TSTP Solution File: RNG008-5 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : RNG008-5 : TPTP v8.1.2. Released v1.0.0.
% 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 : n012.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 : Thu Aug 31 13:58:46 EDT 2023
% Result : Unsatisfiable 13.51s 2.13s
% Output : Proof 15.02s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : RNG008-5 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.33 % Computer : n012.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Sun Aug 27 02:38:10 EDT 2023
% 0.11/0.33 % CPUTime :
% 13.51/2.13 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 13.51/2.13
% 13.51/2.13 % SZS status Unsatisfiable
% 13.51/2.13
% 13.51/2.25 % SZS output start Proof
% 13.51/2.25 Take the following subset of the input axioms:
% 13.51/2.26 fof(a_times_b_is_c, hypothesis, product(a, b, c)).
% 13.51/2.26 fof(addition_is_well_defined, axiom, ![X, Y, U, V]: (~sum(X, Y, U) | (~sum(X, Y, V) | U=V))).
% 13.51/2.26 fof(additive_identity1, axiom, ![X2]: sum(additive_identity, X2, X2)).
% 13.51/2.26 fof(additive_identity2, axiom, ![X2]: sum(X2, additive_identity, X2)).
% 13.51/2.26 fof(additive_inverse_additive_inverse, axiom, ![X2]: sum(additive_inverse(additive_inverse(X2)), additive_identity, X2)).
% 13.51/2.26 fof(associativity_of_addition1, axiom, ![Z, W, X2, Y2, U2, V5]: (~sum(X2, Y2, U2) | (~sum(Y2, Z, V5) | (~sum(U2, Z, W) | sum(X2, V5, W))))).
% 13.51/2.26 fof(associativity_of_addition2, axiom, ![X2, Y2, U2, V5, Z2, W2]: (~sum(X2, Y2, U2) | (~sum(Y2, Z2, V5) | (~sum(X2, V5, W2) | sum(U2, Z2, W2))))).
% 13.51/2.26 fof(associativity_of_multiplication1, axiom, ![X2, Y2, U2, V5, Z2, W2]: (~product(X2, Y2, U2) | (~product(Y2, Z2, V5) | (~product(U2, Z2, W2) | product(X2, V5, W2))))).
% 13.51/2.26 fof(closure_of_addition, axiom, ![X2, Y2]: sum(X2, Y2, add(X2, Y2))).
% 13.51/2.26 fof(closure_of_multiplication, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 13.51/2.26 fof(commutativity_of_addition, axiom, ![X2, Y2, Z2]: (~sum(X2, Y2, Z2) | sum(Y2, X2, Z2))).
% 13.51/2.26 fof(distribute_additive_inverse, axiom, ![X2, Y2]: sum(additive_inverse(X2), additive_inverse(Y2), additive_inverse(add(X2, Y2)))).
% 13.51/2.26 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)))))).
% 13.51/2.26 fof(distributivity3, axiom, ![X2, Y2, Z2, V1_2, V2_2, V3_2, V4_2]: (~product(Y2, X2, V1_2) | (~product(Z2, X2, V2_2) | (~sum(Y2, Z2, V3_2) | (~product(V3_2, X2, V4_2) | sum(V1_2, V2_2, V4_2)))))).
% 13.51/2.26 fof(left_inverse, axiom, ![X2]: sum(additive_inverse(X2), X2, additive_identity)).
% 13.51/2.26 fof(multiplication_is_well_defined, axiom, ![X2, Y2, U2, V5]: (~product(X2, Y2, U2) | (~product(X2, Y2, V5) | U2=V5))).
% 13.51/2.26 fof(multiply_additive_id2, axiom, ![X2]: product(additive_identity, X2, additive_identity)).
% 13.51/2.26 fof(multiply_additive_inverse, axiom, ![X2, Y2]: product(X2, additive_inverse(Y2), additive_inverse(multiply(X2, Y2)))).
% 13.51/2.26 fof(prove_b_times_a_is_c, negated_conjecture, ~product(b, a, c)).
% 13.51/2.26 fof(right_inverse, axiom, ![X2]: sum(X2, additive_inverse(X2), additive_identity)).
% 13.51/2.26 fof(x_squared_is_x, hypothesis, ![X2]: product(X2, X2, X2)).
% 13.51/2.26
% 13.51/2.27 Now clausify the problem and encode Horn clauses using encoding 3 of
% 13.51/2.27 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 13.51/2.27 We repeatedly replace C & s=t => u=v by the two clauses:
% 13.51/2.27 fresh(y, y, x1...xn) = u
% 13.51/2.27 C => fresh(s, t, x1...xn) = v
% 13.51/2.27 where fresh is a fresh function symbol and x1..xn are the free
% 13.51/2.28 variables of u and v.
% 13.51/2.28 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 13.51/2.28 input problem has no model of domain size 1).
% 13.51/2.28
% 13.51/2.28 The encoding turns the above axioms into the following unit equations and goals:
% 13.51/2.28
% 13.51/2.28 Axiom 1 (x_squared_is_x): product(X, X, X) = true.
% 13.51/2.28 Axiom 2 (multiply_additive_id2): product(additive_identity, X, additive_identity) = true.
% 13.51/2.28 Axiom 3 (a_times_b_is_c): product(a, b, c) = true.
% 13.51/2.28 Axiom 4 (additive_identity2): sum(X, additive_identity, X) = true.
% 13.51/2.28 Axiom 5 (additive_identity1): sum(additive_identity, X, X) = true.
% 13.51/2.28 Axiom 6 (right_inverse): sum(X, additive_inverse(X), additive_identity) = true.
% 13.51/2.28 Axiom 7 (left_inverse): sum(additive_inverse(X), X, additive_identity) = true.
% 13.51/2.28 Axiom 8 (multiplication_is_well_defined): fresh(X, X, Y, Z) = Z.
% 13.51/2.28 Axiom 9 (addition_is_well_defined): fresh3(X, X, Y, Z) = Z.
% 13.51/2.28 Axiom 10 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 13.51/2.28 Axiom 11 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 13.51/2.28 Axiom 12 (additive_inverse_additive_inverse): sum(additive_inverse(additive_inverse(X)), additive_identity, X) = true.
% 13.51/2.28 Axiom 13 (associativity_of_addition1): fresh33(X, X, Y, Z, W) = true.
% 13.51/2.28 Axiom 14 (associativity_of_addition2): fresh31(X, X, Y, Z, W) = true.
% 13.51/2.28 Axiom 15 (associativity_of_multiplication1): fresh29(X, X, Y, Z, W) = true.
% 13.51/2.28 Axiom 16 (distributivity1): fresh25(X, X, Y, Z, W) = true.
% 13.51/2.28 Axiom 17 (distributivity3): fresh17(X, X, Y, Z, W) = true.
% 13.51/2.28 Axiom 18 (commutativity_of_addition): fresh5(X, X, Y, Z, W) = true.
% 13.51/2.28 Axiom 19 (addition_is_well_defined): fresh4(X, X, Y, Z, W, V) = W.
% 13.51/2.28 Axiom 20 (multiplication_is_well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 13.51/2.28 Axiom 21 (multiply_additive_inverse): product(X, additive_inverse(Y), additive_inverse(multiply(X, Y))) = true.
% 13.51/2.28 Axiom 22 (associativity_of_addition1): fresh9(X, X, Y, Z, W, V, U) = sum(Y, V, U).
% 13.51/2.28 Axiom 23 (associativity_of_addition2): fresh8(X, X, Y, Z, W, V, U) = sum(W, V, U).
% 13.51/2.28 Axiom 24 (associativity_of_multiplication1): fresh7(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 13.51/2.28 Axiom 25 (distribute_additive_inverse): sum(additive_inverse(X), additive_inverse(Y), additive_inverse(add(X, Y))) = true.
% 13.51/2.28 Axiom 26 (associativity_of_addition1): fresh32(X, X, Y, Z, W, V, U, T) = fresh33(sum(Y, Z, W), true, Y, U, T).
% 13.51/2.28 Axiom 27 (associativity_of_addition2): fresh30(X, X, Y, Z, W, V, U, T) = fresh31(sum(Y, Z, W), true, W, V, T).
% 13.51/2.28 Axiom 28 (associativity_of_multiplication1): fresh28(X, X, Y, Z, W, V, U, T) = fresh29(product(Y, Z, W), true, Y, U, T).
% 13.51/2.28 Axiom 29 (distributivity1): fresh23(X, X, Y, Z, W, V, U, T) = sum(Z, V, T).
% 13.51/2.28 Axiom 30 (distributivity3): fresh15(X, X, Y, Z, W, V, U, T) = sum(Z, V, T).
% 13.51/2.28 Axiom 31 (commutativity_of_addition): fresh5(sum(X, Y, Z), true, X, Y, Z) = sum(Y, X, Z).
% 13.51/2.28 Axiom 32 (distributivity1): fresh24(X, X, Y, Z, W, V, U, T, S) = fresh25(sum(Z, V, T), true, W, U, S).
% 13.51/2.28 Axiom 33 (distributivity3): fresh16(X, X, Y, Z, W, V, U, T, S) = fresh17(sum(Y, V, T), true, W, U, S).
% 13.51/2.28 Axiom 34 (addition_is_well_defined): fresh4(sum(X, Y, Z), true, X, Y, W, Z) = fresh3(sum(X, Y, W), true, W, Z).
% 13.51/2.28 Axiom 35 (multiplication_is_well_defined): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 13.51/2.28 Axiom 36 (associativity_of_addition1): fresh32(sum(X, Y, Z), true, W, V, X, Y, U, Z) = fresh9(sum(V, Y, U), true, W, V, X, U, Z).
% 13.51/2.28 Axiom 37 (associativity_of_addition2): fresh30(sum(X, Y, Z), true, W, X, V, Y, Z, U) = fresh8(sum(W, Z, U), true, W, X, V, Y, U).
% 13.51/2.28 Axiom 38 (associativity_of_multiplication1): fresh28(product(X, Y, Z), true, W, V, X, Y, U, Z) = fresh7(product(V, Y, U), true, W, V, X, U, Z).
% 13.51/2.28 Axiom 39 (distributivity1): fresh22(X, X, Y, Z, W, V, U, T, S) = fresh23(product(Y, Z, W), true, Z, W, V, U, T, S).
% 13.51/2.28 Axiom 40 (distributivity3): fresh14(X, X, Y, Z, W, V, U, T, S) = fresh15(product(Y, Z, W), true, Y, W, V, U, T, S).
% 13.51/2.28 Axiom 41 (distributivity1): fresh22(product(X, Y, Z), true, X, W, V, U, T, Y, Z) = fresh24(product(X, U, T), true, X, W, V, U, T, Y, Z).
% 13.51/2.28 Axiom 42 (distributivity3): fresh14(product(X, Y, Z), true, W, Y, V, U, T, X, Z) = fresh16(product(U, Y, T), true, W, Y, V, U, T, X, Z).
% 13.51/2.28
% 13.51/2.28 Lemma 43: fresh3(sum(X, Y, Z), true, Z, add(X, Y)) = Z.
% 13.51/2.28 Proof:
% 13.51/2.28 fresh3(sum(X, Y, Z), true, Z, add(X, Y))
% 13.51/2.28 = { by axiom 34 (addition_is_well_defined) R->L }
% 13.51/2.28 fresh4(sum(X, Y, add(X, Y)), true, X, Y, Z, add(X, Y))
% 13.51/2.28 = { by axiom 11 (closure_of_addition) }
% 13.51/2.28 fresh4(true, true, X, Y, Z, add(X, Y))
% 13.51/2.28 = { by axiom 19 (addition_is_well_defined) }
% 13.51/2.28 Z
% 13.51/2.28
% 13.51/2.28 Lemma 44: add(X, Y) = add(Y, X).
% 13.51/2.28 Proof:
% 13.51/2.28 add(X, Y)
% 13.51/2.28 = { by lemma 43 R->L }
% 13.51/2.28 fresh3(sum(Y, X, add(X, Y)), true, add(X, Y), add(Y, X))
% 13.51/2.28 = { by axiom 31 (commutativity_of_addition) R->L }
% 13.51/2.28 fresh3(fresh5(sum(X, Y, add(X, Y)), true, X, Y, add(X, Y)), true, add(X, Y), add(Y, X))
% 13.51/2.28 = { by axiom 11 (closure_of_addition) }
% 13.51/2.28 fresh3(fresh5(true, true, X, Y, add(X, Y)), true, add(X, Y), add(Y, X))
% 13.51/2.28 = { by axiom 18 (commutativity_of_addition) }
% 13.51/2.28 fresh3(true, true, add(X, Y), add(Y, X))
% 13.51/2.28 = { by axiom 9 (addition_is_well_defined) }
% 13.51/2.28 add(Y, X)
% 13.51/2.28
% 13.51/2.28 Lemma 45: fresh30(X, X, Y, additive_inverse(Z), W, Z, V, Y) = sum(W, Z, Y).
% 13.51/2.28 Proof:
% 13.51/2.28 fresh30(X, X, Y, additive_inverse(Z), W, Z, V, Y)
% 13.51/2.28 = { by axiom 27 (associativity_of_addition2) }
% 13.51/2.28 fresh31(sum(Y, additive_inverse(Z), W), true, W, Z, Y)
% 13.51/2.28 = { by axiom 27 (associativity_of_addition2) R->L }
% 13.51/2.28 fresh30(true, true, Y, additive_inverse(Z), W, Z, additive_identity, Y)
% 13.51/2.28 = { by axiom 7 (left_inverse) R->L }
% 13.51/2.28 fresh30(sum(additive_inverse(Z), Z, additive_identity), true, Y, additive_inverse(Z), W, Z, additive_identity, Y)
% 13.51/2.28 = { by axiom 37 (associativity_of_addition2) }
% 13.51/2.28 fresh8(sum(Y, additive_identity, Y), true, Y, additive_inverse(Z), W, Z, Y)
% 13.51/2.28 = { by axiom 4 (additive_identity2) }
% 13.51/2.28 fresh8(true, true, Y, additive_inverse(Z), W, Z, Y)
% 13.51/2.28 = { by axiom 23 (associativity_of_addition2) }
% 13.51/2.28 sum(W, Z, Y)
% 13.51/2.28
% 13.51/2.28 Lemma 46: add(X, additive_inverse(add(X, Y))) = additive_inverse(Y).
% 13.51/2.28 Proof:
% 13.51/2.28 add(X, additive_inverse(add(X, Y)))
% 13.51/2.28 = { by lemma 44 R->L }
% 13.51/2.28 add(additive_inverse(add(X, Y)), X)
% 13.51/2.28 = { by axiom 9 (addition_is_well_defined) R->L }
% 13.51/2.28 fresh3(true, true, additive_inverse(Y), add(additive_inverse(add(X, Y)), X))
% 13.51/2.28 = { by axiom 14 (associativity_of_addition2) R->L }
% 13.51/2.28 fresh3(fresh31(true, true, additive_inverse(add(Y, X)), X, additive_inverse(Y)), true, additive_inverse(Y), add(additive_inverse(add(X, Y)), X))
% 13.51/2.28 = { by axiom 25 (distribute_additive_inverse) R->L }
% 13.51/2.28 fresh3(fresh31(sum(additive_inverse(Y), additive_inverse(X), additive_inverse(add(Y, X))), true, additive_inverse(add(Y, X)), X, additive_inverse(Y)), true, additive_inverse(Y), add(additive_inverse(add(X, Y)), X))
% 13.51/2.28 = { by axiom 27 (associativity_of_addition2) R->L }
% 13.51/2.28 fresh3(fresh30(Z, Z, additive_inverse(Y), additive_inverse(X), additive_inverse(add(Y, X)), X, W, additive_inverse(Y)), true, additive_inverse(Y), add(additive_inverse(add(X, Y)), X))
% 13.51/2.28 = { by lemma 45 }
% 13.51/2.28 fresh3(sum(additive_inverse(add(Y, X)), X, additive_inverse(Y)), true, additive_inverse(Y), add(additive_inverse(add(X, Y)), X))
% 13.51/2.28 = { by lemma 44 }
% 13.51/2.28 fresh3(sum(additive_inverse(add(X, Y)), X, additive_inverse(Y)), true, additive_inverse(Y), add(additive_inverse(add(X, Y)), X))
% 13.51/2.28 = { by lemma 43 }
% 13.51/2.28 additive_inverse(Y)
% 13.51/2.28
% 13.51/2.28 Lemma 47: multiply(X, X) = X.
% 13.51/2.28 Proof:
% 13.51/2.28 multiply(X, X)
% 13.51/2.28 = { by axiom 20 (multiplication_is_well_defined) R->L }
% 13.51/2.28 fresh2(true, true, X, X, multiply(X, X), X)
% 13.51/2.28 = { by axiom 1 (x_squared_is_x) R->L }
% 13.51/2.28 fresh2(product(X, X, X), true, X, X, multiply(X, X), X)
% 13.51/2.28 = { by axiom 35 (multiplication_is_well_defined) }
% 13.51/2.28 fresh(product(X, X, multiply(X, X)), true, multiply(X, X), X)
% 13.51/2.28 = { by axiom 10 (closure_of_multiplication) }
% 13.51/2.28 fresh(true, true, multiply(X, X), X)
% 13.51/2.28 = { by axiom 8 (multiplication_is_well_defined) }
% 13.51/2.28 X
% 13.51/2.28
% 13.51/2.28 Lemma 48: fresh14(product(X, Y, Z), true, W, Y, V, Y, Y, X, Z) = fresh17(sum(W, Y, X), true, V, Y, Z).
% 13.51/2.28 Proof:
% 13.51/2.28 fresh14(product(X, Y, Z), true, W, Y, V, Y, Y, X, Z)
% 13.51/2.28 = { by axiom 42 (distributivity3) }
% 13.51/2.28 fresh16(product(Y, Y, Y), true, W, Y, V, Y, Y, X, Z)
% 13.51/2.28 = { by axiom 1 (x_squared_is_x) }
% 13.51/2.28 fresh16(true, true, W, Y, V, Y, Y, X, Z)
% 13.51/2.28 = { by axiom 33 (distributivity3) }
% 13.51/2.28 fresh17(sum(W, Y, X), true, V, Y, Z)
% 13.51/2.28
% 13.51/2.28 Lemma 49: sum(additive_inverse(X), additive_inverse(X), additive_identity) = true.
% 13.51/2.28 Proof:
% 13.51/2.28 sum(additive_inverse(X), additive_inverse(X), additive_identity)
% 13.51/2.28 = { by lemma 47 R->L }
% 13.51/2.28 sum(additive_inverse(multiply(X, X)), additive_inverse(X), additive_identity)
% 13.51/2.28 = { by axiom 30 (distributivity3) R->L }
% 15.02/2.28 fresh15(true, true, X, additive_inverse(multiply(X, X)), additive_inverse(X), additive_inverse(X), additive_identity, additive_identity)
% 15.02/2.28 = { by axiom 21 (multiply_additive_inverse) R->L }
% 15.02/2.28 fresh15(product(X, additive_inverse(X), additive_inverse(multiply(X, X))), true, X, additive_inverse(multiply(X, X)), additive_inverse(X), additive_inverse(X), additive_identity, additive_identity)
% 15.02/2.28 = { by axiom 40 (distributivity3) R->L }
% 15.02/2.28 fresh14(true, true, X, additive_inverse(X), additive_inverse(multiply(X, X)), additive_inverse(X), additive_inverse(X), additive_identity, additive_identity)
% 15.02/2.28 = { by lemma 47 }
% 15.02/2.28 fresh14(true, true, X, additive_inverse(X), additive_inverse(X), additive_inverse(X), additive_inverse(X), additive_identity, additive_identity)
% 15.02/2.28 = { by axiom 2 (multiply_additive_id2) R->L }
% 15.02/2.28 fresh14(product(additive_identity, additive_inverse(X), additive_identity), true, X, additive_inverse(X), additive_inverse(X), additive_inverse(X), additive_inverse(X), additive_identity, additive_identity)
% 15.02/2.28 = { by lemma 48 }
% 15.02/2.28 fresh17(sum(X, additive_inverse(X), additive_identity), true, additive_inverse(X), additive_inverse(X), additive_identity)
% 15.02/2.28 = { by axiom 6 (right_inverse) }
% 15.02/2.28 fresh17(true, true, additive_inverse(X), additive_inverse(X), additive_identity)
% 15.02/2.28 = { by axiom 17 (distributivity3) }
% 15.02/2.28 true
% 15.02/2.28
% 15.02/2.28 Lemma 50: fresh3(sum(X, additive_identity, Y), true, Y, X) = Y.
% 15.02/2.28 Proof:
% 15.02/2.28 fresh3(sum(X, additive_identity, Y), true, Y, X)
% 15.02/2.28 = { by axiom 34 (addition_is_well_defined) R->L }
% 15.02/2.28 fresh4(sum(X, additive_identity, X), true, X, additive_identity, Y, X)
% 15.02/2.28 = { by axiom 4 (additive_identity2) }
% 15.02/2.28 fresh4(true, true, X, additive_identity, Y, X)
% 15.02/2.28 = { by axiom 19 (addition_is_well_defined) }
% 15.02/2.28 Y
% 15.02/2.28
% 15.02/2.28 Lemma 51: add(X, additive_identity) = X.
% 15.02/2.28 Proof:
% 15.02/2.28 add(X, additive_identity)
% 15.02/2.28 = { by lemma 50 R->L }
% 15.02/2.28 fresh3(sum(X, additive_identity, add(X, additive_identity)), true, add(X, additive_identity), X)
% 15.02/2.28 = { by axiom 11 (closure_of_addition) }
% 15.02/2.28 fresh3(true, true, add(X, additive_identity), X)
% 15.02/2.28 = { by axiom 9 (addition_is_well_defined) }
% 15.02/2.28 X
% 15.02/2.28
% 15.02/2.28 Lemma 52: additive_inverse(X) = X.
% 15.02/2.28 Proof:
% 15.02/2.28 additive_inverse(X)
% 15.02/2.28 = { by lemma 46 R->L }
% 15.02/2.28 add(X, additive_inverse(add(X, X)))
% 15.02/2.28 = { by lemma 43 R->L }
% 15.02/2.28 add(X, fresh3(sum(additive_inverse(X), additive_inverse(X), additive_inverse(add(X, X))), true, additive_inverse(add(X, X)), add(additive_inverse(X), additive_inverse(X))))
% 15.02/2.28 = { by axiom 25 (distribute_additive_inverse) }
% 15.02/2.28 add(X, fresh3(true, true, additive_inverse(add(X, X)), add(additive_inverse(X), additive_inverse(X))))
% 15.02/2.28 = { by axiom 9 (addition_is_well_defined) }
% 15.02/2.28 add(X, add(additive_inverse(X), additive_inverse(X)))
% 15.02/2.28 = { by axiom 9 (addition_is_well_defined) R->L }
% 15.02/2.28 add(X, fresh3(true, true, additive_identity, add(additive_inverse(X), additive_inverse(X))))
% 15.02/2.28 = { by lemma 49 R->L }
% 15.02/2.28 add(X, fresh3(sum(additive_inverse(X), additive_inverse(X), additive_identity), true, additive_identity, add(additive_inverse(X), additive_inverse(X))))
% 15.02/2.28 = { by lemma 43 }
% 15.02/2.28 add(X, additive_identity)
% 15.02/2.28 = { by lemma 51 }
% 15.02/2.28 X
% 15.02/2.28
% 15.02/2.28 Lemma 53: multiply(a, b) = c.
% 15.02/2.28 Proof:
% 15.02/2.28 multiply(a, b)
% 15.02/2.28 = { by axiom 20 (multiplication_is_well_defined) R->L }
% 15.02/2.28 fresh2(true, true, a, b, multiply(a, b), c)
% 15.02/2.28 = { by axiom 3 (a_times_b_is_c) R->L }
% 15.02/2.28 fresh2(product(a, b, c), true, a, b, multiply(a, b), c)
% 15.02/2.28 = { by axiom 35 (multiplication_is_well_defined) }
% 15.02/2.28 fresh(product(a, b, multiply(a, b)), true, multiply(a, b), c)
% 15.02/2.28 = { by axiom 10 (closure_of_multiplication) }
% 15.02/2.28 fresh(true, true, multiply(a, b), c)
% 15.02/2.28 = { by axiom 8 (multiplication_is_well_defined) }
% 15.02/2.28 c
% 15.02/2.28
% 15.02/2.28 Lemma 54: sum(add(X, additive_inverse(Y)), Y, X) = true.
% 15.02/2.28 Proof:
% 15.02/2.28 sum(add(X, additive_inverse(Y)), Y, X)
% 15.02/2.28 = { by lemma 45 R->L }
% 15.02/2.28 fresh30(Z, Z, X, additive_inverse(Y), add(X, additive_inverse(Y)), Y, W, X)
% 15.02/2.28 = { by axiom 27 (associativity_of_addition2) }
% 15.02/2.28 fresh31(sum(X, additive_inverse(Y), add(X, additive_inverse(Y))), true, add(X, additive_inverse(Y)), Y, X)
% 15.02/2.28 = { by axiom 11 (closure_of_addition) }
% 15.02/2.28 fresh31(true, true, add(X, additive_inverse(Y)), Y, X)
% 15.02/2.28 = { by axiom 14 (associativity_of_addition2) }
% 15.02/2.29 true
% 15.02/2.29
% 15.02/2.29 Lemma 55: add(X, add(X, Y)) = Y.
% 15.02/2.29 Proof:
% 15.02/2.29 add(X, add(X, Y))
% 15.02/2.29 = { by lemma 44 R->L }
% 15.02/2.29 add(X, add(Y, X))
% 15.02/2.29 = { by lemma 52 R->L }
% 15.02/2.29 add(X, add(Y, additive_inverse(X)))
% 15.02/2.29 = { by lemma 44 R->L }
% 15.02/2.29 add(add(Y, additive_inverse(X)), X)
% 15.02/2.29 = { by axiom 9 (addition_is_well_defined) R->L }
% 15.02/2.29 fresh3(true, true, Y, add(add(Y, additive_inverse(X)), X))
% 15.02/2.29 = { by lemma 54 R->L }
% 15.02/2.29 fresh3(sum(add(Y, additive_inverse(X)), X, Y), true, Y, add(add(Y, additive_inverse(X)), X))
% 15.02/2.29 = { by lemma 43 }
% 15.02/2.29 Y
% 15.02/2.29
% 15.02/2.29 Lemma 56: fresh22(product(X, Y, Z), true, X, W, V, X, X, Y, Z) = fresh25(sum(W, X, Y), true, V, X, Z).
% 15.02/2.29 Proof:
% 15.02/2.29 fresh22(product(X, Y, Z), true, X, W, V, X, X, Y, Z)
% 15.02/2.29 = { by axiom 41 (distributivity1) }
% 15.02/2.29 fresh24(product(X, X, X), true, X, W, V, X, X, Y, Z)
% 15.02/2.29 = { by axiom 1 (x_squared_is_x) }
% 15.02/2.29 fresh24(true, true, X, W, V, X, X, Y, Z)
% 15.02/2.29 = { by axiom 32 (distributivity1) }
% 15.02/2.29 fresh25(sum(W, X, Y), true, V, X, Z)
% 15.02/2.29
% 15.02/2.29 Lemma 57: fresh22(X, X, Y, Z, multiply(Y, Z), W, V, U, T) = sum(multiply(Y, Z), V, T).
% 15.02/2.29 Proof:
% 15.02/2.29 fresh22(X, X, Y, Z, multiply(Y, Z), W, V, U, T)
% 15.02/2.29 = { by axiom 39 (distributivity1) }
% 15.02/2.29 fresh23(product(Y, Z, multiply(Y, Z)), true, Z, multiply(Y, Z), W, V, U, T)
% 15.02/2.29 = { by axiom 10 (closure_of_multiplication) }
% 15.02/2.29 fresh23(true, true, Z, multiply(Y, Z), W, V, U, T)
% 15.02/2.29 = { by axiom 29 (distributivity1) }
% 15.02/2.29 sum(multiply(Y, Z), V, T)
% 15.02/2.29
% 15.02/2.29 Lemma 58: fresh25(sum(X, Y, Z), true, multiply(Y, X), Y, multiply(Y, Z)) = sum(multiply(Y, X), Y, multiply(Y, Z)).
% 15.02/2.29 Proof:
% 15.02/2.29 fresh25(sum(X, Y, Z), true, multiply(Y, X), Y, multiply(Y, Z))
% 15.02/2.29 = { by lemma 56 R->L }
% 15.02/2.29 fresh22(product(Y, Z, multiply(Y, Z)), true, Y, X, multiply(Y, X), Y, Y, Z, multiply(Y, Z))
% 15.02/2.29 = { by axiom 10 (closure_of_multiplication) }
% 15.02/2.29 fresh22(true, true, Y, X, multiply(Y, X), Y, Y, Z, multiply(Y, Z))
% 15.02/2.29 = { by lemma 57 }
% 15.02/2.29 sum(multiply(Y, X), Y, multiply(Y, Z))
% 15.02/2.29
% 15.02/2.29 Lemma 59: add(X, multiply(X, Y)) = multiply(X, add(X, Y)).
% 15.02/2.29 Proof:
% 15.02/2.29 add(X, multiply(X, Y))
% 15.02/2.29 = { by lemma 44 R->L }
% 15.02/2.29 add(multiply(X, Y), X)
% 15.02/2.29 = { by axiom 9 (addition_is_well_defined) R->L }
% 15.02/2.29 fresh3(true, true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 15.02/2.29 = { by axiom 16 (distributivity1) R->L }
% 15.02/2.29 fresh3(fresh25(true, true, multiply(X, Y), X, multiply(X, add(Y, X))), true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 15.02/2.29 = { by axiom 11 (closure_of_addition) R->L }
% 15.02/2.29 fresh3(fresh25(sum(Y, X, add(Y, X)), true, multiply(X, Y), X, multiply(X, add(Y, X))), true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 15.02/2.29 = { by lemma 58 }
% 15.02/2.29 fresh3(sum(multiply(X, Y), X, multiply(X, add(Y, X))), true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 15.02/2.29 = { by lemma 44 }
% 15.02/2.29 fresh3(sum(multiply(X, Y), X, multiply(X, add(X, Y))), true, multiply(X, add(X, Y)), add(multiply(X, Y), X))
% 15.02/2.29 = { by lemma 43 }
% 15.02/2.29 multiply(X, add(X, Y))
% 15.02/2.29
% 15.02/2.29 Lemma 60: multiply(a, add(a, b)) = add(a, c).
% 15.02/2.29 Proof:
% 15.02/2.29 multiply(a, add(a, b))
% 15.02/2.29 = { by lemma 52 R->L }
% 15.02/2.29 additive_inverse(multiply(a, add(a, b)))
% 15.02/2.29 = { by lemma 46 R->L }
% 15.02/2.29 add(a, additive_inverse(add(a, multiply(a, add(a, b)))))
% 15.02/2.29 = { by lemma 44 R->L }
% 15.02/2.29 add(a, additive_inverse(add(multiply(a, add(a, b)), a)))
% 15.02/2.29 = { by axiom 9 (addition_is_well_defined) R->L }
% 15.02/2.29 add(a, additive_inverse(fresh3(true, true, c, add(multiply(a, add(a, b)), a))))
% 15.02/2.29 = { by axiom 16 (distributivity1) R->L }
% 15.02/2.29 add(a, additive_inverse(fresh3(fresh25(true, true, multiply(a, add(b, additive_inverse(a))), a, c), true, c, add(multiply(a, add(a, b)), a))))
% 15.02/2.29 = { by lemma 54 R->L }
% 15.02/2.29 add(a, additive_inverse(fresh3(fresh25(sum(add(b, additive_inverse(a)), a, b), true, multiply(a, add(b, additive_inverse(a))), a, c), true, c, add(multiply(a, add(a, b)), a))))
% 15.02/2.29 = { by lemma 56 R->L }
% 15.02/2.29 add(a, additive_inverse(fresh3(fresh22(product(a, b, c), true, a, add(b, additive_inverse(a)), multiply(a, add(b, additive_inverse(a))), a, a, b, c), true, c, add(multiply(a, add(a, b)), a))))
% 15.02/2.29 = { by axiom 3 (a_times_b_is_c) }
% 15.02/2.29 add(a, additive_inverse(fresh3(fresh22(true, true, a, add(b, additive_inverse(a)), multiply(a, add(b, additive_inverse(a))), a, a, b, c), true, c, add(multiply(a, add(a, b)), a))))
% 15.02/2.29 = { by lemma 57 }
% 15.02/2.29 add(a, additive_inverse(fresh3(sum(multiply(a, add(b, additive_inverse(a))), a, c), true, c, add(multiply(a, add(a, b)), a))))
% 15.02/2.29 = { by lemma 52 }
% 15.02/2.29 add(a, additive_inverse(fresh3(sum(multiply(a, add(b, a)), a, c), true, c, add(multiply(a, add(a, b)), a))))
% 15.02/2.29 = { by lemma 44 }
% 15.02/2.29 add(a, additive_inverse(fresh3(sum(multiply(a, add(a, b)), a, c), true, c, add(multiply(a, add(a, b)), a))))
% 15.02/2.29 = { by lemma 43 }
% 15.02/2.29 add(a, additive_inverse(c))
% 15.02/2.29 = { by lemma 52 }
% 15.02/2.29 add(a, c)
% 15.02/2.29
% 15.02/2.29 Lemma 61: fresh28(X, X, Y, Y, Y, Z, W, V) = true.
% 15.02/2.29 Proof:
% 15.02/2.29 fresh28(X, X, Y, Y, Y, Z, W, V)
% 15.02/2.29 = { by axiom 28 (associativity_of_multiplication1) }
% 15.02/2.29 fresh29(product(Y, Y, Y), true, Y, W, V)
% 15.02/2.29 = { by axiom 1 (x_squared_is_x) }
% 15.02/2.29 fresh29(true, true, Y, W, V)
% 15.02/2.29 = { by axiom 15 (associativity_of_multiplication1) }
% 15.02/2.29 true
% 15.02/2.29
% 15.02/2.29 Lemma 62: fresh28(product(X, Y, Z), true, W, V, X, Y, multiply(V, Y), Z) = product(W, multiply(V, Y), Z).
% 15.02/2.29 Proof:
% 15.02/2.29 fresh28(product(X, Y, Z), true, W, V, X, Y, multiply(V, Y), Z)
% 15.02/2.29 = { by axiom 38 (associativity_of_multiplication1) }
% 15.02/2.29 fresh7(product(V, Y, multiply(V, Y)), true, W, V, X, multiply(V, Y), Z)
% 15.02/2.29 = { by axiom 10 (closure_of_multiplication) }
% 15.02/2.29 fresh7(true, true, W, V, X, multiply(V, Y), Z)
% 15.02/2.29 = { by axiom 24 (associativity_of_multiplication1) }
% 15.02/2.29 product(W, multiply(V, Y), Z)
% 15.02/2.29
% 15.02/2.29 Lemma 63: fresh28(X, X, Y, Z, W, V, multiply(Z, U), multiply(W, U)) = product(Y, multiply(Z, U), multiply(W, U)).
% 15.02/2.29 Proof:
% 15.02/2.29 fresh28(X, X, Y, Z, W, V, multiply(Z, U), multiply(W, U))
% 15.02/2.29 = { by axiom 28 (associativity_of_multiplication1) }
% 15.02/2.29 fresh29(product(Y, Z, W), true, Y, multiply(Z, U), multiply(W, U))
% 15.02/2.29 = { by axiom 28 (associativity_of_multiplication1) R->L }
% 15.02/2.29 fresh28(true, true, Y, Z, W, U, multiply(Z, U), multiply(W, U))
% 15.02/2.29 = { by axiom 10 (closure_of_multiplication) R->L }
% 15.02/2.29 fresh28(product(W, U, multiply(W, U)), true, Y, Z, W, U, multiply(Z, U), multiply(W, U))
% 15.02/2.29 = { by lemma 62 }
% 15.02/2.29 product(Y, multiply(Z, U), multiply(W, U))
% 15.02/2.29
% 15.02/2.29 Lemma 64: fresh(product(X, Y, Z), true, Z, multiply(X, Y)) = Z.
% 15.02/2.29 Proof:
% 15.02/2.29 fresh(product(X, Y, Z), true, Z, multiply(X, Y))
% 15.02/2.29 = { by axiom 35 (multiplication_is_well_defined) R->L }
% 15.02/2.29 fresh2(product(X, Y, multiply(X, Y)), true, X, Y, Z, multiply(X, Y))
% 15.02/2.29 = { by axiom 10 (closure_of_multiplication) }
% 15.02/2.29 fresh2(true, true, X, Y, Z, multiply(X, Y))
% 15.02/2.29 = { by axiom 20 (multiplication_is_well_defined) }
% 15.02/2.29 Z
% 15.02/2.29
% 15.02/2.29 Lemma 65: multiply(X, multiply(X, Y)) = multiply(X, Y).
% 15.02/2.29 Proof:
% 15.02/2.29 multiply(X, multiply(X, Y))
% 15.02/2.29 = { by axiom 8 (multiplication_is_well_defined) R->L }
% 15.02/2.29 fresh(true, true, multiply(X, Y), multiply(X, multiply(X, Y)))
% 15.02/2.29 = { by lemma 61 R->L }
% 15.02/2.29 fresh(fresh28(Z, Z, X, X, X, W, multiply(X, Y), multiply(X, Y)), true, multiply(X, Y), multiply(X, multiply(X, Y)))
% 15.02/2.29 = { by lemma 63 }
% 15.02/2.29 fresh(product(X, multiply(X, Y), multiply(X, Y)), true, multiply(X, Y), multiply(X, multiply(X, Y)))
% 15.02/2.29 = { by lemma 64 }
% 15.02/2.29 multiply(X, Y)
% 15.02/2.29
% 15.02/2.29 Lemma 66: fresh28(X, X, Y, Z, multiply(Y, Z), W, V, U) = true.
% 15.02/2.29 Proof:
% 15.02/2.29 fresh28(X, X, Y, Z, multiply(Y, Z), W, V, U)
% 15.02/2.29 = { by axiom 28 (associativity_of_multiplication1) }
% 15.02/2.29 fresh29(product(Y, Z, multiply(Y, Z)), true, Y, V, U)
% 15.02/2.29 = { by axiom 10 (closure_of_multiplication) }
% 15.02/2.29 fresh29(true, true, Y, V, U)
% 15.02/2.29 = { by axiom 15 (associativity_of_multiplication1) }
% 15.02/2.29 true
% 15.02/2.29
% 15.02/2.29 Lemma 67: product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)) = true.
% 15.02/2.29 Proof:
% 15.02/2.29 product(X, multiply(Y, Z), multiply(multiply(X, Y), Z))
% 15.02/2.29 = { by lemma 63 R->L }
% 15.02/2.29 fresh28(W, W, X, Y, multiply(X, Y), V, multiply(Y, Z), multiply(multiply(X, Y), Z))
% 15.02/2.29 = { by lemma 66 }
% 15.02/2.29 true
% 15.02/2.29
% 15.02/2.29 Lemma 68: multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 15.02/2.29 Proof:
% 15.02/2.29 multiply(multiply(X, Y), Z)
% 15.02/2.29 = { by lemma 64 R->L }
% 15.02/2.29 fresh(product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true, multiply(multiply(X, Y), Z), multiply(X, multiply(Y, Z)))
% 15.02/2.29 = { by lemma 67 }
% 15.02/2.29 fresh(true, true, multiply(multiply(X, Y), Z), multiply(X, multiply(Y, Z)))
% 15.02/2.29 = { by axiom 8 (multiplication_is_well_defined) }
% 15.02/2.29 multiply(X, multiply(Y, Z))
% 15.02/2.29
% 15.02/2.29 Lemma 69: fresh14(X, X, Y, Z, multiply(Y, Z), W, V, U, T) = sum(multiply(Y, Z), V, T).
% 15.02/2.29 Proof:
% 15.02/2.29 fresh14(X, X, Y, Z, multiply(Y, Z), W, V, U, T)
% 15.02/2.29 = { by axiom 40 (distributivity3) }
% 15.02/2.29 fresh15(product(Y, Z, multiply(Y, Z)), true, Y, multiply(Y, Z), W, V, U, T)
% 15.02/2.29 = { by axiom 10 (closure_of_multiplication) }
% 15.02/2.29 fresh15(true, true, Y, multiply(Y, Z), W, V, U, T)
% 15.02/2.29 = { by axiom 30 (distributivity3) }
% 15.02/2.29 sum(multiply(Y, Z), V, T)
% 15.02/2.29
% 15.02/2.29 Lemma 70: sum(multiply(X, Y), Y, multiply(add(X, Y), Y)) = true.
% 15.02/2.29 Proof:
% 15.02/2.29 sum(multiply(X, Y), Y, multiply(add(X, Y), Y))
% 15.02/2.29 = { by lemma 69 R->L }
% 15.02/2.29 fresh14(true, true, X, Y, multiply(X, Y), Y, Y, add(X, Y), multiply(add(X, Y), Y))
% 15.02/2.29 = { by axiom 10 (closure_of_multiplication) R->L }
% 15.02/2.29 fresh14(product(add(X, Y), Y, multiply(add(X, Y), Y)), true, X, Y, multiply(X, Y), Y, Y, add(X, Y), multiply(add(X, Y), Y))
% 15.02/2.29 = { by lemma 48 }
% 15.02/2.29 fresh17(sum(X, Y, add(X, Y)), true, multiply(X, Y), Y, multiply(add(X, Y), Y))
% 15.02/2.29 = { by axiom 11 (closure_of_addition) }
% 15.02/2.29 fresh17(true, true, multiply(X, Y), Y, multiply(add(X, Y), Y))
% 15.02/2.29 = { by axiom 17 (distributivity3) }
% 15.02/2.29 true
% 15.02/2.29
% 15.02/2.29 Lemma 71: additive_inverse(additive_inverse(X)) = X.
% 15.02/2.29 Proof:
% 15.02/2.29 additive_inverse(additive_inverse(X))
% 15.02/2.29 = { by axiom 9 (addition_is_well_defined) R->L }
% 15.02/2.29 fresh3(true, true, X, additive_inverse(additive_inverse(X)))
% 15.02/2.29 = { by axiom 12 (additive_inverse_additive_inverse) R->L }
% 15.02/2.29 fresh3(sum(additive_inverse(additive_inverse(X)), additive_identity, X), true, X, additive_inverse(additive_inverse(X)))
% 15.02/2.29 = { by lemma 50 }
% 15.02/2.29 X
% 15.02/2.29
% 15.02/2.29 Lemma 72: add(X, X) = additive_identity.
% 15.02/2.29 Proof:
% 15.02/2.29 add(X, X)
% 15.02/2.29 = { by axiom 9 (addition_is_well_defined) R->L }
% 15.02/2.29 fresh3(true, true, additive_identity, add(X, X))
% 15.02/2.29 = { by lemma 49 R->L }
% 15.02/2.29 fresh3(sum(additive_inverse(additive_inverse(X)), additive_inverse(additive_inverse(X)), additive_identity), true, additive_identity, add(X, X))
% 15.02/2.29 = { by lemma 71 }
% 15.02/2.29 fresh3(sum(X, additive_inverse(additive_inverse(X)), additive_identity), true, additive_identity, add(X, X))
% 15.02/2.29 = { by lemma 71 }
% 15.02/2.29 fresh3(sum(X, X, additive_identity), true, additive_identity, add(X, X))
% 15.02/2.29 = { by lemma 43 }
% 15.02/2.29 additive_identity
% 15.02/2.29
% 15.02/2.29 Lemma 73: multiply(c, a) = c.
% 15.02/2.29 Proof:
% 15.02/2.29 multiply(c, a)
% 15.02/2.29 = { by lemma 55 R->L }
% 15.02/2.29 multiply(c, add(c, add(c, a)))
% 15.02/2.29 = { by lemma 44 }
% 15.02/2.29 multiply(c, add(c, add(a, c)))
% 15.02/2.29 = { by lemma 59 R->L }
% 15.02/2.29 add(c, multiply(c, add(a, c)))
% 15.02/2.29 = { by lemma 60 R->L }
% 15.02/2.29 add(c, multiply(c, multiply(a, add(a, b))))
% 15.02/2.29 = { by lemma 65 R->L }
% 15.02/2.29 add(c, multiply(c, multiply(a, multiply(a, add(a, b)))))
% 15.02/2.29 = { by lemma 60 }
% 15.02/2.29 add(c, multiply(c, multiply(a, add(a, c))))
% 15.02/2.29 = { by lemma 64 R->L }
% 15.02/2.29 add(c, multiply(fresh(product(a, c, c), true, c, multiply(a, c)), multiply(a, add(a, c))))
% 15.02/2.29 = { by lemma 53 R->L }
% 15.02/2.29 add(c, multiply(fresh(product(a, multiply(a, b), c), true, c, multiply(a, c)), multiply(a, add(a, c))))
% 15.02/2.29 = { by lemma 62 R->L }
% 15.02/2.29 add(c, multiply(fresh(fresh28(product(a, b, c), true, a, a, a, b, multiply(a, b), c), true, c, multiply(a, c)), multiply(a, add(a, c))))
% 15.02/2.29 = { by axiom 3 (a_times_b_is_c) }
% 15.02/2.30 add(c, multiply(fresh(fresh28(true, true, a, a, a, b, multiply(a, b), c), true, c, multiply(a, c)), multiply(a, add(a, c))))
% 15.02/2.30 = { by lemma 61 }
% 15.02/2.30 add(c, multiply(fresh(true, true, c, multiply(a, c)), multiply(a, add(a, c))))
% 15.02/2.30 = { by axiom 8 (multiplication_is_well_defined) }
% 15.02/2.30 add(c, multiply(multiply(a, c), multiply(a, add(a, c))))
% 15.02/2.30 = { by lemma 68 }
% 15.02/2.30 add(c, multiply(a, multiply(c, multiply(a, add(a, c)))))
% 15.02/2.30 = { by lemma 55 R->L }
% 15.02/2.30 add(c, multiply(a, multiply(add(a, add(a, c)), multiply(a, add(a, c)))))
% 15.02/2.30 = { by lemma 68 R->L }
% 15.02/2.30 add(c, multiply(multiply(a, add(a, add(a, c))), multiply(a, add(a, c))))
% 15.02/2.30 = { by lemma 59 R->L }
% 15.02/2.30 add(c, multiply(add(a, multiply(a, add(a, c))), multiply(a, add(a, c))))
% 15.02/2.30 = { by lemma 43 R->L }
% 15.02/2.30 add(c, fresh3(sum(multiply(a, multiply(a, add(a, c))), multiply(a, add(a, c)), multiply(add(a, multiply(a, add(a, c))), multiply(a, add(a, c)))), true, multiply(add(a, multiply(a, add(a, c))), multiply(a, add(a, c))), add(multiply(a, multiply(a, add(a, c))), multiply(a, add(a, c)))))
% 15.02/2.30 = { by lemma 70 }
% 15.02/2.30 add(c, fresh3(true, true, multiply(add(a, multiply(a, add(a, c))), multiply(a, add(a, c))), add(multiply(a, multiply(a, add(a, c))), multiply(a, add(a, c)))))
% 15.02/2.30 = { by axiom 9 (addition_is_well_defined) }
% 15.02/2.30 add(c, add(multiply(a, multiply(a, add(a, c))), multiply(a, add(a, c))))
% 15.02/2.30 = { by lemma 44 }
% 15.02/2.30 add(c, add(multiply(a, add(a, c)), multiply(a, multiply(a, add(a, c)))))
% 15.02/2.30 = { by lemma 65 }
% 15.02/2.30 add(c, add(multiply(a, add(a, c)), multiply(a, add(a, c))))
% 15.02/2.30 = { by lemma 72 }
% 15.02/2.30 add(c, additive_identity)
% 15.02/2.30 = { by lemma 51 }
% 15.02/2.30 c
% 15.02/2.30
% 15.02/2.30 Lemma 74: multiply(a, multiply(b, X)) = multiply(c, X).
% 15.02/2.30 Proof:
% 15.02/2.30 multiply(a, multiply(b, X))
% 15.02/2.30 = { by axiom 8 (multiplication_is_well_defined) R->L }
% 15.02/2.30 fresh(true, true, multiply(c, X), multiply(a, multiply(b, X)))
% 15.02/2.30 = { by lemma 67 R->L }
% 15.02/2.30 fresh(product(a, multiply(b, X), multiply(multiply(a, b), X)), true, multiply(c, X), multiply(a, multiply(b, X)))
% 15.02/2.30 = { by lemma 53 }
% 15.02/2.30 fresh(product(a, multiply(b, X), multiply(c, X)), true, multiply(c, X), multiply(a, multiply(b, X)))
% 15.02/2.30 = { by lemma 64 }
% 15.02/2.30 multiply(c, X)
% 15.02/2.30
% 15.02/2.30 Lemma 75: sum(multiply(add(a, b), b), b, c) = true.
% 15.02/2.30 Proof:
% 15.02/2.30 sum(multiply(add(a, b), b), b, c)
% 15.02/2.30 = { by lemma 52 R->L }
% 15.02/2.30 sum(multiply(add(a, additive_inverse(b)), b), b, c)
% 15.02/2.30 = { by lemma 69 R->L }
% 15.02/2.30 fresh14(true, true, add(a, additive_inverse(b)), b, multiply(add(a, additive_inverse(b)), b), b, b, a, c)
% 15.02/2.30 = { by axiom 3 (a_times_b_is_c) R->L }
% 15.02/2.30 fresh14(product(a, b, c), true, add(a, additive_inverse(b)), b, multiply(add(a, additive_inverse(b)), b), b, b, a, c)
% 15.02/2.30 = { by lemma 48 }
% 15.02/2.30 fresh17(sum(add(a, additive_inverse(b)), b, a), true, multiply(add(a, additive_inverse(b)), b), b, c)
% 15.02/2.30 = { by lemma 54 }
% 15.02/2.30 fresh17(true, true, multiply(add(a, additive_inverse(b)), b), b, c)
% 15.02/2.30 = { by axiom 17 (distributivity3) }
% 15.02/2.30 true
% 15.02/2.30
% 15.02/2.30 Lemma 76: multiply(add(a, b), b) = add(b, c).
% 15.02/2.30 Proof:
% 15.02/2.30 multiply(add(a, b), b)
% 15.02/2.30 = { by lemma 52 R->L }
% 15.02/2.30 additive_inverse(multiply(add(a, b), b))
% 15.02/2.30 = { by lemma 46 R->L }
% 15.02/2.30 add(b, additive_inverse(add(b, multiply(add(a, b), b))))
% 15.02/2.30 = { by lemma 44 R->L }
% 15.02/2.30 add(b, additive_inverse(add(multiply(add(a, b), b), b)))
% 15.02/2.30 = { by axiom 9 (addition_is_well_defined) R->L }
% 15.02/2.30 add(b, additive_inverse(fresh3(true, true, c, add(multiply(add(a, b), b), b))))
% 15.02/2.30 = { by lemma 75 R->L }
% 15.02/2.30 add(b, additive_inverse(fresh3(sum(multiply(add(a, b), b), b, c), true, c, add(multiply(add(a, b), b), b))))
% 15.02/2.30 = { by lemma 43 }
% 15.02/2.30 add(b, additive_inverse(c))
% 15.02/2.30 = { by lemma 52 }
% 15.02/2.30 add(b, c)
% 15.02/2.30
% 15.02/2.30 Lemma 77: add(X, add(Y, Z)) = add(Y, add(X, Z)).
% 15.02/2.30 Proof:
% 15.02/2.30 add(X, add(Y, Z))
% 15.02/2.30 = { by lemma 43 R->L }
% 15.02/2.30 fresh3(sum(Y, add(X, Z), add(X, add(Y, Z))), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 15.02/2.30 = { by lemma 44 R->L }
% 15.02/2.30 fresh3(sum(Y, add(X, Z), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 15.02/2.30 = { by lemma 44 R->L }
% 15.02/2.30 fresh3(sum(Y, add(Z, X), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 15.02/2.30 = { by axiom 22 (associativity_of_addition1) R->L }
% 15.02/2.30 fresh3(fresh9(true, true, Y, Z, add(Y, Z), add(Z, X), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 15.02/2.30 = { by axiom 11 (closure_of_addition) R->L }
% 15.02/2.30 fresh3(fresh9(sum(Z, X, add(Z, X)), true, Y, Z, add(Y, Z), add(Z, X), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 15.02/2.30 = { by axiom 36 (associativity_of_addition1) R->L }
% 15.02/2.30 fresh3(fresh32(sum(add(Y, Z), X, add(add(Y, Z), X)), true, Y, Z, add(Y, Z), X, add(Z, X), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 15.02/2.30 = { by axiom 11 (closure_of_addition) }
% 15.02/2.30 fresh3(fresh32(true, true, Y, Z, add(Y, Z), X, add(Z, X), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 15.02/2.30 = { by axiom 26 (associativity_of_addition1) }
% 15.02/2.30 fresh3(fresh33(sum(Y, Z, add(Y, Z)), true, Y, add(Z, X), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 15.02/2.30 = { by axiom 11 (closure_of_addition) }
% 15.02/2.30 fresh3(fresh33(true, true, Y, add(Z, X), add(add(Y, Z), X)), true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 15.02/2.30 = { by axiom 13 (associativity_of_addition1) }
% 15.02/2.30 fresh3(true, true, add(X, add(Y, Z)), add(Y, add(X, Z)))
% 15.02/2.30 = { by axiom 9 (addition_is_well_defined) }
% 15.02/2.30 add(Y, add(X, Z))
% 15.02/2.30
% 15.02/2.30 Goal 1 (prove_b_times_a_is_c): product(b, a, c) = true.
% 15.02/2.30 Proof:
% 15.02/2.30 product(b, a, c)
% 15.02/2.30 = { by axiom 9 (addition_is_well_defined) R->L }
% 15.02/2.30 product(b, a, fresh3(true, true, multiply(b, a), c))
% 15.02/2.30 = { by axiom 14 (associativity_of_addition2) R->L }
% 15.02/2.30 product(b, a, fresh3(fresh31(true, true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 70 R->L }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, multiply(a, multiply(b, a))), multiply(a, multiply(b, a)), multiply(add(b, multiply(a, multiply(b, a))), multiply(a, multiply(b, a)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by axiom 8 (multiplication_is_well_defined) R->L }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(fresh(true, true, multiply(b, a), multiply(b, multiply(a, multiply(b, a)))), multiply(a, multiply(b, a)), multiply(add(b, multiply(a, multiply(b, a))), multiply(a, multiply(b, a)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 66 R->L }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(fresh(fresh28(true, true, b, a, multiply(b, a), multiply(b, a), multiply(a, multiply(b, a)), multiply(b, a)), true, multiply(b, a), multiply(b, multiply(a, multiply(b, a)))), multiply(a, multiply(b, a)), multiply(add(b, multiply(a, multiply(b, a))), multiply(a, multiply(b, a)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by axiom 1 (x_squared_is_x) R->L }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(fresh(fresh28(product(multiply(b, a), multiply(b, a), multiply(b, a)), true, b, a, multiply(b, a), multiply(b, a), multiply(a, multiply(b, a)), multiply(b, a)), true, multiply(b, a), multiply(b, multiply(a, multiply(b, a)))), multiply(a, multiply(b, a)), multiply(add(b, multiply(a, multiply(b, a))), multiply(a, multiply(b, a)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 62 }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(fresh(product(b, multiply(a, multiply(b, a)), multiply(b, a)), true, multiply(b, a), multiply(b, multiply(a, multiply(b, a)))), multiply(a, multiply(b, a)), multiply(add(b, multiply(a, multiply(b, a))), multiply(a, multiply(b, a)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 64 }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, a), multiply(a, multiply(b, a)), multiply(add(b, multiply(a, multiply(b, a))), multiply(a, multiply(b, a)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 74 }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, a), multiply(c, a), multiply(add(b, multiply(a, multiply(b, a))), multiply(a, multiply(b, a)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 73 }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, multiply(add(b, multiply(a, multiply(b, a))), multiply(a, multiply(b, a)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 74 }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, multiply(add(b, multiply(a, multiply(b, a))), multiply(c, a))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 74 }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, multiply(add(b, multiply(c, a)), multiply(c, a))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 73 }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, multiply(add(b, multiply(c, a)), c)), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 73 }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, multiply(add(b, c), c)), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 43 R->L }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(sum(add(b, c), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 76 R->L }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(sum(multiply(add(a, b), b), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by axiom 8 (multiplication_is_well_defined) R->L }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(sum(fresh(true, true, multiply(multiply(add(a, b), b), b), multiply(add(a, b), b)), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by lemma 66 R->L }
% 15.02/2.30 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(sum(fresh(fresh28(true, true, add(a, b), b, multiply(add(a, b), b), b, b, multiply(multiply(add(a, b), b), b)), true, multiply(multiply(add(a, b), b), b), multiply(add(a, b), b)), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.30 = { by axiom 10 (closure_of_multiplication) R->L }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(sum(fresh(fresh28(product(multiply(add(a, b), b), b, multiply(multiply(add(a, b), b), b)), true, add(a, b), b, multiply(add(a, b), b), b, b, multiply(multiply(add(a, b), b), b)), true, multiply(multiply(add(a, b), b), b), multiply(add(a, b), b)), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 38 (associativity_of_multiplication1) }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(sum(fresh(fresh7(product(b, b, b), true, add(a, b), b, multiply(add(a, b), b), b, multiply(multiply(add(a, b), b), b)), true, multiply(multiply(add(a, b), b), b), multiply(add(a, b), b)), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 1 (x_squared_is_x) }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(sum(fresh(fresh7(true, true, add(a, b), b, multiply(add(a, b), b), b, multiply(multiply(add(a, b), b), b)), true, multiply(multiply(add(a, b), b), b), multiply(add(a, b), b)), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 24 (associativity_of_multiplication1) }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(sum(fresh(product(add(a, b), b, multiply(multiply(add(a, b), b), b)), true, multiply(multiply(add(a, b), b), b), multiply(add(a, b), b)), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 64 }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(sum(multiply(multiply(add(a, b), b), b), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 76 }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(sum(multiply(add(b, c), b), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 58 R->L }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(fresh25(sum(b, add(b, c), c), true, multiply(add(b, c), b), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 76 R->L }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(fresh25(sum(b, multiply(add(a, b), b), c), true, multiply(add(b, c), b), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 31 (commutativity_of_addition) R->L }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(fresh25(fresh5(sum(multiply(add(a, b), b), b, c), true, multiply(add(a, b), b), b, c), true, multiply(add(b, c), b), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 75 }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(fresh25(fresh5(true, true, multiply(add(a, b), b), b, c), true, multiply(add(b, c), b), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 18 (commutativity_of_addition) }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(fresh25(true, true, multiply(add(b, c), b), add(b, c), multiply(add(b, c), c)), true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 16 (distributivity1) }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, fresh3(true, true, multiply(add(b, c), c), add(add(b, c), add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 9 (addition_is_well_defined) }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, add(add(b, c), add(b, c))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 77 }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, add(b, add(add(b, c), c))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 44 R->L }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, add(b, add(c, add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 77 R->L }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, add(c, add(b, add(b, c)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 76 R->L }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, add(c, add(b, multiply(add(a, b), b)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 44 R->L }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, add(c, add(multiply(add(a, b), b), b))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 9 (addition_is_well_defined) R->L }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, add(c, fresh3(true, true, c, add(multiply(add(a, b), b), b)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 75 R->L }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, add(c, fresh3(sum(multiply(add(a, b), b), b, c), true, c, add(multiply(add(a, b), b), b)))), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 43 }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, add(c, c)), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 72 }
% 15.02/2.31 product(b, a, fresh3(fresh31(sum(multiply(b, a), c, additive_identity), true, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 27 (associativity_of_addition2) R->L }
% 15.02/2.31 product(b, a, fresh3(fresh30(true, true, multiply(b, a), c, additive_identity, c, additive_identity, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 52 R->L }
% 15.02/2.31 product(b, a, fresh3(fresh30(true, true, multiply(b, a), c, additive_identity, additive_inverse(c), additive_identity, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 6 (right_inverse) R->L }
% 15.02/2.31 product(b, a, fresh3(fresh30(sum(c, additive_inverse(c), additive_identity), true, multiply(b, a), c, additive_identity, additive_inverse(c), additive_identity, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 37 (associativity_of_addition2) }
% 15.02/2.31 product(b, a, fresh3(fresh8(sum(multiply(b, a), additive_identity, multiply(b, a)), true, multiply(b, a), c, additive_identity, additive_inverse(c), multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by lemma 52 }
% 15.02/2.31 product(b, a, fresh3(fresh8(sum(multiply(b, a), additive_identity, multiply(b, a)), true, multiply(b, a), c, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 4 (additive_identity2) }
% 15.02/2.31 product(b, a, fresh3(fresh8(true, true, multiply(b, a), c, additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 23 (associativity_of_addition2) }
% 15.02/2.31 product(b, a, fresh3(sum(additive_identity, c, multiply(b, a)), true, multiply(b, a), c))
% 15.02/2.31 = { by axiom 34 (addition_is_well_defined) R->L }
% 15.02/2.31 product(b, a, fresh4(sum(additive_identity, c, c), true, additive_identity, c, multiply(b, a), c))
% 15.02/2.31 = { by axiom 5 (additive_identity1) }
% 15.02/2.31 product(b, a, fresh4(true, true, additive_identity, c, multiply(b, a), c))
% 15.02/2.31 = { by axiom 19 (addition_is_well_defined) }
% 15.02/2.31 product(b, a, multiply(b, a))
% 15.02/2.31 = { by axiom 10 (closure_of_multiplication) }
% 15.02/2.31 true
% 15.02/2.31 % SZS output end Proof
% 15.02/2.31
% 15.02/2.31 RESULT: Unsatisfiable (the axioms are contradictory).
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