TSTP Solution File: BOO014-3 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : BOO014-3 : TPTP v8.1.2. Bugfixed v2.2.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Aug 30 18:11:24 EDT 2023
% Result : Unsatisfiable 132.11s 17.25s
% Output : Proof 134.47s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : BOO014-3 : TPTP v8.1.2. Bugfixed v2.2.0.
% 0.00/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.10/0.31 % Computer : n003.cluster.edu
% 0.10/0.31 % Model : x86_64 x86_64
% 0.10/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.31 % Memory : 8042.1875MB
% 0.10/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.31 % CPULimit : 300
% 0.10/0.31 % WCLimit : 300
% 0.10/0.32 % DateTime : Sun Aug 27 08:37:24 EDT 2023
% 0.10/0.32 % CPUTime :
% 132.11/17.24 Command-line arguments: --no-flatten-goal
% 132.11/17.25
% 132.11/17.25 % SZS status Unsatisfiable
% 132.11/17.25
% 133.71/17.45 % SZS output start Proof
% 133.71/17.45 Take the following subset of the input axioms:
% 133.71/17.46 fof(addition_is_well_defined, axiom, ![X, Y, U, V]: (~sum(X, Y, U) | (~sum(X, Y, V) | U=V))).
% 133.71/17.46 fof(additive_identity2, axiom, ![X2]: sum(X2, additive_identity, X2)).
% 133.71/17.46 fof(additive_inverse2, axiom, ![X2]: sum(X2, inverse(X2), multiplicative_identity)).
% 133.71/17.46 fof(closure_of_addition, axiom, ![X2, Y2]: sum(X2, Y2, add(X2, Y2))).
% 133.71/17.46 fof(closure_of_multiplication, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 133.71/17.46 fof(commutativity_of_addition, axiom, ![Z, X2, Y2]: (~sum(X2, Y2, Z) | sum(Y2, X2, Z))).
% 133.71/17.46 fof(commutativity_of_multiplication, axiom, ![X2, Y2, Z2]: (~product(X2, Y2, Z2) | product(Y2, X2, Z2))).
% 133.71/17.46 fof(distributivity1, axiom, ![V1, V2, V3, V4, X2, Y2, Z2]: (~product(X2, Y2, V1) | (~product(X2, Z2, V2) | (~sum(Y2, Z2, V3) | (~product(X2, V3, V4) | sum(V1, V2, V4)))))).
% 133.71/17.46 fof(distributivity5, axiom, ![X2, Y2, Z2, V1_2, V2_2, V3_2, V4_2]: (~sum(X2, Y2, V1_2) | (~sum(X2, Z2, V2_2) | (~product(Y2, Z2, V3_2) | (~sum(X2, V3_2, V4_2) | product(V1_2, V2_2, V4_2)))))).
% 133.71/17.46 fof(distributivity6, axiom, ![X2, Y2, Z2, V1_2, V2_2, V3_2, V4_2]: (~sum(X2, Y2, V1_2) | (~sum(X2, Z2, V2_2) | (~product(Y2, Z2, V3_2) | (~product(V1_2, V2_2, V4_2) | sum(X2, V3_2, V4_2)))))).
% 133.71/17.46 fof(inverse_is_self_cancelling, axiom, ![X2]: inverse(inverse(X2))=X2).
% 133.71/17.46 fof(multiplication_is_well_defined, axiom, ![X2, Y2, U2, V5]: (~product(X2, Y2, U2) | (~product(X2, Y2, V5) | U2=V5))).
% 133.71/17.46 fof(multiplicative_identity1, axiom, ![X2]: product(multiplicative_identity, X2, X2)).
% 133.71/17.46 fof(multiplicative_identity2, axiom, ![X2]: product(X2, multiplicative_identity, X2)).
% 133.71/17.46 fof(multiplicative_inverse1, axiom, ![X2]: product(inverse(X2), X2, additive_identity)).
% 133.71/17.46 fof(multiplicative_inverse2, axiom, ![X2]: product(X2, inverse(X2), additive_identity)).
% 133.71/17.46 fof(prove_equation, negated_conjecture, inverse(x_plus_y)!=x_inverse_times_y_inverse).
% 133.71/17.46 fof(x_inverse_times_y_inverse, hypothesis, product(inverse(x), inverse(y), x_inverse_times_y_inverse)).
% 133.71/17.46 fof(x_plus_y, hypothesis, sum(x, y, x_plus_y)).
% 133.71/17.46
% 133.71/17.46 Now clausify the problem and encode Horn clauses using encoding 3 of
% 133.71/17.46 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 133.71/17.46 We repeatedly replace C & s=t => u=v by the two clauses:
% 133.71/17.46 fresh(y, y, x1...xn) = u
% 133.71/17.46 C => fresh(s, t, x1...xn) = v
% 133.71/17.46 where fresh is a fresh function symbol and x1..xn are the free
% 133.71/17.46 variables of u and v.
% 133.71/17.46 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 133.71/17.46 input problem has no model of domain size 1).
% 133.71/17.46
% 133.71/17.46 The encoding turns the above axioms into the following unit equations and goals:
% 133.71/17.46
% 133.71/17.46 Axiom 1 (inverse_is_self_cancelling): inverse(inverse(X)) = X.
% 133.71/17.46 Axiom 2 (additive_identity2): sum(X, additive_identity, X) = true.
% 133.71/17.46 Axiom 3 (x_plus_y): sum(x, y, x_plus_y) = true.
% 133.71/17.46 Axiom 4 (multiplicative_identity2): product(X, multiplicative_identity, X) = true.
% 133.71/17.46 Axiom 5 (multiplicative_identity1): product(multiplicative_identity, X, X) = true.
% 133.71/17.46 Axiom 6 (multiplication_is_well_defined): fresh(X, X, Y, Z) = Z.
% 133.71/17.46 Axiom 7 (addition_is_well_defined): fresh3(X, X, Y, Z) = Z.
% 133.71/17.46 Axiom 8 (additive_inverse2): sum(X, inverse(X), multiplicative_identity) = true.
% 133.71/17.46 Axiom 9 (multiplicative_inverse2): product(X, inverse(X), additive_identity) = true.
% 133.71/17.46 Axiom 10 (multiplicative_inverse1): product(inverse(X), X, additive_identity) = true.
% 133.71/17.46 Axiom 11 (distributivity1): fresh42(X, X, Y, Z, W) = true.
% 133.71/17.46 Axiom 12 (distributivity5): fresh26(X, X, Y, Z, W) = true.
% 133.71/17.46 Axiom 13 (distributivity6): fresh22(X, X, Y, Z, W) = true.
% 133.71/17.46 Axiom 14 (commutativity_of_multiplication): fresh6(X, X, Y, Z, W) = true.
% 133.71/17.46 Axiom 15 (commutativity_of_addition): fresh5(X, X, Y, Z, W) = true.
% 133.71/17.46 Axiom 16 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 133.71/17.46 Axiom 17 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 133.71/17.46 Axiom 18 (x_inverse_times_y_inverse): product(inverse(x), inverse(y), x_inverse_times_y_inverse) = true.
% 133.71/17.46 Axiom 19 (addition_is_well_defined): fresh4(X, X, Y, Z, W, V) = W.
% 133.71/17.46 Axiom 20 (multiplication_is_well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 133.71/17.46 Axiom 21 (distributivity5): fresh24(X, X, Y, Z, W, V, U) = product(W, V, U).
% 133.71/17.46 Axiom 22 (distributivity6): fresh20(X, X, Y, Z, W, V, U) = sum(Y, V, U).
% 133.71/17.46 Axiom 23 (distributivity1): fresh40(X, X, Y, Z, W, V, U, T) = sum(Z, V, T).
% 133.71/17.46 Axiom 24 (distributivity5): fresh25(X, X, Y, Z, W, V, U, T) = fresh26(sum(Y, Z, W), true, W, U, T).
% 133.71/17.46 Axiom 25 (commutativity_of_multiplication): fresh6(product(X, Y, Z), true, X, Y, Z) = product(Y, X, Z).
% 133.71/17.46 Axiom 26 (commutativity_of_addition): fresh5(sum(X, Y, Z), true, X, Y, Z) = sum(Y, X, Z).
% 133.71/17.46 Axiom 27 (distributivity1): fresh41(X, X, Y, Z, W, V, U, T, S) = fresh42(sum(Z, V, T), true, W, U, S).
% 133.71/17.46 Axiom 28 (distributivity6): fresh21(X, X, Y, Z, W, V, U, T, S) = fresh22(sum(Y, Z, W), true, Y, T, S).
% 133.71/17.46 Axiom 29 (addition_is_well_defined): fresh4(sum(X, Y, Z), true, X, Y, W, Z) = fresh3(sum(X, Y, W), true, W, Z).
% 133.71/17.46 Axiom 30 (multiplication_is_well_defined): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 133.71/17.46 Axiom 31 (distributivity5): fresh23(X, X, Y, Z, W, V, U, T, S) = fresh24(sum(Y, V, U), true, Y, Z, W, U, S).
% 133.71/17.46 Axiom 32 (distributivity6): fresh19(X, X, Y, Z, W, V, U, T, S) = fresh20(sum(Y, V, U), true, Y, Z, W, T, S).
% 133.71/17.46 Axiom 33 (distributivity1): fresh39(X, X, Y, Z, W, V, U, T, S) = fresh40(product(Y, Z, W), true, Z, W, V, U, T, S).
% 133.71/17.46 Axiom 34 (distributivity1): fresh39(product(X, Y, Z), true, X, W, V, U, T, Y, Z) = fresh41(product(X, U, T), true, X, W, V, U, T, Y, Z).
% 133.71/17.46 Axiom 35 (distributivity5): fresh23(product(X, Y, Z), true, W, X, V, Y, U, Z, T) = fresh25(sum(W, Z, T), true, W, X, V, Y, U, T).
% 133.71/17.46 Axiom 36 (distributivity6): fresh19(product(X, Y, Z), true, W, V, X, U, Y, T, Z) = fresh21(product(V, U, T), true, W, V, X, U, Y, T, Z).
% 133.71/17.46
% 133.71/17.46 Lemma 37: fresh3(sum(X, Y, Z), true, Z, add(X, Y)) = Z.
% 133.71/17.46 Proof:
% 133.71/17.46 fresh3(sum(X, Y, Z), true, Z, add(X, Y))
% 133.71/17.46 = { by axiom 29 (addition_is_well_defined) R->L }
% 133.71/17.46 fresh4(sum(X, Y, add(X, Y)), true, X, Y, Z, add(X, Y))
% 133.71/17.46 = { by axiom 16 (closure_of_addition) }
% 133.71/17.46 fresh4(true, true, X, Y, Z, add(X, Y))
% 133.71/17.46 = { by axiom 19 (addition_is_well_defined) }
% 133.71/17.46 Z
% 133.71/17.46
% 133.71/17.46 Lemma 38: sum(X, Y, add(Y, X)) = true.
% 133.71/17.46 Proof:
% 133.71/17.46 sum(X, Y, add(Y, X))
% 133.71/17.46 = { by axiom 26 (commutativity_of_addition) R->L }
% 133.71/17.46 fresh5(sum(Y, X, add(Y, X)), true, Y, X, add(Y, X))
% 133.71/17.46 = { by axiom 16 (closure_of_addition) }
% 133.71/17.46 fresh5(true, true, Y, X, add(Y, X))
% 133.71/17.46 = { by axiom 15 (commutativity_of_addition) }
% 133.71/17.46 true
% 133.71/17.46
% 133.71/17.46 Lemma 39: add(X, Y) = add(Y, X).
% 133.71/17.46 Proof:
% 133.71/17.46 add(X, Y)
% 133.71/17.46 = { by lemma 37 R->L }
% 133.71/17.46 fresh3(sum(Y, X, add(X, Y)), true, add(X, Y), add(Y, X))
% 133.71/17.46 = { by lemma 38 }
% 133.71/17.46 fresh3(true, true, add(X, Y), add(Y, X))
% 133.71/17.46 = { by axiom 7 (addition_is_well_defined) }
% 133.71/17.46 add(Y, X)
% 133.71/17.46
% 133.71/17.46 Lemma 40: fresh3(sum(X, additive_identity, Y), true, Y, X) = Y.
% 133.71/17.46 Proof:
% 133.71/17.46 fresh3(sum(X, additive_identity, Y), true, Y, X)
% 133.71/17.46 = { by axiom 29 (addition_is_well_defined) R->L }
% 133.71/17.46 fresh4(sum(X, additive_identity, X), true, X, additive_identity, Y, X)
% 133.71/17.46 = { by axiom 2 (additive_identity2) }
% 133.71/17.46 fresh4(true, true, X, additive_identity, Y, X)
% 133.71/17.46 = { by axiom 19 (addition_is_well_defined) }
% 133.71/17.46 Y
% 133.71/17.46
% 133.71/17.46 Lemma 41: add(X, additive_identity) = X.
% 133.71/17.46 Proof:
% 133.71/17.46 add(X, additive_identity)
% 133.71/17.46 = { by lemma 40 R->L }
% 133.71/17.46 fresh3(sum(X, additive_identity, add(X, additive_identity)), true, add(X, additive_identity), X)
% 133.71/17.46 = { by axiom 16 (closure_of_addition) }
% 133.71/17.46 fresh3(true, true, add(X, additive_identity), X)
% 133.71/17.46 = { by axiom 7 (addition_is_well_defined) }
% 133.71/17.46 X
% 133.71/17.46
% 133.71/17.46 Lemma 42: fresh(product(X, Y, Z), true, Z, multiply(X, Y)) = Z.
% 133.71/17.46 Proof:
% 133.71/17.46 fresh(product(X, Y, Z), true, Z, multiply(X, Y))
% 133.71/17.46 = { by axiom 30 (multiplication_is_well_defined) R->L }
% 133.71/17.46 fresh2(product(X, Y, multiply(X, Y)), true, X, Y, Z, multiply(X, Y))
% 133.71/17.46 = { by axiom 17 (closure_of_multiplication) }
% 133.71/17.46 fresh2(true, true, X, Y, Z, multiply(X, Y))
% 133.71/17.46 = { by axiom 20 (multiplication_is_well_defined) }
% 133.71/17.46 Z
% 133.71/17.46
% 133.71/17.46 Lemma 43: multiply(X, Y) = multiply(Y, X).
% 133.71/17.46 Proof:
% 133.71/17.46 multiply(X, Y)
% 133.71/17.46 = { by lemma 42 R->L }
% 133.71/17.46 fresh(product(Y, X, multiply(X, Y)), true, multiply(X, Y), multiply(Y, X))
% 133.71/17.46 = { by axiom 25 (commutativity_of_multiplication) R->L }
% 133.71/17.46 fresh(fresh6(product(X, Y, multiply(X, Y)), true, X, Y, multiply(X, Y)), true, multiply(X, Y), multiply(Y, X))
% 133.71/17.46 = { by axiom 17 (closure_of_multiplication) }
% 133.71/17.46 fresh(fresh6(true, true, X, Y, multiply(X, Y)), true, multiply(X, Y), multiply(Y, X))
% 133.71/17.46 = { by axiom 14 (commutativity_of_multiplication) }
% 133.71/17.46 fresh(true, true, multiply(X, Y), multiply(Y, X))
% 133.71/17.46 = { by axiom 6 (multiplication_is_well_defined) }
% 133.71/17.46 multiply(Y, X)
% 133.71/17.46
% 133.71/17.46 Lemma 44: fresh19(X, X, Y, Z, W, inverse(Y), multiplicative_identity, V, U) = sum(Y, V, U).
% 133.71/17.46 Proof:
% 133.71/17.46 fresh19(X, X, Y, Z, W, inverse(Y), multiplicative_identity, V, U)
% 133.71/17.46 = { by axiom 32 (distributivity6) }
% 133.71/17.46 fresh20(sum(Y, inverse(Y), multiplicative_identity), true, Y, Z, W, V, U)
% 133.71/17.46 = { by axiom 8 (additive_inverse2) }
% 133.71/17.46 fresh20(true, true, Y, Z, W, V, U)
% 133.71/17.46 = { by axiom 22 (distributivity6) }
% 133.71/17.46 sum(Y, V, U)
% 133.71/17.46
% 133.71/17.46 Lemma 45: fresh21(product(X, Y, Z), true, W, X, V, Y, multiplicative_identity, Z, V) = fresh19(U, U, W, X, V, Y, multiplicative_identity, Z, V).
% 133.71/17.46 Proof:
% 133.71/17.46 fresh21(product(X, Y, Z), true, W, X, V, Y, multiplicative_identity, Z, V)
% 133.71/17.46 = { by axiom 36 (distributivity6) R->L }
% 133.71/17.46 fresh19(product(V, multiplicative_identity, V), true, W, X, V, Y, multiplicative_identity, Z, V)
% 133.71/17.46 = { by axiom 4 (multiplicative_identity2) }
% 133.71/17.46 fresh19(true, true, W, X, V, Y, multiplicative_identity, Z, V)
% 133.71/17.46 = { by axiom 32 (distributivity6) }
% 133.71/17.46 fresh20(sum(W, Y, multiplicative_identity), true, W, X, V, Z, V)
% 133.71/17.46 = { by axiom 32 (distributivity6) R->L }
% 133.71/17.46 fresh19(U, U, W, X, V, Y, multiplicative_identity, Z, V)
% 133.71/17.46
% 133.71/17.46 Lemma 46: add(x, inverse(y)) = add(x, x_inverse_times_y_inverse).
% 133.71/17.46 Proof:
% 133.71/17.46 add(x, inverse(y))
% 133.71/17.46 = { by lemma 37 R->L }
% 133.71/17.46 fresh3(sum(x, x_inverse_times_y_inverse, add(x, inverse(y))), true, add(x, inverse(y)), add(x, x_inverse_times_y_inverse))
% 133.71/17.46 = { by lemma 44 R->L }
% 133.71/17.46 fresh3(fresh19(X, X, x, inverse(y), add(x, inverse(y)), inverse(x), multiplicative_identity, x_inverse_times_y_inverse, add(x, inverse(y))), true, add(x, inverse(y)), add(x, x_inverse_times_y_inverse))
% 133.71/17.47 = { by lemma 45 R->L }
% 133.71/17.47 fresh3(fresh21(product(inverse(y), inverse(x), x_inverse_times_y_inverse), true, x, inverse(y), add(x, inverse(y)), inverse(x), multiplicative_identity, x_inverse_times_y_inverse, add(x, inverse(y))), true, add(x, inverse(y)), add(x, x_inverse_times_y_inverse))
% 133.71/17.47 = { by axiom 25 (commutativity_of_multiplication) R->L }
% 133.71/17.47 fresh3(fresh21(fresh6(product(inverse(x), inverse(y), x_inverse_times_y_inverse), true, inverse(x), inverse(y), x_inverse_times_y_inverse), true, x, inverse(y), add(x, inverse(y)), inverse(x), multiplicative_identity, x_inverse_times_y_inverse, add(x, inverse(y))), true, add(x, inverse(y)), add(x, x_inverse_times_y_inverse))
% 133.71/17.47 = { by axiom 18 (x_inverse_times_y_inverse) }
% 133.71/17.47 fresh3(fresh21(fresh6(true, true, inverse(x), inverse(y), x_inverse_times_y_inverse), true, x, inverse(y), add(x, inverse(y)), inverse(x), multiplicative_identity, x_inverse_times_y_inverse, add(x, inverse(y))), true, add(x, inverse(y)), add(x, x_inverse_times_y_inverse))
% 133.71/17.47 = { by axiom 14 (commutativity_of_multiplication) }
% 133.71/17.47 fresh3(fresh21(true, true, x, inverse(y), add(x, inverse(y)), inverse(x), multiplicative_identity, x_inverse_times_y_inverse, add(x, inverse(y))), true, add(x, inverse(y)), add(x, x_inverse_times_y_inverse))
% 133.71/17.47 = { by axiom 28 (distributivity6) }
% 133.71/17.47 fresh3(fresh22(sum(x, inverse(y), add(x, inverse(y))), true, x, x_inverse_times_y_inverse, add(x, inverse(y))), true, add(x, inverse(y)), add(x, x_inverse_times_y_inverse))
% 133.71/17.47 = { by axiom 16 (closure_of_addition) }
% 133.71/17.47 fresh3(fresh22(true, true, x, x_inverse_times_y_inverse, add(x, inverse(y))), true, add(x, inverse(y)), add(x, x_inverse_times_y_inverse))
% 133.71/17.47 = { by axiom 13 (distributivity6) }
% 133.71/17.47 fresh3(true, true, add(x, inverse(y)), add(x, x_inverse_times_y_inverse))
% 133.71/17.47 = { by axiom 7 (addition_is_well_defined) }
% 133.71/17.47 add(x, x_inverse_times_y_inverse)
% 133.71/17.47
% 133.71/17.47 Lemma 47: multiply(X, inverse(X)) = additive_identity.
% 133.71/17.47 Proof:
% 133.71/17.47 multiply(X, inverse(X))
% 133.71/17.47 = { by axiom 20 (multiplication_is_well_defined) R->L }
% 133.71/17.47 fresh2(true, true, X, inverse(X), multiply(X, inverse(X)), additive_identity)
% 133.71/17.47 = { by axiom 9 (multiplicative_inverse2) R->L }
% 133.71/17.47 fresh2(product(X, inverse(X), additive_identity), true, X, inverse(X), multiply(X, inverse(X)), additive_identity)
% 133.71/17.47 = { by axiom 30 (multiplication_is_well_defined) }
% 133.71/17.47 fresh(product(X, inverse(X), multiply(X, inverse(X))), true, multiply(X, inverse(X)), additive_identity)
% 133.71/17.47 = { by axiom 17 (closure_of_multiplication) }
% 133.71/17.47 fresh(true, true, multiply(X, inverse(X)), additive_identity)
% 133.71/17.47 = { by axiom 6 (multiplication_is_well_defined) }
% 133.71/17.47 additive_identity
% 133.71/17.47
% 133.71/17.47 Lemma 48: fresh39(X, X, Y, Z, multiply(Y, Z), W, V, U, T) = sum(multiply(Y, Z), V, T).
% 133.71/17.47 Proof:
% 133.71/17.47 fresh39(X, X, Y, Z, multiply(Y, Z), W, V, U, T)
% 133.71/17.47 = { by axiom 33 (distributivity1) }
% 133.71/17.47 fresh40(product(Y, Z, multiply(Y, Z)), true, Z, multiply(Y, Z), W, V, U, T)
% 133.71/17.47 = { by axiom 17 (closure_of_multiplication) }
% 133.71/17.47 fresh40(true, true, Z, multiply(Y, Z), W, V, U, T)
% 133.71/17.47 = { by axiom 23 (distributivity1) }
% 133.71/17.47 sum(multiply(Y, Z), V, T)
% 133.71/17.47
% 133.71/17.47 Lemma 49: fresh(product(X, multiplicative_identity, Y), true, Y, X) = Y.
% 133.71/17.47 Proof:
% 133.71/17.47 fresh(product(X, multiplicative_identity, Y), true, Y, X)
% 133.71/17.47 = { by axiom 30 (multiplication_is_well_defined) R->L }
% 133.71/17.47 fresh2(product(X, multiplicative_identity, X), true, X, multiplicative_identity, Y, X)
% 133.71/17.47 = { by axiom 4 (multiplicative_identity2) }
% 133.71/17.47 fresh2(true, true, X, multiplicative_identity, Y, X)
% 133.71/17.47 = { by axiom 20 (multiplication_is_well_defined) }
% 133.71/17.47 Y
% 133.71/17.47
% 133.71/17.47 Lemma 50: fresh25(X, X, Y, Z, add(Y, Z), W, V, U) = true.
% 133.71/17.47 Proof:
% 133.71/17.47 fresh25(X, X, Y, Z, add(Y, Z), W, V, U)
% 133.71/17.47 = { by axiom 24 (distributivity5) }
% 133.71/17.47 fresh26(sum(Y, Z, add(Y, Z)), true, add(Y, Z), V, U)
% 133.71/17.47 = { by axiom 16 (closure_of_addition) }
% 133.71/17.47 fresh26(true, true, add(Y, Z), V, U)
% 133.71/17.47 = { by axiom 12 (distributivity5) }
% 133.71/17.47 true
% 133.71/17.47
% 133.71/17.47 Lemma 51: fresh39(X, X, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y) = true.
% 133.71/17.47 Proof:
% 133.71/17.47 fresh39(X, X, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 133.71/17.47 = { by axiom 33 (distributivity1) }
% 133.71/17.47 fresh40(product(Y, Z, W), true, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 133.71/17.47 = { by axiom 33 (distributivity1) R->L }
% 133.71/17.47 fresh39(true, true, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 133.71/17.47 = { by axiom 4 (multiplicative_identity2) R->L }
% 133.71/17.47 fresh39(product(Y, multiplicative_identity, Y), true, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 133.71/17.47 = { by axiom 34 (distributivity1) }
% 133.71/17.47 fresh41(product(Y, multiplicative_identity, Y), true, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 133.71/17.47 = { by axiom 4 (multiplicative_identity2) }
% 133.71/17.47 fresh41(true, true, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 133.71/17.47 = { by axiom 27 (distributivity1) }
% 133.71/17.47 fresh42(sum(Z, multiplicative_identity, multiplicative_identity), true, W, Y, Y)
% 133.71/17.47 = { by lemma 49 R->L }
% 133.71/17.47 fresh42(sum(Z, multiplicative_identity, fresh(product(add(Z, multiplicative_identity), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(Z, multiplicative_identity))), true, W, Y, Y)
% 133.71/17.47 = { by axiom 21 (distributivity5) R->L }
% 133.71/17.47 fresh42(sum(Z, multiplicative_identity, fresh(fresh24(true, true, Z, multiplicative_identity, add(Z, multiplicative_identity), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(Z, multiplicative_identity))), true, W, Y, Y)
% 133.71/17.47 = { by axiom 8 (additive_inverse2) R->L }
% 133.71/17.47 fresh42(sum(Z, multiplicative_identity, fresh(fresh24(sum(Z, inverse(Z), multiplicative_identity), true, Z, multiplicative_identity, add(Z, multiplicative_identity), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(Z, multiplicative_identity))), true, W, Y, Y)
% 133.71/17.47 = { by axiom 31 (distributivity5) R->L }
% 133.71/17.47 fresh42(sum(Z, multiplicative_identity, fresh(fresh23(true, true, Z, multiplicative_identity, add(Z, multiplicative_identity), inverse(Z), multiplicative_identity, inverse(Z), multiplicative_identity), true, multiplicative_identity, add(Z, multiplicative_identity))), true, W, Y, Y)
% 133.71/17.47 = { by axiom 5 (multiplicative_identity1) R->L }
% 133.71/17.47 fresh42(sum(Z, multiplicative_identity, fresh(fresh23(product(multiplicative_identity, inverse(Z), inverse(Z)), true, Z, multiplicative_identity, add(Z, multiplicative_identity), inverse(Z), multiplicative_identity, inverse(Z), multiplicative_identity), true, multiplicative_identity, add(Z, multiplicative_identity))), true, W, Y, Y)
% 133.71/17.47 = { by axiom 35 (distributivity5) }
% 133.71/17.47 fresh42(sum(Z, multiplicative_identity, fresh(fresh25(sum(Z, inverse(Z), multiplicative_identity), true, Z, multiplicative_identity, add(Z, multiplicative_identity), inverse(Z), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(Z, multiplicative_identity))), true, W, Y, Y)
% 133.71/17.47 = { by axiom 8 (additive_inverse2) }
% 133.71/17.47 fresh42(sum(Z, multiplicative_identity, fresh(fresh25(true, true, Z, multiplicative_identity, add(Z, multiplicative_identity), inverse(Z), multiplicative_identity, multiplicative_identity), true, multiplicative_identity, add(Z, multiplicative_identity))), true, W, Y, Y)
% 133.71/17.47 = { by lemma 50 }
% 133.71/17.47 fresh42(sum(Z, multiplicative_identity, fresh(true, true, multiplicative_identity, add(Z, multiplicative_identity))), true, W, Y, Y)
% 133.71/17.47 = { by axiom 6 (multiplication_is_well_defined) }
% 133.71/17.47 fresh42(sum(Z, multiplicative_identity, add(Z, multiplicative_identity)), true, W, Y, Y)
% 133.71/17.47 = { by axiom 16 (closure_of_addition) }
% 133.71/17.47 fresh42(true, true, W, Y, Y)
% 133.71/17.47 = { by axiom 11 (distributivity1) }
% 133.71/17.47 true
% 133.71/17.47
% 133.71/17.47 Lemma 52: sum(multiply(X, Y), X, X) = true.
% 133.71/17.47 Proof:
% 133.71/17.47 sum(multiply(X, Y), X, X)
% 133.71/17.47 = { by lemma 48 R->L }
% 133.71/17.47 fresh39(Z, Z, X, Y, multiply(X, Y), multiplicative_identity, X, multiplicative_identity, X)
% 133.71/17.47 = { by lemma 51 }
% 133.71/17.47 true
% 133.71/17.47
% 133.71/17.47 Lemma 53: add(X, multiply(X, Y)) = X.
% 133.71/17.47 Proof:
% 133.71/17.47 add(X, multiply(X, Y))
% 133.71/17.47 = { by lemma 39 R->L }
% 133.71/17.47 add(multiply(X, Y), X)
% 133.71/17.47 = { by axiom 7 (addition_is_well_defined) R->L }
% 133.71/17.47 fresh3(true, true, X, add(multiply(X, Y), X))
% 133.71/17.47 = { by lemma 52 R->L }
% 133.71/17.47 fresh3(sum(multiply(X, Y), X, X), true, X, add(multiply(X, Y), X))
% 133.71/17.47 = { by lemma 37 }
% 133.71/17.47 X
% 133.71/17.47
% 133.71/17.47 Lemma 54: add(X, multiply(Y, X)) = X.
% 133.71/17.47 Proof:
% 133.71/17.47 add(X, multiply(Y, X))
% 133.71/17.47 = { by lemma 43 R->L }
% 133.71/17.47 add(X, multiply(X, Y))
% 133.71/17.47 = { by lemma 53 }
% 133.71/17.47 X
% 133.71/17.47
% 133.71/17.47 Lemma 55: fresh23(product(X, Y, additive_identity), true, Z, X, W, Y, V, additive_identity, Z) = fresh25(U, U, Z, X, W, T, V, Z).
% 133.71/17.47 Proof:
% 133.71/17.47 fresh23(product(X, Y, additive_identity), true, Z, X, W, Y, V, additive_identity, Z)
% 133.71/17.47 = { by axiom 35 (distributivity5) }
% 133.71/17.47 fresh25(sum(Z, additive_identity, Z), true, Z, X, W, Y, V, Z)
% 133.71/17.47 = { by axiom 2 (additive_identity2) }
% 133.71/17.47 fresh25(true, true, Z, X, W, Y, V, Z)
% 133.71/17.47 = { by axiom 24 (distributivity5) }
% 133.71/17.47 fresh26(sum(Z, X, W), true, W, V, Z)
% 133.71/17.47 = { by axiom 24 (distributivity5) R->L }
% 133.71/17.47 fresh25(U, U, Z, X, W, T, V, Z)
% 133.71/17.47
% 133.71/17.47 Lemma 56: fresh25(X, X, Y, Z, W, V, Y, Y) = product(W, Y, Y).
% 133.71/17.47 Proof:
% 133.71/17.47 fresh25(X, X, Y, Z, W, V, Y, Y)
% 133.71/17.47 = { by lemma 55 R->L }
% 133.71/17.47 fresh23(product(Z, additive_identity, additive_identity), true, Y, Z, W, additive_identity, Y, additive_identity, Y)
% 133.71/17.47 = { by lemma 40 R->L }
% 133.71/17.47 fresh23(product(Z, additive_identity, fresh3(sum(multiply(additive_identity, Z), additive_identity, additive_identity), true, additive_identity, multiply(additive_identity, Z))), true, Y, Z, W, additive_identity, Y, additive_identity, Y)
% 133.71/17.47 = { by lemma 52 }
% 133.71/17.47 fresh23(product(Z, additive_identity, fresh3(true, true, additive_identity, multiply(additive_identity, Z))), true, Y, Z, W, additive_identity, Y, additive_identity, Y)
% 133.71/17.47 = { by axiom 7 (addition_is_well_defined) }
% 133.71/17.47 fresh23(product(Z, additive_identity, multiply(additive_identity, Z)), true, Y, Z, W, additive_identity, Y, additive_identity, Y)
% 133.71/17.47 = { by lemma 43 }
% 133.71/17.47 fresh23(product(Z, additive_identity, multiply(Z, additive_identity)), true, Y, Z, W, additive_identity, Y, additive_identity, Y)
% 133.71/17.47 = { by axiom 17 (closure_of_multiplication) }
% 133.71/17.47 fresh23(true, true, Y, Z, W, additive_identity, Y, additive_identity, Y)
% 133.71/17.47 = { by axiom 31 (distributivity5) }
% 133.71/17.47 fresh24(sum(Y, additive_identity, Y), true, Y, Z, W, Y, Y)
% 133.71/17.47 = { by axiom 2 (additive_identity2) }
% 133.71/17.47 fresh24(true, true, Y, Z, W, Y, Y)
% 133.71/17.47 = { by axiom 21 (distributivity5) }
% 133.71/17.47 product(W, Y, Y)
% 133.71/17.47
% 133.71/17.47 Lemma 57: product(add(X, Y), X, X) = true.
% 133.71/17.47 Proof:
% 133.71/17.47 product(add(X, Y), X, X)
% 133.71/17.47 = { by lemma 39 R->L }
% 133.71/17.47 product(add(Y, X), X, X)
% 133.71/17.47 = { by lemma 56 R->L }
% 133.71/17.47 fresh25(Z, Z, X, Y, add(Y, X), W, X, X)
% 133.71/17.47 = { by lemma 39 R->L }
% 133.71/17.47 fresh25(Z, Z, X, Y, add(X, Y), W, X, X)
% 133.71/17.47 = { by lemma 50 }
% 133.71/17.47 true
% 133.71/17.47
% 133.71/17.47 Lemma 58: fresh23(X, X, Y, inverse(Z), W, Z, V, U, Y) = fresh25(T, T, Y, inverse(Z), W, S, V, Y).
% 133.71/17.47 Proof:
% 133.71/17.47 fresh23(X, X, Y, inverse(Z), W, Z, V, U, Y)
% 133.71/17.47 = { by axiom 31 (distributivity5) }
% 133.71/17.47 fresh24(sum(Y, Z, V), true, Y, inverse(Z), W, V, Y)
% 133.71/17.47 = { by axiom 31 (distributivity5) R->L }
% 133.71/17.47 fresh23(true, true, Y, inverse(Z), W, Z, V, additive_identity, Y)
% 133.71/17.47 = { by axiom 10 (multiplicative_inverse1) R->L }
% 133.71/17.47 fresh23(product(inverse(Z), Z, additive_identity), true, Y, inverse(Z), W, Z, V, additive_identity, Y)
% 133.71/17.47 = { by lemma 55 }
% 133.71/17.47 fresh25(T, T, Y, inverse(Z), W, S, V, Y)
% 133.71/17.47
% 133.71/17.47 Lemma 59: fresh23(X, X, Y, Z, W, V, add(Y, V), U, T) = product(W, add(Y, V), T).
% 133.71/17.47 Proof:
% 133.71/17.47 fresh23(X, X, Y, Z, W, V, add(Y, V), U, T)
% 133.71/17.47 = { by axiom 31 (distributivity5) }
% 133.71/17.47 fresh24(sum(Y, V, add(Y, V)), true, Y, Z, W, add(Y, V), T)
% 133.71/17.47 = { by axiom 16 (closure_of_addition) }
% 133.71/17.47 fresh24(true, true, Y, Z, W, add(Y, V), T)
% 133.71/17.47 = { by axiom 21 (distributivity5) }
% 133.71/17.47 product(W, add(Y, V), T)
% 133.71/17.47
% 133.71/17.47 Lemma 60: fresh25(X, X, Y, inverse(Z), W, V, add(Y, Z), Y) = product(W, add(Y, Z), Y).
% 133.71/17.47 Proof:
% 133.71/17.47 fresh25(X, X, Y, inverse(Z), W, V, add(Y, Z), Y)
% 133.71/17.47 = { by lemma 58 R->L }
% 133.71/17.47 fresh23(U, U, Y, inverse(Z), W, Z, add(Y, Z), T, Y)
% 133.71/17.47 = { by lemma 59 }
% 133.71/17.47 product(W, add(Y, Z), Y)
% 133.71/17.47
% 133.71/17.47 Lemma 61: multiply(add(X, Y), add(Y, inverse(X))) = Y.
% 133.71/17.47 Proof:
% 133.71/17.47 multiply(add(X, Y), add(Y, inverse(X)))
% 133.71/17.47 = { by lemma 39 R->L }
% 133.71/17.47 multiply(add(Y, X), add(Y, inverse(X)))
% 133.71/17.47 = { by lemma 43 R->L }
% 133.71/17.48 multiply(add(Y, inverse(X)), add(Y, X))
% 133.71/17.48 = { by axiom 6 (multiplication_is_well_defined) R->L }
% 133.71/17.48 fresh(true, true, Y, multiply(add(Y, inverse(X)), add(Y, X)))
% 133.71/17.48 = { by lemma 50 R->L }
% 133.71/17.48 fresh(fresh25(Z, Z, Y, inverse(X), add(Y, inverse(X)), W, add(Y, X), Y), true, Y, multiply(add(Y, inverse(X)), add(Y, X)))
% 133.71/17.48 = { by lemma 60 }
% 133.71/17.48 fresh(product(add(Y, inverse(X)), add(Y, X), Y), true, Y, multiply(add(Y, inverse(X)), add(Y, X)))
% 133.71/17.48 = { by lemma 42 }
% 133.71/17.48 Y
% 133.71/17.48
% 133.71/17.48 Lemma 62: fresh22(sum(X, Y, Z), true, X, multiply(Y, inverse(X)), Z) = sum(X, multiply(Y, inverse(X)), Z).
% 133.71/17.48 Proof:
% 133.71/17.48 fresh22(sum(X, Y, Z), true, X, multiply(Y, inverse(X)), Z)
% 133.71/17.48 = { by axiom 28 (distributivity6) R->L }
% 133.71/17.48 fresh21(true, true, X, Y, Z, inverse(X), multiplicative_identity, multiply(Y, inverse(X)), Z)
% 133.71/17.48 = { by axiom 17 (closure_of_multiplication) R->L }
% 133.71/17.48 fresh21(product(Y, inverse(X), multiply(Y, inverse(X))), true, X, Y, Z, inverse(X), multiplicative_identity, multiply(Y, inverse(X)), Z)
% 133.71/17.48 = { by lemma 45 }
% 133.71/17.48 fresh19(W, W, X, Y, Z, inverse(X), multiplicative_identity, multiply(Y, inverse(X)), Z)
% 133.71/17.48 = { by lemma 44 }
% 133.71/17.48 sum(X, multiply(Y, inverse(X)), Z)
% 133.71/17.48
% 133.71/17.48 Lemma 63: add(X, multiply(Y, inverse(X))) = add(X, Y).
% 133.71/17.48 Proof:
% 133.71/17.48 add(X, multiply(Y, inverse(X)))
% 133.71/17.48 = { by axiom 7 (addition_is_well_defined) R->L }
% 133.71/17.48 fresh3(true, true, add(X, Y), add(X, multiply(Y, inverse(X))))
% 133.71/17.48 = { by axiom 13 (distributivity6) R->L }
% 133.71/17.48 fresh3(fresh22(true, true, X, multiply(Y, inverse(X)), add(X, Y)), true, add(X, Y), add(X, multiply(Y, inverse(X))))
% 133.71/17.48 = { by axiom 16 (closure_of_addition) R->L }
% 133.71/17.48 fresh3(fresh22(sum(X, Y, add(X, Y)), true, X, multiply(Y, inverse(X)), add(X, Y)), true, add(X, Y), add(X, multiply(Y, inverse(X))))
% 133.71/17.48 = { by lemma 62 }
% 133.71/17.48 fresh3(sum(X, multiply(Y, inverse(X)), add(X, Y)), true, add(X, Y), add(X, multiply(Y, inverse(X))))
% 133.71/17.48 = { by lemma 37 }
% 133.71/17.48 add(X, Y)
% 133.71/17.48
% 133.71/17.48 Lemma 64: fresh25(X, X, multiply(Y, Z), Y, Y, W, V, U) = true.
% 133.71/17.48 Proof:
% 133.71/17.48 fresh25(X, X, multiply(Y, Z), Y, Y, W, V, U)
% 133.71/17.48 = { by axiom 24 (distributivity5) }
% 133.71/17.48 fresh26(sum(multiply(Y, Z), Y, Y), true, Y, V, U)
% 133.71/17.48 = { by lemma 52 }
% 133.71/17.48 fresh26(true, true, Y, V, U)
% 133.71/17.48 = { by axiom 12 (distributivity5) }
% 133.71/17.48 true
% 133.71/17.48
% 133.71/17.48 Lemma 65: multiply(inverse(X), add(X, multiply(Y, inverse(X)))) = multiply(Y, inverse(X)).
% 133.71/17.48 Proof:
% 133.71/17.48 multiply(inverse(X), add(X, multiply(Y, inverse(X))))
% 133.71/17.48 = { by axiom 6 (multiplication_is_well_defined) R->L }
% 133.71/17.48 fresh(true, true, multiply(Y, inverse(X)), multiply(inverse(X), add(X, multiply(Y, inverse(X)))))
% 133.71/17.48 = { by lemma 64 R->L }
% 133.71/17.48 fresh(fresh25(Z, Z, multiply(inverse(X), Y), inverse(X), inverse(X), W, add(multiply(inverse(X), Y), X), multiply(inverse(X), Y)), true, multiply(Y, inverse(X)), multiply(inverse(X), add(X, multiply(Y, inverse(X)))))
% 133.71/17.48 = { by lemma 60 }
% 133.71/17.48 fresh(product(inverse(X), add(multiply(inverse(X), Y), X), multiply(inverse(X), Y)), true, multiply(Y, inverse(X)), multiply(inverse(X), add(X, multiply(Y, inverse(X)))))
% 133.71/17.48 = { by lemma 39 }
% 133.71/17.48 fresh(product(inverse(X), add(X, multiply(inverse(X), Y)), multiply(inverse(X), Y)), true, multiply(Y, inverse(X)), multiply(inverse(X), add(X, multiply(Y, inverse(X)))))
% 133.71/17.48 = { by lemma 43 }
% 133.71/17.48 fresh(product(inverse(X), add(X, multiply(inverse(X), Y)), multiply(Y, inverse(X))), true, multiply(Y, inverse(X)), multiply(inverse(X), add(X, multiply(Y, inverse(X)))))
% 133.71/17.48 = { by lemma 43 }
% 133.71/17.48 fresh(product(inverse(X), add(X, multiply(Y, inverse(X))), multiply(Y, inverse(X))), true, multiply(Y, inverse(X)), multiply(inverse(X), add(X, multiply(Y, inverse(X)))))
% 133.71/17.48 = { by lemma 42 }
% 133.71/17.48 multiply(Y, inverse(X))
% 133.71/17.48
% 133.71/17.48 Lemma 66: multiply(inverse(X), add(X, Y)) = multiply(Y, inverse(X)).
% 133.71/17.48 Proof:
% 133.71/17.48 multiply(inverse(X), add(X, Y))
% 133.71/17.48 = { by lemma 63 R->L }
% 133.71/17.48 multiply(inverse(X), add(X, multiply(Y, inverse(X))))
% 133.71/17.48 = { by lemma 65 }
% 133.71/17.48 multiply(Y, inverse(X))
% 133.71/17.48
% 133.71/17.48 Lemma 67: multiply(inverse(X), add(Y, X)) = multiply(Y, inverse(X)).
% 133.71/17.48 Proof:
% 133.71/17.48 multiply(inverse(X), add(Y, X))
% 133.71/17.48 = { by lemma 39 R->L }
% 133.71/17.48 multiply(inverse(X), add(X, Y))
% 133.71/17.48 = { by lemma 66 }
% 133.71/17.48 multiply(Y, inverse(X))
% 133.71/17.48
% 133.71/17.48 Lemma 68: add(X, multiply(inverse(X), Y)) = add(X, Y).
% 133.71/17.48 Proof:
% 133.71/17.48 add(X, multiply(inverse(X), Y))
% 133.71/17.48 = { by lemma 43 R->L }
% 133.71/17.48 add(X, multiply(Y, inverse(X)))
% 133.71/17.48 = { by lemma 63 }
% 133.71/17.48 add(X, Y)
% 133.71/17.48
% 133.71/17.48 Lemma 69: multiply(X, inverse(add(Y, inverse(X)))) = inverse(add(Y, inverse(X))).
% 133.71/17.48 Proof:
% 133.71/17.48 multiply(X, inverse(add(Y, inverse(X))))
% 133.71/17.48 = { by lemma 67 R->L }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), add(X, add(Y, inverse(X))))
% 133.71/17.48 = { by lemma 39 R->L }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), add(X, add(inverse(X), Y)))
% 133.71/17.48 = { by lemma 68 R->L }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), add(X, multiply(inverse(X), add(inverse(X), Y))))
% 133.71/17.48 = { by lemma 43 R->L }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), add(X, multiply(add(inverse(X), Y), inverse(X))))
% 133.71/17.48 = { by axiom 6 (multiplication_is_well_defined) R->L }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), add(X, fresh(true, true, inverse(X), multiply(add(inverse(X), Y), inverse(X)))))
% 133.71/17.48 = { by lemma 57 R->L }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), add(X, fresh(product(add(inverse(X), Y), inverse(X), inverse(X)), true, inverse(X), multiply(add(inverse(X), Y), inverse(X)))))
% 133.71/17.48 = { by lemma 42 }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), add(X, inverse(X)))
% 133.71/17.48 = { by axiom 19 (addition_is_well_defined) R->L }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), fresh4(true, true, X, inverse(X), add(X, inverse(X)), multiplicative_identity))
% 133.71/17.48 = { by axiom 8 (additive_inverse2) R->L }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), fresh4(sum(X, inverse(X), multiplicative_identity), true, X, inverse(X), add(X, inverse(X)), multiplicative_identity))
% 133.71/17.48 = { by axiom 29 (addition_is_well_defined) }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), fresh3(sum(X, inverse(X), add(X, inverse(X))), true, add(X, inverse(X)), multiplicative_identity))
% 133.71/17.48 = { by axiom 16 (closure_of_addition) }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), fresh3(true, true, add(X, inverse(X)), multiplicative_identity))
% 133.71/17.48 = { by axiom 7 (addition_is_well_defined) }
% 133.71/17.48 multiply(inverse(add(Y, inverse(X))), multiplicative_identity)
% 133.71/17.48 = { by lemma 49 R->L }
% 133.71/17.48 fresh(product(inverse(add(Y, inverse(X))), multiplicative_identity, multiply(inverse(add(Y, inverse(X))), multiplicative_identity)), true, multiply(inverse(add(Y, inverse(X))), multiplicative_identity), inverse(add(Y, inverse(X))))
% 133.71/17.48 = { by axiom 17 (closure_of_multiplication) }
% 133.71/17.48 fresh(true, true, multiply(inverse(add(Y, inverse(X))), multiplicative_identity), inverse(add(Y, inverse(X))))
% 133.71/17.48 = { by axiom 6 (multiplication_is_well_defined) }
% 133.71/17.48 inverse(add(Y, inverse(X)))
% 133.71/17.48
% 133.71/17.48 Lemma 70: fresh26(sum(X, Y, Z), true, Z, add(X, inverse(Y)), X) = product(Z, add(X, inverse(Y)), X).
% 133.71/17.48 Proof:
% 133.71/17.48 fresh26(sum(X, Y, Z), true, Z, add(X, inverse(Y)), X)
% 133.71/17.48 = { by axiom 24 (distributivity5) R->L }
% 133.71/17.48 fresh25(W, W, X, Y, Z, V, add(X, inverse(Y)), X)
% 133.71/17.48 = { by lemma 55 R->L }
% 133.71/17.48 fresh23(product(Y, inverse(Y), additive_identity), true, X, Y, Z, inverse(Y), add(X, inverse(Y)), additive_identity, X)
% 133.71/17.48 = { by axiom 9 (multiplicative_inverse2) }
% 133.71/17.48 fresh23(true, true, X, Y, Z, inverse(Y), add(X, inverse(Y)), additive_identity, X)
% 133.71/17.48 = { by lemma 59 }
% 133.71/17.48 product(Z, add(X, inverse(Y)), X)
% 133.71/17.48
% 133.71/17.48 Lemma 71: add(inverse(X), multiply(X, Y)) = add(Y, inverse(X)).
% 133.71/17.48 Proof:
% 133.71/17.48 add(inverse(X), multiply(X, Y))
% 133.71/17.48 = { by lemma 43 R->L }
% 133.71/17.48 add(inverse(X), multiply(Y, X))
% 133.71/17.48 = { by axiom 1 (inverse_is_self_cancelling) R->L }
% 133.71/17.48 add(inverse(X), multiply(Y, inverse(inverse(X))))
% 133.71/17.48 = { by lemma 63 }
% 133.71/17.48 add(inverse(X), Y)
% 133.71/17.48 = { by lemma 39 }
% 133.71/17.48 add(Y, inverse(X))
% 133.71/17.48
% 133.71/17.48 Lemma 72: add(inverse(X), multiply(Y, X)) = add(Y, inverse(X)).
% 133.71/17.48 Proof:
% 133.71/17.48 add(inverse(X), multiply(Y, X))
% 133.71/17.48 = { by lemma 43 R->L }
% 133.71/17.48 add(inverse(X), multiply(X, Y))
% 133.71/17.48 = { by lemma 71 }
% 133.71/17.48 add(Y, inverse(X))
% 133.71/17.48
% 133.71/17.48 Lemma 73: multiply(inverse(X), multiply(X, Y)) = additive_identity.
% 133.71/17.48 Proof:
% 133.71/17.48 multiply(inverse(X), multiply(X, Y))
% 133.71/17.48 = { by lemma 43 R->L }
% 133.71/17.48 multiply(inverse(X), multiply(Y, X))
% 133.71/17.48 = { by lemma 43 R->L }
% 133.71/17.48 multiply(multiply(Y, X), inverse(X))
% 133.71/17.48 = { by lemma 66 R->L }
% 133.71/17.48 multiply(inverse(X), add(X, multiply(Y, X)))
% 133.71/17.48 = { by lemma 54 }
% 133.71/17.48 multiply(inverse(X), X)
% 133.71/17.48 = { by lemma 43 }
% 133.71/17.48 multiply(X, inverse(X))
% 133.71/17.48 = { by lemma 47 }
% 133.71/17.48 additive_identity
% 133.71/17.48
% 133.71/17.48 Lemma 74: add(X, inverse(multiply(Y, inverse(X)))) = inverse(multiply(Y, inverse(X))).
% 133.71/17.48 Proof:
% 133.71/17.48 add(X, inverse(multiply(Y, inverse(X))))
% 133.71/17.48 = { by lemma 72 R->L }
% 133.71/17.48 add(inverse(multiply(Y, inverse(X))), multiply(X, multiply(Y, inverse(X))))
% 133.71/17.48 = { by lemma 43 R->L }
% 133.71/17.48 add(inverse(multiply(Y, inverse(X))), multiply(X, multiply(inverse(X), Y)))
% 133.71/17.48 = { by axiom 1 (inverse_is_self_cancelling) R->L }
% 133.71/17.48 add(inverse(multiply(Y, inverse(X))), multiply(inverse(inverse(X)), multiply(inverse(X), Y)))
% 133.71/17.48 = { by lemma 73 }
% 133.71/17.48 add(inverse(multiply(Y, inverse(X))), additive_identity)
% 133.71/17.48 = { by lemma 41 }
% 133.71/17.48 inverse(multiply(Y, inverse(X)))
% 133.71/17.48
% 133.71/17.48 Lemma 75: inverse(multiply(x_plus_y, inverse(x))) = add(x, inverse(x_plus_y)).
% 133.71/17.48 Proof:
% 133.71/17.48 inverse(multiply(x_plus_y, inverse(x)))
% 133.71/17.48 = { by lemma 42 R->L }
% 133.71/17.48 inverse(multiply(x_plus_y, inverse(fresh(product(x_plus_y, x, x), true, x, multiply(x_plus_y, x)))))
% 133.71/17.48 = { by axiom 7 (addition_is_well_defined) R->L }
% 133.71/17.48 inverse(multiply(x_plus_y, inverse(fresh(product(fresh3(true, true, add(x, y), x_plus_y), x, x), true, x, multiply(x_plus_y, x)))))
% 133.71/17.48 = { by axiom 16 (closure_of_addition) R->L }
% 133.71/17.48 inverse(multiply(x_plus_y, inverse(fresh(product(fresh3(sum(x, y, add(x, y)), true, add(x, y), x_plus_y), x, x), true, x, multiply(x_plus_y, x)))))
% 133.71/17.48 = { by axiom 29 (addition_is_well_defined) R->L }
% 133.71/17.48 inverse(multiply(x_plus_y, inverse(fresh(product(fresh4(sum(x, y, x_plus_y), true, x, y, add(x, y), x_plus_y), x, x), true, x, multiply(x_plus_y, x)))))
% 133.71/17.48 = { by axiom 3 (x_plus_y) }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(fresh(product(fresh4(true, true, x, y, add(x, y), x_plus_y), x, x), true, x, multiply(x_plus_y, x)))))
% 133.71/17.49 = { by axiom 19 (addition_is_well_defined) }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(fresh(product(add(x, y), x, x), true, x, multiply(x_plus_y, x)))))
% 133.71/17.49 = { by lemma 57 }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(fresh(true, true, x, multiply(x_plus_y, x)))))
% 133.71/17.49 = { by axiom 6 (multiplication_is_well_defined) }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, x))))
% 133.71/17.49 = { by lemma 43 R->L }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(multiply(x, x_plus_y))))
% 133.71/17.49 = { by axiom 1 (inverse_is_self_cancelling) R->L }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(multiply(x, inverse(inverse(x_plus_y))))))
% 133.71/17.49 = { by lemma 61 R->L }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(multiply(add(x_plus_y, multiply(x, inverse(inverse(x_plus_y)))), add(multiply(x, inverse(inverse(x_plus_y))), inverse(x_plus_y))))))
% 133.71/17.49 = { by lemma 39 }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(multiply(add(x_plus_y, multiply(x, inverse(inverse(x_plus_y)))), add(inverse(x_plus_y), multiply(x, inverse(inverse(x_plus_y))))))))
% 133.71/17.49 = { by lemma 63 }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(multiply(add(x_plus_y, multiply(x, inverse(inverse(x_plus_y)))), add(inverse(x_plus_y), x)))))
% 133.71/17.49 = { by axiom 1 (inverse_is_self_cancelling) }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(multiply(add(x_plus_y, multiply(x, x_plus_y)), add(inverse(x_plus_y), x)))))
% 133.71/17.49 = { by lemma 54 }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, add(inverse(x_plus_y), x)))))
% 133.71/17.49 = { by lemma 39 }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, add(x, inverse(x_plus_y))))))
% 133.71/17.49 = { by axiom 1 (inverse_is_self_cancelling) R->L }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(inverse(add(x, inverse(x_plus_y))))))))
% 133.71/17.49 = { by lemma 69 R->L }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))))))))
% 133.71/17.49 = { by axiom 6 (multiplication_is_well_defined) R->L }
% 133.71/17.49 inverse(fresh(true, true, multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))), multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))))))))
% 133.71/17.49 = { by axiom 12 (distributivity5) R->L }
% 133.71/17.49 inverse(fresh(fresh26(true, true, x_plus_y, add(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y), inverse(multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y))))), multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y)), true, multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))), multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))))))))
% 133.71/17.49 = { by axiom 13 (distributivity6) R->L }
% 133.71/17.49 inverse(fresh(fresh26(fresh22(true, true, multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y), multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y))), x_plus_y), true, x_plus_y, add(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y), inverse(multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y))))), multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y)), true, multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))), multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))))))))
% 133.71/17.49 = { by lemma 52 R->L }
% 133.71/17.49 inverse(fresh(fresh26(fresh22(sum(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))), x_plus_y, x_plus_y), true, multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y), multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y))), x_plus_y), true, x_plus_y, add(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y), inverse(multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y))))), multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y)), true, multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))), multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))))))))
% 133.71/17.49 = { by lemma 43 }
% 133.71/17.49 inverse(fresh(fresh26(fresh22(sum(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y), x_plus_y, x_plus_y), true, multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y), multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y))), x_plus_y), true, x_plus_y, add(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y), inverse(multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y))))), multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y)), true, multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))), multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))))))))
% 133.71/17.49 = { by lemma 62 }
% 133.71/17.49 inverse(fresh(fresh26(sum(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y), multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y))), x_plus_y), true, x_plus_y, add(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y), inverse(multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y))))), multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y)), true, multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))), multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))))))))
% 133.71/17.49 = { by lemma 70 }
% 133.71/17.49 inverse(fresh(product(x_plus_y, add(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y), inverse(multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y))))), multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y)), true, multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))), multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))))))))
% 133.71/17.49 = { by lemma 74 }
% 133.71/17.49 inverse(fresh(product(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y)))), multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y)), true, multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))), multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))))))))
% 133.71/17.49 = { by lemma 43 }
% 133.71/17.49 inverse(fresh(product(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(inverse(add(x, inverse(x_plus_y))), x_plus_y)))), multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))), true, multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))), multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))))))))
% 133.71/17.49 = { by lemma 43 }
% 133.71/17.49 inverse(fresh(product(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))))), multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))), true, multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))), multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y))))))))))
% 133.71/17.49 = { by lemma 42 }
% 133.71/17.49 inverse(multiply(x_plus_y, inverse(add(x, inverse(x_plus_y)))))
% 133.71/17.49 = { by lemma 69 }
% 133.71/17.49 inverse(inverse(add(x, inverse(x_plus_y))))
% 133.71/17.49 = { by axiom 1 (inverse_is_self_cancelling) }
% 133.71/17.49 add(x, inverse(x_plus_y))
% 133.71/17.49
% 133.71/17.49 Lemma 76: sum(x_inverse_times_y_inverse, inverse(x), inverse(x)) = true.
% 133.71/17.49 Proof:
% 133.71/17.49 sum(x_inverse_times_y_inverse, inverse(x), inverse(x))
% 133.71/17.49 = { by axiom 23 (distributivity1) R->L }
% 133.71/17.49 fresh40(true, true, inverse(y), x_inverse_times_y_inverse, multiplicative_identity, inverse(x), multiplicative_identity, inverse(x))
% 133.71/17.49 = { by axiom 18 (x_inverse_times_y_inverse) R->L }
% 133.71/17.49 fresh40(product(inverse(x), inverse(y), x_inverse_times_y_inverse), true, inverse(y), x_inverse_times_y_inverse, multiplicative_identity, inverse(x), multiplicative_identity, inverse(x))
% 133.71/17.49 = { by axiom 33 (distributivity1) R->L }
% 133.71/17.49 fresh39(X, X, inverse(x), inverse(y), x_inverse_times_y_inverse, multiplicative_identity, inverse(x), multiplicative_identity, inverse(x))
% 133.71/17.49 = { by lemma 51 }
% 133.71/17.49 true
% 133.71/17.49
% 133.71/17.49 Lemma 77: multiply(x_plus_y, add(x, inverse(y))) = x.
% 133.71/17.49 Proof:
% 133.71/17.49 multiply(x_plus_y, add(x, inverse(y)))
% 133.71/17.49 = { by lemma 43 R->L }
% 133.71/17.49 multiply(add(x, inverse(y)), x_plus_y)
% 133.71/17.49 = { by axiom 6 (multiplication_is_well_defined) R->L }
% 133.71/17.49 fresh(true, true, x, multiply(add(x, inverse(y)), x_plus_y))
% 133.71/17.49 = { by lemma 50 R->L }
% 133.71/17.49 fresh(fresh25(X, X, x, inverse(y), add(x, inverse(y)), Y, x_plus_y, x), true, x, multiply(add(x, inverse(y)), x_plus_y))
% 133.71/17.49 = { by lemma 58 R->L }
% 133.71/17.49 fresh(fresh23(Z, Z, x, inverse(y), add(x, inverse(y)), y, x_plus_y, W, x), true, x, multiply(add(x, inverse(y)), x_plus_y))
% 133.71/17.49 = { by axiom 31 (distributivity5) }
% 133.71/17.49 fresh(fresh24(sum(x, y, x_plus_y), true, x, inverse(y), add(x, inverse(y)), x_plus_y, x), true, x, multiply(add(x, inverse(y)), x_plus_y))
% 133.71/17.49 = { by axiom 3 (x_plus_y) }
% 133.71/17.49 fresh(fresh24(true, true, x, inverse(y), add(x, inverse(y)), x_plus_y, x), true, x, multiply(add(x, inverse(y)), x_plus_y))
% 133.71/17.49 = { by axiom 21 (distributivity5) }
% 133.71/17.49 fresh(product(add(x, inverse(y)), x_plus_y, x), true, x, multiply(add(x, inverse(y)), x_plus_y))
% 133.71/17.49 = { by lemma 42 }
% 133.71/17.49 x
% 133.71/17.49
% 133.71/17.49 Lemma 78: add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse))) = inverse(x_inverse_times_y_inverse).
% 133.71/17.49 Proof:
% 133.71/17.49 add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse)))
% 133.71/17.49 = { by lemma 46 R->L }
% 133.71/17.49 add(x_plus_y, inverse(add(x, inverse(y))))
% 133.71/17.49 = { by lemma 72 R->L }
% 133.71/17.49 add(inverse(add(x, inverse(y))), multiply(x_plus_y, add(x, inverse(y))))
% 133.71/17.49 = { by lemma 77 }
% 133.71/17.49 add(inverse(add(x, inverse(y))), x)
% 133.71/17.49 = { by lemma 46 }
% 133.71/17.49 add(inverse(add(x, x_inverse_times_y_inverse)), x)
% 133.71/17.49 = { by lemma 39 }
% 133.71/17.49 add(x, inverse(add(x, x_inverse_times_y_inverse)))
% 133.71/17.49 = { by lemma 42 R->L }
% 133.71/17.49 add(x, inverse(add(x, fresh(product(inverse(x), inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse))), x_inverse_times_y_inverse), true, x_inverse_times_y_inverse, multiply(inverse(x), inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse))))))))
% 133.71/17.49 = { by lemma 74 R->L }
% 133.71/17.49 add(x, inverse(add(x, fresh(product(inverse(x), add(x_inverse_times_y_inverse, inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse)))), x_inverse_times_y_inverse), true, x_inverse_times_y_inverse, multiply(inverse(x), inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse))))))))
% 133.71/17.49 = { by lemma 70 R->L }
% 133.71/17.49 add(x, inverse(add(x, fresh(fresh26(sum(x_inverse_times_y_inverse, multiply(inverse(x), inverse(x_inverse_times_y_inverse)), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse)))), x_inverse_times_y_inverse), true, x_inverse_times_y_inverse, multiply(inverse(x), inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse))))))))
% 133.71/17.49 = { by lemma 62 R->L }
% 133.71/17.49 add(x, inverse(add(x, fresh(fresh26(fresh22(sum(x_inverse_times_y_inverse, inverse(x), inverse(x)), true, x_inverse_times_y_inverse, multiply(inverse(x), inverse(x_inverse_times_y_inverse)), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse)))), x_inverse_times_y_inverse), true, x_inverse_times_y_inverse, multiply(inverse(x), inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse))))))))
% 133.71/17.49 = { by lemma 76 }
% 133.71/17.49 add(x, inverse(add(x, fresh(fresh26(fresh22(true, true, x_inverse_times_y_inverse, multiply(inverse(x), inverse(x_inverse_times_y_inverse)), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse)))), x_inverse_times_y_inverse), true, x_inverse_times_y_inverse, multiply(inverse(x), inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse))))))))
% 133.71/17.49 = { by axiom 13 (distributivity6) }
% 133.71/17.49 add(x, inverse(add(x, fresh(fresh26(true, true, inverse(x), add(x_inverse_times_y_inverse, inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse)))), x_inverse_times_y_inverse), true, x_inverse_times_y_inverse, multiply(inverse(x), inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse))))))))
% 133.71/17.49 = { by axiom 12 (distributivity5) }
% 133.71/17.49 add(x, inverse(add(x, fresh(true, true, x_inverse_times_y_inverse, multiply(inverse(x), inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse))))))))
% 133.71/17.49 = { by axiom 6 (multiplication_is_well_defined) }
% 133.71/17.49 add(x, inverse(add(x, multiply(inverse(x), inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse)))))))
% 133.71/17.49 = { by lemma 68 }
% 133.71/17.50 add(x, inverse(add(x, inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse))))))
% 133.71/17.50 = { by lemma 43 R->L }
% 133.71/17.50 add(x, inverse(add(x, inverse(multiply(inverse(x_inverse_times_y_inverse), inverse(x))))))
% 133.71/17.50 = { by lemma 74 }
% 133.71/17.50 add(x, inverse(inverse(multiply(inverse(x_inverse_times_y_inverse), inverse(x)))))
% 133.71/17.50 = { by lemma 43 }
% 133.71/17.50 add(x, inverse(inverse(multiply(inverse(x), inverse(x_inverse_times_y_inverse)))))
% 133.71/17.50 = { by axiom 1 (inverse_is_self_cancelling) }
% 133.71/17.50 add(x, multiply(inverse(x), inverse(x_inverse_times_y_inverse)))
% 133.71/17.50 = { by lemma 68 }
% 133.71/17.50 add(x, inverse(x_inverse_times_y_inverse))
% 133.71/17.50 = { by lemma 72 R->L }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(x, x_inverse_times_y_inverse))
% 133.71/17.50 = { by lemma 42 R->L }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), fresh(product(x, multiply(x, x_inverse_times_y_inverse), multiply(x, x_inverse_times_y_inverse)), true, multiply(x, x_inverse_times_y_inverse), multiply(x, multiply(x, x_inverse_times_y_inverse))))
% 133.71/17.50 = { by lemma 56 R->L }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), fresh(fresh25(X, X, multiply(x, x_inverse_times_y_inverse), x, x, Y, multiply(x, x_inverse_times_y_inverse), multiply(x, x_inverse_times_y_inverse)), true, multiply(x, x_inverse_times_y_inverse), multiply(x, multiply(x, x_inverse_times_y_inverse))))
% 133.71/17.50 = { by lemma 64 }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), fresh(true, true, multiply(x, x_inverse_times_y_inverse), multiply(x, multiply(x, x_inverse_times_y_inverse))))
% 133.71/17.50 = { by axiom 6 (multiplication_is_well_defined) }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(x, multiply(x, x_inverse_times_y_inverse)))
% 133.71/17.50 = { by lemma 43 R->L }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(multiply(x, x_inverse_times_y_inverse), x))
% 133.71/17.50 = { by axiom 1 (inverse_is_self_cancelling) R->L }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(multiply(x, x_inverse_times_y_inverse), inverse(inverse(x))))
% 133.71/17.50 = { by lemma 66 R->L }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(inverse(inverse(x)), add(inverse(x), multiply(x, x_inverse_times_y_inverse))))
% 133.71/17.50 = { by axiom 7 (addition_is_well_defined) R->L }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(inverse(inverse(x)), fresh3(true, true, inverse(x), add(inverse(x), multiply(x, x_inverse_times_y_inverse)))))
% 133.71/17.50 = { by axiom 13 (distributivity6) R->L }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(inverse(inverse(x)), fresh3(fresh22(true, true, inverse(x), multiply(x_inverse_times_y_inverse, inverse(inverse(x))), inverse(x)), true, inverse(x), add(inverse(x), multiply(x, x_inverse_times_y_inverse)))))
% 133.71/17.50 = { by axiom 15 (commutativity_of_addition) R->L }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(inverse(inverse(x)), fresh3(fresh22(fresh5(true, true, x_inverse_times_y_inverse, inverse(x), inverse(x)), true, inverse(x), multiply(x_inverse_times_y_inverse, inverse(inverse(x))), inverse(x)), true, inverse(x), add(inverse(x), multiply(x, x_inverse_times_y_inverse)))))
% 133.71/17.50 = { by lemma 76 R->L }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(inverse(inverse(x)), fresh3(fresh22(fresh5(sum(x_inverse_times_y_inverse, inverse(x), inverse(x)), true, x_inverse_times_y_inverse, inverse(x), inverse(x)), true, inverse(x), multiply(x_inverse_times_y_inverse, inverse(inverse(x))), inverse(x)), true, inverse(x), add(inverse(x), multiply(x, x_inverse_times_y_inverse)))))
% 133.71/17.50 = { by axiom 26 (commutativity_of_addition) }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(inverse(inverse(x)), fresh3(fresh22(sum(inverse(x), x_inverse_times_y_inverse, inverse(x)), true, inverse(x), multiply(x_inverse_times_y_inverse, inverse(inverse(x))), inverse(x)), true, inverse(x), add(inverse(x), multiply(x, x_inverse_times_y_inverse)))))
% 133.71/17.50 = { by lemma 62 }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(inverse(inverse(x)), fresh3(sum(inverse(x), multiply(x_inverse_times_y_inverse, inverse(inverse(x))), inverse(x)), true, inverse(x), add(inverse(x), multiply(x, x_inverse_times_y_inverse)))))
% 133.71/17.50 = { by axiom 1 (inverse_is_self_cancelling) }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(inverse(inverse(x)), fresh3(sum(inverse(x), multiply(x_inverse_times_y_inverse, x), inverse(x)), true, inverse(x), add(inverse(x), multiply(x, x_inverse_times_y_inverse)))))
% 133.71/17.50 = { by lemma 43 }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(inverse(inverse(x)), fresh3(sum(inverse(x), multiply(x, x_inverse_times_y_inverse), inverse(x)), true, inverse(x), add(inverse(x), multiply(x, x_inverse_times_y_inverse)))))
% 133.71/17.50 = { by lemma 37 }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(inverse(inverse(x)), inverse(x)))
% 133.71/17.50 = { by axiom 1 (inverse_is_self_cancelling) }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), multiply(x, inverse(x)))
% 133.71/17.50 = { by lemma 47 }
% 133.71/17.50 add(inverse(x_inverse_times_y_inverse), additive_identity)
% 133.71/17.50 = { by lemma 41 }
% 133.71/17.50 inverse(x_inverse_times_y_inverse)
% 133.71/17.50
% 133.71/17.50 Lemma 79: multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y))) = inverse(x_plus_y).
% 133.71/17.50 Proof:
% 133.71/17.50 multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y)))
% 133.71/17.50 = { by lemma 75 R->L }
% 133.71/17.50 multiply(x_inverse_times_y_inverse, inverse(multiply(x_plus_y, inverse(x))))
% 133.71/17.50 = { by lemma 67 R->L }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x))))
% 133.71/17.50 = { by axiom 7 (addition_is_well_defined) R->L }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(true, true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by axiom 11 (distributivity1) R->L }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(fresh42(true, true, multiply(inverse(y), inverse(x)), multiply(inverse(inverse(y)), inverse(x)), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by axiom 8 (additive_inverse2) R->L }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(fresh42(sum(inverse(y), inverse(inverse(y)), multiplicative_identity), true, multiply(inverse(y), inverse(x)), multiply(inverse(inverse(y)), inverse(x)), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by lemma 43 R->L }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(fresh42(sum(inverse(y), inverse(inverse(y)), multiplicative_identity), true, multiply(inverse(y), inverse(x)), multiply(inverse(x), inverse(inverse(y))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by axiom 27 (distributivity1) R->L }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(fresh41(true, true, inverse(x), inverse(y), multiply(inverse(y), inverse(x)), inverse(inverse(y)), multiply(inverse(x), inverse(inverse(y))), multiplicative_identity, inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by axiom 17 (closure_of_multiplication) R->L }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(fresh41(product(inverse(x), inverse(inverse(y)), multiply(inverse(x), inverse(inverse(y)))), true, inverse(x), inverse(y), multiply(inverse(y), inverse(x)), inverse(inverse(y)), multiply(inverse(x), inverse(inverse(y))), multiplicative_identity, inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by axiom 34 (distributivity1) R->L }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(fresh39(product(inverse(x), multiplicative_identity, inverse(x)), true, inverse(x), inverse(y), multiply(inverse(y), inverse(x)), inverse(inverse(y)), multiply(inverse(x), inverse(inverse(y))), multiplicative_identity, inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by axiom 4 (multiplicative_identity2) }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(fresh39(true, true, inverse(x), inverse(y), multiply(inverse(y), inverse(x)), inverse(inverse(y)), multiply(inverse(x), inverse(inverse(y))), multiplicative_identity, inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by lemma 43 }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(fresh39(true, true, inverse(x), inverse(y), multiply(inverse(y), inverse(x)), inverse(inverse(y)), multiply(inverse(inverse(y)), inverse(x)), multiplicative_identity, inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by lemma 43 R->L }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(fresh39(true, true, inverse(x), inverse(y), multiply(inverse(x), inverse(y)), inverse(inverse(y)), multiply(inverse(inverse(y)), inverse(x)), multiplicative_identity, inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by lemma 48 }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(multiply(inverse(x), inverse(y)), multiply(inverse(inverse(y)), inverse(x)), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by lemma 43 }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(multiply(inverse(x), inverse(y)), multiply(inverse(x), inverse(inverse(y))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by axiom 6 (multiplication_is_well_defined) R->L }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(fresh(true, true, x_inverse_times_y_inverse, multiply(inverse(x), inverse(y))), multiply(inverse(x), inverse(inverse(y))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by axiom 18 (x_inverse_times_y_inverse) R->L }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(fresh(product(inverse(x), inverse(y), x_inverse_times_y_inverse), true, x_inverse_times_y_inverse, multiply(inverse(x), inverse(y))), multiply(inverse(x), inverse(inverse(y))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by lemma 42 }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), inverse(inverse(y))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by axiom 1 (inverse_is_self_cancelling) }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), y), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by lemma 43 }
% 133.71/17.50 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(y, inverse(x)), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.50 = { by lemma 65 R->L }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), add(x, multiply(y, inverse(x)))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by axiom 7 (addition_is_well_defined) R->L }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), fresh3(true, true, x_plus_y, add(x, multiply(y, inverse(x))))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by axiom 13 (distributivity6) R->L }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), fresh3(fresh22(true, true, x, multiply(inverse(x), y), x_plus_y), true, x_plus_y, add(x, multiply(y, inverse(x))))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by axiom 8 (additive_inverse2) R->L }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), fresh3(fresh22(sum(x, inverse(x), multiplicative_identity), true, x, multiply(inverse(x), y), x_plus_y), true, x_plus_y, add(x, multiply(y, inverse(x))))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by axiom 28 (distributivity6) R->L }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), fresh3(fresh21(true, true, x, inverse(x), multiplicative_identity, y, x_plus_y, multiply(inverse(x), y), x_plus_y), true, x_plus_y, add(x, multiply(y, inverse(x))))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by axiom 17 (closure_of_multiplication) R->L }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), fresh3(fresh21(product(inverse(x), y, multiply(inverse(x), y)), true, x, inverse(x), multiplicative_identity, y, x_plus_y, multiply(inverse(x), y), x_plus_y), true, x_plus_y, add(x, multiply(y, inverse(x))))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by axiom 36 (distributivity6) R->L }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), fresh3(fresh19(product(multiplicative_identity, x_plus_y, x_plus_y), true, x, inverse(x), multiplicative_identity, y, x_plus_y, multiply(inverse(x), y), x_plus_y), true, x_plus_y, add(x, multiply(y, inverse(x))))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by axiom 5 (multiplicative_identity1) }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), fresh3(fresh19(true, true, x, inverse(x), multiplicative_identity, y, x_plus_y, multiply(inverse(x), y), x_plus_y), true, x_plus_y, add(x, multiply(y, inverse(x))))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by axiom 32 (distributivity6) }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), fresh3(fresh20(sum(x, y, x_plus_y), true, x, inverse(x), multiplicative_identity, multiply(inverse(x), y), x_plus_y), true, x_plus_y, add(x, multiply(y, inverse(x))))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by axiom 3 (x_plus_y) }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), fresh3(fresh20(true, true, x, inverse(x), multiplicative_identity, multiply(inverse(x), y), x_plus_y), true, x_plus_y, add(x, multiply(y, inverse(x))))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by axiom 22 (distributivity6) }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), fresh3(sum(x, multiply(inverse(x), y), x_plus_y), true, x_plus_y, add(x, multiply(y, inverse(x))))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by lemma 43 }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), fresh3(sum(x, multiply(y, inverse(x)), x_plus_y), true, x_plus_y, add(x, multiply(y, inverse(x))))), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by lemma 37 }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(inverse(x), x_plus_y), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 133.71/17.51 = { by lemma 43 }
% 133.71/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), fresh3(sum(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)), inverse(x)), true, inverse(x), add(x_inverse_times_y_inverse, multiply(x_plus_y, inverse(x)))))
% 134.47/17.51 = { by lemma 37 }
% 134.47/17.51 multiply(inverse(multiply(x_plus_y, inverse(x))), inverse(x))
% 134.47/17.51 = { by lemma 75 }
% 134.47/17.51 multiply(add(x, inverse(x_plus_y)), inverse(x))
% 134.47/17.51 = { by lemma 41 R->L }
% 134.47/17.51 multiply(add(x, inverse(x_plus_y)), add(inverse(x), additive_identity))
% 134.47/17.51 = { by lemma 47 R->L }
% 134.47/17.51 multiply(add(x, inverse(x_plus_y)), add(inverse(x), multiply(x_plus_y, inverse(x_plus_y))))
% 134.47/17.51 = { by lemma 43 R->L }
% 134.47/17.51 multiply(add(x, inverse(x_plus_y)), add(inverse(x), multiply(inverse(x_plus_y), x_plus_y)))
% 134.47/17.51 = { by lemma 53 R->L }
% 134.47/17.51 multiply(add(x, inverse(x_plus_y)), add(inverse(x), multiply(inverse(x_plus_y), add(x_plus_y, multiply(x_plus_y, add(x, inverse(y)))))))
% 134.47/17.51 = { by lemma 77 }
% 134.47/17.51 multiply(add(x, inverse(x_plus_y)), add(inverse(x), multiply(inverse(x_plus_y), add(x_plus_y, x))))
% 134.47/17.51 = { by lemma 66 }
% 134.47/17.51 multiply(add(x, inverse(x_plus_y)), add(inverse(x), multiply(x, inverse(x_plus_y))))
% 134.47/17.51 = { by lemma 71 }
% 134.47/17.51 multiply(add(x, inverse(x_plus_y)), add(inverse(x_plus_y), inverse(x)))
% 134.47/17.51 = { by lemma 61 }
% 134.47/17.51 inverse(x_plus_y)
% 134.47/17.51
% 134.47/17.51 Lemma 80: multiply(inverse(X), multiply(Y, X)) = additive_identity.
% 134.47/17.51 Proof:
% 134.47/17.51 multiply(inverse(X), multiply(Y, X))
% 134.47/17.51 = { by lemma 43 R->L }
% 134.47/17.51 multiply(inverse(X), multiply(X, Y))
% 134.47/17.51 = { by lemma 73 }
% 134.47/17.51 additive_identity
% 134.47/17.51
% 134.47/17.51 Goal 1 (prove_equation): inverse(x_plus_y) = x_inverse_times_y_inverse.
% 134.47/17.51 Proof:
% 134.47/17.51 inverse(x_plus_y)
% 134.47/17.51 = { by axiom 19 (addition_is_well_defined) R->L }
% 134.47/17.51 inverse(fresh4(true, true, inverse(x_inverse_times_y_inverse), x_plus_y, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by lemma 38 R->L }
% 134.47/17.51 inverse(fresh4(sum(inverse(x_inverse_times_y_inverse), x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))), true, inverse(x_inverse_times_y_inverse), x_plus_y, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by axiom 29 (addition_is_well_defined) }
% 134.47/17.51 inverse(fresh3(sum(inverse(x_inverse_times_y_inverse), x_plus_y, x_plus_y), true, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by axiom 1 (inverse_is_self_cancelling) R->L }
% 134.47/17.51 inverse(fresh3(sum(inverse(x_inverse_times_y_inverse), x_plus_y, inverse(inverse(x_plus_y))), true, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by lemma 79 R->L }
% 134.47/17.51 inverse(fresh3(sum(inverse(x_inverse_times_y_inverse), x_plus_y, inverse(multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y))))), true, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by axiom 1 (inverse_is_self_cancelling) R->L }
% 134.47/17.51 inverse(fresh3(sum(inverse(x_inverse_times_y_inverse), inverse(inverse(x_plus_y)), inverse(multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y))))), true, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by lemma 79 R->L }
% 134.47/17.51 inverse(fresh3(sum(inverse(x_inverse_times_y_inverse), inverse(multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y)))), inverse(multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y))))), true, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by lemma 43 R->L }
% 134.47/17.51 inverse(fresh3(sum(inverse(x_inverse_times_y_inverse), inverse(multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y)))), inverse(multiply(add(x, inverse(x_plus_y)), x_inverse_times_y_inverse))), true, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by lemma 41 R->L }
% 134.47/17.51 inverse(fresh3(sum(inverse(x_inverse_times_y_inverse), inverse(multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y)))), add(inverse(multiply(add(x, inverse(x_plus_y)), x_inverse_times_y_inverse)), additive_identity)), true, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by lemma 80 R->L }
% 134.47/17.51 inverse(fresh3(sum(inverse(x_inverse_times_y_inverse), inverse(multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y)))), add(inverse(multiply(add(x, inverse(x_plus_y)), x_inverse_times_y_inverse)), multiply(inverse(x_inverse_times_y_inverse), multiply(add(x, inverse(x_plus_y)), x_inverse_times_y_inverse)))), true, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by lemma 72 }
% 134.47/17.51 inverse(fresh3(sum(inverse(x_inverse_times_y_inverse), inverse(multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y)))), add(inverse(x_inverse_times_y_inverse), inverse(multiply(add(x, inverse(x_plus_y)), x_inverse_times_y_inverse)))), true, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by lemma 43 }
% 134.47/17.51 inverse(fresh3(sum(inverse(x_inverse_times_y_inverse), inverse(multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y)))), add(inverse(x_inverse_times_y_inverse), inverse(multiply(x_inverse_times_y_inverse, add(x, inverse(x_plus_y)))))), true, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by axiom 16 (closure_of_addition) }
% 134.47/17.51 inverse(fresh3(true, true, x_plus_y, add(x_plus_y, inverse(x_inverse_times_y_inverse))))
% 134.47/17.51 = { by axiom 7 (addition_is_well_defined) }
% 134.47/17.51 inverse(add(x_plus_y, inverse(x_inverse_times_y_inverse)))
% 134.47/17.51 = { by lemma 78 R->L }
% 134.47/17.51 inverse(add(x_plus_y, add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse)))))
% 134.47/17.51 = { by lemma 39 R->L }
% 134.47/17.51 inverse(add(add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse))), x_plus_y))
% 134.47/17.51 = { by lemma 63 R->L }
% 134.47/17.51 inverse(add(add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse))), multiply(x_plus_y, inverse(add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse)))))))
% 134.47/17.51 = { by lemma 43 R->L }
% 134.47/17.51 inverse(add(add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse))), multiply(inverse(add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse)))), x_plus_y)))
% 134.47/17.51 = { by lemma 61 R->L }
% 134.47/17.51 inverse(add(add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse))), multiply(inverse(add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse)))), multiply(add(add(x, x_inverse_times_y_inverse), x_plus_y), add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse)))))))
% 134.47/17.51 = { by lemma 80 }
% 134.47/17.51 inverse(add(add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse))), additive_identity))
% 134.47/17.51 = { by lemma 41 }
% 134.47/17.51 inverse(add(x_plus_y, inverse(add(x, x_inverse_times_y_inverse))))
% 134.47/17.51 = { by lemma 78 }
% 134.47/17.51 inverse(inverse(x_inverse_times_y_inverse))
% 134.47/17.51 = { by axiom 1 (inverse_is_self_cancelling) }
% 134.47/17.51 x_inverse_times_y_inverse
% 134.47/17.51 % SZS output end Proof
% 134.47/17.51
% 134.47/17.51 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------