TSTP Solution File: BOO015-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : BOO015-1 : TPTP v8.1.2. Bugfixed v1.2.1.
% 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 : n016.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 18:11:25 EDT 2023
% Result : Unsatisfiable 45.06s 6.21s
% Output : Proof 46.45s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : BOO015-1 : TPTP v8.1.2. Bugfixed v1.2.1.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n016.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sun Aug 27 08:39:28 EDT 2023
% 0.13/0.35 % CPUTime :
% 45.06/6.21 Command-line arguments: --no-flatten-goal
% 45.06/6.21
% 45.06/6.21 % SZS status Unsatisfiable
% 45.06/6.21
% 46.18/6.32 % SZS output start Proof
% 46.18/6.32 Take the following subset of the input axioms:
% 46.18/6.33 fof(addition_is_well_defined, axiom, ![X, Y, U, V]: (~sum(X, Y, U) | (~sum(X, Y, V) | U=V))).
% 46.18/6.33 fof(additive_identity2, axiom, ![X2]: sum(X2, additive_identity, X2)).
% 46.18/6.33 fof(additive_inverse1, axiom, ![X2]: sum(inverse(X2), X2, multiplicative_identity)).
% 46.18/6.33 fof(additive_inverse2, axiom, ![X2]: sum(X2, inverse(X2), multiplicative_identity)).
% 46.18/6.33 fof(closure_of_addition, axiom, ![X2, Y2]: sum(X2, Y2, add(X2, Y2))).
% 46.18/6.33 fof(closure_of_multiplication, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 46.18/6.33 fof(commutativity_of_addition, axiom, ![Z, X2, Y2]: (~sum(X2, Y2, Z) | sum(Y2, X2, Z))).
% 46.18/6.33 fof(commutativity_of_multiplication, axiom, ![X2, Y2, Z2]: (~product(X2, Y2, Z2) | product(Y2, X2, Z2))).
% 46.18/6.33 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)))))).
% 46.18/6.33 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)))))).
% 46.18/6.33 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)))))).
% 46.18/6.33 fof(multiplication_is_well_defined, axiom, ![X2, Y2, U2, V5]: (~product(X2, Y2, U2) | (~product(X2, Y2, V5) | U2=V5))).
% 46.18/6.33 fof(multiplicative_identity1, axiom, ![X2]: product(multiplicative_identity, X2, X2)).
% 46.18/6.33 fof(multiplicative_identity2, axiom, ![X2]: product(X2, multiplicative_identity, X2)).
% 46.18/6.33 fof(multiplicative_inverse1, axiom, ![X2]: product(inverse(X2), X2, additive_identity)).
% 46.18/6.33 fof(multiplicative_inverse2, axiom, ![X2]: product(X2, inverse(X2), additive_identity)).
% 46.18/6.33 fof(prove_equation, negated_conjecture, inverse(x_times_y)!=x_inverse_plus_y_inverse).
% 46.18/6.33 fof(x_inverse_plus_y_inverse, negated_conjecture, sum(inverse(x), inverse(y), x_inverse_plus_y_inverse)).
% 46.18/6.33 fof(x_times_y, negated_conjecture, product(x, y, x_times_y)).
% 46.18/6.33
% 46.18/6.33 Now clausify the problem and encode Horn clauses using encoding 3 of
% 46.18/6.33 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 46.18/6.33 We repeatedly replace C & s=t => u=v by the two clauses:
% 46.18/6.33 fresh(y, y, x1...xn) = u
% 46.18/6.33 C => fresh(s, t, x1...xn) = v
% 46.18/6.33 where fresh is a fresh function symbol and x1..xn are the free
% 46.18/6.33 variables of u and v.
% 46.18/6.33 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 46.18/6.33 input problem has no model of domain size 1).
% 46.18/6.33
% 46.18/6.33 The encoding turns the above axioms into the following unit equations and goals:
% 46.18/6.33
% 46.18/6.33 Axiom 1 (additive_identity2): sum(X, additive_identity, X) = true.
% 46.18/6.33 Axiom 2 (multiplicative_identity2): product(X, multiplicative_identity, X) = true.
% 46.18/6.33 Axiom 3 (x_times_y): product(x, y, x_times_y) = true.
% 46.18/6.33 Axiom 4 (multiplicative_identity1): product(multiplicative_identity, X, X) = true.
% 46.18/6.33 Axiom 5 (multiplication_is_well_defined): fresh(X, X, Y, Z) = Z.
% 46.18/6.33 Axiom 6 (addition_is_well_defined): fresh3(X, X, Y, Z) = Z.
% 46.18/6.33 Axiom 7 (additive_inverse2): sum(X, inverse(X), multiplicative_identity) = true.
% 46.18/6.33 Axiom 8 (additive_inverse1): sum(inverse(X), X, multiplicative_identity) = true.
% 46.18/6.33 Axiom 9 (multiplicative_inverse2): product(X, inverse(X), additive_identity) = true.
% 46.18/6.33 Axiom 10 (multiplicative_inverse1): product(inverse(X), X, additive_identity) = true.
% 46.18/6.33 Axiom 11 (distributivity1): fresh38(X, X, Y, Z, W) = true.
% 46.18/6.33 Axiom 12 (distributivity5): fresh22(X, X, Y, Z, W) = true.
% 46.18/6.33 Axiom 13 (distributivity6): fresh18(X, X, Y, Z, W) = true.
% 46.18/6.33 Axiom 14 (commutativity_of_multiplication): fresh6(X, X, Y, Z, W) = true.
% 46.18/6.33 Axiom 15 (commutativity_of_addition): fresh5(X, X, Y, Z, W) = true.
% 46.18/6.33 Axiom 16 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 46.18/6.33 Axiom 17 (x_inverse_plus_y_inverse): sum(inverse(x), inverse(y), x_inverse_plus_y_inverse) = true.
% 46.18/6.33 Axiom 18 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 46.18/6.33 Axiom 19 (addition_is_well_defined): fresh4(X, X, Y, Z, W, V) = W.
% 46.18/6.33 Axiom 20 (multiplication_is_well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 46.18/6.33 Axiom 21 (distributivity5): fresh20(X, X, Y, Z, W, V, U) = product(W, V, U).
% 46.18/6.33 Axiom 22 (distributivity6): fresh16(X, X, Y, Z, W, V, U) = sum(Y, V, U).
% 46.18/6.33 Axiom 23 (distributivity1): fresh36(X, X, Y, Z, W, V, U, T) = sum(Z, V, T).
% 46.18/6.33 Axiom 24 (distributivity5): fresh21(X, X, Y, Z, W, V, U, T) = fresh22(sum(Y, Z, W), true, W, U, T).
% 46.18/6.33 Axiom 25 (commutativity_of_multiplication): fresh6(product(X, Y, Z), true, X, Y, Z) = product(Y, X, Z).
% 46.18/6.33 Axiom 26 (commutativity_of_addition): fresh5(sum(X, Y, Z), true, X, Y, Z) = sum(Y, X, Z).
% 46.18/6.33 Axiom 27 (distributivity1): fresh37(X, X, Y, Z, W, V, U, T, S) = fresh38(sum(Z, V, T), true, W, U, S).
% 46.18/6.33 Axiom 28 (distributivity6): fresh17(X, X, Y, Z, W, V, U, T, S) = fresh18(sum(Y, Z, W), true, Y, T, S).
% 46.18/6.33 Axiom 29 (addition_is_well_defined): fresh4(sum(X, Y, Z), true, X, Y, W, Z) = fresh3(sum(X, Y, W), true, W, Z).
% 46.18/6.33 Axiom 30 (multiplication_is_well_defined): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 46.18/6.33 Axiom 31 (distributivity5): fresh19(X, X, Y, Z, W, V, U, T, S) = fresh20(sum(Y, V, U), true, Y, Z, W, U, S).
% 46.18/6.33 Axiom 32 (distributivity6): fresh15(X, X, Y, Z, W, V, U, T, S) = fresh16(sum(Y, V, U), true, Y, Z, W, T, S).
% 46.18/6.33 Axiom 33 (distributivity1): fresh35(X, X, Y, Z, W, V, U, T, S) = fresh36(product(Y, Z, W), true, Z, W, V, U, T, S).
% 46.18/6.33 Axiom 34 (distributivity1): fresh35(product(X, Y, Z), true, X, W, V, U, T, Y, Z) = fresh37(product(X, U, T), true, X, W, V, U, T, Y, Z).
% 46.18/6.33 Axiom 35 (distributivity5): fresh19(product(X, Y, Z), true, W, X, V, Y, U, Z, T) = fresh21(sum(W, Z, T), true, W, X, V, Y, U, T).
% 46.18/6.33 Axiom 36 (distributivity6): fresh15(product(X, Y, Z), true, W, V, X, U, Y, T, Z) = fresh17(product(V, U, T), true, W, V, X, U, Y, T, Z).
% 46.18/6.33
% 46.18/6.33 Lemma 37: fresh3(sum(X, Y, Z), true, Z, add(X, Y)) = Z.
% 46.18/6.33 Proof:
% 46.18/6.33 fresh3(sum(X, Y, Z), true, Z, add(X, Y))
% 46.18/6.33 = { by axiom 29 (addition_is_well_defined) R->L }
% 46.18/6.33 fresh4(sum(X, Y, add(X, Y)), true, X, Y, Z, add(X, Y))
% 46.18/6.33 = { by axiom 16 (closure_of_addition) }
% 46.18/6.33 fresh4(true, true, X, Y, Z, add(X, Y))
% 46.18/6.33 = { by axiom 19 (addition_is_well_defined) }
% 46.18/6.33 Z
% 46.18/6.33
% 46.18/6.33 Lemma 38: add(X, Y) = add(Y, X).
% 46.18/6.33 Proof:
% 46.18/6.33 add(X, Y)
% 46.18/6.33 = { by lemma 37 R->L }
% 46.18/6.33 fresh3(sum(Y, X, add(X, Y)), true, add(X, Y), add(Y, X))
% 46.18/6.33 = { by axiom 26 (commutativity_of_addition) R->L }
% 46.18/6.33 fresh3(fresh5(sum(X, Y, add(X, Y)), true, X, Y, add(X, Y)), true, add(X, Y), add(Y, X))
% 46.18/6.33 = { by axiom 16 (closure_of_addition) }
% 46.18/6.33 fresh3(fresh5(true, true, X, Y, add(X, Y)), true, add(X, Y), add(Y, X))
% 46.18/6.33 = { by axiom 15 (commutativity_of_addition) }
% 46.18/6.33 fresh3(true, true, add(X, Y), add(Y, X))
% 46.18/6.33 = { by axiom 6 (addition_is_well_defined) }
% 46.18/6.33 add(Y, X)
% 46.18/6.33
% 46.18/6.33 Lemma 39: fresh3(sum(X, additive_identity, Y), true, Y, X) = Y.
% 46.18/6.33 Proof:
% 46.18/6.33 fresh3(sum(X, additive_identity, Y), true, Y, X)
% 46.18/6.33 = { by axiom 29 (addition_is_well_defined) R->L }
% 46.18/6.33 fresh4(sum(X, additive_identity, X), true, X, additive_identity, Y, X)
% 46.18/6.33 = { by axiom 1 (additive_identity2) }
% 46.18/6.33 fresh4(true, true, X, additive_identity, Y, X)
% 46.18/6.33 = { by axiom 19 (addition_is_well_defined) }
% 46.18/6.33 Y
% 46.18/6.33
% 46.18/6.33 Lemma 40: add(X, additive_identity) = X.
% 46.18/6.33 Proof:
% 46.18/6.33 add(X, additive_identity)
% 46.18/6.33 = { by lemma 39 R->L }
% 46.18/6.33 fresh3(sum(X, additive_identity, add(X, additive_identity)), true, add(X, additive_identity), X)
% 46.18/6.33 = { by axiom 16 (closure_of_addition) }
% 46.18/6.33 fresh3(true, true, add(X, additive_identity), X)
% 46.18/6.33 = { by axiom 6 (addition_is_well_defined) }
% 46.18/6.33 X
% 46.18/6.33
% 46.18/6.33 Lemma 41: fresh(product(X, Y, Z), true, Z, multiply(X, Y)) = Z.
% 46.18/6.33 Proof:
% 46.18/6.33 fresh(product(X, Y, Z), true, Z, multiply(X, Y))
% 46.18/6.33 = { by axiom 30 (multiplication_is_well_defined) R->L }
% 46.18/6.33 fresh2(product(X, Y, multiply(X, Y)), true, X, Y, Z, multiply(X, Y))
% 46.18/6.33 = { by axiom 18 (closure_of_multiplication) }
% 46.18/6.33 fresh2(true, true, X, Y, Z, multiply(X, Y))
% 46.18/6.33 = { by axiom 20 (multiplication_is_well_defined) }
% 46.18/6.33 Z
% 46.18/6.33
% 46.18/6.33 Lemma 42: multiply(X, Y) = multiply(Y, X).
% 46.18/6.33 Proof:
% 46.18/6.33 multiply(X, Y)
% 46.18/6.33 = { by lemma 41 R->L }
% 46.18/6.33 fresh(product(Y, X, multiply(X, Y)), true, multiply(X, Y), multiply(Y, X))
% 46.18/6.33 = { by axiom 25 (commutativity_of_multiplication) R->L }
% 46.18/6.33 fresh(fresh6(product(X, Y, multiply(X, Y)), true, X, Y, multiply(X, Y)), true, multiply(X, Y), multiply(Y, X))
% 46.18/6.33 = { by axiom 18 (closure_of_multiplication) }
% 46.18/6.33 fresh(fresh6(true, true, X, Y, multiply(X, Y)), true, multiply(X, Y), multiply(Y, X))
% 46.18/6.33 = { by axiom 14 (commutativity_of_multiplication) }
% 46.18/6.33 fresh(true, true, multiply(X, Y), multiply(Y, X))
% 46.18/6.33 = { by axiom 5 (multiplication_is_well_defined) }
% 46.18/6.33 multiply(Y, X)
% 46.18/6.33
% 46.18/6.33 Lemma 43: fresh(product(X, multiplicative_identity, Y), true, Y, X) = Y.
% 46.18/6.33 Proof:
% 46.18/6.33 fresh(product(X, multiplicative_identity, Y), true, Y, X)
% 46.18/6.33 = { by axiom 30 (multiplication_is_well_defined) R->L }
% 46.18/6.33 fresh2(product(X, multiplicative_identity, X), true, X, multiplicative_identity, Y, X)
% 46.18/6.33 = { by axiom 2 (multiplicative_identity2) }
% 46.18/6.33 fresh2(true, true, X, multiplicative_identity, Y, X)
% 46.18/6.33 = { by axiom 20 (multiplication_is_well_defined) }
% 46.18/6.33 Y
% 46.18/6.33
% 46.18/6.33 Lemma 44: fresh19(X, X, Y, Z, W, inverse(Y), multiplicative_identity, V, U) = product(W, multiplicative_identity, U).
% 46.18/6.33 Proof:
% 46.18/6.33 fresh19(X, X, Y, Z, W, inverse(Y), multiplicative_identity, V, U)
% 46.18/6.33 = { by axiom 31 (distributivity5) }
% 46.18/6.33 fresh20(sum(Y, inverse(Y), multiplicative_identity), true, Y, Z, W, multiplicative_identity, U)
% 46.18/6.33 = { by axiom 7 (additive_inverse2) }
% 46.18/6.33 fresh20(true, true, Y, Z, W, multiplicative_identity, U)
% 46.18/6.33 = { by axiom 21 (distributivity5) }
% 46.18/6.33 product(W, multiplicative_identity, U)
% 46.18/6.33
% 46.18/6.33 Lemma 45: fresh19(X2, X2, Y, Z, W, V, U, Y2, S) = fresh19(X, X, Y, Z, W, V, U, T, S).
% 46.18/6.33 Proof:
% 46.18/6.33 fresh19(X2, X2, Y, Z, W, V, U, Y2, S)
% 46.18/6.33 = { by axiom 31 (distributivity5) }
% 46.18/6.33 fresh20(sum(Y, V, U), true, Y, Z, W, U, S)
% 46.18/6.33 = { by axiom 31 (distributivity5) R->L }
% 46.18/6.33 fresh19(X, X, Y, Z, W, V, U, T, S)
% 46.18/6.33
% 46.18/6.33 Lemma 46: fresh19(product(X, Y, additive_identity), true, Z, X, W, Y, V, additive_identity, Z) = fresh21(U, U, Z, X, W, T, V, Z).
% 46.18/6.33 Proof:
% 46.18/6.34 fresh19(product(X, Y, additive_identity), true, Z, X, W, Y, V, additive_identity, Z)
% 46.18/6.34 = { by axiom 35 (distributivity5) }
% 46.18/6.34 fresh21(sum(Z, additive_identity, Z), true, Z, X, W, Y, V, Z)
% 46.18/6.34 = { by axiom 1 (additive_identity2) }
% 46.18/6.34 fresh21(true, true, Z, X, W, Y, V, Z)
% 46.18/6.34 = { by axiom 24 (distributivity5) }
% 46.18/6.34 fresh22(sum(Z, X, W), true, W, V, Z)
% 46.18/6.34 = { by axiom 24 (distributivity5) R->L }
% 46.18/6.34 fresh21(U, U, Z, X, W, T, V, Z)
% 46.18/6.34
% 46.18/6.34 Lemma 47: fresh19(X, X, Y, inverse(Z), W, Z, V, U, Y) = fresh21(T, T, Y, inverse(Z), W, S, V, Y).
% 46.18/6.34 Proof:
% 46.18/6.34 fresh19(X, X, Y, inverse(Z), W, Z, V, U, Y)
% 46.18/6.34 = { by lemma 45 }
% 46.18/6.34 fresh19(true, true, Y, inverse(Z), W, Z, V, additive_identity, Y)
% 46.18/6.34 = { by axiom 10 (multiplicative_inverse1) R->L }
% 46.18/6.34 fresh19(product(inverse(Z), Z, additive_identity), true, Y, inverse(Z), W, Z, V, additive_identity, Y)
% 46.18/6.34 = { by lemma 46 }
% 46.18/6.34 fresh21(T, T, Y, inverse(Z), W, S, V, Y)
% 46.18/6.34
% 46.18/6.34 Lemma 48: fresh21(X, X, Y, Z, add(Y, Z), W, V, U) = true.
% 46.18/6.34 Proof:
% 46.18/6.34 fresh21(X, X, Y, Z, add(Y, Z), W, V, U)
% 46.18/6.34 = { by axiom 24 (distributivity5) }
% 46.18/6.34 fresh22(sum(Y, Z, add(Y, Z)), true, add(Y, Z), V, U)
% 46.18/6.34 = { by axiom 16 (closure_of_addition) }
% 46.18/6.34 fresh22(true, true, add(Y, Z), V, U)
% 46.18/6.34 = { by axiom 12 (distributivity5) }
% 46.18/6.34 true
% 46.18/6.34
% 46.18/6.34 Lemma 49: fresh19(X, X, Y, Z, W, inverse(Z), V, U, Y) = fresh21(T, T, Y, Z, W, S, V, Y).
% 46.18/6.34 Proof:
% 46.18/6.34 fresh19(X, X, Y, Z, W, inverse(Z), V, U, Y)
% 46.18/6.34 = { by lemma 45 }
% 46.18/6.34 fresh19(true, true, Y, Z, W, inverse(Z), V, additive_identity, Y)
% 46.18/6.34 = { by axiom 9 (multiplicative_inverse2) R->L }
% 46.18/6.34 fresh19(product(Z, inverse(Z), additive_identity), true, Y, Z, W, inverse(Z), V, additive_identity, Y)
% 46.18/6.34 = { by lemma 46 }
% 46.18/6.34 fresh21(T, T, Y, Z, W, S, V, Y)
% 46.18/6.34
% 46.18/6.34 Lemma 50: fresh21(X, X, Y, Z, add(Z, Y), W, V, U) = true.
% 46.18/6.34 Proof:
% 46.18/6.34 fresh21(X, X, Y, Z, add(Z, Y), W, V, U)
% 46.18/6.34 = { by lemma 38 R->L }
% 46.18/6.34 fresh21(X, X, Y, Z, add(Y, Z), W, V, U)
% 46.18/6.34 = { by lemma 48 }
% 46.18/6.34 true
% 46.18/6.34
% 46.18/6.34 Lemma 51: inverse(inverse(X)) = X.
% 46.18/6.34 Proof:
% 46.18/6.34 inverse(inverse(X))
% 46.18/6.34 = { by lemma 43 R->L }
% 46.18/6.34 fresh(product(X, multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 46.18/6.34 = { by lemma 43 R->L }
% 46.18/6.34 fresh(product(fresh(product(add(X, inverse(inverse(X))), multiplicative_identity, X), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 46.18/6.34 = { by lemma 44 R->L }
% 46.18/6.34 fresh(product(fresh(fresh19(Y, Y, X, inverse(inverse(X)), add(X, inverse(inverse(X))), inverse(X), multiplicative_identity, Z, X), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 46.18/6.34 = { by lemma 47 }
% 46.18/6.34 fresh(product(fresh(fresh21(W, W, X, inverse(inverse(X)), add(X, inverse(inverse(X))), V, multiplicative_identity, X), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 46.18/6.34 = { by lemma 48 }
% 46.18/6.34 fresh(product(fresh(true, true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 46.18/6.34 = { by axiom 5 (multiplication_is_well_defined) }
% 46.18/6.34 fresh(product(add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 46.18/6.34 = { by axiom 21 (distributivity5) R->L }
% 46.18/6.34 fresh(fresh20(true, true, inverse(inverse(X)), X, add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 46.18/6.34 = { by axiom 8 (additive_inverse1) R->L }
% 46.18/6.34 fresh(fresh20(sum(inverse(inverse(X)), inverse(X), multiplicative_identity), true, inverse(inverse(X)), X, add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 46.18/6.34 = { by axiom 31 (distributivity5) R->L }
% 46.18/6.34 fresh(fresh19(U, U, inverse(inverse(X)), X, add(X, inverse(inverse(X))), inverse(X), multiplicative_identity, T, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 46.18/6.34 = { by lemma 49 }
% 46.18/6.34 fresh(fresh21(S, S, inverse(inverse(X)), X, add(X, inverse(inverse(X))), X2, multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 46.18/6.34 = { by lemma 50 }
% 46.18/6.34 fresh(true, true, inverse(inverse(X)), X)
% 46.18/6.34 = { by axiom 5 (multiplication_is_well_defined) }
% 46.18/6.34 X
% 46.18/6.34
% 46.18/6.34 Lemma 52: fresh35(X, X, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y) = true.
% 46.18/6.34 Proof:
% 46.18/6.34 fresh35(X, X, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 46.18/6.34 = { by axiom 33 (distributivity1) }
% 46.18/6.34 fresh36(product(Y, Z, W), true, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 46.18/6.34 = { by axiom 33 (distributivity1) R->L }
% 46.18/6.34 fresh35(true, true, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 46.18/6.34 = { by axiom 2 (multiplicative_identity2) R->L }
% 46.18/6.34 fresh35(product(Y, multiplicative_identity, Y), true, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 46.18/6.34 = { by axiom 34 (distributivity1) }
% 46.18/6.34 fresh37(product(Y, multiplicative_identity, Y), true, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 46.18/6.34 = { by axiom 2 (multiplicative_identity2) }
% 46.18/6.34 fresh37(true, true, Y, Z, W, multiplicative_identity, Y, multiplicative_identity, Y)
% 46.18/6.34 = { by axiom 27 (distributivity1) }
% 46.18/6.34 fresh38(sum(Z, multiplicative_identity, multiplicative_identity), true, W, Y, Y)
% 46.18/6.34 = { by lemma 43 R->L }
% 46.18/6.34 fresh38(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)
% 46.18/6.34 = { by lemma 44 R->L }
% 46.18/6.34 fresh38(sum(Z, multiplicative_identity, fresh(fresh19(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)
% 46.18/6.34 = { by axiom 4 (multiplicative_identity1) R->L }
% 46.18/6.34 fresh38(sum(Z, multiplicative_identity, fresh(fresh19(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)
% 46.18/6.34 = { by axiom 35 (distributivity5) }
% 46.18/6.34 fresh38(sum(Z, multiplicative_identity, fresh(fresh21(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)
% 46.18/6.34 = { by axiom 7 (additive_inverse2) }
% 46.18/6.34 fresh38(sum(Z, multiplicative_identity, fresh(fresh21(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)
% 46.18/6.34 = { by lemma 48 }
% 46.18/6.34 fresh38(sum(Z, multiplicative_identity, fresh(true, true, multiplicative_identity, add(Z, multiplicative_identity))), true, W, Y, Y)
% 46.18/6.34 = { by axiom 5 (multiplication_is_well_defined) }
% 46.18/6.34 fresh38(sum(Z, multiplicative_identity, add(Z, multiplicative_identity)), true, W, Y, Y)
% 46.18/6.34 = { by axiom 16 (closure_of_addition) }
% 46.18/6.34 fresh38(true, true, W, Y, Y)
% 46.18/6.34 = { by axiom 11 (distributivity1) }
% 46.18/6.34 true
% 46.18/6.34
% 46.18/6.34 Lemma 53: sum(x_times_y, y, y) = true.
% 46.18/6.34 Proof:
% 46.18/6.34 sum(x_times_y, y, y)
% 46.18/6.34 = { by axiom 23 (distributivity1) R->L }
% 46.18/6.34 fresh36(true, true, x, x_times_y, multiplicative_identity, y, multiplicative_identity, y)
% 46.18/6.34 = { by axiom 14 (commutativity_of_multiplication) R->L }
% 46.18/6.34 fresh36(fresh6(true, true, x, y, x_times_y), true, x, x_times_y, multiplicative_identity, y, multiplicative_identity, y)
% 46.18/6.34 = { by axiom 3 (x_times_y) R->L }
% 46.18/6.34 fresh36(fresh6(product(x, y, x_times_y), true, x, y, x_times_y), true, x, x_times_y, multiplicative_identity, y, multiplicative_identity, y)
% 46.18/6.34 = { by axiom 25 (commutativity_of_multiplication) }
% 46.18/6.34 fresh36(product(y, x, x_times_y), true, x, x_times_y, multiplicative_identity, y, multiplicative_identity, y)
% 46.18/6.34 = { by axiom 33 (distributivity1) R->L }
% 46.18/6.34 fresh35(X, X, y, x, x_times_y, multiplicative_identity, y, multiplicative_identity, y)
% 46.18/6.34 = { by lemma 52 }
% 46.18/6.34 true
% 46.18/6.34
% 46.18/6.34 Lemma 54: multiply(X, inverse(X)) = additive_identity.
% 46.18/6.34 Proof:
% 46.18/6.34 multiply(X, inverse(X))
% 46.18/6.34 = { by axiom 20 (multiplication_is_well_defined) R->L }
% 46.18/6.34 fresh2(true, true, X, inverse(X), multiply(X, inverse(X)), additive_identity)
% 46.18/6.34 = { by axiom 9 (multiplicative_inverse2) R->L }
% 46.18/6.34 fresh2(product(X, inverse(X), additive_identity), true, X, inverse(X), multiply(X, inverse(X)), additive_identity)
% 46.18/6.34 = { by axiom 30 (multiplication_is_well_defined) }
% 46.18/6.34 fresh(product(X, inverse(X), multiply(X, inverse(X))), true, multiply(X, inverse(X)), additive_identity)
% 46.18/6.34 = { by axiom 18 (closure_of_multiplication) }
% 46.18/6.34 fresh(true, true, multiply(X, inverse(X)), additive_identity)
% 46.18/6.34 = { by axiom 5 (multiplication_is_well_defined) }
% 46.18/6.34 additive_identity
% 46.18/6.34
% 46.18/6.34 Lemma 55: fresh35(X, X, Y, Z, multiply(Y, Z), W, V, U, T) = sum(multiply(Y, Z), V, T).
% 46.18/6.34 Proof:
% 46.18/6.34 fresh35(X, X, Y, Z, multiply(Y, Z), W, V, U, T)
% 46.18/6.34 = { by axiom 33 (distributivity1) }
% 46.18/6.34 fresh36(product(Y, Z, multiply(Y, Z)), true, Z, multiply(Y, Z), W, V, U, T)
% 46.18/6.34 = { by axiom 18 (closure_of_multiplication) }
% 46.18/6.34 fresh36(true, true, Z, multiply(Y, Z), W, V, U, T)
% 46.18/6.34 = { by axiom 23 (distributivity1) }
% 46.18/6.34 sum(multiply(Y, Z), V, T)
% 46.18/6.34
% 46.18/6.34 Lemma 56: sum(multiply(X, Y), X, X) = true.
% 46.18/6.34 Proof:
% 46.18/6.34 sum(multiply(X, Y), X, X)
% 46.18/6.34 = { by lemma 55 R->L }
% 46.18/6.34 fresh35(Z, Z, X, Y, multiply(X, Y), multiplicative_identity, X, multiplicative_identity, X)
% 46.18/6.34 = { by lemma 52 }
% 46.18/6.34 true
% 46.18/6.34
% 46.18/6.34 Lemma 57: add(X, multiply(X, Y)) = X.
% 46.18/6.34 Proof:
% 46.18/6.34 add(X, multiply(X, Y))
% 46.18/6.34 = { by lemma 38 R->L }
% 46.18/6.34 add(multiply(X, Y), X)
% 46.18/6.34 = { by axiom 6 (addition_is_well_defined) R->L }
% 46.18/6.34 fresh3(true, true, X, add(multiply(X, Y), X))
% 46.18/6.34 = { by lemma 56 R->L }
% 46.18/6.34 fresh3(sum(multiply(X, Y), X, X), true, X, add(multiply(X, Y), X))
% 46.18/6.34 = { by lemma 37 }
% 46.18/6.34 X
% 46.18/6.34
% 46.18/6.34 Lemma 58: add(X, multiply(Y, X)) = X.
% 46.18/6.34 Proof:
% 46.18/6.34 add(X, multiply(Y, X))
% 46.18/6.34 = { by lemma 42 R->L }
% 46.18/6.34 add(X, multiply(X, Y))
% 46.18/6.34 = { by lemma 57 }
% 46.18/6.34 X
% 46.18/6.34
% 46.18/6.34 Lemma 59: multiply(X, add(Y, X)) = X.
% 46.18/6.34 Proof:
% 46.18/6.34 multiply(X, add(Y, X))
% 46.18/6.34 = { by lemma 38 R->L }
% 46.18/6.34 multiply(X, add(X, Y))
% 46.18/6.34 = { by lemma 42 R->L }
% 46.18/6.34 multiply(add(X, Y), X)
% 46.18/6.34 = { by axiom 5 (multiplication_is_well_defined) R->L }
% 46.18/6.34 fresh(true, true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by lemma 50 R->L }
% 46.18/6.34 fresh(fresh21(Z, Z, X, Y, add(Y, X), W, X, X), true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by lemma 46 R->L }
% 46.18/6.34 fresh(fresh19(product(Y, additive_identity, additive_identity), true, X, Y, add(Y, X), additive_identity, X, additive_identity, X), true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by lemma 39 R->L }
% 46.18/6.34 fresh(fresh19(product(Y, additive_identity, fresh3(sum(multiply(additive_identity, Y), additive_identity, additive_identity), true, additive_identity, multiply(additive_identity, Y))), true, X, Y, add(Y, X), additive_identity, X, additive_identity, X), true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by lemma 56 }
% 46.18/6.34 fresh(fresh19(product(Y, additive_identity, fresh3(true, true, additive_identity, multiply(additive_identity, Y))), true, X, Y, add(Y, X), additive_identity, X, additive_identity, X), true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by axiom 6 (addition_is_well_defined) }
% 46.18/6.34 fresh(fresh19(product(Y, additive_identity, multiply(additive_identity, Y)), true, X, Y, add(Y, X), additive_identity, X, additive_identity, X), true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by lemma 42 }
% 46.18/6.34 fresh(fresh19(product(Y, additive_identity, multiply(Y, additive_identity)), true, X, Y, add(Y, X), additive_identity, X, additive_identity, X), true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by axiom 18 (closure_of_multiplication) }
% 46.18/6.34 fresh(fresh19(true, true, X, Y, add(Y, X), additive_identity, X, additive_identity, X), true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by axiom 31 (distributivity5) }
% 46.18/6.34 fresh(fresh20(sum(X, additive_identity, X), true, X, Y, add(Y, X), X, X), true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by axiom 1 (additive_identity2) }
% 46.18/6.34 fresh(fresh20(true, true, X, Y, add(Y, X), X, X), true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by axiom 21 (distributivity5) }
% 46.18/6.34 fresh(product(add(Y, X), X, X), true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by lemma 38 }
% 46.18/6.34 fresh(product(add(X, Y), X, X), true, X, multiply(add(X, Y), X))
% 46.18/6.34 = { by lemma 41 }
% 46.18/6.34 X
% 46.18/6.34
% 46.18/6.34 Lemma 60: fresh21(X, X, Y, inverse(Z), W, V, add(Y, Z), Y) = product(W, add(Z, Y), Y).
% 46.18/6.34 Proof:
% 46.18/6.34 fresh21(X, X, Y, inverse(Z), W, V, add(Y, Z), Y)
% 46.18/6.34 = { by lemma 47 R->L }
% 46.18/6.34 fresh19(U, U, Y, inverse(Z), W, Z, add(Y, Z), T, Y)
% 46.18/6.34 = { by axiom 31 (distributivity5) }
% 46.18/6.34 fresh20(sum(Y, Z, add(Y, Z)), true, Y, inverse(Z), W, add(Y, Z), Y)
% 46.18/6.34 = { by axiom 16 (closure_of_addition) }
% 46.18/6.34 fresh20(true, true, Y, inverse(Z), W, add(Y, Z), Y)
% 46.18/6.34 = { by axiom 21 (distributivity5) }
% 46.18/6.34 product(W, add(Y, Z), Y)
% 46.18/6.34 = { by lemma 38 }
% 46.18/6.34 product(W, add(Z, Y), Y)
% 46.18/6.34
% 46.18/6.34 Lemma 61: multiply(add(X, Y), add(Y, inverse(X))) = Y.
% 46.18/6.34 Proof:
% 46.18/6.34 multiply(add(X, Y), add(Y, inverse(X)))
% 46.18/6.34 = { by lemma 38 R->L }
% 46.18/6.34 multiply(add(Y, X), add(Y, inverse(X)))
% 46.18/6.34 = { by lemma 42 R->L }
% 46.18/6.34 multiply(add(Y, inverse(X)), add(Y, X))
% 46.18/6.34 = { by axiom 5 (multiplication_is_well_defined) R->L }
% 46.18/6.34 fresh(true, true, Y, multiply(add(Y, inverse(X)), add(Y, X)))
% 46.18/6.34 = { by lemma 48 R->L }
% 46.18/6.34 fresh(fresh21(Z, Z, Y, inverse(X), add(Y, inverse(X)), W, add(Y, X), Y), true, Y, multiply(add(Y, inverse(X)), add(Y, X)))
% 46.18/6.34 = { by lemma 60 }
% 46.18/6.34 fresh(product(add(Y, inverse(X)), add(X, Y), Y), true, Y, multiply(add(Y, inverse(X)), add(Y, X)))
% 46.18/6.34 = { by lemma 38 }
% 46.18/6.34 fresh(product(add(Y, inverse(X)), add(Y, X), Y), true, Y, multiply(add(Y, inverse(X)), add(Y, X)))
% 46.18/6.34 = { by lemma 41 }
% 46.18/6.34 Y
% 46.18/6.34
% 46.18/6.34 Lemma 62: multiply(add(X, Y), add(inverse(X), Y)) = Y.
% 46.18/6.34 Proof:
% 46.18/6.34 multiply(add(X, Y), add(inverse(X), Y))
% 46.18/6.34 = { by lemma 38 R->L }
% 46.18/6.34 multiply(add(X, Y), add(Y, inverse(X)))
% 46.18/6.34 = { by lemma 61 }
% 46.18/6.34 Y
% 46.18/6.34
% 46.18/6.34 Lemma 63: fresh17(product(X, Y, Z), true, W, X, V, Y, multiplicative_identity, Z, V) = fresh15(U, U, W, X, V, Y, multiplicative_identity, Z, V).
% 46.18/6.34 Proof:
% 46.18/6.34 fresh17(product(X, Y, Z), true, W, X, V, Y, multiplicative_identity, Z, V)
% 46.18/6.34 = { by axiom 36 (distributivity6) R->L }
% 46.18/6.34 fresh15(product(V, multiplicative_identity, V), true, W, X, V, Y, multiplicative_identity, Z, V)
% 46.18/6.34 = { by axiom 2 (multiplicative_identity2) }
% 46.18/6.34 fresh15(true, true, W, X, V, Y, multiplicative_identity, Z, V)
% 46.18/6.34 = { by axiom 32 (distributivity6) }
% 46.18/6.34 fresh16(sum(W, Y, multiplicative_identity), true, W, X, V, Z, V)
% 46.18/6.34 = { by axiom 32 (distributivity6) R->L }
% 46.18/6.34 fresh15(U, U, W, X, V, Y, multiplicative_identity, Z, V)
% 46.18/6.34
% 46.18/6.34 Lemma 64: fresh18(sum(X, Y, Z), true, X, multiply(Y, inverse(X)), Z) = sum(X, multiply(Y, inverse(X)), Z).
% 46.18/6.34 Proof:
% 46.18/6.34 fresh18(sum(X, Y, Z), true, X, multiply(Y, inverse(X)), Z)
% 46.18/6.34 = { by axiom 28 (distributivity6) R->L }
% 46.18/6.34 fresh17(true, true, X, Y, Z, inverse(X), multiplicative_identity, multiply(Y, inverse(X)), Z)
% 46.18/6.34 = { by axiom 18 (closure_of_multiplication) R->L }
% 46.18/6.34 fresh17(product(Y, inverse(X), multiply(Y, inverse(X))), true, X, Y, Z, inverse(X), multiplicative_identity, multiply(Y, inverse(X)), Z)
% 46.18/6.34 = { by lemma 63 }
% 46.18/6.34 fresh15(W, W, X, Y, Z, inverse(X), multiplicative_identity, multiply(Y, inverse(X)), Z)
% 46.18/6.34 = { by axiom 32 (distributivity6) }
% 46.18/6.34 fresh16(sum(X, inverse(X), multiplicative_identity), true, X, Y, Z, multiply(Y, inverse(X)), Z)
% 46.18/6.34 = { by axiom 7 (additive_inverse2) }
% 46.18/6.34 fresh16(true, true, X, Y, Z, multiply(Y, inverse(X)), Z)
% 46.18/6.34 = { by axiom 22 (distributivity6) }
% 46.18/6.34 sum(X, multiply(Y, inverse(X)), Z)
% 46.18/6.34
% 46.18/6.34 Lemma 65: sum(X, multiply(Y, inverse(X)), add(X, Y)) = true.
% 46.18/6.34 Proof:
% 46.18/6.34 sum(X, multiply(Y, inverse(X)), add(X, Y))
% 46.18/6.34 = { by lemma 64 R->L }
% 46.18/6.34 fresh18(sum(X, Y, add(X, Y)), true, X, multiply(Y, inverse(X)), add(X, Y))
% 46.18/6.34 = { by axiom 16 (closure_of_addition) }
% 46.18/6.34 fresh18(true, true, X, multiply(Y, inverse(X)), add(X, Y))
% 46.18/6.34 = { by axiom 13 (distributivity6) }
% 46.18/6.34 true
% 46.18/6.34
% 46.18/6.34 Lemma 66: add(X, multiply(Y, inverse(X))) = add(X, Y).
% 46.18/6.34 Proof:
% 46.18/6.34 add(X, multiply(Y, inverse(X)))
% 46.18/6.34 = { by axiom 6 (addition_is_well_defined) R->L }
% 46.18/6.34 fresh3(true, true, add(X, Y), add(X, multiply(Y, inverse(X))))
% 46.18/6.34 = { by lemma 65 R->L }
% 46.18/6.35 fresh3(sum(X, multiply(Y, inverse(X)), add(X, Y)), true, add(X, Y), add(X, multiply(Y, inverse(X))))
% 46.18/6.35 = { by lemma 37 }
% 46.18/6.35 add(X, Y)
% 46.18/6.35
% 46.18/6.35 Lemma 67: add(X, multiply(inverse(X), Y)) = add(X, Y).
% 46.18/6.35 Proof:
% 46.18/6.35 add(X, multiply(inverse(X), Y))
% 46.18/6.35 = { by lemma 42 R->L }
% 46.18/6.35 add(X, multiply(Y, inverse(X)))
% 46.18/6.35 = { by lemma 66 }
% 46.18/6.35 add(X, Y)
% 46.18/6.35
% 46.18/6.35 Lemma 68: multiply(inverse(X), add(X, Y)) = multiply(Y, inverse(X)).
% 46.18/6.35 Proof:
% 46.18/6.35 multiply(inverse(X), add(X, Y))
% 46.18/6.35 = { by lemma 66 R->L }
% 46.18/6.35 multiply(inverse(X), add(X, multiply(Y, inverse(X))))
% 46.18/6.35 = { by axiom 5 (multiplication_is_well_defined) R->L }
% 46.18/6.35 fresh(true, true, multiply(Y, inverse(X)), multiply(inverse(X), add(X, multiply(Y, inverse(X)))))
% 46.18/6.35 = { by axiom 12 (distributivity5) R->L }
% 46.18/6.35 fresh(fresh22(true, true, 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)))))
% 46.18/6.35 = { by lemma 56 R->L }
% 46.18/6.35 fresh(fresh22(sum(multiply(inverse(X), Y), inverse(X), inverse(X)), true, 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)))))
% 46.18/6.35 = { by axiom 24 (distributivity5) R->L }
% 46.18/6.35 fresh(fresh21(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)))))
% 46.18/6.35 = { by lemma 60 }
% 46.18/6.35 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)))))
% 46.18/6.35 = { by lemma 42 }
% 46.18/6.35 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)))))
% 46.18/6.35 = { by lemma 42 }
% 46.18/6.35 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)))))
% 46.18/6.35 = { by lemma 41 }
% 46.18/6.35 multiply(Y, inverse(X))
% 46.18/6.35
% 46.18/6.35 Lemma 69: multiply(X, inverse(add(X, Y))) = additive_identity.
% 46.18/6.35 Proof:
% 46.18/6.35 multiply(X, inverse(add(X, Y)))
% 46.18/6.35 = { by lemma 38 R->L }
% 46.18/6.35 multiply(X, inverse(add(Y, X)))
% 46.18/6.35 = { by lemma 42 R->L }
% 46.18/6.35 multiply(inverse(add(Y, X)), X)
% 46.18/6.35 = { by lemma 61 R->L }
% 46.18/6.35 multiply(inverse(add(Y, X)), multiply(add(Y, X), add(X, inverse(Y))))
% 46.18/6.35 = { by lemma 42 R->L }
% 46.18/6.35 multiply(inverse(add(Y, X)), multiply(add(X, inverse(Y)), add(Y, X)))
% 46.18/6.35 = { by lemma 42 R->L }
% 46.18/6.35 multiply(multiply(add(X, inverse(Y)), add(Y, X)), inverse(add(Y, X)))
% 46.18/6.35 = { by lemma 68 R->L }
% 46.18/6.35 multiply(inverse(add(Y, X)), add(add(Y, X), multiply(add(X, inverse(Y)), add(Y, X))))
% 46.18/6.35 = { by lemma 58 }
% 46.18/6.35 multiply(inverse(add(Y, X)), add(Y, X))
% 46.18/6.35 = { by lemma 42 }
% 46.18/6.35 multiply(add(Y, X), inverse(add(Y, X)))
% 46.18/6.35 = { by lemma 54 }
% 46.18/6.35 additive_identity
% 46.18/6.35
% 46.18/6.35 Lemma 70: add(X, inverse(add(Y, inverse(X)))) = X.
% 46.18/6.35 Proof:
% 46.18/6.35 add(X, inverse(add(Y, inverse(X))))
% 46.18/6.35 = { by lemma 38 R->L }
% 46.18/6.35 add(X, inverse(add(inverse(X), Y)))
% 46.18/6.35 = { by lemma 67 R->L }
% 46.18/6.35 add(X, multiply(inverse(X), inverse(add(inverse(X), Y))))
% 46.18/6.35 = { by lemma 69 }
% 46.18/6.35 add(X, additive_identity)
% 46.18/6.35 = { by lemma 40 }
% 46.18/6.35 X
% 46.18/6.35
% 46.18/6.35 Lemma 71: add(inverse(X), multiply(X, Y)) = add(Y, inverse(X)).
% 46.18/6.35 Proof:
% 46.18/6.35 add(inverse(X), multiply(X, Y))
% 46.18/6.35 = { by lemma 42 R->L }
% 46.18/6.35 add(inverse(X), multiply(Y, X))
% 46.18/6.35 = { by lemma 51 R->L }
% 46.18/6.35 add(inverse(X), multiply(Y, inverse(inverse(X))))
% 46.18/6.35 = { by lemma 66 }
% 46.18/6.35 add(inverse(X), Y)
% 46.18/6.35 = { by lemma 38 }
% 46.18/6.35 add(Y, inverse(X))
% 46.18/6.35
% 46.18/6.35 Lemma 72: inverse(add(x_times_y, inverse(y))) = multiply(y, inverse(x_times_y)).
% 46.18/6.35 Proof:
% 46.18/6.35 inverse(add(x_times_y, inverse(y)))
% 46.18/6.35 = { by lemma 62 R->L }
% 46.18/6.35 multiply(add(x_times_y, inverse(add(x_times_y, inverse(y)))), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.18/6.35 = { by lemma 38 R->L }
% 46.18/6.35 multiply(add(inverse(add(x_times_y, inverse(y))), x_times_y), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.18/6.35 = { by axiom 6 (addition_is_well_defined) R->L }
% 46.18/6.35 multiply(fresh3(true, true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.18/6.35 = { by lemma 65 R->L }
% 46.18/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), multiply(y, inverse(inverse(add(x_times_y, inverse(y))))), add(inverse(add(x_times_y, inverse(y))), y)), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.18/6.35 = { by lemma 51 }
% 46.18/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), multiply(y, add(x_times_y, inverse(y))), add(inverse(add(x_times_y, inverse(y))), y)), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.18/6.35 = { by lemma 42 }
% 46.18/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), multiply(add(x_times_y, inverse(y)), y), add(inverse(add(x_times_y, inverse(y))), y)), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.18/6.35 = { by lemma 38 }
% 46.18/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), multiply(add(x_times_y, inverse(y)), y), add(y, inverse(add(x_times_y, inverse(y))))), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.18/6.35 = { by axiom 5 (multiplication_is_well_defined) R->L }
% 46.45/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), fresh(true, true, x_times_y, multiply(add(x_times_y, inverse(y)), y)), add(y, inverse(add(x_times_y, inverse(y))))), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.45/6.35 = { by lemma 48 R->L }
% 46.45/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), fresh(fresh21(X, X, x_times_y, inverse(y), add(x_times_y, inverse(y)), Y, y, x_times_y), true, x_times_y, multiply(add(x_times_y, inverse(y)), y)), add(y, inverse(add(x_times_y, inverse(y))))), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.45/6.35 = { by lemma 47 R->L }
% 46.45/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), fresh(fresh19(Z, Z, x_times_y, inverse(y), add(x_times_y, inverse(y)), y, y, W, x_times_y), true, x_times_y, multiply(add(x_times_y, inverse(y)), y)), add(y, inverse(add(x_times_y, inverse(y))))), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.45/6.35 = { by axiom 31 (distributivity5) }
% 46.45/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), fresh(fresh20(sum(x_times_y, y, y), true, x_times_y, inverse(y), add(x_times_y, inverse(y)), y, x_times_y), true, x_times_y, multiply(add(x_times_y, inverse(y)), y)), add(y, inverse(add(x_times_y, inverse(y))))), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.45/6.35 = { by lemma 53 }
% 46.45/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), fresh(fresh20(true, true, x_times_y, inverse(y), add(x_times_y, inverse(y)), y, x_times_y), true, x_times_y, multiply(add(x_times_y, inverse(y)), y)), add(y, inverse(add(x_times_y, inverse(y))))), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.45/6.35 = { by axiom 21 (distributivity5) }
% 46.45/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), fresh(product(add(x_times_y, inverse(y)), y, x_times_y), true, x_times_y, multiply(add(x_times_y, inverse(y)), y)), add(y, inverse(add(x_times_y, inverse(y))))), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.45/6.35 = { by lemma 41 }
% 46.45/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), x_times_y, add(y, inverse(add(x_times_y, inverse(y))))), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.45/6.35 = { by lemma 70 }
% 46.45/6.35 multiply(fresh3(sum(inverse(add(x_times_y, inverse(y))), x_times_y, y), true, y, add(inverse(add(x_times_y, inverse(y))), x_times_y)), add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.45/6.35 = { by lemma 37 }
% 46.45/6.35 multiply(y, add(inverse(x_times_y), inverse(add(x_times_y, inverse(y)))))
% 46.45/6.35 = { by lemma 38 R->L }
% 46.45/6.35 multiply(y, add(inverse(add(x_times_y, inverse(y))), inverse(x_times_y)))
% 46.45/6.35 = { by lemma 71 R->L }
% 46.45/6.35 multiply(y, add(inverse(x_times_y), multiply(x_times_y, inverse(add(x_times_y, inverse(y))))))
% 46.45/6.35 = { by lemma 69 }
% 46.45/6.35 multiply(y, add(inverse(x_times_y), additive_identity))
% 46.45/6.35 = { by lemma 40 }
% 46.45/6.35 multiply(y, inverse(x_times_y))
% 46.45/6.35
% 46.45/6.35 Lemma 73: multiply(x_inverse_plus_y_inverse, add(x, inverse(y))) = inverse(y).
% 46.45/6.35 Proof:
% 46.45/6.35 multiply(x_inverse_plus_y_inverse, add(x, inverse(y)))
% 46.45/6.35 = { by lemma 42 R->L }
% 46.45/6.35 multiply(add(x, inverse(y)), x_inverse_plus_y_inverse)
% 46.45/6.35 = { by axiom 5 (multiplication_is_well_defined) R->L }
% 46.45/6.35 fresh(true, true, inverse(y), multiply(add(x, inverse(y)), x_inverse_plus_y_inverse))
% 46.45/6.35 = { by lemma 48 R->L }
% 46.45/6.35 fresh(fresh21(X, X, inverse(y), x, add(inverse(y), x), Y, x_inverse_plus_y_inverse, inverse(y)), true, inverse(y), multiply(add(x, inverse(y)), x_inverse_plus_y_inverse))
% 46.45/6.35 = { by lemma 49 R->L }
% 46.45/6.35 fresh(fresh19(Z, Z, inverse(y), x, add(inverse(y), x), inverse(x), x_inverse_plus_y_inverse, W, inverse(y)), true, inverse(y), multiply(add(x, inverse(y)), x_inverse_plus_y_inverse))
% 46.45/6.35 = { by axiom 31 (distributivity5) }
% 46.45/6.35 fresh(fresh20(sum(inverse(y), inverse(x), x_inverse_plus_y_inverse), true, inverse(y), x, add(inverse(y), x), x_inverse_plus_y_inverse, inverse(y)), true, inverse(y), multiply(add(x, inverse(y)), x_inverse_plus_y_inverse))
% 46.45/6.35 = { by axiom 26 (commutativity_of_addition) R->L }
% 46.45/6.35 fresh(fresh20(fresh5(sum(inverse(x), inverse(y), x_inverse_plus_y_inverse), true, inverse(x), inverse(y), x_inverse_plus_y_inverse), true, inverse(y), x, add(inverse(y), x), x_inverse_plus_y_inverse, inverse(y)), true, inverse(y), multiply(add(x, inverse(y)), x_inverse_plus_y_inverse))
% 46.45/6.35 = { by axiom 17 (x_inverse_plus_y_inverse) }
% 46.45/6.35 fresh(fresh20(fresh5(true, true, inverse(x), inverse(y), x_inverse_plus_y_inverse), true, inverse(y), x, add(inverse(y), x), x_inverse_plus_y_inverse, inverse(y)), true, inverse(y), multiply(add(x, inverse(y)), x_inverse_plus_y_inverse))
% 46.45/6.35 = { by axiom 15 (commutativity_of_addition) }
% 46.45/6.35 fresh(fresh20(true, true, inverse(y), x, add(inverse(y), x), x_inverse_plus_y_inverse, inverse(y)), true, inverse(y), multiply(add(x, inverse(y)), x_inverse_plus_y_inverse))
% 46.45/6.35 = { by axiom 21 (distributivity5) }
% 46.45/6.35 fresh(product(add(inverse(y), x), x_inverse_plus_y_inverse, inverse(y)), true, inverse(y), multiply(add(x, inverse(y)), x_inverse_plus_y_inverse))
% 46.45/6.35 = { by lemma 38 }
% 46.45/6.35 fresh(product(add(x, inverse(y)), x_inverse_plus_y_inverse, inverse(y)), true, inverse(y), multiply(add(x, inverse(y)), x_inverse_plus_y_inverse))
% 46.45/6.35 = { by lemma 41 }
% 46.45/6.35 inverse(y)
% 46.45/6.35
% 46.45/6.35 Goal 1 (prove_equation): inverse(x_times_y) = x_inverse_plus_y_inverse.
% 46.45/6.35 Proof:
% 46.45/6.35 inverse(x_times_y)
% 46.45/6.35 = { by lemma 40 R->L }
% 46.45/6.35 add(inverse(x_times_y), additive_identity)
% 46.45/6.35 = { by lemma 69 R->L }
% 46.45/6.35 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(x_inverse_plus_y_inverse, multiply(y, inverse(x_times_y))))))
% 46.45/6.35 = { by lemma 72 R->L }
% 46.45/6.35 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(x_inverse_plus_y_inverse, inverse(add(x_times_y, inverse(y)))))))
% 46.45/6.35 = { by lemma 71 R->L }
% 46.45/6.35 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(add(x_times_y, inverse(y)), x_inverse_plus_y_inverse)))))
% 46.45/6.35 = { by lemma 42 }
% 46.45/6.35 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, add(x_times_y, inverse(y)))))))
% 46.45/6.35 = { by lemma 38 R->L }
% 46.45/6.35 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, add(inverse(y), x_times_y))))))
% 46.45/6.35 = { by axiom 6 (addition_is_well_defined) R->L }
% 46.45/6.35 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, fresh3(true, true, add(x, inverse(y)), add(inverse(y), x_times_y)))))))
% 46.45/6.35 = { by axiom 13 (distributivity6) R->L }
% 46.45/6.35 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, fresh3(fresh18(true, true, inverse(y), x_times_y, add(inverse(y), x)), true, add(x, inverse(y)), add(inverse(y), x_times_y)))))))
% 46.45/6.35 = { by axiom 16 (closure_of_addition) R->L }
% 46.45/6.35 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, fresh3(fresh18(sum(inverse(y), x, add(inverse(y), x)), true, inverse(y), x_times_y, add(inverse(y), x)), true, add(x, inverse(y)), add(inverse(y), x_times_y)))))))
% 46.45/6.35 = { by axiom 28 (distributivity6) R->L }
% 46.45/6.35 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, fresh3(fresh17(true, true, inverse(y), x, add(inverse(y), x), y, multiplicative_identity, x_times_y, add(inverse(y), x)), true, add(x, inverse(y)), add(inverse(y), x_times_y)))))))
% 46.45/6.36 = { by axiom 3 (x_times_y) R->L }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, fresh3(fresh17(product(x, y, x_times_y), true, inverse(y), x, add(inverse(y), x), y, multiplicative_identity, x_times_y, add(inverse(y), x)), true, add(x, inverse(y)), add(inverse(y), x_times_y)))))))
% 46.45/6.36 = { by lemma 63 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, fresh3(fresh15(X, X, inverse(y), x, add(inverse(y), x), y, multiplicative_identity, x_times_y, add(inverse(y), x)), true, add(x, inverse(y)), add(inverse(y), x_times_y)))))))
% 46.45/6.36 = { by axiom 32 (distributivity6) }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, fresh3(fresh16(sum(inverse(y), y, multiplicative_identity), true, inverse(y), x, add(inverse(y), x), x_times_y, add(inverse(y), x)), true, add(x, inverse(y)), add(inverse(y), x_times_y)))))))
% 46.45/6.36 = { by axiom 8 (additive_inverse1) }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, fresh3(fresh16(true, true, inverse(y), x, add(inverse(y), x), x_times_y, add(inverse(y), x)), true, add(x, inverse(y)), add(inverse(y), x_times_y)))))))
% 46.45/6.36 = { by axiom 22 (distributivity6) }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, fresh3(sum(inverse(y), x_times_y, add(inverse(y), x)), true, add(x, inverse(y)), add(inverse(y), x_times_y)))))))
% 46.45/6.36 = { by lemma 38 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, fresh3(sum(inverse(y), x_times_y, add(x, inverse(y))), true, add(x, inverse(y)), add(inverse(y), x_times_y)))))))
% 46.45/6.36 = { by lemma 37 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), multiply(x_inverse_plus_y_inverse, add(x, inverse(y)))))))
% 46.45/6.36 = { by lemma 73 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(add(x_times_y, inverse(y))), inverse(y)))))
% 46.45/6.36 = { by lemma 72 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(multiply(y, inverse(x_times_y)), inverse(y)))))
% 46.45/6.36 = { by lemma 38 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(y), multiply(y, inverse(x_times_y))))))
% 46.45/6.36 = { by lemma 71 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), inverse(y)))))
% 46.45/6.36 = { by lemma 38 R->L }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(y), inverse(x_times_y)))))
% 46.45/6.36 = { by lemma 71 R->L }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), multiply(x_times_y, inverse(y))))))
% 46.45/6.36 = { by lemma 62 R->L }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), multiply(add(y, multiply(x_times_y, inverse(y))), add(inverse(y), multiply(x_times_y, inverse(y))))))))
% 46.45/6.36 = { by axiom 6 (addition_is_well_defined) R->L }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), multiply(fresh3(true, true, y, add(y, multiply(x_times_y, inverse(y)))), add(inverse(y), multiply(x_times_y, inverse(y))))))))
% 46.45/6.36 = { by axiom 13 (distributivity6) R->L }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), multiply(fresh3(fresh18(true, true, y, multiply(x_times_y, inverse(y)), y), true, y, add(y, multiply(x_times_y, inverse(y)))), add(inverse(y), multiply(x_times_y, inverse(y))))))))
% 46.45/6.36 = { by axiom 15 (commutativity_of_addition) R->L }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), multiply(fresh3(fresh18(fresh5(true, true, x_times_y, y, y), true, y, multiply(x_times_y, inverse(y)), y), true, y, add(y, multiply(x_times_y, inverse(y)))), add(inverse(y), multiply(x_times_y, inverse(y))))))))
% 46.45/6.36 = { by lemma 53 R->L }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), multiply(fresh3(fresh18(fresh5(sum(x_times_y, y, y), true, x_times_y, y, y), true, y, multiply(x_times_y, inverse(y)), y), true, y, add(y, multiply(x_times_y, inverse(y)))), add(inverse(y), multiply(x_times_y, inverse(y))))))))
% 46.45/6.36 = { by axiom 26 (commutativity_of_addition) }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), multiply(fresh3(fresh18(sum(y, x_times_y, y), true, y, multiply(x_times_y, inverse(y)), y), true, y, add(y, multiply(x_times_y, inverse(y)))), add(inverse(y), multiply(x_times_y, inverse(y))))))))
% 46.45/6.36 = { by lemma 64 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), multiply(fresh3(sum(y, multiply(x_times_y, inverse(y)), y), true, y, add(y, multiply(x_times_y, inverse(y)))), add(inverse(y), multiply(x_times_y, inverse(y))))))))
% 46.45/6.36 = { by lemma 37 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), multiply(y, add(inverse(y), multiply(x_times_y, inverse(y))))))))
% 46.45/6.36 = { by lemma 58 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), multiply(y, inverse(y))))))
% 46.45/6.36 = { by lemma 54 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(add(inverse(x_times_y), additive_identity))))
% 46.45/6.36 = { by lemma 40 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, inverse(inverse(x_times_y))))
% 46.45/6.36 = { by lemma 51 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_inverse_plus_y_inverse, x_times_y))
% 46.45/6.36 = { by lemma 42 }
% 46.45/6.36 add(inverse(x_times_y), multiply(x_times_y, x_inverse_plus_y_inverse))
% 46.45/6.36 = { by lemma 71 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(x_times_y))
% 46.45/6.36 = { by lemma 37 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(fresh3(sum(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y, x_times_y), true, x_times_y, add(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y))))
% 46.45/6.36 = { by lemma 55 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(fresh3(fresh35(true, true, x, multiply(inverse(x_inverse_plus_y_inverse), y), multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), y, x_times_y, y, x_times_y), true, x_times_y, add(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y))))
% 46.45/6.36 = { by lemma 42 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(fresh3(fresh35(true, true, x, multiply(inverse(x_inverse_plus_y_inverse), y), multiply(multiply(inverse(x_inverse_plus_y_inverse), y), x), y, x_times_y, y, x_times_y), true, x_times_y, add(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y))))
% 46.45/6.36 = { by axiom 3 (x_times_y) R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(fresh3(fresh35(product(x, y, x_times_y), true, x, multiply(inverse(x_inverse_plus_y_inverse), y), multiply(multiply(inverse(x_inverse_plus_y_inverse), y), x), y, x_times_y, y, x_times_y), true, x_times_y, add(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y))))
% 46.45/6.36 = { by axiom 34 (distributivity1) }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(fresh3(fresh37(product(x, y, x_times_y), true, x, multiply(inverse(x_inverse_plus_y_inverse), y), multiply(multiply(inverse(x_inverse_plus_y_inverse), y), x), y, x_times_y, y, x_times_y), true, x_times_y, add(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y))))
% 46.45/6.36 = { by axiom 3 (x_times_y) }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(fresh3(fresh37(true, true, x, multiply(inverse(x_inverse_plus_y_inverse), y), multiply(multiply(inverse(x_inverse_plus_y_inverse), y), x), y, x_times_y, y, x_times_y), true, x_times_y, add(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y))))
% 46.45/6.36 = { by axiom 27 (distributivity1) }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(fresh3(fresh38(sum(multiply(inverse(x_inverse_plus_y_inverse), y), y, y), true, multiply(multiply(inverse(x_inverse_plus_y_inverse), y), x), x_times_y, x_times_y), true, x_times_y, add(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y))))
% 46.45/6.36 = { by lemma 42 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(fresh3(fresh38(sum(multiply(y, inverse(x_inverse_plus_y_inverse)), y, y), true, multiply(multiply(inverse(x_inverse_plus_y_inverse), y), x), x_times_y, x_times_y), true, x_times_y, add(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y))))
% 46.45/6.36 = { by lemma 56 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(fresh3(fresh38(true, true, multiply(multiply(inverse(x_inverse_plus_y_inverse), y), x), x_times_y, x_times_y), true, x_times_y, add(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y))))
% 46.45/6.36 = { by axiom 11 (distributivity1) }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(fresh3(true, true, x_times_y, add(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y))))
% 46.45/6.36 = { by axiom 6 (addition_is_well_defined) }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)), x_times_y)))
% 46.45/6.36 = { by lemma 38 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(x, multiply(inverse(x_inverse_plus_y_inverse), y)))))
% 46.45/6.36 = { by lemma 40 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(x, multiply(inverse(x_inverse_plus_y_inverse), add(y, additive_identity))))))
% 46.45/6.36 = { by lemma 54 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(x, multiply(inverse(x_inverse_plus_y_inverse), add(y, multiply(x_inverse_plus_y_inverse, inverse(x_inverse_plus_y_inverse))))))))
% 46.45/6.36 = { by lemma 42 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(x, multiply(inverse(x_inverse_plus_y_inverse), add(y, multiply(inverse(x_inverse_plus_y_inverse), x_inverse_plus_y_inverse)))))))
% 46.45/6.36 = { by lemma 57 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(x, multiply(inverse(x_inverse_plus_y_inverse), add(y, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, add(x, inverse(y)))))))))))
% 46.45/6.36 = { by lemma 73 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(x, multiply(inverse(x_inverse_plus_y_inverse), add(y, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, inverse(y)))))))))
% 46.45/6.36 = { by lemma 68 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(x, multiply(inverse(x_inverse_plus_y_inverse), add(y, multiply(inverse(y), inverse(x_inverse_plus_y_inverse))))))))
% 46.45/6.36 = { by lemma 67 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(x, multiply(inverse(x_inverse_plus_y_inverse), add(y, inverse(x_inverse_plus_y_inverse)))))))
% 46.45/6.36 = { by lemma 59 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(x, inverse(x_inverse_plus_y_inverse)))))
% 46.45/6.36 = { by lemma 42 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), x))))
% 46.45/6.36 = { by lemma 40 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, additive_identity)))))
% 46.45/6.36 = { by lemma 54 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(x_inverse_plus_y_inverse, inverse(x_inverse_plus_y_inverse)))))))
% 46.45/6.36 = { by lemma 42 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x_inverse_plus_y_inverse), x_inverse_plus_y_inverse))))))
% 46.45/6.36 = { by lemma 57 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, add(y, inverse(x))))))))))
% 46.45/6.36 = { by lemma 42 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, multiply(add(y, inverse(x)), x_inverse_plus_y_inverse))))))))
% 46.45/6.36 = { by axiom 5 (multiplication_is_well_defined) R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, fresh(true, true, inverse(x), multiply(add(y, inverse(x)), x_inverse_plus_y_inverse)))))))))
% 46.45/6.36 = { by lemma 48 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, fresh(fresh21(Y, Y, inverse(x), y, add(inverse(x), y), Z, x_inverse_plus_y_inverse, inverse(x)), true, inverse(x), multiply(add(y, inverse(x)), x_inverse_plus_y_inverse)))))))))
% 46.45/6.36 = { by lemma 49 R->L }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, fresh(fresh19(W, W, inverse(x), y, add(inverse(x), y), inverse(y), x_inverse_plus_y_inverse, V, inverse(x)), true, inverse(x), multiply(add(y, inverse(x)), x_inverse_plus_y_inverse)))))))))
% 46.45/6.36 = { by axiom 31 (distributivity5) }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, fresh(fresh20(sum(inverse(x), inverse(y), x_inverse_plus_y_inverse), true, inverse(x), y, add(inverse(x), y), x_inverse_plus_y_inverse, inverse(x)), true, inverse(x), multiply(add(y, inverse(x)), x_inverse_plus_y_inverse)))))))))
% 46.45/6.36 = { by axiom 17 (x_inverse_plus_y_inverse) }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, fresh(fresh20(true, true, inverse(x), y, add(inverse(x), y), x_inverse_plus_y_inverse, inverse(x)), true, inverse(x), multiply(add(y, inverse(x)), x_inverse_plus_y_inverse)))))))))
% 46.45/6.36 = { by axiom 21 (distributivity5) }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, fresh(product(add(inverse(x), y), x_inverse_plus_y_inverse, inverse(x)), true, inverse(x), multiply(add(y, inverse(x)), x_inverse_plus_y_inverse)))))))))
% 46.45/6.36 = { by lemma 38 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, fresh(product(add(y, inverse(x)), x_inverse_plus_y_inverse, inverse(x)), true, inverse(x), multiply(add(y, inverse(x)), x_inverse_plus_y_inverse)))))))))
% 46.45/6.36 = { by lemma 41 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x_inverse_plus_y_inverse), add(x_inverse_plus_y_inverse, inverse(x))))))))
% 46.45/6.36 = { by lemma 68 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, multiply(inverse(x), inverse(x_inverse_plus_y_inverse)))))))
% 46.45/6.36 = { by lemma 67 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, multiply(inverse(x_inverse_plus_y_inverse), add(x, inverse(x_inverse_plus_y_inverse))))))
% 46.45/6.36 = { by lemma 59 }
% 46.45/6.36 add(x_inverse_plus_y_inverse, inverse(add(x_times_y, inverse(x_inverse_plus_y_inverse))))
% 46.45/6.36 = { by lemma 70 }
% 46.45/6.36 x_inverse_plus_y_inverse
% 46.45/6.36 % SZS output end Proof
% 46.45/6.36
% 46.45/6.36 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------