0.12/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.12/0.14 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.15/0.35 % Computer : n006.cluster.edu 0.15/0.35 % Model : x86_64 x86_64 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.15/0.35 % Memory : 8042.1875MB 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64 0.15/0.35 % CPULimit : 180 0.15/0.35 % DateTime : Thu Aug 29 11:26:39 EDT 2019 0.15/0.36 % CPUTime : 0.49/0.69 % SZS status Unsatisfiable 0.49/0.69 0.49/0.69 % SZS output start Proof 0.49/0.69 Take the following subset of the input axioms: 0.56/0.71 fof(prove_these_axioms_2, negated_conjecture, multiply(multiply(inverse(b2), b2), a2)!=a2). 0.56/0.71 fof(single_axiom, axiom, ![C, A, B]: C=multiply(A, inverse(multiply(inverse(multiply(inverse(multiply(A, B)), C)), multiply(inverse(B), multiply(inverse(B), B)))))). 0.56/0.71 0.56/0.71 Now clausify the problem and encode Horn clauses using encoding 3 of 0.56/0.71 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 0.56/0.71 We repeatedly replace C & s=t => u=v by the two clauses: 0.56/0.71 fresh(y, y, x1...xn) = u 0.56/0.71 C => fresh(s, t, x1...xn) = v 0.56/0.71 where fresh is a fresh function symbol and x1..xn are the free 0.56/0.71 variables of u and v. 0.56/0.71 A predicate p(X) is encoded as p(X)=true (this is sound, because the 0.56/0.71 input problem has no model of domain size 1). 0.56/0.71 0.56/0.71 The encoding turns the above axioms into the following unit equations and goals: 0.56/0.71 0.56/0.74 Axiom 1 (single_axiom): X = multiply(Y, inverse(multiply(inverse(multiply(inverse(multiply(Y, Z)), X)), multiply(inverse(Z), multiply(inverse(Z), Z))))). 0.56/0.74 0.56/0.74 Lemma 2: inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(multiply(X, Y)), Z))), multiply(inverse(?), multiply(inverse(?), ?)))) = Z. 0.56/0.74 Proof: 0.56/0.74 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(multiply(X, Y)), Z))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.56/0.74 = { by axiom 1 (single_axiom) } 0.56/0.74 multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(multiply(X, Y)), Z))), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(inverse(Y), multiply(inverse(Y), Y))))) 0.56/0.74 = { by axiom 1 (single_axiom) } 0.56/0.74 multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), multiply(inverse(Y), multiply(inverse(Y), Y))))) 0.56/0.74 = { by axiom 1 (single_axiom) } 0.56/0.74 Z 0.56/0.74 0.56/0.74 Lemma 3: multiply(inverse(multiply(Y, Z)), multiply(Y, inverse(multiply(inverse(X), multiply(inverse(Z), multiply(inverse(Z), Z)))))) = X. 0.56/0.74 Proof: 0.56/0.74 multiply(inverse(multiply(Y, Z)), multiply(Y, inverse(multiply(inverse(X), multiply(inverse(Z), multiply(inverse(Z), Z)))))) 0.56/0.74 = { by axiom 1 (single_axiom) } 0.56/0.74 multiply(inverse(multiply(Y, Z)), multiply(Y, inverse(multiply(inverse(multiply(inverse(multiply(Y, Z)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(Y, Z)), ?)), X)), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(inverse(Z), multiply(inverse(Z), Z)))))) 0.56/0.74 = { by axiom 1 (single_axiom) } 0.56/0.74 multiply(inverse(multiply(Y, Z)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(Y, Z)), ?)), X)), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.74 = { by axiom 1 (single_axiom) } 0.56/0.74 X 0.56/0.74 0.56/0.74 Lemma 4: multiply(inverse(multiply(W, ?)), multiply(W, Z)) = multiply(inverse(multiply(?, ?)), multiply(?, Z)). 0.56/0.74 Proof: 0.56/0.74 multiply(inverse(multiply(W, ?)), multiply(W, Z)) 0.56/0.74 = { by lemma 2 } 0.56/0.74 multiply(inverse(multiply(W, ?)), multiply(W, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(multiply(X, Y)), Z))), multiply(inverse(?), multiply(inverse(?), ?)))))) 0.56/0.74 = { by lemma 3 } 0.56/0.74 multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(multiply(X, Y)), Z)) 0.56/0.74 = { by lemma 3 } 0.56/0.74 multiply(inverse(multiply(?, ?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(multiply(X, Y)), Z))), multiply(inverse(?), multiply(inverse(?), ?)))))) 0.56/0.74 = { by lemma 2 } 0.56/0.74 multiply(inverse(multiply(?, ?)), multiply(?, Z)) 0.56/0.74 0.56/0.74 Lemma 5: multiply(inverse(multiply(Y, X)), multiply(Y, inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(X), X)))))) = multiply(inverse(X), ?). 0.56/0.74 Proof: 0.56/0.74 multiply(inverse(multiply(Y, X)), multiply(Y, inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(X), X)))))) 0.56/0.74 = { by lemma 4 } 0.56/0.74 multiply(inverse(multiply(Y, X)), multiply(Y, inverse(multiply(inverse(multiply(inverse(X), ?)), multiply(inverse(X), multiply(inverse(X), X)))))) 0.56/0.74 = { by lemma 3 } 0.56/0.75 multiply(inverse(X), ?) 0.56/0.75 0.56/0.75 Lemma 6: inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))) = X. 0.56/0.75 Proof: 0.56/0.75 inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.56/0.75 = { by lemma 2 } 0.56/0.75 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), ?))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.56/0.75 = { by lemma 2 } 0.56/0.75 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), ?))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), ?))), multiply(inverse(?), multiply(inverse(?), ?)))), X))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.56/0.75 = { by lemma 2 } 0.56/0.75 X 0.56/0.75 0.56/0.75 Lemma 7: multiply(inverse(multiply(?, ?)), inverse(multiply(X, multiply(inverse(multiply(?, X)), multiply(inverse(multiply(?, X)), multiply(?, X)))))) = multiply(inverse(?), multiply(inverse(?), ?)). 0.56/0.75 Proof: 0.56/0.75 multiply(inverse(multiply(?, ?)), inverse(multiply(X, multiply(inverse(multiply(?, X)), multiply(inverse(multiply(?, X)), multiply(?, X)))))) 0.56/0.75 = { by lemma 6 } 0.56/0.75 multiply(inverse(multiply(?, ?)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(multiply(?, X)), multiply(inverse(multiply(?, X)), multiply(?, X)))))) 0.56/0.75 = { by axiom 1 (single_axiom) } 0.56/0.75 multiply(inverse(?), multiply(inverse(?), ?)) 0.56/0.75 0.56/0.75 Lemma 8: multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, inverse(X)))), multiply(inverse(?), multiply(inverse(?), ?))) = X. 0.56/0.75 Proof: 0.56/0.75 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, inverse(X)))), multiply(inverse(?), multiply(inverse(?), ?))) 0.56/0.75 = { by lemma 7 } 0.56/0.75 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, inverse(X)))), multiply(inverse(multiply(?, ?)), inverse(multiply(inverse(X), multiply(inverse(multiply(?, inverse(X))), multiply(inverse(multiply(?, inverse(X))), multiply(?, inverse(X)))))))) 0.56/0.75 = { by lemma 3 } 0.56/0.75 X 0.56/0.75 0.56/0.75 Lemma 9: inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), W)), multiply(inverse(Z), multiply(inverse(Z), Z)))) = inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), W)), multiply(inverse(?), multiply(inverse(?), ?)))). 0.56/0.75 Proof: 0.56/0.75 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), W)), multiply(inverse(Z), multiply(inverse(Z), Z)))) 0.56/0.75 = { by axiom 1 (single_axiom) } 0.56/0.75 multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), W)), multiply(inverse(Z), multiply(inverse(Z), Z)))))), multiply(inverse(Y), multiply(inverse(Y), Y))))) 0.56/0.75 = { by axiom 1 (single_axiom) } 0.56/0.75 multiply(X, inverse(multiply(inverse(W), multiply(inverse(Y), multiply(inverse(Y), Y))))) 0.56/0.75 = { by axiom 1 (single_axiom) } 0.56/0.75 multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), W)), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(inverse(Y), multiply(inverse(Y), Y))))) 0.56/0.75 = { by axiom 1 (single_axiom) } 0.56/0.75 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), W)), multiply(inverse(?), multiply(inverse(?), ?)))) 0.56/0.75 0.56/0.75 Lemma 10: inverse(multiply(inverse(multiply(inverse(multiply(Z, X)), Y)), multiply(inverse(X), multiply(inverse(X), X)))) = inverse(multiply(inverse(multiply(inverse(multiply(Z, ?)), Y)), multiply(inverse(?), multiply(inverse(?), ?)))). 0.56/0.75 Proof: 0.56/0.75 inverse(multiply(inverse(multiply(inverse(multiply(Z, X)), Y)), multiply(inverse(X), multiply(inverse(X), X)))) 0.56/0.75 = { by lemma 2 } 0.56/0.75 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(V, W)), ?)), multiply(inverse(multiply(V, W)), Z))), multiply(inverse(?), multiply(inverse(?), ?)))), X)), Y)), multiply(inverse(X), multiply(inverse(X), X)))) 0.56/0.75 = { by lemma 9 } 0.56/0.75 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(V, W)), ?)), multiply(inverse(multiply(V, W)), Z))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), Y)), multiply(inverse(?), multiply(inverse(?), ?)))) 0.56/0.75 = { by lemma 2 } 0.56/0.75 inverse(multiply(inverse(multiply(inverse(multiply(Z, ?)), Y)), multiply(inverse(?), multiply(inverse(?), ?)))) 0.56/0.75 0.56/0.75 Lemma 11: multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(multiply(X, Y)), Z)) = multiply(inverse(multiply(?, ?)), multiply(?, Z)). 0.56/0.75 Proof: 0.56/0.75 multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(multiply(X, Y)), Z)) 0.56/0.75 = { by lemma 3 } 0.56/0.75 multiply(inverse(multiply(?, ?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(multiply(X, Y)), Z))), multiply(inverse(?), multiply(inverse(?), ?)))))) 0.56/0.75 = { by lemma 2 } 0.56/0.75 multiply(inverse(multiply(?, ?)), multiply(?, Z)) 0.56/0.75 0.56/0.75 Lemma 12: multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), inverse(multiply(Z, multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), multiply(inverse(multiply(X, Y)), Z)))))) = multiply(inverse(?), multiply(inverse(?), ?)). 0.56/0.75 Proof: 0.56/0.75 multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), inverse(multiply(Z, multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), multiply(inverse(multiply(X, Y)), Z)))))) 0.56/0.75 = { by lemma 2 } 0.56/0.75 multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(multiply(X, Y)), Z))), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), multiply(inverse(multiply(X, Y)), Z)))))) 0.56/0.75 = { by axiom 1 (single_axiom) } 0.56/0.75 multiply(inverse(?), multiply(inverse(?), ?)) 0.56/0.75 0.56/0.75 Lemma 13: multiply(inverse(multiply(X, ?)), inverse(multiply(Y, multiply(inverse(multiply(X, Y)), multiply(inverse(multiply(X, Y)), multiply(X, Y)))))) = multiply(inverse(?), multiply(inverse(?), ?)). 0.56/0.75 Proof: 0.56/0.75 multiply(inverse(multiply(X, ?)), inverse(multiply(Y, multiply(inverse(multiply(X, Y)), multiply(inverse(multiply(X, Y)), multiply(X, Y)))))) 0.56/0.75 = { by lemma 2 } 0.56/0.75 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), inverse(multiply(Y, multiply(inverse(multiply(X, Y)), multiply(inverse(multiply(X, Y)), multiply(X, Y)))))) 0.56/0.75 = { by lemma 2 } 0.56/0.75 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), inverse(multiply(Y, multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))), Y)), multiply(inverse(multiply(X, Y)), multiply(X, Y)))))) 0.56/0.75 = { by lemma 2 } 0.56/0.75 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), inverse(multiply(Y, multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))), Y)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))), Y)), multiply(X, Y)))))) 0.56/0.75 = { by lemma 2 } 0.56/0.75 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), inverse(multiply(Y, multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))), Y)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))), Y)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))), Y)))))) 0.56/0.75 = { by lemma 12 } 0.56/0.75 multiply(inverse(?), multiply(inverse(?), ?)) 0.56/0.75 0.56/0.75 Lemma 14: inverse(multiply(X, multiply(inverse(multiply(Y, X)), multiply(inverse(multiply(Y, X)), multiply(Y, X))))) = inverse(multiply(?, multiply(inverse(multiply(Y, ?)), multiply(inverse(multiply(Y, ?)), multiply(Y, ?))))). 0.56/0.75 Proof: 0.56/0.75 inverse(multiply(X, multiply(inverse(multiply(Y, X)), multiply(inverse(multiply(Y, X)), multiply(Y, X))))) 0.56/0.75 = { by axiom 1 (single_axiom) } 0.56/0.75 multiply(Y, inverse(multiply(inverse(multiply(inverse(multiply(Y, ?)), inverse(multiply(X, multiply(inverse(multiply(Y, X)), multiply(inverse(multiply(Y, X)), multiply(Y, X))))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.75 = { by lemma 13 } 0.56/0.75 multiply(Y, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.75 = { by lemma 13 } 0.56/0.75 multiply(Y, inverse(multiply(inverse(multiply(inverse(multiply(Y, ?)), inverse(multiply(?, multiply(inverse(multiply(Y, ?)), multiply(inverse(multiply(Y, ?)), multiply(Y, ?))))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.75 = { by axiom 1 (single_axiom) } 0.56/0.76 inverse(multiply(?, multiply(inverse(multiply(Y, ?)), multiply(inverse(multiply(Y, ?)), multiply(Y, ?))))) 0.56/0.76 0.56/0.76 Lemma 15: multiply(X, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))) = inverse(multiply(?, multiply(inverse(multiply(X, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))). 0.56/0.76 Proof: 0.56/0.76 multiply(X, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.76 = { by lemma 12 } 0.56/0.76 multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?))))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.76 = { by lemma 6 } 0.56/0.76 multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(X, ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?))))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.76 = { by lemma 6 } 0.56/0.76 multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(X, ?)), inverse(multiply(?, multiply(inverse(multiply(X, ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))), ?))))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.76 = { by lemma 11 } 0.56/0.76 multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(X, ?)), inverse(multiply(?, multiply(inverse(multiply(X, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.76 = { by axiom 1 (single_axiom) } 0.56/0.76 inverse(multiply(?, multiply(inverse(multiply(X, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.56/0.76 0.56/0.76 Lemma 16: multiply(X, inverse(multiply(inverse(multiply(inverse(Y), multiply(inverse(Y), Y))), multiply(inverse(Y), multiply(inverse(Y), Y))))) = inverse(multiply(?, multiply(inverse(multiply(X, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))). 0.56/0.76 Proof: 0.56/0.76 multiply(X, inverse(multiply(inverse(multiply(inverse(Y), multiply(inverse(Y), Y))), multiply(inverse(Y), multiply(inverse(Y), Y))))) 0.56/0.76 = { by axiom 1 (single_axiom) } 0.56/0.76 multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(Y), multiply(inverse(Y), Y)))), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(inverse(Y), multiply(inverse(Y), Y))))) 0.56/0.76 = { by axiom 1 (single_axiom) } 0.56/0.76 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), multiply(inverse(Y), multiply(inverse(Y), Y)))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.56/0.76 = { by lemma 10 } 0.56/0.76 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, ?)), ?)), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.56/0.76 = { by axiom 1 (single_axiom) } 0.56/0.76 multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(X, ?)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, ?)), ?)), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.76 = { by axiom 1 (single_axiom) } 0.56/0.76 multiply(X, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.76 = { by lemma 15 } 0.56/0.76 inverse(multiply(?, multiply(inverse(multiply(X, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.56/0.76 0.56/0.76 Lemma 17: multiply(inverse(multiply(?, ?)), multiply(?, multiply(X, inverse(multiply(inverse(Z), multiply(inverse(Y), multiply(inverse(Y), Y))))))) = multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), Z). 0.56/0.76 Proof: 0.56/0.76 multiply(inverse(multiply(?, ?)), multiply(?, multiply(X, inverse(multiply(inverse(Z), multiply(inverse(Y), multiply(inverse(Y), Y))))))) 0.56/0.76 = { by axiom 1 (single_axiom) } 0.56/0.76 multiply(inverse(multiply(?, ?)), multiply(?, multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), Z)), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(inverse(Y), multiply(inverse(Y), Y))))))) 0.56/0.76 = { by axiom 1 (single_axiom) } 0.56/0.76 multiply(inverse(multiply(?, ?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), Z)), multiply(inverse(?), multiply(inverse(?), ?)))))) 0.56/0.76 = { by lemma 3 } 0.56/0.76 multiply(inverse(multiply(inverse(multiply(X, Y)), ?)), Z) 0.56/0.76 0.56/0.76 Lemma 18: inverse(multiply(X, multiply(inverse(multiply(?, X)), multiply(inverse(multiply(?, X)), multiply(?, X))))) = inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))). 0.56/0.76 Proof: 0.56/0.76 inverse(multiply(X, multiply(inverse(multiply(?, X)), multiply(inverse(multiply(?, X)), multiply(?, X))))) 0.56/0.76 = { by axiom 1 (single_axiom) } 0.56/0.76 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), inverse(multiply(X, multiply(inverse(multiply(?, X)), multiply(inverse(multiply(?, X)), multiply(?, X))))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.76 = { by lemma 7 } 0.56/0.76 multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.76 = { by lemma 7 } 0.56/0.76 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.56/0.76 = { by axiom 1 (single_axiom) } 0.56/0.76 inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.56/0.76 0.56/0.76 Lemma 19: multiply(inverse(multiply(inverse(multiply(X, multiply(?, inverse(Y)))), ?)), Y) = multiply(inverse(multiply(inverse(multiply(X, multiply(?, inverse(?)))), ?)), ?). 0.56/0.76 Proof: 0.56/0.76 multiply(inverse(multiply(inverse(multiply(X, multiply(?, inverse(Y)))), ?)), Y) 0.56/0.76 = { by lemma 17 } 0.56/0.76 multiply(inverse(multiply(?, ?)), multiply(?, multiply(X, inverse(multiply(inverse(Y), multiply(inverse(multiply(?, inverse(Y))), multiply(inverse(multiply(?, inverse(Y))), multiply(?, inverse(Y))))))))) 0.62/0.76 = { by lemma 18 } 0.62/0.76 multiply(inverse(multiply(?, ?)), multiply(?, multiply(X, inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))))) 0.62/0.76 = { by lemma 18 } 0.62/0.76 multiply(inverse(multiply(?, ?)), multiply(?, multiply(X, inverse(multiply(inverse(?), multiply(inverse(multiply(?, inverse(?))), multiply(inverse(multiply(?, inverse(?))), multiply(?, inverse(?))))))))) 0.62/0.76 = { by lemma 17 } 0.62/0.76 multiply(inverse(multiply(inverse(multiply(X, multiply(?, inverse(?)))), ?)), ?) 0.62/0.76 0.62/0.76 Lemma 20: multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))) = multiply(inverse(multiply(?, ?)), multiply(?, ?)). 0.62/0.76 Proof: 0.62/0.76 multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.76 = { by lemma 8 } 0.62/0.76 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))))), multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.76 = { by lemma 16 } 0.62/0.76 multiply(inverse(multiply(inverse(multiply(?, ?)), inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))), multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.76 = { by lemma 16 } 0.62/0.76 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)))), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)))))))), multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.76 = { by lemma 8 } 0.62/0.76 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)))), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)))) 0.62/0.76 = { by lemma 4 } 0.62/0.76 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)))) 0.62/0.76 = { by lemma 19 } 0.62/0.76 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(?)))), ?)), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)))) 0.62/0.76 = { by lemma 4 } 0.62/0.76 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(?)))), ?)), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.76 = { by lemma 19 } 0.62/0.76 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(?)))), ?)), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(?)))), ?)), ?)) 0.62/0.76 = { by lemma 11 } 0.62/0.76 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.76 0.62/0.76 Lemma 21: inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) = multiply(inverse(?), ?). 0.62/0.76 Proof: 0.62/0.76 inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.76 = { by lemma 20 } 0.62/0.76 inverse(multiply(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.76 = { by lemma 4 } 0.62/0.76 inverse(multiply(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))) 0.62/0.76 = { by lemma 10 } 0.62/0.76 inverse(multiply(inverse(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.76 = { by lemma 4 } 0.62/0.76 inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(?), ?)))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.76 = { by lemma 6 } 0.62/0.76 multiply(inverse(?), ?) 0.62/0.76 0.62/0.76 Lemma 22: multiply(inverse(multiply(X, multiply(inverse(?), ?))), multiply(X, multiply(inverse(?), ?))) = multiply(inverse(multiply(?, ?)), multiply(?, ?)). 0.62/0.76 Proof: 0.62/0.76 multiply(inverse(multiply(X, multiply(inverse(?), ?))), multiply(X, multiply(inverse(?), ?))) 0.62/0.76 = { by lemma 21 } 0.62/0.76 multiply(inverse(multiply(X, multiply(inverse(?), ?))), multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))) 0.62/0.76 = { by lemma 4 } 0.62/0.76 multiply(inverse(multiply(X, multiply(inverse(?), ?))), multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))))) 0.62/0.76 = { by lemma 3 } 0.62/0.77 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.77 0.62/0.77 Lemma 23: multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?))) = multiply(inverse(multiply(?, ?)), multiply(?, ?)). 0.62/0.77 Proof: 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.77 = { by lemma 4 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.77 = { by lemma 4 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))) 0.62/0.77 = { by lemma 22 } 0.62/0.77 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.77 0.62/0.77 Lemma 24: multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(X), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) = X. 0.62/0.77 Proof: 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(X), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(X), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(X), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(X), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))))) 0.62/0.77 = { by lemma 3 } 0.62/0.77 X 0.62/0.77 0.62/0.77 Lemma 25: multiply(X, multiply(inverse(multiply(Y, X)), multiply(inverse(multiply(Y, X)), multiply(Y, X)))) = multiply(?, multiply(inverse(multiply(Y, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))). 0.62/0.77 Proof: 0.62/0.77 multiply(X, multiply(inverse(multiply(Y, X)), multiply(inverse(multiply(Y, X)), multiply(Y, X)))) 0.62/0.77 = { by lemma 3 } 0.62/0.77 multiply(inverse(multiply(?, ?)), multiply(?, inverse(multiply(inverse(multiply(X, multiply(inverse(multiply(Y, X)), multiply(inverse(multiply(Y, X)), multiply(Y, X))))), multiply(inverse(?), multiply(inverse(?), ?)))))) 0.62/0.77 = { by lemma 14 } 0.62/0.77 multiply(inverse(multiply(?, ?)), multiply(?, inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(Y, ?)), multiply(inverse(multiply(Y, ?)), multiply(Y, ?))))), multiply(inverse(?), multiply(inverse(?), ?)))))) 0.62/0.77 = { by lemma 3 } 0.62/0.77 multiply(?, multiply(inverse(multiply(Y, ?)), multiply(inverse(multiply(Y, ?)), multiply(Y, ?)))) 0.62/0.77 = { by lemma 4 } 0.62/0.77 multiply(?, multiply(inverse(multiply(Y, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.77 0.62/0.77 Lemma 26: inverse(multiply(?, multiply(inverse(multiply(X, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) = multiply(X, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))). 0.62/0.77 Proof: 0.62/0.77 inverse(multiply(?, multiply(inverse(multiply(X, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.77 = { by lemma 25 } 0.62/0.77 inverse(multiply(?, multiply(inverse(multiply(X, ?)), multiply(inverse(multiply(X, ?)), multiply(X, ?))))) 0.62/0.77 = { by axiom 1 (single_axiom) } 0.62/0.77 multiply(X, inverse(multiply(inverse(multiply(inverse(multiply(X, ?)), inverse(multiply(?, multiply(inverse(multiply(X, ?)), multiply(inverse(multiply(X, ?)), multiply(X, ?))))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.77 = { by lemma 13 } 0.62/0.77 multiply(X, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.77 = { by lemma 20 } 0.62/0.77 multiply(X, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.77 0.62/0.77 Lemma 27: inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) = multiply(inverse(multiply(?, ?)), multiply(?, ?)). 0.62/0.77 Proof: 0.62/0.77 inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))) 0.62/0.77 = { by lemma 14 } 0.62/0.77 inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), ?))))) 0.62/0.77 = { by lemma 25 } 0.62/0.77 inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.77 = { by lemma 26 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))) 0.62/0.77 = { by axiom 1 (single_axiom) } 0.62/0.77 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.77 0.62/0.77 Lemma 28: multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))) = multiply(inverse(multiply(?, ?)), multiply(?, ?)). 0.62/0.77 Proof: 0.62/0.77 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.77 = { by lemma 24 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.77 = { by lemma 27 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.77 = { by lemma 27 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.77 = { by lemma 23 } 0.62/0.77 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.77 0.62/0.77 Lemma 29: inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))) = multiply(inverse(multiply(?, ?)), multiply(?, ?)). 0.62/0.77 Proof: 0.62/0.77 inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.77 = { by lemma 28 } 0.62/0.77 inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.77 = { by lemma 27 } 0.62/0.78 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.78 0.62/0.78 Lemma 30: inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) = multiply(inverse(multiply(?, ?)), multiply(?, ?)). 0.62/0.78 Proof: 0.62/0.78 inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.78 = { by lemma 2 } 0.62/0.78 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.78 = { by lemma 9 } 0.62/0.78 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.78 = { by lemma 23 } 0.62/0.78 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.78 = { by lemma 9 } 0.62/0.78 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.78 = { by lemma 2 } 0.62/0.78 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.78 0.62/0.78 Lemma 31: multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))) = multiply(inverse(?), multiply(inverse(?), ?)). 0.62/0.78 Proof: 0.62/0.78 multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 = { by lemma 4 } 0.62/0.78 multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 = { by lemma 29 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 = { by lemma 28 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 = { by lemma 29 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 = { by lemma 28 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 = { by lemma 29 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 = { by lemma 27 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 = { by lemma 30 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 = { by lemma 24 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 = { by lemma 30 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 = { by lemma 29 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.78 = { by lemma 26 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.78 = { by lemma 25 } 0.62/0.78 multiply(inverse(multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(multiply(?, ?), multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)))))) 0.62/0.78 = { by lemma 12 } 0.62/0.78 multiply(inverse(?), multiply(inverse(?), ?)) 0.62/0.78 0.62/0.78 Lemma 32: multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))) = inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))). 0.62/0.78 Proof: 0.62/0.78 multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.78 = { by lemma 7 } 0.62/0.78 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.78 = { by axiom 1 (single_axiom) } 0.62/0.78 inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.78 0.62/0.78 Lemma 33: inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) = multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))). 0.62/0.78 Proof: 0.62/0.78 inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.78 = { by lemma 32 } 0.62/0.78 multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.78 = { by lemma 20 } 0.62/0.78 multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.78 0.62/0.78 Lemma 34: multiply(inverse(multiply(?, X)), inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) = multiply(inverse(X), multiply(inverse(X), X)). 0.62/0.78 Proof: 0.62/0.78 multiply(inverse(multiply(?, X)), inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.78 = { by lemma 16 } 0.62/0.78 multiply(inverse(multiply(?, X)), multiply(?, inverse(multiply(inverse(multiply(inverse(X), multiply(inverse(X), X))), multiply(inverse(X), multiply(inverse(X), X)))))) 0.62/0.78 = { by lemma 3 } 0.62/0.79 multiply(inverse(X), multiply(inverse(X), X)) 0.62/0.79 0.62/0.79 Lemma 35: multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))) = multiply(inverse(multiply(?, ?)), multiply(?, ?)). 0.62/0.79 Proof: 0.62/0.79 multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.79 = { by lemma 29 } 0.62/0.79 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.79 = { by lemma 33 } 0.62/0.79 multiply(inverse(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.79 = { by lemma 32 } 0.62/0.79 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.79 = { by lemma 29 } 0.62/0.79 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.79 = { by lemma 33 } 0.62/0.79 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))), inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.79 = { by lemma 18 } 0.62/0.79 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))), inverse(multiply(inverse(?), multiply(inverse(multiply(?, inverse(?))), multiply(inverse(multiply(?, inverse(?))), multiply(?, inverse(?))))))) 0.62/0.79 = { by axiom 1 (single_axiom) } 0.62/0.79 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(?, inverse(?)))), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(?, inverse(?)))), ?)), ?)), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(inverse(multiply(?, inverse(?))), multiply(inverse(multiply(?, inverse(?))), multiply(?, inverse(?))))))) 0.62/0.79 = { by axiom 1 (single_axiom) } 0.62/0.79 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(?, inverse(?)))), ?)), ?)), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.79 = { by lemma 17 } 0.62/0.79 inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))), inverse(multiply(inverse(?), multiply(inverse(multiply(?, inverse(?))), multiply(inverse(multiply(?, inverse(?))), multiply(?, inverse(?)))))))))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.79 = { by lemma 18 } 0.62/0.79 inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))), inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.79 = { by lemma 34 } 0.62/0.79 inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), multiply(inverse(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))))))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.79 = { by lemma 6 } 0.62/0.79 multiply(inverse(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), multiply(inverse(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))) 0.62/0.79 = { by lemma 20 } 0.62/0.79 multiply(inverse(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))) 0.62/0.79 = { by lemma 29 } 0.62/0.79 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))) 0.62/0.79 = { by lemma 29 } 0.62/0.79 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))) 0.62/0.79 = { by lemma 15 } 0.62/0.79 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.79 = { by lemma 26 } 0.62/0.79 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.79 = { by lemma 20 } 0.62/0.79 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))), inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.79 = { by lemma 29 } 0.62/0.79 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.79 = { by lemma 29 } 0.62/0.79 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.79 = { by lemma 23 } 0.62/0.79 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.79 = { by lemma 28 } 0.62/0.80 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.80 0.62/0.80 Lemma 36: inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?)))) = multiply(inverse(multiply(?, ?)), multiply(?, ?)). 0.62/0.80 Proof: 0.62/0.80 inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.80 = { by lemma 12 } 0.62/0.80 inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)))))))) 0.62/0.80 = { by lemma 35 } 0.62/0.80 inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)))))))) 0.62/0.80 = { by lemma 29 } 0.62/0.80 inverse(multiply(?, multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)))))))) 0.62/0.80 = { by lemma 25 } 0.62/0.80 inverse(multiply(?, multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))))) 0.62/0.80 = { by lemma 26 } 0.62/0.80 inverse(multiply(?, multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))), inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))))) 0.62/0.80 = { by lemma 35 } 0.62/0.80 inverse(multiply(?, multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))))) 0.62/0.80 = { by lemma 29 } 0.62/0.80 inverse(multiply(?, multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.80 = { by lemma 23 } 0.62/0.80 inverse(multiply(?, multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.80 = { by lemma 26 } 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.80 = { by lemma 29 } 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.80 = { by lemma 28 } 0.62/0.80 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.80 0.62/0.80 Lemma 37: inverse(multiply(inverse(?), multiply(inverse(?), ?))) = ?. 0.62/0.80 Proof: 0.62/0.80 inverse(multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.80 = { by lemma 31 } 0.62/0.80 inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.80 = { by lemma 36 } 0.62/0.80 inverse(multiply(inverse(multiply(?, ?)), multiply(?, inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?))))))) 0.62/0.80 = { by lemma 6 } 0.62/0.80 inverse(multiply(inverse(multiply(?, ?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(?), multiply(inverse(?), ?))))))) 0.62/0.80 = { by lemma 3 } 0.62/0.80 inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.80 = { by lemma 6 } 0.62/0.80 ? 0.62/0.80 0.62/0.80 Lemma 38: multiply(inverse(multiply(inverse(?), ?)), ?) = multiply(?, multiply(inverse(?), ?)). 0.62/0.80 Proof: 0.62/0.80 multiply(inverse(multiply(inverse(?), ?)), ?) 0.62/0.80 = { by lemma 5 } 0.62/0.80 multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))))) 0.62/0.80 = { by lemma 37 } 0.62/0.80 multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))))) 0.62/0.80 = { by lemma 4 } 0.62/0.80 multiply(?, multiply(inverse(?), inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))))) 0.62/0.80 = { by lemma 31 } 0.62/0.80 multiply(?, multiply(inverse(?), inverse(multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.80 = { by lemma 37 } 0.62/0.80 multiply(?, multiply(inverse(?), ?)) 0.62/0.80 0.62/0.80 Lemma 39: multiply(?, multiply(inverse(?), multiply(inverse(?), ?))) = multiply(inverse(multiply(?, ?)), multiply(?, ?)). 0.62/0.80 Proof: 0.62/0.80 multiply(?, multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.80 = { by lemma 24 } 0.62/0.80 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.80 = { by lemma 29 } 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.80 = { by lemma 29 } 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), inverse(multiply(inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.80 = { by lemma 36 } 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.80 = { by lemma 27 } 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.80 = { by lemma 28 } 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.80 = { by lemma 28 } 0.62/0.80 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.80 0.62/0.80 Lemma 40: multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(?, ?)) = multiply(?, ?). 0.62/0.80 Proof: 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(?, ?)) 0.62/0.80 = { by lemma 29 } 0.62/0.80 multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(?, ?)) 0.62/0.80 = { by lemma 39 } 0.62/0.80 multiply(inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?)))), multiply(?, ?)) 0.62/0.80 = { by lemma 37 } 0.62/0.80 multiply(inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?)))), multiply(?, inverse(multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.80 = { by lemma 31 } 0.62/0.80 multiply(inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?)))), multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))))) 0.62/0.80 = { by lemma 20 } 0.62/0.80 multiply(inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?)))), multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))))) 0.62/0.80 = { by lemma 5 } 0.62/0.80 multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), ?) 0.62/0.80 = { by lemma 37 } 0.62/0.80 multiply(?, ?) 0.62/0.80 0.62/0.80 Lemma 41: multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?) = ?. 0.62/0.80 Proof: 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?) 0.62/0.80 = { by axiom 1 (single_axiom) } 0.62/0.80 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.80 = { by lemma 40 } 0.62/0.80 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(?, ?))), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.80 = { by lemma 37 } 0.62/0.80 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(?, ?))), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), inverse(multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.80 = { by lemma 31 } 0.62/0.80 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(?, ?))), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.80 = { by lemma 5 } 0.62/0.80 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.80 = { by axiom 1 (single_axiom) } 0.62/0.80 ? 0.62/0.80 0.62/0.80 Lemma 42: multiply(inverse(multiply(?, ?)), multiply(?, ?)) = multiply(inverse(?), ?). 0.62/0.80 Proof: 0.62/0.80 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.80 = { by lemma 4 } 0.62/0.80 multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?)) 0.62/0.80 = { by lemma 41 } 0.62/0.80 multiply(inverse(?), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?)) 0.62/0.80 = { by lemma 41 } 0.62/0.80 multiply(inverse(?), ?) 0.62/0.80 0.62/0.80 Lemma 43: multiply(multiply(inverse(?), ?), multiply(inverse(?), ?)) = multiply(inverse(?), ?). 0.62/0.80 Proof: 0.62/0.80 multiply(multiply(inverse(?), ?), multiply(inverse(?), ?)) 0.62/0.80 = { by lemma 42 } 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(?), ?)) 0.62/0.80 = { by lemma 42 } 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.80 = { by lemma 28 } 0.62/0.80 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.80 = { by lemma 42 } 0.62/0.80 multiply(inverse(?), ?) 0.62/0.80 0.62/0.80 Lemma 44: inverse(multiply(inverse(?), ?)) = multiply(inverse(?), ?). 0.62/0.80 Proof: 0.62/0.80 inverse(multiply(inverse(?), ?)) 0.62/0.80 = { by lemma 42 } 0.62/0.80 inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.80 = { by lemma 29 } 0.62/0.80 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.80 = { by lemma 42 } 0.62/0.80 multiply(inverse(?), ?) 0.62/0.80 0.62/0.80 Lemma 45: multiply(multiply(inverse(?), ?), ?) = ?. 0.62/0.80 Proof: 0.62/0.80 multiply(multiply(inverse(?), ?), ?) 0.62/0.80 = { by lemma 42 } 0.62/0.80 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?) 0.62/0.80 = { by lemma 41 } 0.62/0.80 ? 0.62/0.80 0.62/0.80 Lemma 46: multiply(?, multiply(inverse(?), ?)) = ?. 0.62/0.80 Proof: 0.62/0.80 multiply(?, multiply(inverse(?), ?)) 0.62/0.80 = { by lemma 38 } 0.62/0.80 multiply(inverse(multiply(inverse(?), ?)), ?) 0.62/0.80 = { by lemma 5 } 0.62/0.80 multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), ?))), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))))) 0.62/0.80 = { by lemma 43 } 0.62/0.80 multiply(inverse(multiply(inverse(?), ?)), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))))) 0.62/0.80 = { by lemma 44 } 0.62/0.80 multiply(multiply(inverse(?), ?), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))))) 0.62/0.80 = { by lemma 4 } 0.62/0.80 multiply(multiply(inverse(?), ?), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))))) 0.62/0.80 = { by lemma 31 } 0.62/0.80 multiply(multiply(inverse(?), ?), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.80 = { by lemma 37 } 0.62/0.80 multiply(multiply(inverse(?), ?), multiply(multiply(inverse(?), ?), ?)) 0.62/0.80 = { by lemma 45 } 0.62/0.80 multiply(multiply(inverse(?), ?), ?) 0.62/0.80 = { by lemma 45 } 0.62/0.80 ? 0.62/0.80 0.62/0.80 Lemma 47: multiply(X, multiply(inverse(multiply(?, X)), multiply(inverse(multiply(?, X)), multiply(?, X)))) = multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))). 0.62/0.80 Proof: 0.62/0.80 multiply(X, multiply(inverse(multiply(?, X)), multiply(inverse(multiply(?, X)), multiply(?, X)))) 0.62/0.80 = { by lemma 3 } 0.62/0.80 multiply(inverse(multiply(Y, Z)), multiply(Y, inverse(multiply(inverse(multiply(X, multiply(inverse(multiply(?, X)), multiply(inverse(multiply(?, X)), multiply(?, X))))), multiply(inverse(Z), multiply(inverse(Z), Z)))))) 0.62/0.80 = { by lemma 18 } 0.62/0.80 multiply(inverse(multiply(Y, Z)), multiply(Y, inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), multiply(inverse(Z), multiply(inverse(Z), Z)))))) 0.62/0.80 = { by lemma 3 } 0.62/0.81 multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.81 0.62/0.81 Lemma 48: multiply(multiply(inverse(?), ?), multiply(inverse(?), multiply(inverse(?), ?))) = multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(?), ?))). 0.62/0.81 Proof: 0.62/0.81 multiply(multiply(inverse(?), ?), multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.81 = { by lemma 46 } 0.62/0.81 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(inverse(?), ?))) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(?), ?))) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.81 = { by lemma 35 } 0.62/0.81 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.81 = { by lemma 47 } 0.62/0.81 multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(?), ?))) 0.62/0.81 0.62/0.81 Lemma 49: multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(?), ?)))), multiply(inverse(?), ?)) = ?. 0.62/0.81 Proof: 0.62/0.81 multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(?), ?)))), multiply(inverse(?), ?)) 0.62/0.81 = { by lemma 48 } 0.62/0.81 multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(?), ?)) 0.62/0.81 = { by lemma 43 } 0.62/0.81 multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(multiply(inverse(?), ?), multiply(inverse(?), ?))) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(multiply(inverse(?), ?), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.81 = { by lemma 29 } 0.62/0.81 multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.81 = { by lemma 26 } 0.62/0.81 multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), multiply(inverse(?), ?)))), inverse(multiply(?, multiply(inverse(multiply(multiply(inverse(?), ?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.81 = { by lemma 16 } 0.62/0.81 multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))))) 0.62/0.81 = { by lemma 3 } 0.62/0.81 multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 3 } 0.62/0.81 multiply(inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?)))), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))))) 0.62/0.81 = { by lemma 36 } 0.62/0.81 multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))))) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(multiply(inverse(?), ?), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?)))))))) 0.62/0.81 = { by lemma 16 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.81 = { by lemma 33 } 0.62/0.81 multiply(multiply(inverse(?), ?), multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.81 = { by lemma 29 } 0.62/0.81 multiply(multiply(inverse(?), ?), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(multiply(inverse(?), ?), multiply(?, multiply(inverse(?), ?))) 0.62/0.81 = { by lemma 46 } 0.62/0.81 multiply(multiply(inverse(?), ?), ?) 0.62/0.81 = { by lemma 45 } 0.62/0.81 ? 0.62/0.81 0.62/0.81 Lemma 50: multiply(multiply(inverse(?), ?), multiply(inverse(?), X)) = multiply(inverse(multiply(?, ?)), multiply(?, X)). 0.62/0.81 Proof: 0.62/0.81 multiply(multiply(inverse(?), ?), multiply(inverse(?), X)) 0.62/0.81 = { by lemma 44 } 0.62/0.81 multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), X)) 0.62/0.81 = { by lemma 49 } 0.62/0.81 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(?), ?)))), multiply(inverse(?), ?))), ?)), multiply(inverse(?), X)) 0.62/0.81 = { by lemma 49 } 0.62/0.81 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(?), ?)))), multiply(inverse(?), ?))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(?), ?)))), multiply(inverse(?), ?))), X)) 0.62/0.81 = { by lemma 11 } 0.62/0.81 multiply(inverse(multiply(?, ?)), multiply(?, X)) 0.62/0.81 0.62/0.81 Lemma 51: multiply(Y, inverse(multiply(inverse(multiply(inverse(multiply(Y, multiply(Z, ?))), X)), multiply(inverse(multiply(Z, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) = X. 0.62/0.81 Proof: 0.62/0.81 multiply(Y, inverse(multiply(inverse(multiply(inverse(multiply(Y, multiply(Z, ?))), X)), multiply(inverse(multiply(Z, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.81 = { by lemma 4 } 0.62/0.81 multiply(Y, inverse(multiply(inverse(multiply(inverse(multiply(Y, multiply(Z, ?))), X)), multiply(inverse(multiply(Z, ?)), multiply(inverse(multiply(Z, ?)), multiply(Z, ?)))))) 0.62/0.81 = { by axiom 1 (single_axiom) } 0.62/0.81 X 0.62/0.81 0.62/0.81 Lemma 52: multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(multiply(inverse(?), ?), X)), multiply(inverse(?), ?)))) = X. 0.62/0.81 Proof: 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(multiply(inverse(?), ?), X)), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), X)), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 29 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), X)), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 50 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), ?))), X)), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 43 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), ?))), X)), multiply(multiply(inverse(?), ?), multiply(inverse(?), ?))))) 0.62/0.81 = { by lemma 44 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), ?))), X)), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), ?))), X)), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.81 = { by lemma 51 } 0.62/0.81 X 0.62/0.81 0.62/0.81 Lemma 53: multiply(multiply(inverse(?), ?), inverse(?)) = multiply(inverse(?), multiply(inverse(?), ?)). 0.62/0.81 Proof: 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(?)) 0.62/0.81 = { by lemma 46 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 46 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(multiply(?, multiply(inverse(?), ?)), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 29 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 33 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 18 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(?, multiply(inverse(?), ?)))))), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 52 } 0.62/0.81 multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(?, multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 46 } 0.62/0.81 multiply(inverse(?), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(?, multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 22 } 0.62/0.81 multiply(inverse(?), multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(inverse(?), multiply(inverse(?), ?)) 0.62/0.81 0.62/0.81 Lemma 54: multiply(inverse(?), multiply(inverse(?), ?)) = inverse(?). 0.62/0.81 Proof: 0.62/0.81 multiply(inverse(?), multiply(inverse(?), ?)) 0.62/0.81 = { by lemma 53 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(?)) 0.62/0.81 = { by lemma 46 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 37 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 53 } 0.62/0.81 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(?))), multiply(inverse(?), ?)))) 0.62/0.81 = { by lemma 52 } 0.62/0.81 inverse(?) 0.62/0.81 0.62/0.81 Lemma 55: inverse(inverse(?)) = ?. 0.62/0.81 Proof: 0.62/0.81 inverse(inverse(?)) 0.62/0.81 = { by lemma 54 } 0.62/0.81 inverse(multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.81 = { by lemma 37 } 0.62/0.81 ? 0.62/0.81 0.62/0.81 Lemma 56: multiply(inverse(?), ?) = multiply(?, inverse(?)). 0.62/0.81 Proof: 0.62/0.81 multiply(inverse(?), ?) 0.62/0.81 = { by lemma 42 } 0.62/0.81 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.81 = { by lemma 39 } 0.62/0.81 multiply(?, multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.81 = { by lemma 54 } 0.62/0.82 multiply(?, inverse(?)) 0.62/0.82 0.62/0.82 Lemma 57: multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?))))) = inverse(?). 0.62/0.82 Proof: 0.62/0.82 multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?))))) 0.62/0.82 = { by lemma 56 } 0.62/0.82 multiply(multiply(inverse(?), ?), inverse(multiply(?, multiply(?, inverse(?))))) 0.62/0.82 = { by lemma 46 } 0.62/0.82 multiply(multiply(inverse(?), ?), inverse(multiply(multiply(?, multiply(inverse(?), ?)), multiply(?, inverse(?))))) 0.62/0.82 = { by lemma 42 } 0.62/0.82 multiply(multiply(inverse(?), ?), inverse(multiply(multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(?, inverse(?))))) 0.62/0.82 = { by lemma 29 } 0.62/0.82 multiply(multiply(inverse(?), ?), inverse(multiply(multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?)))), multiply(?, inverse(?))))) 0.62/0.82 = { by lemma 33 } 0.62/0.82 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), multiply(?, inverse(?))))) 0.62/0.82 = { by lemma 47 } 0.62/0.82 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(?, multiply(inverse(?), ?)))))), multiply(?, inverse(?))))) 0.62/0.82 = { by lemma 56 } 0.62/0.82 multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(?, multiply(inverse(?), ?)))))), multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 52 } 0.62/0.82 multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(?, multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 46 } 0.62/0.82 multiply(inverse(?), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(?, multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 22 } 0.62/0.82 multiply(inverse(?), multiply(inverse(multiply(?, ?)), multiply(?, ?))) 0.62/0.82 = { by lemma 42 } 0.62/0.82 multiply(inverse(?), multiply(inverse(?), ?)) 0.62/0.82 = { by lemma 54 } 0.62/0.82 inverse(?) 0.62/0.82 0.62/0.82 Lemma 58: multiply(inverse(multiply(?, ?)), ?) = inverse(?). 0.62/0.82 Proof: 0.62/0.82 multiply(inverse(multiply(?, ?)), ?) 0.62/0.82 = { by lemma 55 } 0.62/0.82 multiply(inverse(multiply(inverse(inverse(?)), ?)), ?) 0.62/0.82 = { by lemma 57 } 0.62/0.82 multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)), ?) 0.62/0.82 = { by lemma 46 } 0.62/0.82 multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)), multiply(?, multiply(inverse(?), ?))) 0.62/0.82 = { by lemma 55 } 0.62/0.82 multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)), multiply(inverse(inverse(?)), multiply(inverse(?), ?))) 0.62/0.82 = { by lemma 57 } 0.62/0.82 multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)), multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), multiply(inverse(?), ?))) 0.62/0.82 = { by lemma 42 } 0.62/0.82 multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)), multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), multiply(inverse(multiply(?, ?)), multiply(?, ?)))) 0.62/0.82 = { by lemma 29 } 0.62/0.82 multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)), multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.82 = { by lemma 26 } 0.62/0.82 multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.82 = { by lemma 25 } 0.62/0.82 multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)), multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)), multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(?, multiply(?, inverse(?)))))), ?)))))) 0.62/0.82 = { by lemma 12 } 0.62/0.82 multiply(inverse(?), multiply(inverse(?), ?)) 0.62/0.82 = { by lemma 54 } 0.62/0.82 inverse(?) 0.62/0.82 0.62/0.82 Lemma 59: multiply(inverse(multiply(?, ?)), multiply(?, X)) = multiply(?, multiply(inverse(multiply(?, ?)), X)). 0.62/0.82 Proof: 0.62/0.82 multiply(inverse(multiply(?, ?)), multiply(?, X)) 0.62/0.82 = { by lemma 11 } 0.62/0.82 multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X)) 0.62/0.82 = { by lemma 58 } 0.62/0.82 multiply(inverse(inverse(?)), multiply(inverse(multiply(?, ?)), X)) 0.62/0.82 = { by lemma 55 } 0.62/0.82 multiply(?, multiply(inverse(multiply(?, ?)), X)) 0.62/0.82 0.62/0.82 Lemma 60: multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), X)), inverse(?)))) = X. 0.62/0.82 Proof: 0.62/0.82 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), X)), inverse(?)))) 0.62/0.82 = { by axiom 1 (single_axiom) } 0.62/0.82 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))))), inverse(?)))) 0.62/0.82 = { by lemma 54 } 0.62/0.82 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))))), multiply(inverse(?), multiply(inverse(?), ?))))) 0.62/0.82 = { by axiom 1 (single_axiom) } 0.62/0.82 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 58 } 0.62/0.82 inverse(multiply(inverse(multiply(inverse(inverse(?)), multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 55 } 0.62/0.82 inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), X))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 59 } 0.62/0.82 inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, X))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 50 } 0.62/0.82 inverse(multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), X))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 44 } 0.62/0.82 inverse(multiply(inverse(multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), X))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 49 } 0.62/0.82 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(?), ?)))), multiply(inverse(?), ?))), ?)), multiply(inverse(?), X))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 49 } 0.62/0.82 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(?), ?)))), multiply(inverse(?), ?))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(?), ?)))), multiply(inverse(?), ?))), X))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 2 } 0.62/0.82 X 0.62/0.82 0.62/0.82 Lemma 61: inverse(multiply(?, inverse(?))) = multiply(?, inverse(?)). 0.62/0.82 Proof: 0.62/0.82 inverse(multiply(?, inverse(?))) 0.62/0.82 = { by lemma 54 } 0.62/0.82 inverse(multiply(?, multiply(inverse(?), multiply(inverse(?), ?)))) 0.62/0.82 = { by lemma 36 } 0.62/0.82 multiply(inverse(multiply(?, ?)), multiply(?, ?)) 0.62/0.82 = { by lemma 42 } 0.62/0.82 multiply(inverse(?), ?) 0.62/0.82 = { by lemma 56 } 0.62/0.82 multiply(?, inverse(?)) 0.62/0.82 0.62/0.82 Lemma 62: multiply(inverse(X), multiply(inverse(X), X)) = multiply(inverse(multiply(?, X)), ?). 0.62/0.82 Proof: 0.62/0.82 multiply(inverse(X), multiply(inverse(X), X)) 0.62/0.82 = { by lemma 34 } 0.62/0.82 multiply(inverse(multiply(?, X)), inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))) 0.62/0.82 = { by lemma 33 } 0.62/0.82 multiply(inverse(multiply(?, X)), multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))))) 0.62/0.82 = { by lemma 59 } 0.62/0.82 multiply(inverse(multiply(?, X)), multiply(?, inverse(multiply(?, multiply(inverse(multiply(?, ?)), ?))))) 0.62/0.82 = { by lemma 55 } 0.62/0.82 multiply(inverse(multiply(?, X)), multiply(?, inverse(multiply(inverse(inverse(?)), multiply(inverse(multiply(?, ?)), ?))))) 0.62/0.82 = { by lemma 58 } 0.62/0.82 multiply(inverse(multiply(?, X)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), ?))))) 0.62/0.82 = { by lemma 58 } 0.62/0.82 multiply(inverse(multiply(?, X)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), inverse(?))))) 0.62/0.82 = { by lemma 54 } 0.62/0.82 multiply(inverse(multiply(?, X)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(?), multiply(inverse(?), ?)))))) 0.62/0.82 = { by axiom 1 (single_axiom) } 0.62/0.82 multiply(inverse(multiply(?, X)), ?) 0.62/0.82 0.62/0.82 Lemma 63: multiply(?, inverse(multiply(inverse(multiply(inverse(Y), X)), multiply(inverse(Y), ?)))) = X. 0.62/0.82 Proof: 0.62/0.82 multiply(?, inverse(multiply(inverse(multiply(inverse(Y), X)), multiply(inverse(Y), ?)))) 0.62/0.82 = { by lemma 60 } 0.62/0.82 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?))))), X)), multiply(inverse(Y), ?)))) 0.62/0.82 = { by lemma 60 } 0.62/0.82 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?))))), X)), multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?))))), ?)))) 0.62/0.82 = { by lemma 62 } 0.62/0.82 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?))))), X)), multiply(inverse(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?)))), multiply(inverse(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?)))), inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?)))))))) 0.62/0.82 = { by axiom 1 (single_axiom) } 0.62/0.83 X 0.62/0.83 0.62/0.83 Lemma 64: multiply(?, inverse(multiply(inverse(multiply(Y, X)), multiply(Y, ?)))) = X. 0.62/0.83 Proof: 0.62/0.83 multiply(?, inverse(multiply(inverse(multiply(Y, X)), multiply(Y, ?)))) 0.62/0.83 = { by lemma 2 } 0.62/0.83 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), Y))), multiply(inverse(?), multiply(inverse(?), ?)))), X)), multiply(Y, ?)))) 0.62/0.83 = { by lemma 2 } 0.62/0.83 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), Y))), multiply(inverse(?), multiply(inverse(?), ?)))), X)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), Y))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)))) 0.62/0.83 = { by lemma 63 } 0.62/0.83 X 0.62/0.83 0.62/0.83 Lemma 65: multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), multiply(inverse(Y), multiply(inverse(Y), Y))) = multiply(inverse(multiply(inverse(multiply(X, ?)), Z)), multiply(inverse(?), multiply(inverse(?), ?))). 0.62/0.83 Proof: 0.62/0.83 multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), multiply(inverse(Y), multiply(inverse(Y), Y))) 0.62/0.83 = { by lemma 3 } 0.62/0.83 multiply(inverse(multiply(W, V)), multiply(W, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, Y)), Z)), multiply(inverse(Y), multiply(inverse(Y), Y)))), multiply(inverse(V), multiply(inverse(V), V)))))) 0.62/0.83 = { by lemma 10 } 0.62/0.83 multiply(inverse(multiply(W, V)), multiply(W, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(X, ?)), Z)), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(V), multiply(inverse(V), V)))))) 0.62/0.83 = { by lemma 3 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(X, ?)), Z)), multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.83 0.62/0.83 Lemma 66: multiply(inverse(multiply(inverse(Y), X)), multiply(inverse(Y), ?)) = multiply(inverse(multiply(inverse(?), X)), multiply(inverse(?), ?)). 0.62/0.83 Proof: 0.62/0.83 multiply(inverse(multiply(inverse(Y), X)), multiply(inverse(Y), ?)) 0.62/0.83 = { by lemma 60 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?))))), X)), multiply(inverse(Y), ?)) 0.62/0.83 = { by lemma 60 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?))))), X)), multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?))))), ?)) 0.62/0.83 = { by lemma 62 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?))))), X)), multiply(inverse(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?)))), multiply(inverse(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?)))), inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), Y)), inverse(?)))))) 0.62/0.83 = { by lemma 65 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(?, ?)), X)), multiply(inverse(?), multiply(inverse(?), ?))) 0.62/0.83 = { by lemma 65 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), inverse(?))))), X)), multiply(inverse(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), inverse(?)))), multiply(inverse(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), inverse(?)))), inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), inverse(?)))))) 0.62/0.83 = { by lemma 60 } 0.62/0.83 multiply(inverse(multiply(inverse(?), X)), multiply(inverse(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), inverse(?)))), multiply(inverse(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), inverse(?)))), inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), inverse(?)))))) 0.62/0.83 = { by lemma 62 } 0.62/0.83 multiply(inverse(multiply(inverse(?), X)), multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), inverse(?))))), ?)) 0.62/0.83 = { by lemma 60 } 0.62/0.83 multiply(inverse(multiply(inverse(?), X)), multiply(inverse(?), ?)) 0.62/0.83 0.62/0.83 Lemma 67: multiply(inverse(multiply(Y, X)), multiply(Y, ?)) = multiply(inverse(multiply(?, X)), multiply(?, ?)). 0.62/0.83 Proof: 0.62/0.83 multiply(inverse(multiply(Y, X)), multiply(Y, ?)) 0.62/0.83 = { by lemma 2 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(Z, W)), ?)), multiply(inverse(multiply(Z, W)), Y))), multiply(inverse(?), multiply(inverse(?), ?)))), X)), multiply(Y, ?)) 0.62/0.83 = { by lemma 2 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(Z, W)), ?)), multiply(inverse(multiply(Z, W)), Y))), multiply(inverse(?), multiply(inverse(?), ?)))), X)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(Z, W)), ?)), multiply(inverse(multiply(Z, W)), Y))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)) 0.62/0.83 = { by lemma 66 } 0.62/0.83 multiply(inverse(multiply(inverse(?), X)), multiply(inverse(?), ?)) 0.62/0.83 = { by lemma 66 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(Z, W)), ?)), multiply(inverse(multiply(Z, W)), ?))), multiply(inverse(?), multiply(inverse(?), ?)))), X)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(Z, W)), ?)), multiply(inverse(multiply(Z, W)), ?))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)) 0.62/0.83 = { by lemma 2 } 0.62/0.83 multiply(inverse(multiply(?, X)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(Z, W)), ?)), multiply(inverse(multiply(Z, W)), ?))), multiply(inverse(?), multiply(inverse(?), ?)))), ?)) 0.62/0.83 = { by lemma 2 } 0.62/0.83 multiply(inverse(multiply(?, X)), multiply(?, ?)) 0.62/0.83 0.62/0.83 Lemma 68: inverse(multiply(inverse(multiply(?, X)), multiply(?, ?))) = multiply(?, inverse(multiply(inverse(X), multiply(?, ?)))). 0.62/0.83 Proof: 0.62/0.83 inverse(multiply(inverse(multiply(?, X)), multiply(?, ?))) 0.62/0.83 = { by lemma 64 } 0.62/0.83 multiply(?, inverse(multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(?, X)), multiply(?, ?))))), multiply(?, ?)))) 0.62/0.83 = { by lemma 64 } 0.62/0.83 multiply(?, inverse(multiply(inverse(X), multiply(?, ?)))) 0.62/0.83 0.62/0.83 Lemma 69: multiply(inverse(multiply(multiply(?, inverse(?)), X)), ?) = multiply(inverse(multiply(?, X)), multiply(?, ?)). 0.62/0.83 Proof: 0.62/0.83 multiply(inverse(multiply(multiply(?, inverse(?)), X)), ?) 0.62/0.83 = { by lemma 56 } 0.62/0.83 multiply(inverse(multiply(multiply(inverse(?), ?), X)), ?) 0.62/0.83 = { by lemma 21 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), X)), ?) 0.62/0.83 = { by lemma 45 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), X)), multiply(multiply(inverse(?), ?), ?)) 0.62/0.83 = { by lemma 21 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), X)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)) 0.62/0.83 = { by lemma 66 } 0.62/0.83 multiply(inverse(multiply(inverse(?), X)), multiply(inverse(?), ?)) 0.62/0.83 = { by lemma 67 } 0.62/0.83 multiply(inverse(multiply(?, X)), multiply(?, ?)) 0.62/0.83 0.62/0.83 Lemma 70: multiply(inverse(X), X) = multiply(?, inverse(?)). 0.62/0.83 Proof: 0.62/0.83 multiply(inverse(X), X) 0.62/0.83 = { by lemma 64 } 0.62/0.83 multiply(multiply(?, inverse(multiply(inverse(multiply(?, inverse(X))), multiply(?, ?)))), X) 0.62/0.83 = { by lemma 68 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(X)))), multiply(?, ?))), X) 0.62/0.83 = { by lemma 69 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(?, inverse(X)))), ?)), X) 0.62/0.83 = { by lemma 19 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(?, inverse(?)))), ?)), ?) 0.62/0.83 = { by lemma 69 } 0.62/0.83 multiply(inverse(multiply(inverse(multiply(?, multiply(?, inverse(?)))), multiply(?, ?))), ?) 0.62/0.83 = { by lemma 68 } 0.62/0.83 multiply(multiply(?, inverse(multiply(inverse(multiply(?, inverse(?))), multiply(?, ?)))), ?) 0.62/0.83 = { by lemma 64 } 0.62/0.83 multiply(inverse(?), ?) 0.62/0.83 = { by lemma 56 } 0.62/0.83 multiply(?, inverse(?)) 0.62/0.83 0.62/0.83 Lemma 71: multiply(?, inverse(multiply(inverse(multiply(multiply(?, inverse(?)), X)), ?))) = X. 0.62/0.83 Proof: 0.62/0.83 multiply(?, inverse(multiply(inverse(multiply(multiply(?, inverse(?)), X)), ?))) 0.62/0.83 = { by lemma 56 } 0.62/0.83 multiply(?, inverse(multiply(inverse(multiply(multiply(inverse(?), ?), X)), ?))) 0.62/0.83 = { by lemma 21 } 0.62/0.83 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), X)), ?))) 0.62/0.83 = { by lemma 45 } 0.62/0.83 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), X)), multiply(multiply(inverse(?), ?), ?)))) 0.62/0.83 = { by lemma 21 } 0.62/0.83 multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), X)), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?)))) 0.62/0.83 = { by lemma 63 } 0.62/0.84 X 0.62/0.84 0.62/0.84 Lemma 72: inverse(multiply(inverse(multiply(?, X)), ?)) = multiply(?, inverse(multiply(inverse(X), ?))). 0.62/0.84 Proof: 0.62/0.84 inverse(multiply(inverse(multiply(?, X)), ?)) 0.62/0.84 = { by lemma 55 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(inverse(?)), X)), ?)) 0.62/0.84 = { by lemma 54 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), X)), ?)) 0.62/0.84 = { by lemma 52 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(inverse(?), ?), inverse(multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(?), ?))))), X)), ?)) 0.62/0.84 = { by lemma 56 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(inverse(multiply(multiply(inverse(?), ?), multiply(inverse(?), multiply(inverse(?), ?)))), multiply(inverse(?), ?))))), X)), ?)) 0.62/0.84 = { by lemma 48 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, ?)), multiply(inverse(?), ?)))), multiply(inverse(?), ?))))), X)), ?)) 0.62/0.84 = { by lemma 49 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(?))), X)), ?)) 0.62/0.84 = { by lemma 58 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), ?)) 0.62/0.84 = { by lemma 46 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(?, multiply(inverse(?), ?)))) 0.62/0.84 = { by lemma 38 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(multiply(inverse(?), ?)), ?))) 0.62/0.84 = { by lemma 5 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(multiply(?, multiply(inverse(?), ?))), multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))))))) 0.62/0.84 = { by lemma 46 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(?), multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))))))) 0.62/0.84 = { by lemma 45 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(multiply(multiply(inverse(?), ?), ?)), multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(?), ?))))))))) 0.62/0.84 = { by lemma 4 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(multiply(multiply(inverse(?), ?), ?)), multiply(?, inverse(multiply(inverse(multiply(?, ?)), multiply(?, multiply(inverse(multiply(?, ?)), multiply(?, ?))))))))) 0.62/0.84 = { by lemma 31 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(multiply(multiply(inverse(?), ?), ?)), multiply(?, inverse(multiply(inverse(?), multiply(inverse(?), ?))))))) 0.62/0.84 = { by lemma 37 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(multiply(multiply(inverse(?), ?), ?)), multiply(?, ?)))) 0.62/0.84 = { by lemma 40 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(multiply(multiply(inverse(?), ?), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(?, ?))))) 0.62/0.84 = { by lemma 42 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(multiply(multiply(inverse(?), ?), ?)), multiply(multiply(inverse(?), ?), multiply(?, ?))))) 0.62/0.84 = { by lemma 4 } 0.62/0.84 inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(multiply(?, ?)), multiply(?, multiply(?, ?))))) 0.62/0.84 = { by lemma 71 } 0.62/0.84 multiply(?, inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(multiply(?, ?)), multiply(?, multiply(?, ?))))))), ?))) 0.62/0.84 = { by lemma 11 } 0.62/0.84 multiply(?, inverse(multiply(inverse(multiply(multiply(?, inverse(?)), inverse(multiply(inverse(multiply(inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, ?)), ?))), X)), multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))), ?))) 0.62/0.84 = { by lemma 51 } 0.62/0.84 multiply(?, inverse(multiply(inverse(X), ?))) 0.62/0.84 0.62/0.84 Lemma 73: multiply(multiply(?, inverse(?)), multiply(?, inverse(multiply(inverse(X), ?)))) = X. 0.62/0.84 Proof: 0.62/0.84 multiply(multiply(?, inverse(?)), multiply(?, inverse(multiply(inverse(X), ?)))) 0.62/0.84 = { by lemma 56 } 0.62/0.84 multiply(multiply(inverse(?), ?), multiply(?, inverse(multiply(inverse(X), ?)))) 0.62/0.84 = { by lemma 21 } 0.62/0.84 multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), multiply(?, inverse(multiply(inverse(X), ?)))) 0.62/0.84 = { by lemma 60 } 0.62/0.84 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), inverse(?))))), multiply(?, inverse(multiply(inverse(X), ?)))) 0.62/0.84 = { by lemma 45 } 0.62/0.84 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), inverse(?))))), multiply(?, inverse(multiply(inverse(X), multiply(multiply(inverse(?), ?), ?))))) 0.62/0.84 = { by lemma 21 } 0.62/0.84 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), inverse(?))))), multiply(?, inverse(multiply(inverse(X), multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), ?))))) 0.62/0.84 = { by lemma 60 } 0.62/0.84 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), inverse(?))))), multiply(?, inverse(multiply(inverse(X), multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), inverse(?))))), ?))))) 0.62/0.84 = { by lemma 62 } 0.62/0.84 multiply(inverse(multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), inverse(?))))), multiply(?, inverse(multiply(inverse(X), multiply(inverse(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), inverse(?)))), multiply(inverse(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), inverse(?)))), inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(?), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?)))))), inverse(?))))))))) 0.62/0.84 = { by lemma 3 } 0.71/0.86 X 0.71/0.86 0.71/0.86 Goal 1 (prove_these_axioms_2): multiply(multiply(inverse(b2), b2), a2) = a2. 0.71/0.86 Proof: 0.71/0.86 multiply(multiply(inverse(b2), b2), a2) 0.71/0.86 = { by lemma 70 } 0.71/0.86 multiply(multiply(?, inverse(?)), a2) 0.71/0.86 = { by lemma 2 } 0.71/0.86 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), ?)), multiply(inverse(multiply(?, ?)), multiply(multiply(?, inverse(?)), a2)))), multiply(inverse(?), multiply(inverse(?), ?)))) 0.71/0.86 = { by lemma 9 } 0.71/0.86 inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(multiply(?, inverse(?)), a2))), multiply(inverse(multiply(?, ?)), multiply(multiply(?, inverse(?)), a2)))), multiply(inverse(multiply(multiply(?, inverse(?)), a2)), multiply(inverse(multiply(multiply(?, inverse(?)), a2)), multiply(multiply(?, inverse(?)), a2))))) 0.71/0.86 = { by lemma 70 } 0.71/0.86 inverse(multiply(inverse(multiply(?, inverse(?))), multiply(inverse(multiply(multiply(?, inverse(?)), a2)), multiply(inverse(multiply(multiply(?, inverse(?)), a2)), multiply(multiply(?, inverse(?)), a2))))) 0.71/0.86 = { by lemma 61 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(multiply(?, inverse(?)), a2)), multiply(inverse(multiply(multiply(?, inverse(?)), a2)), multiply(multiply(?, inverse(?)), a2))))) 0.71/0.86 = { by lemma 62 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?))) 0.71/0.86 = { by lemma 73 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(multiply(?, inverse(?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), ?)))))) 0.71/0.86 = { by lemma 56 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(multiply(inverse(?), ?), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), ?)))))) 0.71/0.86 = { by lemma 42 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), ?)))))) 0.71/0.86 = { by lemma 29 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), ?)))))) 0.71/0.86 = { by lemma 72 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?))), ?))))) 0.71/0.86 = { by lemma 62 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?))))))) 0.71/0.86 = { by lemma 64 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(?, inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?))))), multiply(?, ?)))))))) 0.71/0.86 = { by axiom 1 (single_axiom) } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(?, inverse(multiply(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?))))))), multiply(inverse(?), multiply(inverse(?), ?))))), multiply(?, ?)))))))) 0.71/0.86 = { by lemma 13 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(?, inverse(multiply(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), inverse(multiply(inverse(multiply(inverse(?), multiply(inverse(?), ?))), multiply(inverse(?), multiply(inverse(?), ?))))), multiply(?, ?)))))))) 0.71/0.86 = { by lemma 15 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(?, inverse(multiply(inverse(multiply(?, multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))), multiply(?, ?)))))))) 0.71/0.86 = { by lemma 64 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))) 0.71/0.86 = { by lemma 63 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), multiply(inverse(multiply(?, ?)), multiply(?, ?))))))), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), ?)))))) 0.71/0.86 = { by lemma 24 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), multiply(inverse(multiply(inverse(multiply(?, ?)), multiply(?, ?))), ?)))))) 0.71/0.86 = { by lemma 29 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), multiply(multiply(inverse(multiply(?, ?)), multiply(?, ?)), ?)))))) 0.71/0.86 = { by lemma 41 } 0.71/0.86 inverse(multiply(multiply(?, inverse(?)), multiply(?, inverse(multiply(inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)), ?))))) 0.71/0.86 = { by lemma 73 } 0.71/0.86 inverse(multiply(inverse(multiply(?, multiply(multiply(?, inverse(?)), a2))), ?)) 0.71/0.86 = { by lemma 72 } 0.71/0.86 multiply(?, inverse(multiply(inverse(multiply(multiply(?, inverse(?)), a2)), ?))) 0.71/0.86 = { by lemma 71 } 0.71/0.86 a2 0.71/0.86 % SZS output end Proof 0.71/0.86 0.71/0.86 RESULT: Unsatisfiable (the axioms are contradictory). 0.71/0.87 EOF