0.07/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.34	% Computer   : n011.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 180
0.12/0.34	% DateTime   : Thu Aug 29 11:15:14 EDT 2019
0.12/0.34	% CPUTime    : 
0.20/0.47	% SZS status Unsatisfiable
0.20/0.47	
0.20/0.48	% SZS output start Proof
0.20/0.48	Take the following subset of the input axioms:
0.20/0.48	  fof(multiply, axiom, ![A, B]: multiply(A, B)=inverse(double_divide(B, A))).
0.20/0.48	  fof(prove_these_axioms_3, negated_conjecture, multiply(multiply(a3, b3), c3)!=multiply(a3, multiply(b3, c3))).
0.20/0.48	  fof(single_axiom, axiom, ![A, B, C, D]: double_divide(double_divide(A, inverse(double_divide(B, C))), double_divide(inverse(B), inverse(double_divide(D, double_divide(A, D)))))=C).
0.20/0.48	
0.20/0.48	Now clausify the problem and encode Horn clauses using encoding 3 of
0.20/0.48	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.20/0.48	We repeatedly replace C & s=t => u=v by the two clauses:
0.20/0.48	  fresh(y, y, x1...xn) = u
0.20/0.48	  C => fresh(s, t, x1...xn) = v
0.20/0.48	where fresh is a fresh function symbol and x1..xn are the free
0.20/0.48	variables of u and v.
0.20/0.48	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.20/0.48	input problem has no model of domain size 1).
0.20/0.48	
0.20/0.48	The encoding turns the above axioms into the following unit equations and goals:
0.20/0.48	
0.20/0.48	Axiom 1 (single_axiom): double_divide(double_divide(X, inverse(double_divide(Y, Z))), double_divide(inverse(Y), inverse(double_divide(W, double_divide(X, W))))) = Z.
0.20/0.49	Axiom 2 (multiply): multiply(X, Y) = inverse(double_divide(Y, X)).
0.20/0.49	
0.20/0.49	Lemma 3: double_divide(double_divide(Y, multiply(X, Z)), double_divide(inverse(Z), multiply(double_divide(Y, W), W))) = X.
0.20/0.49	Proof:
0.20/0.49	  double_divide(double_divide(Y, multiply(X, Z)), double_divide(inverse(Z), multiply(double_divide(Y, W), W)))
0.20/0.49	= { by axiom 2 (multiply) }
0.20/0.49	  double_divide(double_divide(Y, inverse(double_divide(Z, X))), double_divide(inverse(Z), multiply(double_divide(Y, W), W)))
0.20/0.49	= { by axiom 2 (multiply) }
0.20/0.49	  double_divide(double_divide(Y, inverse(double_divide(Z, X))), double_divide(inverse(Z), inverse(double_divide(W, double_divide(Y, W)))))
0.20/0.49	= { by axiom 1 (single_axiom) }
0.20/0.49	  X
0.20/0.49	
0.20/0.49	Lemma 4: double_divide(inverse(X), multiply(double_divide(Y, Z), Z)) = double_divide(inverse(X), multiply(double_divide(Y, ?), ?)).
0.20/0.49	Proof:
0.20/0.49	  double_divide(inverse(X), multiply(double_divide(Y, Z), Z))
0.20/0.49	= { by axiom 1 (single_axiom) }
0.20/0.49	  double_divide(double_divide(W, inverse(double_divide(double_divide(Y, multiply(V, X)), double_divide(inverse(X), multiply(double_divide(Y, Z), Z))))), double_divide(inverse(double_divide(Y, multiply(V, X))), inverse(double_divide(U, double_divide(W, U)))))
0.20/0.49	= { by lemma 3 }
0.20/0.49	  double_divide(double_divide(W, inverse(V)), double_divide(inverse(double_divide(Y, multiply(V, X))), inverse(double_divide(U, double_divide(W, U)))))
0.20/0.49	= { by lemma 3 }
0.20/0.49	  double_divide(double_divide(W, inverse(double_divide(double_divide(Y, multiply(V, X)), double_divide(inverse(X), multiply(double_divide(Y, ?), ?))))), double_divide(inverse(double_divide(Y, multiply(V, X))), inverse(double_divide(U, double_divide(W, U)))))
0.20/0.49	= { by axiom 1 (single_axiom) }
0.20/0.50	  double_divide(inverse(X), multiply(double_divide(Y, ?), ?))
0.20/0.50	
0.20/0.50	Lemma 5: multiply(double_divide(X, Y), Y) = multiply(double_divide(X, ?), ?).
0.20/0.50	Proof:
0.20/0.50	  multiply(double_divide(X, Y), Y)
0.20/0.50	= { by axiom 1 (single_axiom) }
0.20/0.50	  double_divide(double_divide(Z, inverse(double_divide(inverse(W), multiply(double_divide(X, Y), Y)))), double_divide(inverse(inverse(W)), inverse(double_divide(V, double_divide(Z, V)))))
0.20/0.50	= { by lemma 4 }
0.20/0.50	  double_divide(double_divide(Z, inverse(double_divide(inverse(W), multiply(double_divide(X, ?), ?)))), double_divide(inverse(inverse(W)), inverse(double_divide(V, double_divide(Z, V)))))
0.20/0.50	= { by axiom 1 (single_axiom) }
0.20/0.50	  multiply(double_divide(X, ?), ?)
0.20/0.50	
0.20/0.50	Lemma 6: double_divide(double_divide(Y, multiply(X, double_divide(Z, W))), double_divide(multiply(W, Z), multiply(double_divide(Y, V), V))) = X.
0.20/0.50	Proof:
0.20/0.50	  double_divide(double_divide(Y, multiply(X, double_divide(Z, W))), double_divide(multiply(W, Z), multiply(double_divide(Y, V), V)))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(double_divide(Y, inverse(double_divide(double_divide(Z, W), X))), double_divide(multiply(W, Z), multiply(double_divide(Y, V), V)))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(double_divide(Y, inverse(double_divide(double_divide(Z, W), X))), double_divide(inverse(double_divide(Z, W)), multiply(double_divide(Y, V), V)))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(double_divide(Y, inverse(double_divide(double_divide(Z, W), X))), double_divide(inverse(double_divide(Z, W)), inverse(double_divide(V, double_divide(Y, V)))))
0.20/0.50	= { by axiom 1 (single_axiom) }
0.20/0.50	  X
0.20/0.50	
0.20/0.50	Lemma 7: double_divide(double_divide(inverse(W), multiply(double_divide(X, ?), ?)), double_divide(multiply(Z, Y), multiply(double_divide(inverse(W), ?), ?))) = double_divide(X, double_divide(Y, Z)).
0.20/0.50	Proof:
0.20/0.50	  double_divide(double_divide(inverse(W), multiply(double_divide(X, ?), ?)), double_divide(multiply(Z, Y), multiply(double_divide(inverse(W), ?), ?)))
0.20/0.50	= { by lemma 4 }
0.20/0.50	  double_divide(double_divide(inverse(W), multiply(double_divide(X, double_divide(Y, Z)), double_divide(Y, Z))), double_divide(multiply(Z, Y), multiply(double_divide(inverse(W), ?), ?)))
0.20/0.50	= { by lemma 6 }
0.20/0.50	  double_divide(X, double_divide(Y, Z))
0.20/0.50	
0.20/0.50	Lemma 8: double_divide(X, double_divide(Y, double_divide(Z, Y))) = double_divide(X, double_divide(?, double_divide(Z, ?))).
0.20/0.50	Proof:
0.20/0.50	  double_divide(X, double_divide(Y, double_divide(Z, Y)))
0.20/0.50	= { by lemma 7 }
0.20/0.50	  double_divide(double_divide(inverse(W), multiply(double_divide(X, ?), ?)), double_divide(multiply(double_divide(Z, Y), Y), multiply(double_divide(inverse(W), ?), ?)))
0.20/0.50	= { by lemma 5 }
0.20/0.50	  double_divide(double_divide(inverse(W), multiply(double_divide(X, ?), ?)), double_divide(multiply(double_divide(Z, ?), ?), multiply(double_divide(inverse(W), ?), ?)))
0.20/0.50	= { by lemma 7 }
0.20/0.50	  double_divide(X, double_divide(?, double_divide(Z, ?)))
0.20/0.50	
0.20/0.50	Lemma 9: double_divide(X, double_divide(Y, X)) = double_divide(?, double_divide(Y, ?)).
0.20/0.50	Proof:
0.20/0.50	  double_divide(X, double_divide(Y, X))
0.20/0.50	= { by axiom 1 (single_axiom) }
0.20/0.50	  double_divide(double_divide(Z, inverse(double_divide(W, double_divide(X, double_divide(Y, X))))), double_divide(inverse(W), inverse(double_divide(V, double_divide(Z, V)))))
0.20/0.50	= { by lemma 8 }
0.20/0.50	  double_divide(double_divide(Z, inverse(double_divide(W, double_divide(?, double_divide(Y, ?))))), double_divide(inverse(W), inverse(double_divide(V, double_divide(Z, V)))))
0.20/0.50	= { by axiom 1 (single_axiom) }
0.20/0.50	  double_divide(?, double_divide(Y, ?))
0.20/0.50	
0.20/0.50	Lemma 10: double_divide(double_divide(Y, X), double_divide(?, double_divide(Y, ?))) = double_divide(?, double_divide(X, ?)).
0.20/0.50	Proof:
0.20/0.50	  double_divide(double_divide(Y, X), double_divide(?, double_divide(Y, ?)))
0.20/0.50	= { by lemma 8 }
0.20/0.50	  double_divide(double_divide(Y, X), double_divide(X, double_divide(Y, X)))
0.20/0.50	= { by lemma 9 }
0.20/0.50	  double_divide(?, double_divide(X, ?))
0.20/0.50	
0.20/0.50	Lemma 11: multiply(double_divide(inverse(X), multiply(double_divide(Z, Y), Y)), double_divide(Z, multiply(W, X))) = inverse(W).
0.20/0.50	Proof:
0.20/0.50	  multiply(double_divide(inverse(X), multiply(double_divide(Z, Y), Y)), double_divide(Z, multiply(W, X)))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  multiply(double_divide(inverse(X), inverse(double_divide(Y, double_divide(Z, Y)))), double_divide(Z, multiply(W, X)))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  multiply(double_divide(inverse(X), inverse(double_divide(Y, double_divide(Z, Y)))), double_divide(Z, inverse(double_divide(X, W))))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  inverse(double_divide(double_divide(Z, inverse(double_divide(X, W))), double_divide(inverse(X), inverse(double_divide(Y, double_divide(Z, Y))))))
0.20/0.50	= { by axiom 1 (single_axiom) }
0.20/0.50	  inverse(W)
0.20/0.50	
0.20/0.50	Lemma 12: multiply(double_divide(double_divide(X, multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), ?), ?)), Z)), ?), ?) = multiply(Y, X).
0.20/0.50	Proof:
0.20/0.50	  multiply(double_divide(double_divide(X, multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), ?), ?)), Z)), ?), ?)
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  multiply(double_divide(double_divide(X, inverse(double_divide(Z, double_divide(inverse(Y), multiply(double_divide(inverse(Z), ?), ?))))), ?), ?)
0.20/0.50	= { by lemma 5 }
0.20/0.50	  multiply(double_divide(double_divide(X, inverse(double_divide(Z, double_divide(inverse(Y), multiply(double_divide(inverse(Z), ?), ?))))), double_divide(inverse(Z), inverse(double_divide(?, double_divide(X, ?))))), double_divide(inverse(Z), inverse(double_divide(?, double_divide(X, ?)))))
0.20/0.50	= { by axiom 1 (single_axiom) }
0.20/0.50	  multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), ?), ?)), double_divide(inverse(Z), inverse(double_divide(?, double_divide(X, ?)))))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), ?), ?)), double_divide(inverse(Z), multiply(double_divide(X, ?), ?)))
0.20/0.50	= { by lemma 4 }
0.20/0.50	  multiply(double_divide(inverse(Y), multiply(double_divide(inverse(Z), ?), ?)), double_divide(inverse(Z), multiply(double_divide(X, Y), Y)))
0.20/0.50	= { by lemma 11 }
0.20/0.50	  inverse(double_divide(X, Y))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  multiply(Y, X)
0.20/0.50	
0.20/0.50	Lemma 13: double_divide(double_divide(W, inverse(V)), double_divide(multiply(multiply(V, X), Y), multiply(double_divide(W, U), U))) = double_divide(inverse(X), multiply(double_divide(Y, ?), ?)).
0.20/0.50	Proof:
0.20/0.50	  double_divide(double_divide(W, inverse(V)), double_divide(multiply(multiply(V, X), Y), multiply(double_divide(W, U), U)))
0.20/0.50	= { by lemma 3 }
0.20/0.50	  double_divide(double_divide(W, inverse(double_divide(double_divide(Y, multiply(V, X)), double_divide(inverse(X), multiply(double_divide(Y, ?), ?))))), double_divide(multiply(multiply(V, X), Y), multiply(double_divide(W, U), U)))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(double_divide(W, inverse(double_divide(double_divide(Y, multiply(V, X)), double_divide(inverse(X), multiply(double_divide(Y, ?), ?))))), double_divide(inverse(double_divide(Y, multiply(V, X))), multiply(double_divide(W, U), U)))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(double_divide(W, inverse(double_divide(double_divide(Y, multiply(V, X)), double_divide(inverse(X), multiply(double_divide(Y, ?), ?))))), double_divide(inverse(double_divide(Y, multiply(V, X))), inverse(double_divide(U, double_divide(W, U)))))
0.20/0.50	= { by axiom 1 (single_axiom) }
0.20/0.50	  double_divide(inverse(X), multiply(double_divide(Y, ?), ?))
0.20/0.50	
0.20/0.50	Lemma 14: double_divide(W, double_divide(Y, multiply(double_divide(?, double_divide(W, ?)), X))) = double_divide(inverse(X), multiply(double_divide(Y, ?), ?)).
0.20/0.50	Proof:
0.20/0.50	  double_divide(W, double_divide(Y, multiply(double_divide(?, double_divide(W, ?)), X)))
0.20/0.50	= { by lemma 7 }
0.20/0.50	  double_divide(double_divide(inverse(?), multiply(double_divide(W, ?), ?)), double_divide(multiply(multiply(double_divide(?, double_divide(W, ?)), X), Y), multiply(double_divide(inverse(?), ?), ?)))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(double_divide(inverse(?), inverse(double_divide(?, double_divide(W, ?)))), double_divide(multiply(multiply(double_divide(?, double_divide(W, ?)), X), Y), multiply(double_divide(inverse(?), ?), ?)))
0.20/0.50	= { by lemma 13 }
0.20/0.50	  double_divide(inverse(X), multiply(double_divide(Y, ?), ?))
0.20/0.50	
0.20/0.50	Lemma 15: multiply(double_divide(X, multiply(double_divide(?, double_divide(Y, ?)), Z)), Y) = multiply(multiply(double_divide(X, ?), ?), inverse(Z)).
0.20/0.50	Proof:
0.20/0.50	  multiply(double_divide(X, multiply(double_divide(?, double_divide(Y, ?)), Z)), Y)
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  inverse(double_divide(Y, double_divide(X, multiply(double_divide(?, double_divide(Y, ?)), Z))))
0.20/0.50	= { by lemma 14 }
0.20/0.50	  inverse(double_divide(inverse(Z), multiply(double_divide(X, ?), ?)))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  multiply(multiply(double_divide(X, ?), ?), inverse(Z))
0.20/0.50	
0.20/0.50	Lemma 16: multiply(double_divide(X, multiply(double_divide(?, ?), ?)), Y) = multiply(multiply(double_divide(X, ?), ?), multiply(?, Y)).
0.20/0.50	Proof:
0.20/0.50	  multiply(double_divide(X, multiply(double_divide(?, ?), ?)), Y)
0.20/0.50	= { by lemma 5 }
0.20/0.50	  multiply(double_divide(X, multiply(double_divide(?, double_divide(Y, ?)), double_divide(Y, ?))), Y)
0.20/0.50	= { by lemma 15 }
0.20/0.50	  multiply(multiply(double_divide(X, ?), ?), inverse(double_divide(Y, ?)))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  multiply(multiply(double_divide(X, ?), ?), multiply(?, Y))
0.20/0.50	
0.20/0.50	Lemma 17: multiply(multiply(double_divide(X, ?), ?), multiply(?, multiply(double_divide(?, ?), ?))) = multiply(double_divide(X, ?), ?).
0.20/0.50	Proof:
0.20/0.50	  multiply(multiply(double_divide(X, ?), ?), multiply(?, multiply(double_divide(?, ?), ?)))
0.20/0.50	= { by lemma 16 }
0.20/0.50	  multiply(double_divide(X, multiply(double_divide(?, ?), ?)), multiply(double_divide(?, ?), ?))
0.20/0.50	= { by lemma 5 }
0.20/0.50	  multiply(double_divide(X, ?), ?)
0.20/0.50	
0.20/0.50	Lemma 18: double_divide(X, double_divide(multiply(?, multiply(double_divide(?, ?), ?)), inverse(Y))) = double_divide(X, Y).
0.20/0.50	Proof:
0.20/0.50	  double_divide(X, double_divide(multiply(?, multiply(double_divide(?, ?), ?)), inverse(Y)))
0.20/0.50	= { by lemma 7 }
0.20/0.50	  double_divide(double_divide(inverse(?), multiply(double_divide(X, ?), ?)), double_divide(multiply(inverse(Y), multiply(?, multiply(double_divide(?, ?), ?))), multiply(double_divide(inverse(?), ?), ?)))
0.20/0.50	= { by lemma 4 }
0.20/0.50	  double_divide(double_divide(inverse(?), multiply(double_divide(X, Y), Y)), double_divide(multiply(inverse(Y), multiply(?, multiply(double_divide(?, ?), ?))), multiply(double_divide(inverse(?), ?), ?)))
0.20/0.50	= { by lemma 11 }
0.20/0.50	  double_divide(double_divide(inverse(?), multiply(double_divide(X, Y), Y)), double_divide(multiply(multiply(double_divide(inverse(?), multiply(double_divide(?, ?), ?)), double_divide(?, multiply(Y, ?))), multiply(?, multiply(double_divide(?, ?), ?))), multiply(double_divide(inverse(?), ?), ?)))
0.20/0.50	= { by lemma 12 }
0.20/0.50	  double_divide(double_divide(inverse(?), multiply(double_divide(X, Y), Y)), double_divide(multiply(multiply(double_divide(double_divide(double_divide(?, multiply(Y, ?)), multiply(double_divide(inverse(double_divide(inverse(?), multiply(double_divide(?, ?), ?))), multiply(double_divide(inverse(?), ?), ?)), ?)), ?), ?), multiply(?, multiply(double_divide(?, ?), ?))), multiply(double_divide(inverse(?), ?), ?)))
0.20/0.50	= { by lemma 17 }
0.20/0.50	  double_divide(double_divide(inverse(?), multiply(double_divide(X, Y), Y)), double_divide(multiply(double_divide(double_divide(double_divide(?, multiply(Y, ?)), multiply(double_divide(inverse(double_divide(inverse(?), multiply(double_divide(?, ?), ?))), multiply(double_divide(inverse(?), ?), ?)), ?)), ?), ?), multiply(double_divide(inverse(?), ?), ?)))
0.20/0.50	= { by lemma 12 }
0.20/0.50	  double_divide(double_divide(inverse(?), multiply(double_divide(X, Y), Y)), double_divide(multiply(double_divide(inverse(?), multiply(double_divide(?, ?), ?)), double_divide(?, multiply(Y, ?))), multiply(double_divide(inverse(?), ?), ?)))
0.20/0.50	= { by lemma 11 }
0.20/0.50	  double_divide(double_divide(inverse(?), multiply(double_divide(X, Y), Y)), double_divide(inverse(Y), multiply(double_divide(inverse(?), ?), ?)))
0.20/0.50	= { by lemma 3 }
0.20/0.50	  double_divide(X, Y)
0.20/0.50	
0.20/0.50	Lemma 19: double_divide(inverse(X), X) = double_divide(inverse(?), ?).
0.20/0.50	Proof:
0.20/0.50	  double_divide(inverse(X), X)
0.20/0.50	= { by lemma 18 }
0.20/0.50	  double_divide(inverse(X), double_divide(multiply(?, multiply(double_divide(?, ?), ?)), inverse(X)))
0.20/0.50	= { by lemma 9 }
0.20/0.50	  double_divide(?, double_divide(multiply(?, multiply(double_divide(?, ?), ?)), ?))
0.20/0.50	= { by lemma 9 }
0.20/0.50	  double_divide(inverse(?), double_divide(multiply(?, multiply(double_divide(?, ?), ?)), inverse(?)))
0.20/0.50	= { by lemma 18 }
0.20/0.50	  double_divide(inverse(?), ?)
0.20/0.50	
0.20/0.50	Lemma 20: double_divide(multiply(X, Y), double_divide(Y, X)) = double_divide(inverse(?), ?).
0.20/0.50	Proof:
0.20/0.50	  double_divide(multiply(X, Y), double_divide(Y, X))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(inverse(double_divide(Y, X)), double_divide(Y, X))
0.20/0.50	= { by lemma 19 }
0.20/0.50	  double_divide(inverse(?), ?)
0.20/0.50	
0.20/0.50	Lemma 21: multiply(X, inverse(X)) = multiply(?, inverse(?)).
0.20/0.50	Proof:
0.20/0.50	  multiply(X, inverse(X))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  inverse(double_divide(inverse(X), X))
0.20/0.50	= { by lemma 19 }
0.20/0.50	  inverse(double_divide(inverse(?), ?))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  multiply(?, inverse(?))
0.20/0.50	
0.20/0.50	Lemma 22: double_divide(X, double_divide(inverse(?), ?)) = double_divide(?, double_divide(inverse(X), ?)).
0.20/0.50	Proof:
0.20/0.50	  double_divide(X, double_divide(inverse(?), ?))
0.20/0.50	= { by lemma 19 }
0.20/0.50	  double_divide(X, double_divide(inverse(X), X))
0.20/0.50	= { by lemma 9 }
0.20/0.50	  double_divide(?, double_divide(inverse(X), ?))
0.20/0.50	
0.20/0.50	Lemma 23: double_divide(multiply(?, multiply(double_divide(?, ?), ?)), inverse(X)) = X.
0.20/0.50	Proof:
0.20/0.50	  double_divide(multiply(?, multiply(double_divide(?, ?), ?)), inverse(X))
0.20/0.50	= { by axiom 1 (single_axiom) }
0.20/0.50	  double_divide(double_divide(?, inverse(double_divide(?, double_divide(multiply(?, multiply(double_divide(?, ?), ?)), inverse(X))))), double_divide(inverse(?), inverse(double_divide(?, double_divide(?, ?)))))
0.20/0.50	= { by lemma 18 }
0.20/0.50	  double_divide(double_divide(?, inverse(double_divide(?, X))), double_divide(inverse(?), inverse(double_divide(?, double_divide(?, ?)))))
0.20/0.50	= { by axiom 1 (single_axiom) }
0.20/0.50	  X
0.20/0.50	
0.20/0.50	Lemma 24: double_divide(double_divide(Y, X), double_divide(inverse(Y), multiply(Z, inverse(Z)))) = X.
0.20/0.50	Proof:
0.20/0.50	  double_divide(double_divide(Y, X), double_divide(inverse(Y), multiply(Z, inverse(Z))))
0.20/0.50	= { by lemma 23 }
0.20/0.50	  double_divide(double_divide(multiply(?, multiply(double_divide(?, ?), ?)), inverse(double_divide(Y, X))), double_divide(inverse(Y), multiply(Z, inverse(Z))))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(double_divide(multiply(?, multiply(double_divide(?, ?), ?)), inverse(double_divide(Y, X))), double_divide(inverse(Y), inverse(double_divide(inverse(Z), Z))))
0.20/0.50	= { by lemma 23 }
0.20/0.50	  double_divide(double_divide(multiply(?, multiply(double_divide(?, ?), ?)), inverse(double_divide(Y, X))), double_divide(inverse(Y), inverse(double_divide(inverse(Z), double_divide(multiply(?, multiply(double_divide(?, ?), ?)), inverse(Z))))))
0.20/0.50	= { by axiom 1 (single_axiom) }
0.20/0.50	  X
0.20/0.50	
0.20/0.50	Lemma 25: double_divide(?, double_divide(multiply(X, multiply(?, inverse(?))), ?)) = X.
0.20/0.50	Proof:
0.20/0.50	  double_divide(?, double_divide(multiply(X, multiply(?, inverse(?))), ?))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(?, double_divide(inverse(double_divide(multiply(?, inverse(?)), X)), ?))
0.20/0.50	= { by lemma 22 }
0.20/0.50	  double_divide(double_divide(multiply(?, inverse(?)), X), double_divide(inverse(?), ?))
0.20/0.50	= { by lemma 19 }
0.20/0.50	  double_divide(double_divide(multiply(?, inverse(?)), X), double_divide(inverse(multiply(?, inverse(?))), multiply(?, inverse(?))))
0.20/0.50	= { by lemma 24 }
0.20/0.50	  X
0.20/0.50	
0.20/0.50	Lemma 26: double_divide(?, double_divide(multiply(?, inverse(?)), ?)) = double_divide(inverse(?), ?).
0.20/0.50	Proof:
0.20/0.50	  double_divide(?, double_divide(multiply(?, inverse(?)), ?))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(?, double_divide(inverse(double_divide(inverse(?), ?)), ?))
0.20/0.50	= { by lemma 20 }
0.20/0.50	  double_divide(?, double_divide(inverse(double_divide(multiply(?, inverse(?)), double_divide(inverse(?), ?))), ?))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(?, double_divide(multiply(double_divide(inverse(?), ?), multiply(?, inverse(?))), ?))
0.20/0.50	= { by lemma 25 }
0.20/0.50	  double_divide(inverse(?), ?)
0.20/0.50	
0.20/0.50	Lemma 27: double_divide(?, double_divide(X, ?)) = X.
0.20/0.50	Proof:
0.20/0.50	  double_divide(?, double_divide(X, ?))
0.20/0.50	= { by lemma 10 }
0.20/0.50	  double_divide(double_divide(multiply(?, inverse(?)), X), double_divide(?, double_divide(multiply(?, inverse(?)), ?)))
0.20/0.50	= { by lemma 26 }
0.20/0.50	  double_divide(double_divide(multiply(?, inverse(?)), X), double_divide(inverse(?), ?))
0.20/0.50	= { by lemma 22 }
0.20/0.50	  double_divide(?, double_divide(inverse(double_divide(multiply(?, inverse(?)), X)), ?))
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  double_divide(?, double_divide(multiply(X, multiply(?, inverse(?))), ?))
0.20/0.50	= { by lemma 25 }
0.20/0.50	  X
0.20/0.50	
0.20/0.50	Lemma 28: multiply(double_divide(X, ?), ?) = inverse(X).
0.20/0.50	Proof:
0.20/0.50	  multiply(double_divide(X, ?), ?)
0.20/0.50	= { by axiom 2 (multiply) }
0.20/0.50	  inverse(double_divide(?, double_divide(X, ?)))
0.20/0.50	= { by lemma 27 }
0.20/0.50	  inverse(X)
0.20/0.50	
0.20/0.50	Lemma 29: multiply(double_divide(?, double_divide(Y, ?)), double_divide(Y, X)) = multiply(double_divide(X, ?), ?).
0.20/0.50	Proof:
0.20/0.50	  multiply(double_divide(?, double_divide(Y, ?)), double_divide(Y, X))
0.20/0.50	= { by lemma 9 }
0.20/0.50	  multiply(double_divide(X, double_divide(Y, X)), double_divide(Y, X))
0.20/0.50	= { by lemma 5 }
0.20/0.50	  multiply(double_divide(X, ?), ?)
0.20/0.50	
0.20/0.50	Lemma 30: multiply(?, inverse(?)) = double_divide(inverse(?), ?).
0.20/0.50	Proof:
0.20/0.50	  multiply(?, inverse(?))
0.20/0.50	= { by lemma 27 }
0.20/0.50	  double_divide(?, double_divide(multiply(?, inverse(?)), ?))
0.20/0.50	= { by lemma 26 }
0.20/0.50	  double_divide(inverse(?), ?)
0.20/0.50	
0.20/0.50	Lemma 31: double_divide(inverse(inverse(X)), Y) = double_divide(X, Y).
0.20/0.50	Proof:
0.20/0.50	  double_divide(inverse(inverse(X)), Y)
0.20/0.50	= { by lemma 27 }
0.20/0.50	  double_divide(double_divide(?, double_divide(inverse(inverse(X)), ?)), Y)
0.20/0.50	= { by lemma 22 }
0.20/0.50	  double_divide(double_divide(inverse(X), double_divide(inverse(?), ?)), Y)
0.20/0.50	= { by lemma 30 }
0.20/0.50	  double_divide(double_divide(inverse(X), multiply(?, inverse(?))), Y)
0.20/0.50	= { by lemma 24 }
0.20/0.50	  double_divide(double_divide(inverse(X), multiply(?, inverse(?))), double_divide(double_divide(X, Y), double_divide(inverse(X), multiply(?, inverse(?)))))
0.20/0.50	= { by lemma 9 }
0.20/0.50	  double_divide(?, double_divide(double_divide(X, Y), ?))
0.20/0.50	= { by lemma 27 }
0.20/0.50	  double_divide(X, Y)
0.20/0.50	
0.20/0.50	Lemma 32: inverse(inverse(X)) = X.
0.20/0.50	Proof:
0.20/0.50	  inverse(inverse(X))
0.20/0.50	= { by lemma 27 }
0.20/0.50	  double_divide(?, double_divide(inverse(inverse(X)), ?))
0.20/0.50	= { by lemma 31 }
0.20/0.50	  double_divide(?, double_divide(X, ?))
0.20/0.50	= { by lemma 27 }
0.20/0.50	  X
0.20/0.50	
0.20/0.50	Lemma 33: double_divide(double_divide(Z, multiply(double_divide(X, ?), ?)), double_divide(inverse(Y), multiply(double_divide(Z, ?), ?))) = double_divide(X, Y).
0.20/0.50	Proof:
0.20/0.51	  double_divide(double_divide(Z, multiply(double_divide(X, ?), ?)), double_divide(inverse(Y), multiply(double_divide(Z, ?), ?)))
0.20/0.51	= { by lemma 5 }
0.20/0.51	  double_divide(double_divide(Z, multiply(double_divide(X, Y), Y)), double_divide(inverse(Y), multiply(double_divide(Z, ?), ?)))
0.20/0.51	= { by lemma 3 }
0.20/0.51	  double_divide(X, Y)
0.20/0.51	
0.20/0.51	Lemma 34: multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), ?), ?) = inverse(?).
0.20/0.51	Proof:
0.20/0.51	  multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), ?), ?)
0.20/0.51	= { by lemma 17 }
0.20/0.51	  multiply(multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), ?), ?), multiply(?, multiply(double_divide(?, ?), ?)))
0.20/0.51	= { by lemma 3 }
0.20/0.51	  multiply(multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), ?), ?), multiply(?, double_divide(double_divide(?, multiply(multiply(double_divide(?, ?), ?), multiply(?, multiply(double_divide(?, ?), ?)))), double_divide(inverse(multiply(?, multiply(double_divide(?, ?), ?))), multiply(double_divide(?, ?), ?)))))
0.20/0.51	= { by lemma 17 }
0.20/0.51	  multiply(multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), ?), ?), multiply(?, double_divide(double_divide(?, multiply(double_divide(?, ?), ?)), double_divide(inverse(multiply(?, multiply(double_divide(?, ?), ?))), multiply(double_divide(?, ?), ?)))))
0.20/0.51	= { by lemma 33 }
0.20/0.51	  multiply(multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), ?), ?), multiply(?, double_divide(?, multiply(?, multiply(double_divide(?, ?), ?)))))
0.20/0.51	= { by axiom 2 (multiply) }
0.20/0.51	  multiply(multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), ?), ?), inverse(double_divide(double_divide(?, multiply(?, multiply(double_divide(?, ?), ?))), ?)))
0.20/0.51	= { by lemma 15 }
0.20/0.51	  multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), multiply(double_divide(?, double_divide(double_divide(?, multiply(?, multiply(double_divide(?, ?), ?))), ?)), double_divide(double_divide(?, multiply(?, multiply(double_divide(?, ?), ?))), ?))), double_divide(?, multiply(?, multiply(double_divide(?, ?), ?))))
0.20/0.51	= { by lemma 11 }
0.20/0.51	  inverse(?)
0.20/0.51	
0.20/0.51	Lemma 35: multiply(double_divide(inverse(?), ?), X) = multiply(inverse(?), multiply(?, X)).
0.20/0.51	Proof:
0.20/0.51	  multiply(double_divide(inverse(?), ?), X)
0.20/0.51	= { by lemma 19 }
0.20/0.51	  multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), multiply(double_divide(?, ?), ?)), X)
0.20/0.51	= { by lemma 16 }
0.20/0.51	  multiply(multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), ?), ?), multiply(?, X))
0.20/0.51	= { by lemma 34 }
0.20/0.51	  multiply(inverse(?), multiply(?, X))
0.20/0.51	
0.20/0.51	Lemma 36: multiply(inverse(?), multiply(?, double_divide(Y, X))) = multiply(double_divide(multiply(X, Y), ?), ?).
0.20/0.51	Proof:
0.20/0.51	  multiply(inverse(?), multiply(?, double_divide(Y, X)))
0.20/0.51	= { by lemma 35 }
0.20/0.51	  multiply(double_divide(inverse(?), ?), double_divide(Y, X))
0.20/0.51	= { by lemma 20 }
0.20/0.51	  multiply(double_divide(multiply(X, Y), double_divide(Y, X)), double_divide(Y, X))
0.20/0.51	= { by lemma 5 }
0.20/0.51	  multiply(double_divide(multiply(X, Y), ?), ?)
0.20/0.51	
0.20/0.51	Lemma 37: double_divide(multiply(Y, Z), multiply(?, double_divide(Z, Y))) = inverse(?).
0.20/0.51	Proof:
0.20/0.51	  double_divide(multiply(Y, Z), multiply(?, double_divide(Z, Y)))
0.20/0.51	= { by lemma 33 }
0.20/0.51	  double_divide(double_divide(?, multiply(double_divide(multiply(Y, Z), ?), ?)), double_divide(inverse(multiply(?, double_divide(Z, Y))), multiply(double_divide(?, ?), ?)))
0.20/0.51	= { by lemma 36 }
0.20/0.51	  double_divide(double_divide(?, multiply(inverse(?), multiply(?, double_divide(Z, Y)))), double_divide(inverse(multiply(?, double_divide(Z, Y))), multiply(double_divide(?, ?), ?)))
0.20/0.51	= { by lemma 3 }
0.20/0.51	  inverse(?)
0.20/0.51	
0.20/0.51	Lemma 38: double_divide(inverse(X), multiply(?, X)) = inverse(?).
0.20/0.51	Proof:
0.20/0.51	  double_divide(inverse(X), multiply(?, X))
0.20/0.51	= { by lemma 11 }
0.20/0.51	  double_divide(multiply(double_divide(inverse(?), multiply(double_divide(?, ?), ?)), double_divide(?, multiply(X, ?))), multiply(?, X))
0.20/0.51	= { by axiom 2 (multiply) }
0.20/0.51	  double_divide(multiply(double_divide(inverse(?), inverse(double_divide(?, double_divide(?, ?)))), double_divide(?, multiply(X, ?))), multiply(?, X))
0.20/0.51	= { by axiom 2 (multiply) }
0.20/0.51	  double_divide(multiply(double_divide(inverse(?), inverse(double_divide(?, double_divide(?, ?)))), double_divide(?, inverse(double_divide(?, X)))), multiply(?, X))
0.20/0.51	= { by axiom 1 (single_axiom) }
0.20/0.51	  double_divide(multiply(double_divide(inverse(?), inverse(double_divide(?, double_divide(?, ?)))), double_divide(?, inverse(double_divide(?, X)))), multiply(?, double_divide(double_divide(?, inverse(double_divide(?, X))), double_divide(inverse(?), inverse(double_divide(?, double_divide(?, ?)))))))
0.20/0.51	= { by lemma 37 }
0.20/0.51	  inverse(?)
0.20/0.51	
0.20/0.51	Lemma 39: double_divide(double_divide(Y, X), Y) = X.
0.20/0.51	Proof:
0.20/0.51	  double_divide(double_divide(Y, X), Y)
0.20/0.51	= { by lemma 27 }
0.20/0.51	  double_divide(double_divide(Y, X), double_divide(?, double_divide(Y, ?)))
0.20/0.51	= { by lemma 10 }
0.20/0.51	  double_divide(?, double_divide(X, ?))
0.20/0.51	= { by lemma 27 }
0.20/0.51	  X
0.20/0.51	
0.20/0.51	Lemma 40: double_divide(X, double_divide(inverse(multiply(double_divide(?, ?), ?)), multiply(double_divide(?, ?), ?))) = double_divide(multiply(?, X), inverse(?)).
0.20/0.51	Proof:
0.20/0.51	  double_divide(X, double_divide(inverse(multiply(double_divide(?, ?), ?)), multiply(double_divide(?, ?), ?)))
0.20/0.51	= { by lemma 5 }
0.20/0.51	  double_divide(X, double_divide(inverse(multiply(double_divide(?, ?), ?)), multiply(double_divide(?, double_divide(X, ?)), double_divide(X, ?))))
0.20/0.51	= { by lemma 14 }
0.20/0.51	  double_divide(inverse(double_divide(X, ?)), multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), ?), ?))
0.20/0.51	= { by axiom 2 (multiply) }
0.20/0.51	  double_divide(multiply(?, X), multiply(double_divide(inverse(multiply(double_divide(?, ?), ?)), ?), ?))
0.20/0.51	= { by lemma 34 }
0.20/0.51	  double_divide(multiply(?, X), inverse(?))
0.20/0.51	
0.20/0.51	Lemma 41: double_divide(multiply(?, X), inverse(?)) = double_divide(?, double_divide(inverse(X), ?)).
0.20/0.51	Proof:
0.20/0.51	  double_divide(multiply(?, X), inverse(?))
0.20/0.51	= { by lemma 40 }
0.20/0.51	  double_divide(X, double_divide(inverse(multiply(double_divide(?, ?), ?)), multiply(double_divide(?, ?), ?)))
0.20/0.51	= { by lemma 19 }
0.20/0.51	  double_divide(X, double_divide(inverse(?), ?))
0.20/0.51	= { by lemma 22 }
0.20/0.51	  double_divide(?, double_divide(inverse(X), ?))
0.20/0.51	
0.20/0.51	Lemma 42: multiply(X, double_divide(?, inverse(?))) = X.
0.20/0.51	Proof:
0.20/0.51	  multiply(X, double_divide(?, inverse(?)))
0.20/0.51	= { by lemma 27 }
0.20/0.51	  multiply(X, double_divide(double_divide(?, double_divide(?, ?)), inverse(?)))
0.20/0.51	= { by lemma 9 }
0.20/0.51	  multiply(X, double_divide(double_divide(double_divide(multiply(?, inverse(?)), ?), double_divide(?, double_divide(multiply(?, inverse(?)), ?))), inverse(?)))
0.20/0.51	= { by lemma 26 }
0.20/0.51	  multiply(X, double_divide(double_divide(double_divide(multiply(?, inverse(?)), ?), double_divide(inverse(?), ?)), inverse(?)))
0.20/0.51	= { by lemma 22 }
0.20/0.51	  multiply(X, double_divide(double_divide(?, double_divide(inverse(double_divide(multiply(?, inverse(?)), ?)), ?)), inverse(?)))
0.20/0.51	= { by lemma 27 }
0.20/0.51	  multiply(X, double_divide(inverse(double_divide(multiply(?, inverse(?)), ?)), inverse(?)))
0.20/0.51	= { by axiom 2 (multiply) }
0.20/0.51	  multiply(X, double_divide(multiply(?, multiply(?, inverse(?))), inverse(?)))
0.20/0.51	= { by lemma 41 }
0.20/0.51	  multiply(X, double_divide(?, double_divide(inverse(multiply(?, inverse(?))), ?)))
0.20/0.51	= { by lemma 21 }
0.20/0.51	  multiply(X, double_divide(?, double_divide(inverse(multiply(multiply(double_divide(?, ?), ?), inverse(multiply(double_divide(?, ?), ?)))), ?)))
0.20/0.51	= { by lemma 41 }
0.20/0.51	  multiply(X, double_divide(multiply(?, multiply(multiply(double_divide(?, ?), ?), inverse(multiply(double_divide(?, ?), ?)))), inverse(?)))
0.20/0.51	= { by lemma 40 }
0.20/0.51	  multiply(X, double_divide(multiply(multiply(double_divide(?, ?), ?), inverse(multiply(double_divide(?, ?), ?))), double_divide(inverse(multiply(double_divide(?, ?), ?)), multiply(double_divide(?, ?), ?))))
0.20/0.51	= { by lemma 20 }
0.20/0.51	  multiply(X, double_divide(inverse(?), ?))
0.20/0.51	= { by lemma 30 }
0.20/0.51	  multiply(X, multiply(?, inverse(?)))
0.20/0.51	= { by lemma 27 }
0.20/0.51	  double_divide(?, double_divide(multiply(X, multiply(?, inverse(?))), ?))
0.20/0.51	= { by lemma 25 }
0.20/0.51	  X
0.20/0.51	
0.20/0.51	Lemma 43: multiply(double_divide(X, multiply(double_divide(Z, ?), ?)), Y) = multiply(multiply(double_divide(X, ?), ?), multiply(Z, Y)).
0.20/0.51	Proof:
0.20/0.51	  multiply(double_divide(X, multiply(double_divide(Z, ?), ?)), Y)
0.20/0.51	= { by lemma 29 }
0.20/0.51	  multiply(double_divide(X, multiply(double_divide(?, double_divide(Y, ?)), double_divide(Y, Z))), Y)
0.20/0.51	= { by lemma 15 }
0.20/0.51	  multiply(multiply(double_divide(X, ?), ?), inverse(double_divide(Y, Z)))
0.20/0.51	= { by axiom 2 (multiply) }
0.20/0.51	  multiply(multiply(double_divide(X, ?), ?), multiply(Z, Y))
0.20/0.51	
0.20/0.51	Lemma 44: multiply(inverse(X), Y) = double_divide(X, inverse(Y)).
0.20/0.51	Proof:
0.20/0.51	  multiply(inverse(X), Y)
0.20/0.51	= { by lemma 28 }
0.20/0.51	  multiply(multiply(double_divide(X, ?), ?), Y)
0.20/0.51	= { by lemma 42 }
0.20/0.51	  multiply(multiply(double_divide(X, ?), ?), multiply(Y, double_divide(?, inverse(?))))
0.20/0.51	= { by lemma 43 }
0.20/0.51	  multiply(double_divide(X, multiply(double_divide(Y, ?), ?)), double_divide(?, inverse(?)))
0.20/0.51	= { by lemma 42 }
0.20/0.51	  double_divide(X, multiply(double_divide(Y, ?), ?))
0.20/0.51	= { by lemma 28 }
0.20/0.51	  double_divide(X, inverse(Y))
0.20/0.51	
0.20/0.51	Lemma 45: inverse(inverse(?)) = ?.
0.20/0.51	Proof:
0.20/0.51	  inverse(inverse(?))
0.20/0.51	= { by lemma 27 }
0.20/0.51	  double_divide(?, double_divide(inverse(inverse(?)), ?))
0.20/0.51	= { by lemma 41 }
0.20/0.51	  double_divide(multiply(?, inverse(?)), inverse(?))
0.20/0.51	= { by lemma 30 }
0.20/0.51	  double_divide(double_divide(inverse(?), ?), inverse(?))
0.20/0.51	= { by lemma 38 }
0.20/0.51	  double_divide(double_divide(inverse(?), ?), double_divide(inverse(inverse(?)), multiply(?, inverse(?))))
0.20/0.51	= { by lemma 24 }
0.20/0.51	  ?
0.20/0.51	
0.20/0.51	Lemma 46: multiply(double_divide(?, double_divide(Y, ?)), double_divide(?, double_divide(X, ?))) = multiply(double_divide(double_divide(X, Y), ?), ?).
0.20/0.51	Proof:
0.20/0.51	  multiply(double_divide(?, double_divide(Y, ?)), double_divide(?, double_divide(X, ?)))
0.20/0.51	= { by lemma 9 }
0.20/0.51	  multiply(double_divide(?, double_divide(Y, ?)), double_divide(Y, double_divide(X, Y)))
0.20/0.51	= { by lemma 29 }
0.20/0.51	  multiply(double_divide(double_divide(X, Y), ?), ?)
0.20/0.51	
0.20/0.51	Lemma 47: double_divide(inverse(X), inverse(Y)) = multiply(X, Y).
0.20/0.51	Proof:
0.20/0.51	  double_divide(inverse(X), inverse(Y))
0.20/0.51	= { by lemma 44 }
0.20/0.51	  multiply(inverse(inverse(X)), Y)
0.20/0.51	= { by axiom 2 (multiply) }
0.20/0.51	  inverse(double_divide(Y, inverse(inverse(X))))
0.20/0.51	= { by lemma 28 }
0.20/0.51	  multiply(double_divide(double_divide(Y, inverse(inverse(X))), ?), ?)
0.20/0.51	= { by lemma 46 }
0.20/0.51	  multiply(double_divide(?, double_divide(inverse(inverse(X)), ?)), double_divide(?, double_divide(Y, ?)))
0.20/0.51	= { by lemma 31 }
0.20/0.51	  multiply(double_divide(?, double_divide(X, ?)), double_divide(?, double_divide(Y, ?)))
0.20/0.51	= { by lemma 46 }
0.20/0.51	  multiply(double_divide(double_divide(Y, X), ?), ?)
0.20/0.51	= { by lemma 28 }
0.20/0.51	  inverse(double_divide(Y, X))
0.20/0.51	= { by axiom 2 (multiply) }
0.20/0.51	  multiply(X, Y)
0.20/0.51	
0.20/0.51	Lemma 48: multiply(X, inverse(Y)) = double_divide(inverse(X), Y).
0.20/0.51	Proof:
0.20/0.51	  multiply(X, inverse(Y))
0.20/0.51	= { by lemma 47 }
0.20/0.51	  double_divide(inverse(X), inverse(inverse(Y)))
0.20/0.51	= { by lemma 32 }
0.20/0.53	  double_divide(inverse(X), Y)
0.20/0.53	
0.20/0.53	Lemma 49: double_divide(double_divide(?, inverse(X)), Y) = double_divide(X, double_divide(inverse(Y), ?)).
0.20/0.53	Proof:
0.20/0.53	  double_divide(double_divide(?, inverse(X)), Y)
0.20/0.53	= { by lemma 45 }
0.20/0.53	  double_divide(double_divide(inverse(inverse(?)), inverse(X)), Y)
0.20/0.53	= { by lemma 11 }
0.20/0.53	  double_divide(double_divide(multiply(double_divide(inverse(multiply(?, double_divide(?, ?))), multiply(double_divide(?, ?), ?)), double_divide(?, multiply(inverse(?), multiply(?, double_divide(?, ?))))), inverse(X)), Y)
0.20/0.53	= { by lemma 36 }
0.20/0.53	  double_divide(double_divide(multiply(double_divide(inverse(multiply(?, double_divide(?, ?))), multiply(double_divide(?, ?), ?)), double_divide(?, multiply(double_divide(multiply(?, ?), ?), ?))), inverse(X)), Y)
0.20/0.53	= { by lemma 28 }
0.20/0.53	  double_divide(double_divide(multiply(double_divide(inverse(multiply(?, double_divide(?, ?))), multiply(double_divide(?, ?), ?)), double_divide(?, inverse(multiply(?, ?)))), inverse(X)), Y)
0.20/0.53	= { by lemma 43 }
0.20/0.53	  double_divide(double_divide(multiply(multiply(double_divide(inverse(multiply(?, double_divide(?, ?))), ?), ?), multiply(?, double_divide(?, inverse(multiply(?, ?))))), inverse(X)), Y)
0.20/0.53	= { by lemma 5 }
0.20/0.53	  double_divide(double_divide(multiply(multiply(double_divide(inverse(multiply(?, double_divide(?, ?))), multiply(?, double_divide(?, ?))), multiply(?, double_divide(?, ?))), multiply(?, double_divide(?, inverse(multiply(?, ?))))), inverse(X)), Y)
0.20/0.53	= { by lemma 19 }
0.20/0.53	  double_divide(double_divide(multiply(multiply(double_divide(inverse(?), ?), multiply(?, double_divide(?, ?))), multiply(?, double_divide(?, inverse(multiply(?, ?))))), inverse(X)), Y)
0.20/0.53	= { by axiom 2 (multiply) }
0.20/0.53	  double_divide(double_divide(multiply(multiply(double_divide(inverse(?), ?), multiply(?, double_divide(?, ?))), inverse(double_divide(double_divide(?, inverse(multiply(?, ?))), ?))), inverse(X)), Y)
0.20/0.53	= { by lemma 39 }
0.20/0.53	  double_divide(double_divide(multiply(multiply(double_divide(inverse(?), ?), multiply(?, double_divide(?, ?))), inverse(inverse(multiply(?, ?)))), inverse(X)), Y)
0.20/0.53	= { by lemma 48 }
0.20/0.53	  double_divide(double_divide(double_divide(inverse(multiply(double_divide(inverse(?), ?), multiply(?, double_divide(?, ?)))), inverse(multiply(?, ?))), inverse(X)), Y)
0.20/0.53	= { by lemma 47 }
0.20/0.53	  double_divide(double_divide(multiply(multiply(double_divide(inverse(?), ?), multiply(?, double_divide(?, ?))), multiply(?, ?)), inverse(X)), Y)
0.20/0.53	= { by lemma 35 }
0.20/0.53	  double_divide(double_divide(multiply(multiply(inverse(?), multiply(?, multiply(?, double_divide(?, ?)))), multiply(?, ?)), inverse(X)), Y)
0.20/0.53	= { by lemma 44 }
0.20/0.53	  double_divide(double_divide(multiply(double_divide(?, inverse(multiply(?, multiply(?, double_divide(?, ?))))), multiply(?, ?)), inverse(X)), Y)
0.20/0.53	= { by lemma 39 }
0.20/0.53	  double_divide(double_divide(multiply(double_divide(?, inverse(double_divide(double_divide(inverse(multiply(?, double_divide(?, ?))), multiply(?, multiply(?, double_divide(?, ?)))), inverse(multiply(?, double_divide(?, ?)))))), multiply(?, ?)), inverse(X)), Y)
0.20/0.53	= { by lemma 38 }
0.20/0.53	  double_divide(double_divide(multiply(double_divide(?, inverse(double_divide(inverse(?), inverse(multiply(?, double_divide(?, ?)))))), multiply(?, ?)), inverse(X)), Y)
0.20/0.53	= { by axiom 2 (multiply) }
0.20/0.53	  double_divide(double_divide(multiply(double_divide(?, multiply(inverse(multiply(?, double_divide(?, ?))), inverse(?))), multiply(?, ?)), inverse(X)), Y)
0.20/0.53	= { by lemma 44 }
0.20/0.53	  double_divide(double_divide(multiply(double_divide(?, double_divide(multiply(?, double_divide(?, ?)), inverse(inverse(?)))), multiply(?, ?)), inverse(X)), Y)
0.20/0.53	= { by lemma 45 }
0.20/0.53	  double_divide(double_divide(multiply(double_divide(?, double_divide(multiply(?, double_divide(?, ?)), ?)), multiply(?, ?)), inverse(X)), Y)
0.20/0.53	= { by lemma 27 }
0.20/0.53	  double_divide(double_divide(multiply(multiply(?, double_divide(?, ?)), multiply(?, ?)), inverse(X)), Y)
0.20/0.53	= { by lemma 28 }
0.20/0.53	  double_divide(double_divide(multiply(multiply(?, double_divide(?, ?)), multiply(?, ?)), multiply(double_divide(X, ?), ?)), Y)
0.20/0.53	= { by lemma 6 }
0.20/0.53	  double_divide(double_divide(multiply(multiply(?, double_divide(?, ?)), multiply(?, ?)), multiply(double_divide(X, ?), ?)), double_divide(double_divide(X, multiply(Y, double_divide(multiply(?, ?), multiply(?, double_divide(?, ?))))), double_divide(multiply(multiply(?, double_divide(?, ?)), multiply(?, ?)), multiply(double_divide(X, ?), ?))))
0.20/0.53	= { by lemma 9 }
0.20/0.53	  double_divide(?, double_divide(double_divide(X, multiply(Y, double_divide(multiply(?, ?), multiply(?, double_divide(?, ?))))), ?))
0.20/0.53	= { by lemma 27 }
0.20/0.53	  double_divide(X, multiply(Y, double_divide(multiply(?, ?), multiply(?, double_divide(?, ?)))))
0.20/0.53	= { by lemma 37 }
0.20/0.53	  double_divide(X, multiply(Y, inverse(?)))
0.20/0.53	= { by lemma 48 }
0.37/0.53	  double_divide(X, double_divide(inverse(Y), ?))
0.37/0.53	
0.37/0.53	Goal 1 (prove_these_axioms_3): multiply(multiply(a3, b3), c3) = multiply(a3, multiply(b3, c3)).
0.37/0.53	Proof:
0.37/0.53	  multiply(multiply(a3, b3), c3)
0.37/0.53	= { by axiom 2 (multiply) }
0.37/0.53	  inverse(double_divide(c3, multiply(a3, b3)))
0.37/0.53	= { by lemma 28 }
0.37/0.53	  multiply(double_divide(double_divide(c3, multiply(a3, b3)), ?), ?)
0.37/0.53	= { by lemma 29 }
0.37/0.53	  multiply(double_divide(?, double_divide(a3, ?)), double_divide(a3, double_divide(c3, multiply(a3, b3))))
0.37/0.53	= { by lemma 27 }
0.37/0.53	  multiply(a3, double_divide(a3, double_divide(c3, multiply(a3, b3))))
0.37/0.53	= { by lemma 32 }
0.37/0.53	  multiply(a3, double_divide(a3, inverse(inverse(double_divide(c3, multiply(a3, b3))))))
0.37/0.53	= { by axiom 2 (multiply) }
0.37/0.53	  multiply(a3, double_divide(a3, inverse(multiply(multiply(a3, b3), c3))))
0.37/0.53	= { by lemma 27 }
0.37/0.53	  multiply(a3, double_divide(a3, double_divide(?, double_divide(inverse(multiply(multiply(a3, b3), c3)), ?))))
0.37/0.53	= { by lemma 22 }
0.37/0.53	  multiply(a3, double_divide(a3, double_divide(multiply(multiply(a3, b3), c3), double_divide(inverse(?), ?))))
0.37/0.53	= { by lemma 49 }
0.37/0.53	  multiply(a3, double_divide(a3, double_divide(double_divide(?, inverse(multiply(multiply(a3, b3), c3))), ?)))
0.37/0.53	= { by lemma 44 }
0.37/0.53	  multiply(a3, double_divide(a3, double_divide(multiply(inverse(?), multiply(multiply(a3, b3), c3)), ?)))
0.37/0.53	= { by lemma 28 }
0.37/0.53	  multiply(a3, double_divide(a3, double_divide(multiply(multiply(double_divide(?, ?), ?), multiply(multiply(a3, b3), c3)), ?)))
0.37/0.53	= { by axiom 2 (multiply) }
0.37/0.53	  multiply(a3, double_divide(a3, double_divide(inverse(double_divide(multiply(multiply(a3, b3), c3), multiply(double_divide(?, ?), ?))), ?)))
0.37/0.53	= { by lemma 49 }
0.37/0.53	  multiply(a3, double_divide(double_divide(?, inverse(a3)), double_divide(multiply(multiply(a3, b3), c3), multiply(double_divide(?, ?), ?))))
0.37/0.53	= { by lemma 13 }
0.37/0.53	  multiply(a3, double_divide(inverse(b3), multiply(double_divide(c3, ?), ?)))
0.37/0.53	= { by lemma 28 }
0.37/0.53	  multiply(a3, double_divide(inverse(b3), inverse(c3)))
0.37/0.53	= { by lemma 47 }
0.37/0.53	  multiply(a3, multiply(b3, c3))
0.37/0.53	% SZS output end Proof
0.37/0.53	
0.37/0.53	RESULT: Unsatisfiable (the axioms are contradictory).
0.37/0.53	EOF