/43.94 quotient(product(difference(X, Y), product(X, j(X))), X) 43.76/43.94 = { by axiom 6 (sos11) } 43.76/43.94 quotient(product(difference(X, Y), product(i(X), X)), X) 43.76/43.94 = { by lemma 42 } 43.76/43.94 difference(X, product(Y, i(X))) 43.76/43.94 43.76/43.94 Lemma 129: product(difference(eta(X), product(i(X), Y)), i(X)) = difference(X, product(Y, i(X))). 43.76/43.94 Proof: 43.76/43.94 product(difference(eta(X), product(i(X), Y)), i(X)) 43.76/43.94 = { by lemma 127 } 43.76/43.94 quotient(product(difference(difference(i(X), eta(X)), Y), eta(X)), difference(i(X), eta(X))) 43.76/43.94 = { by lemma 70 } 43.76/43.94 quotient(product(difference(X, Y), eta(X)), difference(i(X), eta(X))) 43.76/43.94 = { by lemma 70 } 43.76/43.94 quotient(product(difference(X, Y), eta(X)), X) 43.76/43.94 = { by lemma 128 } 43.76/43.94 difference(X, product(Y, i(X))) 43.76/43.94 43.76/43.94 Lemma 130: product(product(product(j(X), Y), X), i(X)) = difference(X, Y). 43.76/43.94 Proof: 43.76/43.94 product(product(product(j(X), Y), X), i(X)) 43.76/43.94 = { by lemma 89 } 43.76/43.94 product(quotient(difference(X, Y), i(X)), i(X)) 43.76/43.94 = { by lemma 15 } 43.76/43.94 product(quotient(difference(j(i(X)), Y), i(X)), i(X)) 43.76/43.94 = { by axiom 1 (sos06) } 43.76/43.94 product(quotient(difference(j(i(X)), product(quotient(Y, i(X)), i(X))), i(X)), i(X)) 43.76/43.94 = { by lemma 91 } 43.76/43.94 product(difference(eta(i(X)), product(i(X), quotient(Y, i(X)))), i(X)) 43.76/43.94 = { by lemma 16 } 43.76/43.94 product(difference(eta(X), product(i(X), quotient(Y, i(X)))), i(X)) 43.76/43.94 = { by lemma 129 } 43.76/43.94 difference(X, product(quotient(Y, i(X)), i(X))) 43.76/43.94 = { by axiom 1 (sos06) } 43.76/43.94 difference(X, Y) 43.76/43.94 43.76/43.94 Lemma 131: difference(Y, eta(product(X, Y))) = product(i(X), quotient(X, Y)). 43.76/43.94 Proof: 43.76/43.94 difference(Y, eta(product(X, Y))) 43.76/43.94 = { by lemma 51 } 43.76/43.94 difference(difference(quotient(X, Y), X), eta(product(X, Y))) 43.76/43.94 = { by lemma 118 } 43.76/43.94 difference(difference(quotient(X, Y), X), eta(quotient(X, Y))) 43.76/43.94 = { by lemma 74 } 43.76/43.94 product(i(X), quotient(X, Y)) 43.76/43.94 43.76/43.94 Lemma 132: product(i(X), quotient(X, Y)) = product(X, quotient(j(X), Y)). 43.76/43.94 Proof: 43.76/43.94 product(i(X), quotient(X, Y)) 43.76/43.94 = { by lemma 131 } 43.76/43.94 difference(Y, eta(product(X, Y))) 43.76/43.94 = { by lemma 119 } 43.76/43.94 difference(Y, eta(product(Y, X))) 43.76/43.94 = { by lemma 118 } 43.76/43.94 difference(Y, eta(quotient(Y, X))) 43.76/43.94 = { by lemma 61 } 43.76/43.94 difference(Y, eta(quotient(j(X), Y))) 43.76/43.94 = { by lemma 44 } 43.76/43.94 difference(Y, product(difference(quotient(j(X), Y), j(X)), product(X, quotient(j(X), Y)))) 43.76/43.94 = { by lemma 51 } 43.76/43.94 difference(Y, product(Y, product(X, quotient(j(X), Y)))) 43.76/43.94 = { by axiom 9 (sos04) } 43.76/43.94 product(X, quotient(j(X), Y)) 43.76/43.94 43.76/43.94 Lemma 133: product(X, quotient(j(X), difference(Y, X))) = product(i(X), Y). 43.76/43.94 Proof: 43.76/43.94 product(X, quotient(j(X), difference(Y, X))) 43.76/43.94 = { by lemma 132 } 43.76/43.94 product(i(X), quotient(X, difference(Y, X))) 43.76/43.94 = { by lemma 131 } 43.76/43.94 difference(difference(Y, X), eta(product(X, difference(Y, X)))) 43.76/43.94 = { by lemma 119 } 43.76/43.94 difference(difference(Y, X), eta(product(difference(Y, X), X))) 43.76/43.94 = { by lemma 13 } 43.76/43.94 difference(difference(Y, X), product(product(difference(Y, X), X), j(product(difference(Y, X), X)))) 43.76/43.94 = { by axiom 7 (sos07) } 43.76/43.94 quotient(product(X, product(j(product(difference(Y, X), X)), difference(Y, X))), difference(Y, X)) 43.76/43.94 = { by axiom 1 (sos06) } 43.76/43.94 quotient(product(quotient(product(X, product(j(product(difference(Y, X), X)), difference(Y, X))), difference(Y, X)), difference(Y, X)), difference(Y, X)) 43.76/43.94 = { by lemma 107 } 43.76/43.94 quotient(product(difference(difference(Y, X), eta(product(difference(Y, X), X))), difference(Y, X)), difference(Y, X)) 43.76/43.94 = { by lemma 119 } 43.76/43.94 quotient(product(difference(difference(Y, X), eta(product(X, difference(Y, X)))), difference(Y, X)), difference(Y, X)) 43.76/43.94 = { by lemma 131 } 43.76/43.94 quotient(product(product(i(X), quotient(X, difference(Y, X))), difference(Y, X)), difference(Y, X)) 43.76/43.94 = { by lemma 74 } 43.76/43.94 quotient(product(difference(difference(quotient(X, difference(Y, X)), X), eta(quotient(X, difference(Y, X)))), difference(Y, X)), difference(Y, X)) 43.76/43.94 = { by lemma 51 } 43.76/43.94 quotient(product(difference(difference(Y, X), eta(quotient(X, difference(Y, X)))), difference(Y, X)), difference(Y, X)) 43.76/43.94 = { by lemma 13 } 43.76/43.94 quotient(product(difference(difference(Y, X), product(quotient(X, difference(Y, X)), j(quotient(X, difference(Y, X))))), difference(Y, X)), difference(Y, X)) 43.76/43.94 = { by axiom 6 (sos11) } 43.76/43.94 quotient(product(difference(difference(Y, X), product(i(quotient(X, difference(Y, X))), quotient(X, difference(Y, X)))), difference(Y, X)), difference(Y, X)) 43.76/43.94 = { by lemma 43 } 43.76/43.94 quotient(product(difference(difference(Y, X), i(quotient(X, difference(Y, X)))), product(quotient(X, difference(Y, X)), difference(Y, X))), difference(Y, X)) 43.76/43.94 = { by axiom 1 (sos06) } 43.76/43.94 quotient(product(difference(difference(Y, X), i(quotient(X, difference(Y, X)))), X), difference(Y, X)) 43.76/43.94 = { by lemma 127 } 43.76/43.94 product(difference(X, product(Y, i(quotient(X, difference(Y, X))))), Y) 43.76/43.94 = { by lemma 111 } 43.76/43.94 product(difference(X, product(Y, i(Y))), Y) 43.76/43.94 = { by lemma 14 } 43.76/43.94 product(difference(X, one), Y) 43.76/43.94 = { by axiom 4 (sos09) } 43.76/43.94 product(i(X), Y) 43.76/43.94 43.76/43.94 Lemma 134: quotient(difference(X, i(Y)), i(Y)) = quotient(difference(X, Y), Y). 43.76/43.94 Proof: 43.76/43.94 quotient(difference(X, i(Y)), i(Y)) 43.76/43.94 = { by lemma 125 } 43.76/43.94 difference(product(i(Y), X), i(Y)) 43.76/43.94 = { by lemma 133 } 43.76/43.94 difference(product(Y, quotient(j(Y), difference(X, Y))), i(Y)) 43.76/43.94 = { by lemma 41 } 43.76/43.94 quotient(difference(quotient(j(Y), difference(X, Y)), j(Y)), Y) 43.76/43.94 = { by lemma 51 } 43.76/43.94 quotient(difference(X, Y), Y) 43.76/43.94 43.76/43.94 Lemma 135: quotient(difference(X, eta(Y)), eta(Y)) = i(X). 43.76/43.94 Proof: 43.76/43.94 quotient(difference(X, eta(Y)), eta(Y)) 43.76/43.94 = { by lemma 108 } 43.76/43.94 quotient(product(i(X), eta(Y)), eta(Y)) 43.76/43.94 = { by axiom 11 (sos05) } 43.76/43.94 i(X) 43.76/43.94 43.76/43.94 Lemma 136: quotient(product(X, Y), quotient(Z, X)) = product(X, quotient(product(Y, X), Z)). 43.76/43.94 Proof: 43.76/43.94 quotient(product(X, Y), quotient(Z, X)) 43.76/43.94 = { by axiom 11 (sos05) } 43.76/43.94 quotient(product(X, quotient(product(Y, X), X)), quotient(Z, X)) 43.76/43.94 = { by lemma 57 } 43.76/43.94 product(X, quotient(product(Y, X), product(quotient(Z, X), X))) 43.76/43.94 = { by axiom 1 (sos06) } 43.76/43.95 product(X, quotient(product(Y, X), Z)) 43.76/43.95 43.76/43.95 Lemma 137: product(i(X), product(X, Y)) = product(X, product(j(X), Y)). 43.76/43.95 Proof: 43.76/43.95 product(i(X), product(X, Y)) 43.76/43.95 = { by lemma 133 } 43.76/43.95 product(X, quotient(j(X), difference(product(X, Y), X))) 43.76/43.95 = { by axiom 8 (sos03) } 43.76/43.95 product(X, quotient(product(X, difference(X, j(X))), difference(product(X, Y), X))) 43.76/43.95 = { by lemma 115 } 43.76/43.95 product(X, quotient(product(X, quotient(i(X), X)), difference(product(X, Y), X))) 43.76/43.95 = { by lemma 125 } 43.76/43.95 product(X, quotient(product(X, quotient(i(X), X)), quotient(difference(Y, X), X))) 43.76/43.95 = { by lemma 57 } 43.76/43.95 product(X, product(X, quotient(i(X), product(quotient(difference(Y, X), X), X)))) 43.76/43.95 = { by axiom 1 (sos06) } 43.76/43.95 product(X, product(X, quotient(i(X), difference(Y, X)))) 43.76/43.95 = { by lemma 58 } 43.76/43.95 product(X, product(X, product(X, quotient(j(X), product(difference(Y, X), X))))) 43.76/43.95 = { by lemma 132 } 43.76/43.95 product(X, product(X, product(i(X), quotient(X, product(difference(Y, X), X))))) 43.76/43.95 = { by axiom 8 (sos03) } 43.76/43.95 product(X, product(X, product(i(X), quotient(product(Y, difference(Y, X)), product(difference(Y, X), X))))) 43.76/43.95 = { by axiom 8 (sos03) } 43.76/43.95 product(X, product(X, product(i(X), quotient(product(Y, difference(Y, X)), product(difference(Y, X), product(Y, difference(Y, X))))))) 43.76/43.95 = { by lemma 123 } 43.76/43.95 product(X, product(X, product(i(X), difference(product(Y, difference(Y, X)), Y)))) 43.76/43.95 = { by axiom 8 (sos03) } 43.76/43.95 product(X, product(X, product(i(X), difference(X, Y)))) 43.76/43.95 = { by lemma 87 } 43.76/43.95 product(X, product(j(X), product(X, difference(X, Y)))) 43.76/43.95 = { by lemma 31 } 43.76/43.95 product(X, product(j(X), product(i(j(X)), difference(X, Y)))) 43.76/43.95 = { by lemma 87 } 43.76/43.95 product(X, product(j(j(X)), product(j(X), difference(X, Y)))) 43.76/43.95 = { by lemma 30 } 43.76/43.95 product(X, quotient(quotient(difference(X, Y), j(X)), j(j(X)))) 43.76/43.95 = { by lemma 31 } 43.76/43.95 product(X, quotient(quotient(difference(i(j(X)), Y), j(X)), j(j(X)))) 43.76/43.95 = { by axiom 1 (sos06) } 43.76/43.95 product(X, quotient(quotient(difference(i(j(X)), product(quotient(Y, eta(X)), eta(X))), j(X)), j(j(X)))) 43.76/43.95 = { by lemma 69 } 43.76/43.95 product(X, quotient(product(j(X), quotient(product(quotient(Y, eta(X)), eta(X)), j(X))), j(j(X)))) 43.76/43.95 = { by lemma 13 } 43.76/43.95 product(X, quotient(product(j(X), quotient(product(quotient(Y, eta(X)), product(X, j(X))), j(X))), j(j(X)))) 43.76/43.95 = { by lemma 72 } 43.76/43.95 product(X, product(j(X), product(quotient(Y, eta(X)), product(X, j(X))))) 43.76/43.95 = { by lemma 13 } 43.76/43.95 product(X, product(j(X), product(quotient(Y, eta(X)), eta(X)))) 43.76/43.95 = { by axiom 1 (sos06) } 43.76/43.95 product(X, product(j(X), Y)) 43.76/43.95 43.76/43.95 Lemma 138: j(difference(Y, eta(X))) = difference(eta(X), Y). 43.76/43.95 Proof: 43.76/43.95 j(difference(Y, eta(X))) 43.76/43.95 = { by axiom 2 (sos10) } 43.76/43.95 quotient(one, difference(Y, eta(X))) 43.76/43.95 = { by lemma 65 } 43.76/43.95 quotient(eta(eta(X)), difference(Y, eta(X))) 43.76/43.95 = { by lemma 108 } 43.76/43.95 quotient(eta(eta(X)), product(i(Y), eta(X))) 43.76/43.95 = { by lemma 79 } 43.76/43.95 difference(eta(X), j(i(Y))) 43.76/43.95 = { by lemma 15 } 43.76/43.95 difference(eta(X), Y) 43.76/43.95 43.76/43.95 Lemma 139: quotient(i(difference(X, Y)), i(X)) = product(i(product(j(X), Y)), X). 43.76/43.95 Proof: 43.76/43.95 quotient(i(difference(X, Y)), i(X)) 43.76/43.95 = { by lemma 19 } 43.76/43.95 product(product(X, product(i(X), i(difference(X, Y)))), X) 43.76/43.95 = { by lemma 102 } 43.76/43.95 product(i(product(j(X), Y)), X) 43.76/43.95 43.76/43.95 Lemma 140: quotient(eta(X), product(X, product(i(X), Y))) = product(X, j(quotient(Y, i(X)))). 43.76/43.95 Proof: 43.76/43.95 quotient(eta(X), product(X, product(i(X), Y))) 43.76/43.95 = { by lemma 67 } 43.76/43.95 product(X, j(product(product(X, product(i(X), Y)), X))) 43.76/43.95 = { by lemma 19 } 43.76/43.95 product(X, j(quotient(Y, i(X)))) 43.76/43.95 43.76/43.95 Lemma 141: difference(i(X), product(eta(X), X)) = quotient(X, i(X)). 43.76/43.95 Proof: 43.76/43.95 difference(i(X), product(eta(X), X)) 43.76/43.95 = { by lemma 16 } 43.76/43.95 difference(i(X), product(eta(i(X)), X)) 43.76/43.95 = { by lemma 24 } 43.76/43.95 quotient(product(j(i(X)), product(X, i(X))), i(X)) 43.76/43.95 = { by lemma 14 } 43.76/43.95 quotient(product(j(i(X)), one), i(X)) 43.76/43.95 = { by axiom 12 (sos01) } 43.76/43.95 quotient(j(i(X)), i(X)) 43.76/43.95 = { by lemma 15 } 43.76/43.95 quotient(X, i(X)) 43.76/43.95 43.76/43.95 Lemma 142: product(eta(X), X) = i(i(X)). 43.76/43.95 Proof: 43.76/43.95 product(eta(X), X) 43.76/43.95 = { by lemma 32 } 43.76/43.95 product(quotient(i(i(X)), X), X) 43.76/43.95 = { by axiom 1 (sos06) } 43.76/43.95 i(i(X)) 43.76/43.95 43.76/43.95 Lemma 143: eta(quotient(X, i(X))) = one. 43.76/43.95 Proof: 43.76/43.95 eta(quotient(X, i(X))) 43.76/43.95 = { by lemma 22 } 43.76/43.95 product(product(quotient(X, i(X)), j(product(i(X), quotient(X, i(X))))), i(X)) 43.76/43.95 = { by lemma 141 } 43.76/43.95 product(product(quotient(X, i(X)), j(product(i(X), difference(i(X), product(eta(X), X))))), i(X)) 43.76/43.95 = { by axiom 8 (sos03) } 43.76/43.95 product(product(quotient(X, i(X)), j(product(eta(X), X))), i(X)) 43.76/43.95 = { by lemma 142 } 43.76/43.95 product(product(quotient(X, i(X)), j(i(i(X)))), i(X)) 43.76/43.95 = { by lemma 15 } 43.76/43.95 product(product(quotient(X, i(X)), i(X)), i(X)) 43.76/43.95 = { by axiom 1 (sos06) } 43.76/43.95 product(X, i(X)) 43.76/43.95 = { by lemma 14 } 43.76/43.95 one 43.76/43.95 43.76/43.95 Lemma 144: i(quotient(X, i(X))) = quotient(i(X), X). 43.76/43.95 Proof: 43.76/43.95 i(quotient(X, i(X))) 43.76/43.95 = { by axiom 4 (sos09) } 43.76/43.95 difference(quotient(X, i(X)), one) 43.76/43.95 = { by lemma 143 } 43.76/43.95 difference(quotient(X, i(X)), eta(quotient(X, i(X)))) 43.76/43.95 = { by lemma 35 } 43.76/43.95 j(quotient(X, i(X))) 43.76/43.95 = { by lemma 141 } 43.76/43.95 j(difference(i(X), product(eta(X), X))) 43.76/43.95 = { by lemma 50 } 43.76/43.95 j(product(product(X, eta(X)), X)) 43.76/43.95 = { by lemma 36 } 43.76/43.95 j(product(j(j(X)), X)) 43.76/43.95 = { by lemma 85 } 43.76/43.95 difference(X, quotient(eta(X), j(j(X)))) 43.76/43.95 = { by lemma 33 } 43.76/43.95 difference(X, quotient(eta(j(X)), j(j(X)))) 43.76/43.95 = { by lemma 13 } 43.76/43.95 difference(X, quotient(product(j(X), j(j(X))), j(j(X)))) 43.76/43.95 = { by axiom 11 (sos05) } 43.76/43.95 difference(X, j(X)) 43.76/43.95 = { by lemma 115 } 43.76/43.95 quotient(i(X), X) 43.76/43.95 43.76/43.95 Lemma 145: j(quotient(X, i(X))) = quotient(i(X), X). 43.76/43.95 Proof: 43.76/43.95 j(quotient(X, i(X))) 43.76/43.95 = { by lemma 35 } 43.76/43.95 difference(quotient(X, i(X)), eta(quotient(X, i(X)))) 43.76/43.95 = { by lemma 143 } 43.76/43.95 difference(quotient(X, i(X)), one) 43.76/43.95 = { by axiom 4 (sos09) } 43.76/43.95 i(quotient(X, i(X))) 43.76/43.95 = { by lemma 144 } 43.76/43.95 quotient(i(X), X) 43.76/43.95 43.76/43.95 Lemma 146: difference(product(W, X), product(product(W, Y), Z)) = quotient(difference(X, product(Y, product(Z, W))), W). 43.76/43.95 Proof: 43.76/43.95 difference(product(W, X), product(product(W, Y), Z)) 43.76/43.95 = { by lemma 20 } 43.76/43.95 difference(product(W, X), product(W, quotient(product(Y, product(Z, W)), W))) 43.76/43.95 = { by lemma 101 } 43.76/43.95 quotient(difference(X, product(quotient(product(Y, product(Z, W)), W), W)), W) 43.76/43.95 = { by axiom 1 (sos06) } 46.79/47.02 quotient(difference(X, product(Y, product(Z, W))), W) 46.79/47.02 46.79/47.02 Goal 1 (goals): product(eta(x0), product(x0, x1)) = product(i(i(x0)), x1). 46.79/47.02 Proof: 46.79/47.02 product(eta(x0), product(x0, x1)) 46.79/47.02 = { by lemma 16 } 46.79/47.02 product(eta(i(x0)), product(x0, x1)) 46.79/47.02 = { by lemma 109 } 46.79/47.02 quotient(eta(i(x0)), i(product(x0, x1))) 46.79/47.02 = { by lemma 67 } 46.79/47.02 product(i(x0), j(product(i(product(x0, x1)), i(x0)))) 46.79/47.02 = { by lemma 76 } 46.79/47.02 product(i(x0), j(product(quotient(difference(x1, eta(x0)), x0), i(x0)))) 46.79/47.02 = { by axiom 9 (sos04) } 46.79/47.02 product(i(x0), j(product(difference(eta(x0), product(eta(x0), quotient(difference(x1, eta(x0)), x0))), i(x0)))) 46.79/47.02 = { by lemma 13 } 46.79/47.02 product(i(x0), j(product(difference(product(x0, j(x0)), product(eta(x0), quotient(difference(x1, eta(x0)), x0))), i(x0)))) 46.79/47.02 = { by axiom 6 (sos11) } 46.79/47.02 product(i(x0), j(product(difference(product(i(x0), x0), product(eta(x0), quotient(difference(x1, eta(x0)), x0))), i(x0)))) 46.79/47.02 = { by lemma 39 } 46.79/47.02 product(i(x0), j(difference(x0, product(difference(i(x0), product(eta(x0), quotient(difference(x1, eta(x0)), x0))), i(x0))))) 46.79/47.02 = { by lemma 128 } 46.79/47.02 product(i(x0), j(quotient(product(difference(x0, difference(i(x0), product(eta(x0), quotient(difference(x1, eta(x0)), x0)))), eta(x0)), x0))) 46.79/47.02 = { by lemma 27 } 46.79/47.02 product(i(x0), j(quotient(product(difference(x0, difference(quotient(eta(x0), x0), product(eta(x0), quotient(difference(x1, eta(x0)), x0)))), eta(x0)), x0))) 46.79/47.02 = { by axiom 1 (sos06) } 46.79/47.02 product(i(x0), j(quotient(product(difference(x0, difference(quotient(eta(x0), x0), product(product(quotient(eta(x0), x0), x0), quotient(difference(x1, eta(x0)), x0)))), eta(x0)), x0))) 46.79/47.02 = { by axiom 7 (sos07) } 46.79/47.02 product(i(x0), j(quotient(product(difference(x0, quotient(product(x0, product(quotient(difference(x1, eta(x0)), x0), quotient(eta(x0), x0))), quotient(eta(x0), x0))), eta(x0)), x0))) 46.79/47.02 = { by lemma 136 } 46.79/47.02 product(i(x0), j(quotient(product(difference(x0, product(x0, quotient(product(product(quotient(difference(x1, eta(x0)), x0), quotient(eta(x0), x0)), x0), eta(x0)))), eta(x0)), x0))) 46.79/47.02 = { by axiom 9 (sos04) } 46.79/47.02 product(i(x0), j(quotient(product(quotient(product(product(quotient(difference(x1, eta(x0)), x0), quotient(eta(x0), x0)), x0), eta(x0)), eta(x0)), x0))) 46.79/47.02 = { by lemma 27 } 46.79/47.02 product(i(x0), j(quotient(product(quotient(product(product(quotient(difference(x1, eta(x0)), x0), i(x0)), x0), eta(x0)), eta(x0)), x0))) 46.79/47.02 = { by lemma 20 } 46.79/47.02 product(i(x0), j(quotient(product(quotient(product(quotient(difference(x1, eta(x0)), x0), quotient(product(i(x0), product(x0, quotient(difference(x1, eta(x0)), x0))), quotient(difference(x1, eta(x0)), x0))), eta(x0)), eta(x0)), x0))) 46.79/47.02 = { by lemma 137 } 46.79/47.02 product(i(x0), j(quotient(product(quotient(product(quotient(difference(x1, eta(x0)), x0), quotient(product(x0, product(j(x0), quotient(difference(x1, eta(x0)), x0))), quotient(difference(x1, eta(x0)), x0))), eta(x0)), eta(x0)), x0))) 46.79/47.02 = { by lemma 20 } 46.79/47.02 product(i(x0), j(quotient(product(quotient(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)), eta(x0)), eta(x0)), x0))) 46.79/47.02 = { by lemma 31 } 46.79/47.02 product(i(x0), j(quotient(product(i(j(quotient(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)), eta(x0)))), eta(x0)), x0))) 46.79/47.02 = { by lemma 108 } 46.79/47.02 product(i(x0), j(quotient(difference(j(quotient(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)), eta(x0))), eta(x0)), x0))) 46.79/47.02 = { by lemma 76 } 46.79/47.02 product(i(x0), j(i(product(x0, j(quotient(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)), eta(x0))))))) 46.79/47.02 = { by axiom 2 (sos10) } 46.79/47.02 product(i(x0), j(i(product(x0, quotient(one, quotient(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)), eta(x0))))))) 46.79/47.02 = { by lemma 65 } 46.79/47.02 product(i(x0), j(i(product(x0, quotient(eta(eta(x0)), quotient(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)), eta(x0))))))) 46.79/47.02 = { by lemma 68 } 46.79/47.02 product(i(x0), j(i(product(x0, product(eta(x0), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))))))) 46.79/47.02 = { by lemma 121 } 46.79/47.02 product(i(x0), j(i(product(x0, difference(i(product(product(eta(x0), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))))), difference(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(product(eta(x0), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))))))))) 46.79/47.02 = { by lemma 56 } 46.79/47.02 product(i(x0), j(i(product(x0, difference(i(product(product(eta(x0), j(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), difference(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(product(eta(x0), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))))))))) 46.79/47.02 = { by lemma 55 } 46.79/47.02 product(i(x0), j(i(product(x0, difference(i(product(product(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), quotient(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), difference(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(product(eta(x0), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))))))))) 46.79/47.02 = { by lemma 53 } 46.79/47.02 product(i(x0), j(i(product(x0, difference(i(difference(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)))), difference(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(product(eta(x0), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))))))))) 46.79/47.02 = { by lemma 108 } 46.79/47.02 product(i(x0), j(i(product(x0, difference(i(difference(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), product(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0)))), difference(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(product(eta(x0), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))))))))) 46.79/47.02 = { by axiom 9 (sos04) } 46.79/47.02 product(i(x0), j(i(product(x0, difference(i(eta(x0)), difference(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(product(eta(x0), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))))))))) 46.79/47.02 = { by lemma 66 } 46.79/47.02 product(i(x0), j(i(product(x0, difference(j(eta(x0)), difference(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(product(eta(x0), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))))))))) 46.79/47.02 = { by lemma 120 } 46.79/47.02 product(i(x0), j(i(product(x0, difference(j(eta(x0)), difference(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))))))))) 46.79/47.02 = { by lemma 70 } 46.79/47.02 product(i(x0), j(i(product(x0, difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))))))) 46.79/47.02 = { by lemma 76 } 46.79/47.02 product(i(x0), j(quotient(difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0)), x0))) 46.79/47.02 = { by axiom 11 (sos05) } 46.79/47.02 product(i(x0), j(quotient(quotient(product(difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0)), eta(x0)), eta(x0)), x0))) 46.79/47.02 = { by axiom 8 (sos03) } 46.79/47.02 product(i(x0), j(quotient(quotient(product(difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0)), product(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0)))), eta(x0)), x0))) 46.79/47.02 = { by axiom 9 (sos04) } 46.79/47.02 product(i(x0), j(quotient(quotient(difference(difference(product(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0))), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), product(difference(product(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0))), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), product(difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0)), product(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0)))))), eta(x0)), x0))) 46.79/47.02 = { by axiom 8 (sos03) } 46.79/47.02 product(i(x0), j(quotient(quotient(difference(difference(eta(x0), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), product(difference(product(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0))), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), product(difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0)), product(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0)))))), eta(x0)), x0))) 46.79/47.02 = { by lemma 122 } 46.79/47.02 product(i(x0), j(quotient(quotient(difference(difference(eta(x0), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), product(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), difference(difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), eta(x0)))), eta(x0)), x0))) 46.79/47.02 = { by axiom 8 (sos03) } 46.79/47.02 product(i(x0), j(quotient(quotient(difference(difference(eta(x0), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0)), eta(x0)), x0))) 46.79/47.02 = { by lemma 134 } 46.79/47.02 product(i(x0), j(quotient(quotient(difference(difference(eta(x0), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), i(eta(x0))), i(eta(x0))), x0))) 46.79/47.02 = { by lemma 134 } 46.79/47.02 product(i(x0), j(quotient(quotient(difference(difference(eta(x0), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), i(i(eta(x0)))), i(i(eta(x0)))), x0))) 46.79/47.02 = { by lemma 130 } 46.79/47.02 product(i(x0), j(quotient(quotient(difference(product(product(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0)), i(eta(x0))), i(i(eta(x0)))), i(i(eta(x0)))), x0))) 46.79/47.02 = { by lemma 125 } 46.79/47.02 product(i(x0), j(quotient(difference(product(i(i(eta(x0))), product(product(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0)), i(eta(x0)))), i(i(eta(x0)))), x0))) 46.79/47.02 = { by lemma 94 } 46.79/47.02 product(i(x0), j(quotient(difference(product(difference(i(eta(x0)), product(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0))), i(eta(x0))), i(i(eta(x0)))), x0))) 46.79/47.02 = { by axiom 1 (sos06) } 46.79/47.03 product(i(x0), j(quotient(difference(product(difference(i(eta(x0)), product(quotient(product(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0)), eta(x0)), eta(x0))), i(eta(x0))), i(i(eta(x0)))), x0))) 46.79/47.03 = { by lemma 43 } 46.79/47.03 product(i(x0), j(quotient(difference(product(difference(i(eta(x0)), quotient(product(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0)), eta(x0))), product(eta(x0), i(eta(x0)))), i(i(eta(x0)))), x0))) 46.79/47.03 = { by lemma 14 } 46.79/47.03 product(i(x0), j(quotient(difference(product(difference(i(eta(x0)), quotient(product(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0)), eta(x0))), one), i(i(eta(x0)))), x0))) 46.79/47.03 = { by axiom 12 (sos01) } 46.79/47.03 product(i(x0), j(quotient(difference(difference(i(eta(x0)), quotient(product(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0)), eta(x0))), i(i(eta(x0)))), x0))) 46.79/47.03 = { by axiom 1 (sos06) } 46.79/47.03 product(i(x0), j(quotient(difference(difference(i(eta(x0)), quotient(product(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0)), eta(x0))), product(quotient(i(i(eta(x0))), i(eta(x0))), i(eta(x0)))), x0))) 46.79/47.03 = { by lemma 115 } 46.79/47.03 product(i(x0), j(quotient(difference(difference(i(eta(x0)), quotient(product(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0)), eta(x0))), product(difference(i(eta(x0)), j(i(eta(x0)))), i(eta(x0)))), x0))) 46.79/47.03 = { by lemma 39 } 46.79/47.03 product(i(x0), j(quotient(product(difference(product(i(eta(x0)), difference(i(eta(x0)), quotient(product(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0)), eta(x0)))), j(i(eta(x0)))), i(eta(x0))), x0))) 46.79/47.03 = { by axiom 8 (sos03) } 46.79/47.03 product(i(x0), j(quotient(product(difference(quotient(product(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), eta(x0)), eta(x0)), j(i(eta(x0)))), i(eta(x0))), x0))) 46.79/47.03 = { by axiom 11 (sos05) } 46.79/47.03 product(i(x0), j(quotient(product(difference(product(j(eta(x0)), difference(j(eta(x0)), j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))))), j(i(eta(x0)))), i(eta(x0))), x0))) 46.79/47.03 = { by axiom 8 (sos03) } 46.79/47.03 product(i(x0), j(quotient(product(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), j(i(eta(x0)))), i(eta(x0))), x0))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 product(i(x0), j(quotient(product(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), i(eta(x0))), x0))) 46.79/47.03 = { by lemma 31 } 46.79/47.03 product(i(x0), j(quotient(product(i(j(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)))), i(eta(x0))), x0))) 46.79/47.03 = { by lemma 35 } 46.79/47.03 product(i(x0), j(quotient(product(i(difference(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), eta(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0))))), i(eta(x0))), x0))) 46.79/47.03 = { by lemma 120 } 46.79/47.03 product(i(x0), j(quotient(product(i(difference(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), eta(product(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)))))), i(eta(x0))), x0))) 46.79/47.03 = { by lemma 66 } 46.79/47.03 product(i(x0), j(quotient(product(i(difference(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), eta(product(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)))))), j(eta(x0))), x0))) 46.79/47.03 = { by lemma 74 } 46.79/47.03 product(i(x0), j(quotient(difference(difference(j(eta(x0)), difference(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), eta(product(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)))))), eta(j(eta(x0)))), x0))) 46.79/47.03 = { by lemma 33 } 46.79/47.03 product(i(x0), j(quotient(difference(difference(j(eta(x0)), difference(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), eta(product(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)))))), eta(eta(x0))), x0))) 46.79/47.03 = { by lemma 65 } 46.79/47.03 product(i(x0), j(quotient(difference(difference(j(eta(x0)), difference(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), eta(product(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)))))), one), x0))) 46.79/47.03 = { by axiom 4 (sos09) } 46.79/47.03 product(i(x0), j(quotient(i(difference(j(eta(x0)), difference(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), eta(product(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0))))))), x0))) 46.79/47.03 = { by lemma 66 } 46.79/47.03 product(i(x0), j(quotient(i(difference(i(eta(x0)), difference(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), eta(product(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0))))))), x0))) 46.79/47.03 = { by axiom 1 (sos06) } 46.79/47.03 product(i(x0), j(quotient(i(difference(i(product(quotient(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0))), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)))), difference(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), eta(product(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0))))))), x0))) 46.79/47.03 = { by lemma 118 } 46.79/47.03 product(i(x0), j(quotient(i(difference(i(product(quotient(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0))), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)))), difference(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), eta(quotient(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0))))))), x0))) 46.79/47.03 = { by lemma 121 } 46.79/47.03 product(i(x0), j(quotient(i(quotient(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)))), x0))) 46.79/47.03 = { by lemma 135 } 46.79/47.03 product(i(x0), j(quotient(quotient(difference(quotient(eta(x0), difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0))), eta(x0)), eta(x0)), x0))) 46.79/47.03 = { by lemma 51 } 46.79/47.03 product(i(x0), j(quotient(quotient(difference(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0))), eta(x0)), eta(x0)), x0))) 46.79/47.03 = { by lemma 135 } 46.79/47.03 product(i(x0), j(quotient(i(j(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)))), x0))) 46.79/47.03 = { by lemma 31 } 46.79/47.03 product(i(x0), j(quotient(product(product(quotient(difference(x1, eta(x0)), x0), x0), j(x0)), x0))) 46.79/47.03 = { by axiom 1 (sos06) } 46.79/47.03 product(i(x0), j(quotient(product(difference(x1, eta(x0)), j(x0)), x0))) 46.79/47.03 = { by lemma 110 } 46.79/47.03 product(i(x0), j(quotient(product(quotient(i(difference(x0, x1)), x0), j(x0)), x0))) 46.79/47.03 = { by lemma 55 } 46.79/47.03 product(i(x0), j(quotient(product(x0, quotient(difference(x0, quotient(i(difference(x0, x1)), x0)), x0)), x0))) 46.79/47.03 = { by lemma 29 } 46.79/47.03 product(i(x0), j(quotient(product(x0, quotient(quotient(product(i(x0), i(difference(x0, x1))), x0), x0)), x0))) 46.79/47.03 = { by lemma 103 } 46.79/47.03 product(i(x0), j(quotient(product(x0, quotient(quotient(difference(x0, i(product(j(x0), x1))), x0), x0)), x0))) 46.79/47.03 = { by lemma 54 } 46.79/47.03 product(i(x0), j(quotient(product(product(x0, quotient(difference(x0, i(product(j(x0), x1))), x0)), j(x0)), x0))) 46.79/47.03 = { by lemma 55 } 46.79/47.03 product(i(x0), j(quotient(product(product(i(product(j(x0), x1)), j(x0)), j(x0)), x0))) 46.79/47.03 = { by lemma 75 } 46.79/47.03 product(i(x0), j(quotient(product(difference(x1, eta(j(x0))), j(x0)), x0))) 46.79/47.03 = { by lemma 31 } 46.79/47.03 product(i(x0), j(quotient(i(j(product(difference(x1, eta(j(x0))), j(x0)))), x0))) 46.79/47.03 = { by lemma 85 } 46.79/47.03 product(i(x0), j(quotient(i(difference(j(x0), quotient(eta(j(x0)), difference(x1, eta(j(x0)))))), x0))) 46.79/47.03 = { by lemma 111 } 46.79/47.03 product(i(x0), j(quotient(i(difference(j(x0), x1)), x0))) 46.79/47.03 = { by axiom 11 (sos05) } 46.79/47.03 quotient(product(product(i(x0), j(quotient(i(difference(j(x0), x1)), x0))), x0), x0) 46.79/47.03 = { by axiom 8 (sos03) } 46.79/47.03 quotient(product(product(i(x0), product(x0, difference(x0, j(quotient(i(difference(j(x0), x1)), x0))))), x0), x0) 46.79/47.03 = { by lemma 126 } 46.79/47.03 quotient(product(i(x0), quotient(product(x0, difference(x0, j(quotient(i(difference(j(x0), x1)), x0)))), i(x0))), x0) 46.79/47.03 = { by lemma 59 } 46.79/47.03 quotient(product(product(i(x0), x0), quotient(difference(x0, j(quotient(i(difference(j(x0), x1)), x0))), i(x0))), x0) 46.79/47.03 = { by axiom 6 (sos11) } 46.79/47.03 quotient(product(product(x0, j(x0)), quotient(difference(x0, j(quotient(i(difference(j(x0), x1)), x0))), i(x0))), x0) 46.79/47.03 = { by lemma 13 } 46.79/47.03 quotient(product(eta(x0), quotient(difference(x0, j(quotient(i(difference(j(x0), x1)), x0))), i(x0))), x0) 46.79/47.03 = { by lemma 16 } 46.79/47.03 quotient(product(eta(i(x0)), quotient(difference(x0, j(quotient(i(difference(j(x0), x1)), x0))), i(x0))), x0) 46.79/47.03 = { by lemma 89 } 46.79/47.03 quotient(product(eta(i(x0)), product(product(j(x0), j(quotient(i(difference(j(x0), x1)), x0))), x0)), x0) 46.79/47.03 = { by lemma 15 } 46.79/47.03 quotient(product(eta(i(x0)), product(product(j(x0), j(quotient(i(difference(j(x0), x1)), x0))), x0)), j(i(x0))) 46.79/47.03 = { by lemma 78 } 46.79/47.03 product(i(x0), product(difference(i(x0), product(eta(i(x0)), product(product(j(x0), j(quotient(i(difference(j(x0), x1)), x0))), x0))), i(x0))) 46.79/47.03 = { by lemma 73 } 46.79/47.03 quotient(product(i(x0), difference(i(x0), product(eta(i(x0)), product(product(j(x0), j(quotient(i(difference(j(x0), x1)), x0))), x0)))), j(i(x0))) 46.79/47.03 = { by lemma 24 } 46.79/47.03 quotient(product(i(x0), quotient(product(j(i(x0)), product(product(product(j(x0), j(quotient(i(difference(j(x0), x1)), x0))), x0), i(x0))), i(x0))), j(i(x0))) 46.79/47.03 = { by lemma 72 } 46.79/47.03 product(i(x0), product(j(i(x0)), product(product(product(j(x0), j(quotient(i(difference(j(x0), x1)), x0))), x0), i(x0)))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 product(i(x0), product(x0, product(product(product(j(x0), j(quotient(i(difference(j(x0), x1)), x0))), x0), i(x0)))) 46.79/47.03 = { by lemma 130 } 46.79/47.03 product(i(x0), product(x0, difference(x0, j(quotient(i(difference(j(x0), x1)), x0))))) 46.79/47.03 = { by lemma 137 } 46.79/47.03 product(x0, product(j(x0), difference(x0, j(quotient(i(difference(j(x0), x1)), x0))))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 product(j(i(x0)), product(j(x0), difference(x0, j(quotient(i(difference(j(x0), x1)), x0))))) 46.79/47.03 = { by lemma 80 } 46.79/47.03 product(j(i(x0)), product(j(x0), quotient(eta(x0), i(difference(j(x0), x1))))) 46.79/47.03 = { by lemma 13 } 46.79/47.03 product(j(i(x0)), product(j(x0), quotient(product(x0, j(x0)), i(difference(j(x0), x1))))) 46.79/47.03 = { by lemma 136 } 46.79/47.03 product(j(i(x0)), quotient(product(j(x0), x0), quotient(i(difference(j(x0), x1)), j(x0)))) 46.79/47.03 = { by lemma 21 } 46.79/47.03 product(j(i(x0)), quotient(one, quotient(i(difference(j(x0), x1)), j(x0)))) 46.79/47.03 = { by axiom 2 (sos10) } 46.79/47.03 product(j(i(x0)), j(quotient(i(difference(j(x0), x1)), j(x0)))) 46.79/47.03 = { by lemma 110 } 46.79/47.03 product(j(i(x0)), j(difference(x1, eta(j(x0))))) 46.79/47.03 = { by lemma 138 } 46.79/47.03 product(j(i(x0)), difference(eta(j(x0)), x1)) 46.79/47.03 = { by lemma 33 } 46.79/47.03 product(j(i(x0)), difference(eta(x0), x1)) 46.79/47.03 = { by lemma 16 } 46.79/47.03 product(j(i(x0)), difference(eta(i(x0)), x1)) 46.79/47.03 = { by lemma 138 } 46.79/47.03 product(j(i(x0)), j(difference(x1, eta(i(x0))))) 46.79/47.03 = { by lemma 110 } 46.79/47.03 product(j(i(x0)), j(quotient(i(difference(i(x0), x1)), i(x0)))) 46.79/47.03 = { by lemma 31 } 46.79/47.03 product(j(i(x0)), j(quotient(i(difference(i(x0), x1)), i(j(i(x0)))))) 46.79/47.03 = { by lemma 140 } 46.79/47.03 quotient(eta(j(i(x0))), product(j(i(x0)), product(i(j(i(x0))), i(difference(i(x0), x1))))) 46.79/47.03 = { by lemma 33 } 46.79/47.03 quotient(eta(i(x0)), product(j(i(x0)), product(i(j(i(x0))), i(difference(i(x0), x1))))) 46.79/47.03 = { by lemma 31 } 46.79/47.03 quotient(eta(i(x0)), product(j(i(x0)), product(i(x0), i(difference(i(x0), x1))))) 46.79/47.03 = { by lemma 87 } 46.79/47.03 quotient(eta(i(x0)), product(i(x0), product(i(i(x0)), i(difference(i(x0), x1))))) 46.79/47.03 = { by lemma 140 } 46.79/47.03 product(i(x0), j(quotient(i(difference(i(x0), x1)), i(i(x0))))) 46.79/47.03 = { by axiom 11 (sos05) } 46.79/47.03 product(i(x0), quotient(product(j(quotient(i(difference(i(x0), x1)), i(i(x0)))), i(i(x0))), i(i(x0)))) 46.79/47.03 = { by lemma 51 } 46.79/47.03 product(i(x0), quotient(product(difference(quotient(quotient(eta(i(x0)), quotient(i(difference(i(x0), x1)), i(i(x0)))), j(quotient(i(difference(i(x0), x1)), i(i(x0))))), quotient(eta(i(x0)), quotient(i(difference(i(x0), x1)), i(i(x0))))), i(i(x0))), i(i(x0)))) 46.79/47.03 = { by lemma 30 } 46.79/47.03 product(i(x0), quotient(product(difference(product(j(quotient(i(difference(i(x0), x1)), i(i(x0)))), product(quotient(i(difference(i(x0), x1)), i(i(x0))), eta(i(x0)))), quotient(eta(i(x0)), quotient(i(difference(i(x0), x1)), i(i(x0))))), i(i(x0))), i(i(x0)))) 46.79/47.03 = { by lemma 87 } 46.79/47.03 product(i(x0), quotient(product(difference(product(quotient(i(difference(i(x0), x1)), i(i(x0))), product(i(quotient(i(difference(i(x0), x1)), i(i(x0)))), eta(i(x0)))), quotient(eta(i(x0)), quotient(i(difference(i(x0), x1)), i(i(x0))))), i(i(x0))), i(i(x0)))) 46.79/47.03 = { by lemma 108 } 46.79/47.03 product(i(x0), quotient(product(difference(product(quotient(i(difference(i(x0), x1)), i(i(x0))), difference(quotient(i(difference(i(x0), x1)), i(i(x0))), eta(i(x0)))), quotient(eta(i(x0)), quotient(i(difference(i(x0), x1)), i(i(x0))))), i(i(x0))), i(i(x0)))) 46.79/47.03 = { by axiom 8 (sos03) } 46.79/47.03 product(i(x0), quotient(product(difference(eta(i(x0)), quotient(eta(i(x0)), quotient(i(difference(i(x0), x1)), i(i(x0))))), i(i(x0))), i(i(x0)))) 46.79/47.03 = { by lemma 16 } 46.79/47.03 product(i(x0), quotient(product(difference(eta(i(x0)), quotient(eta(i(i(x0))), quotient(i(difference(i(x0), x1)), i(i(x0))))), i(i(x0))), i(i(x0)))) 46.79/47.03 = { by lemma 67 } 46.79/47.03 product(i(x0), quotient(product(difference(eta(i(x0)), product(i(i(x0)), j(product(quotient(i(difference(i(x0), x1)), i(i(x0))), i(i(x0)))))), i(i(x0))), i(i(x0)))) 46.79/47.03 = { by lemma 129 } 46.79/47.03 product(i(x0), quotient(difference(i(x0), product(j(product(quotient(i(difference(i(x0), x1)), i(i(x0))), i(i(x0)))), i(i(x0)))), i(i(x0)))) 46.79/47.03 = { by axiom 1 (sos06) } 46.79/47.03 product(i(x0), quotient(difference(i(x0), product(j(i(difference(i(x0), x1))), i(i(x0)))), i(i(x0)))) 46.79/47.03 = { by lemma 89 } 46.79/47.03 product(i(x0), product(product(j(i(x0)), product(j(i(difference(i(x0), x1))), i(i(x0)))), i(x0))) 46.79/47.03 = { by lemma 92 } 46.79/47.03 product(i(x0), difference(eta(i(x0)), product(i(i(x0)), j(i(difference(i(x0), x1)))))) 46.79/47.03 = { by lemma 16 } 46.79/47.03 product(i(x0), difference(eta(i(i(x0))), product(i(i(x0)), j(i(difference(i(x0), x1)))))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 product(i(x0), difference(eta(i(i(x0))), product(i(i(x0)), difference(i(x0), x1)))) 46.79/47.03 = { by lemma 91 } 46.79/47.03 product(i(x0), quotient(difference(j(i(i(x0))), product(difference(i(x0), x1), i(i(x0)))), i(i(x0)))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 product(i(x0), quotient(difference(i(x0), product(difference(i(x0), x1), i(i(x0)))), i(i(x0)))) 46.79/47.03 = { by lemma 19 } 46.79/47.03 product(i(x0), product(product(i(x0), product(i(i(x0)), difference(i(x0), product(difference(i(x0), x1), i(i(x0)))))), i(x0))) 46.79/47.03 = { by lemma 98 } 46.79/47.03 product(i(x0), product(quotient(quotient(difference(i(x0), product(difference(i(x0), x1), i(i(x0)))), i(x0)), j(i(x0))), i(x0))) 46.79/47.03 = { by axiom 1 (sos06) } 46.79/47.03 product(i(x0), product(quotient(quotient(difference(i(x0), product(quotient(product(difference(i(x0), x1), i(i(x0))), i(x0)), i(x0))), i(x0)), j(i(x0))), i(x0))) 46.79/47.03 = { by lemma 101 } 46.79/47.03 product(i(x0), product(quotient(difference(product(i(x0), i(x0)), product(i(x0), quotient(product(difference(i(x0), x1), i(i(x0))), i(x0)))), j(i(x0))), i(x0))) 46.79/47.03 = { by lemma 106 } 46.79/47.03 product(i(x0), product(quotient(difference(product(i(x0), i(x0)), product(x1, quotient(i(i(x0)), i(x0)))), j(i(x0))), i(x0))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 product(i(x0), product(quotient(difference(product(i(x0), i(x0)), product(x1, quotient(i(i(x0)), i(x0)))), j(j(i(i(x0))))), i(x0))) 46.79/47.03 = { by lemma 88 } 46.79/47.03 product(i(x0), product(product(i(i(x0)), product(i(i(i(x0))), product(difference(product(i(x0), i(x0)), product(x1, quotient(i(i(x0)), i(x0)))), j(i(i(x0)))))), i(x0))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 product(i(x0), product(product(i(i(x0)), product(i(i(i(x0))), product(difference(product(i(x0), i(x0)), product(x1, quotient(i(i(x0)), i(x0)))), j(i(i(x0)))))), j(i(i(x0))))) 46.79/47.03 = { by lemma 54 } 46.79/47.03 product(i(x0), product(i(i(x0)), quotient(product(i(i(i(x0))), product(difference(product(i(x0), i(x0)), product(x1, quotient(i(i(x0)), i(x0)))), j(i(i(x0))))), i(i(x0))))) 46.79/47.03 = { by lemma 29 } 46.79/47.03 product(i(x0), product(i(i(x0)), difference(i(i(x0)), quotient(product(difference(product(i(x0), i(x0)), product(x1, quotient(i(i(x0)), i(x0)))), j(i(i(x0)))), i(i(x0)))))) 46.79/47.03 = { by axiom 8 (sos03) } 46.79/47.03 product(i(x0), quotient(product(difference(product(i(x0), i(x0)), product(x1, quotient(i(i(x0)), i(x0)))), j(i(i(x0)))), i(i(x0)))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 product(i(x0), quotient(product(difference(product(i(x0), i(x0)), product(x1, quotient(i(i(x0)), i(x0)))), i(x0)), i(i(x0)))) 46.79/47.03 = { by lemma 136 } 46.79/47.03 quotient(product(i(x0), difference(product(i(x0), i(x0)), product(x1, quotient(i(i(x0)), i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 quotient(product(j(i(i(x0))), difference(product(i(x0), i(x0)), product(x1, quotient(i(i(x0)), i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 114 } 46.79/47.03 quotient(product(j(i(i(x0))), difference(difference(i(i(x0)), i(x0)), product(x1, quotient(i(i(x0)), i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 28 } 46.79/47.03 quotient(product(j(i(i(x0))), difference(quotient(product(i(i(i(x0))), product(i(x0), i(i(x0)))), i(i(x0))), product(x1, quotient(i(i(x0)), i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 14 } 46.79/47.03 quotient(product(j(i(i(x0))), difference(quotient(product(i(i(i(x0))), one), i(i(x0))), product(x1, quotient(i(i(x0)), i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by axiom 12 (sos01) } 46.79/47.03 quotient(product(j(i(i(x0))), difference(quotient(i(i(i(x0))), i(i(x0))), product(x1, quotient(i(i(x0)), i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 116 } 46.79/47.03 quotient(product(j(i(i(x0))), difference(product(j(i(i(x0))), j(i(i(x0)))), product(x1, quotient(i(i(x0)), i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 145 } 46.79/47.03 quotient(product(j(i(i(x0))), difference(product(j(i(i(x0))), j(i(i(x0)))), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by axiom 5 (sos02) } 46.79/47.03 quotient(product(j(i(i(x0))), difference(product(j(i(i(x0))), j(i(i(x0)))), product(one, product(x1, j(quotient(i(x0), i(i(x0)))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 21 } 46.79/47.03 quotient(product(j(i(i(x0))), difference(product(j(i(i(x0))), j(i(i(x0)))), product(product(j(i(i(x0))), i(i(x0))), product(x1, j(quotient(i(x0), i(i(x0)))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 146 } 46.79/47.03 quotient(product(j(i(i(x0))), quotient(difference(j(i(i(x0))), product(i(i(x0)), product(product(x1, j(quotient(i(x0), i(i(x0))))), j(i(i(x0)))))), j(i(i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 55 } 46.79/47.03 quotient(product(product(i(i(x0)), product(product(x1, j(quotient(i(x0), i(i(x0))))), j(i(i(x0))))), j(j(i(i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 quotient(product(product(i(i(x0)), product(product(x1, j(quotient(i(x0), i(i(x0))))), i(x0))), j(j(i(i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by axiom 9 (sos04) } 46.79/47.03 quotient(product(difference(i(x0), product(i(x0), product(i(i(x0)), product(product(x1, j(quotient(i(x0), i(i(x0))))), i(x0))))), j(j(i(i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 18 } 46.79/47.03 quotient(product(difference(i(x0), quotient(quotient(product(product(x1, j(quotient(i(x0), i(i(x0))))), i(x0)), i(i(x0))), i(x0))), j(j(i(i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 29 } 46.79/47.03 quotient(product(quotient(product(i(i(x0)), quotient(product(product(x1, j(quotient(i(x0), i(i(x0))))), i(x0)), i(i(x0)))), i(x0)), j(j(i(i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by axiom 11 (sos05) } 46.79/47.03 quotient(product(quotient(quotient(product(product(i(i(x0)), quotient(product(product(x1, j(quotient(i(x0), i(i(x0))))), i(x0)), i(i(x0)))), i(i(x0))), i(i(x0))), i(x0)), j(j(i(i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 18 } 46.79/47.03 quotient(product(product(i(x0), product(i(i(x0)), product(product(i(i(x0)), quotient(product(product(x1, j(quotient(i(x0), i(i(x0))))), i(x0)), i(i(x0)))), i(i(x0))))), j(j(i(i(x0))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 quotient(product(product(i(x0), product(i(i(x0)), product(product(i(i(x0)), quotient(product(product(x1, j(quotient(i(x0), i(i(x0))))), i(x0)), i(i(x0)))), i(i(x0))))), j(i(x0))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 54 } 46.79/47.03 quotient(product(i(x0), quotient(product(i(i(x0)), product(product(i(i(x0)), quotient(product(product(x1, j(quotient(i(x0), i(i(x0))))), i(x0)), i(i(x0)))), i(i(x0)))), i(x0))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 29 } 46.79/47.03 quotient(product(i(x0), difference(i(x0), quotient(product(product(i(i(x0)), quotient(product(product(x1, j(quotient(i(x0), i(i(x0))))), i(x0)), i(i(x0)))), i(i(x0))), i(x0)))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by axiom 8 (sos03) } 46.79/47.03 quotient(quotient(product(product(i(i(x0)), quotient(product(product(x1, j(quotient(i(x0), i(i(x0))))), i(x0)), i(i(x0)))), i(i(x0))), i(x0)), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 53 } 46.79/47.03 quotient(quotient(difference(i(i(i(x0))), product(product(x1, j(quotient(i(x0), i(i(x0))))), i(x0))), i(x0)), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 101 } 46.79/47.03 quotient(difference(product(i(x0), i(i(i(x0)))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 142 } 46.79/47.03 quotient(difference(product(i(x0), product(eta(i(x0)), i(x0))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 73 } 46.79/47.03 quotient(difference(quotient(product(i(x0), eta(i(x0))), j(i(x0))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 64 } 46.79/47.03 quotient(difference(difference(j(i(x0)), i(x0)), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 31 } 46.79/47.03 quotient(difference(i(j(difference(j(i(x0)), i(x0)))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 64 } 46.79/47.03 quotient(difference(i(j(quotient(product(i(x0), eta(i(x0))), j(i(x0))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 36 } 46.79/47.03 quotient(difference(i(j(quotient(j(j(i(x0))), j(i(x0))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 35 } 46.79/47.03 quotient(difference(i(difference(quotient(j(j(i(x0))), j(i(x0))), eta(quotient(j(j(i(x0))), j(i(x0)))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 22 } 46.79/47.03 quotient(difference(i(difference(quotient(j(j(i(x0))), j(i(x0))), product(product(quotient(j(j(i(x0))), j(i(x0))), j(product(j(i(x0)), quotient(j(j(i(x0))), j(i(x0)))))), j(i(x0))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 26 } 46.79/47.03 quotient(difference(i(difference(quotient(j(j(i(x0))), j(i(x0))), product(product(quotient(j(j(i(x0))), j(i(x0))), j(product(eta(j(i(x0))), j(j(i(x0)))))), j(i(x0))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 38 } 46.79/47.03 quotient(difference(i(difference(quotient(j(j(i(x0))), j(i(x0))), product(product(quotient(j(j(i(x0))), j(i(x0))), j(i(j(i(x0))))), j(i(x0))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 15 } 46.79/47.03 quotient(difference(i(difference(quotient(j(j(i(x0))), j(i(x0))), product(product(quotient(j(j(i(x0))), j(i(x0))), j(i(x0))), j(i(x0))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by axiom 1 (sos06) } 46.79/47.03 quotient(difference(i(difference(quotient(j(j(i(x0))), j(i(x0))), product(j(j(i(x0))), j(i(x0))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 21 } 46.79/47.03 quotient(difference(i(difference(quotient(j(j(i(x0))), j(i(x0))), one)), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by axiom 4 (sos09) } 46.79/47.03 quotient(difference(i(i(quotient(j(j(i(x0))), j(i(x0))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 37 } 46.79/47.03 quotient(difference(i(i(difference(j(i(x0)), i(j(i(x0)))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 27 } 46.79/47.03 quotient(difference(i(i(difference(j(i(x0)), quotient(eta(j(i(x0))), j(i(x0)))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 85 } 46.79/47.03 quotient(difference(i(i(j(product(j(i(x0)), j(i(x0)))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 31 } 46.79/47.03 quotient(difference(i(product(j(i(x0)), j(i(x0)))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 116 } 46.79/47.03 quotient(difference(i(quotient(i(i(x0)), i(x0))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by axiom 4 (sos09) } 46.79/47.03 quotient(difference(difference(quotient(i(i(x0)), i(x0)), one), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 144 } 46.79/47.03 quotient(difference(difference(i(quotient(i(x0), i(i(x0)))), one), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 143 } 46.79/47.03 quotient(difference(difference(i(quotient(i(x0), i(i(x0)))), eta(quotient(i(x0), i(i(x0))))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 70 } 46.79/47.03 quotient(difference(quotient(i(x0), i(i(x0))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), quotient(i(i(x0)), i(x0))) 46.79/47.03 = { by lemma 145 } 46.79/47.03 quotient(difference(quotient(i(x0), i(i(x0))), product(i(x0), product(x1, j(quotient(i(x0), i(i(x0))))))), j(quotient(i(x0), i(i(x0))))) 46.79/47.03 = { by lemma 146 } 46.79/47.03 difference(product(j(quotient(i(x0), i(i(x0)))), quotient(i(x0), i(i(x0)))), product(product(j(quotient(i(x0), i(i(x0)))), i(x0)), x1)) 46.79/47.03 = { by lemma 21 } 46.79/47.03 difference(one, product(product(j(quotient(i(x0), i(i(x0)))), i(x0)), x1)) 46.79/47.03 = { by lemma 17 } 46.79/47.03 product(product(j(quotient(i(x0), i(i(x0)))), i(x0)), x1) 46.79/47.03 = { by lemma 145 } 46.79/47.03 product(product(quotient(i(i(x0)), i(x0)), i(x0)), x1) 46.79/47.03 = { by axiom 1 (sos06) } 46.79/47.03 product(i(i(x0)), x1) 46.79/47.03 % SZS output end Proof 46.79/47.03 46.79/47.03 RESULT: Unsatisfiable (the axioms are contradictory). 46.79/47.04 EOF