TSTP Solution File: SET951+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET951+1 : TPTP v8.1.2. Released v3.2.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n002.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 15:33:49 EDT 2023
% Result : Theorem 0.20s 0.69s
% Output : Proof 2.81s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SET951+1 : TPTP v8.1.2. Released v3.2.0.
% 0.11/0.13 % 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 : n002.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 : Sat Aug 26 10:02:48 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.69 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.20/0.69
% 0.20/0.69 % SZS status Theorem
% 0.20/0.69
% 2.81/0.72 % SZS output start Proof
% 2.81/0.72 Take the following subset of the input axioms:
% 2.81/0.72 fof(antisymmetry_r2_hidden, axiom, ![A, B]: (in(A, B) => ~in(B, A))).
% 2.81/0.72 fof(d2_zfmisc_1, axiom, ![C, A2, B2]: (C=cartesian_product2(A2, B2) <=> ![D]: (in(D, C) <=> ?[E, F]: (in(E, A2) & (in(F, B2) & D=ordered_pair(E, F)))))).
% 2.81/0.72 fof(d3_xboole_0, axiom, ![B2, C2, A2_2]: (C2=set_intersection2(A2_2, B2) <=> ![D2]: (in(D2, C2) <=> (in(D2, A2_2) & in(D2, B2))))).
% 2.81/0.72 fof(fc1_zfmisc_1, axiom, ![B2, A3]: ~empty(ordered_pair(A3, B2))).
% 2.81/0.72 fof(t104_zfmisc_1, conjecture, ![B2, C2, D2, A3, E2]: ~(in(A3, set_intersection2(cartesian_product2(B2, C2), cartesian_product2(D2, E2))) & ![G, F2]: ~(A3=ordered_pair(F2, G) & (in(F2, set_intersection2(B2, D2)) & in(G, set_intersection2(C2, E2)))))).
% 2.81/0.72 fof(t33_zfmisc_1, axiom, ![B2, C2, D2, A2_2]: (ordered_pair(A2_2, B2)=ordered_pair(C2, D2) => (A2_2=C2 & B2=D2))).
% 2.81/0.72
% 2.81/0.72 Now clausify the problem and encode Horn clauses using encoding 3 of
% 2.81/0.72 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 2.81/0.72 We repeatedly replace C & s=t => u=v by the two clauses:
% 2.81/0.72 fresh(y, y, x1...xn) = u
% 2.81/0.72 C => fresh(s, t, x1...xn) = v
% 2.81/0.72 where fresh is a fresh function symbol and x1..xn are the free
% 2.81/0.72 variables of u and v.
% 2.81/0.72 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 2.81/0.72 input problem has no model of domain size 1).
% 2.81/0.72
% 2.81/0.72 The encoding turns the above axioms into the following unit equations and goals:
% 2.81/0.72
% 2.81/0.72 Axiom 1 (t33_zfmisc_1_1): fresh(X, X, Y, Z) = Z.
% 2.81/0.72 Axiom 2 (d3_xboole_0_2): fresh11(X, X, Y, Z) = true2.
% 2.81/0.72 Axiom 3 (d3_xboole_0_5): fresh7(X, X, Y, Z) = true2.
% 2.81/0.72 Axiom 4 (d3_xboole_0_6): fresh6(X, X, Y, Z) = true2.
% 2.81/0.72 Axiom 5 (t33_zfmisc_1): fresh2(X, X, Y, Z) = Z.
% 2.81/0.72 Axiom 6 (t104_zfmisc_1): in(a, set_intersection2(cartesian_product2(b, c), cartesian_product2(d, e))) = true2.
% 2.81/0.72 Axiom 7 (d2_zfmisc_1_3): fresh24(X, X, Y, Z, W) = true2.
% 2.81/0.72 Axiom 8 (d2_zfmisc_1_1): fresh22(X, X, Y, Z, W) = true2.
% 2.81/0.72 Axiom 9 (d2_zfmisc_1_6): fresh16(X, X, Y, Z, W) = true2.
% 2.81/0.72 Axiom 10 (d2_zfmisc_1_7): fresh15(X, X, Y, Z, W) = true2.
% 2.81/0.72 Axiom 11 (d3_xboole_0_1): fresh13(X, X, Y, Z, W) = true2.
% 2.81/0.72 Axiom 12 (d3_xboole_0_3): fresh10(X, X, Y, Z, W) = equiv(Y, Z, W).
% 2.81/0.72 Axiom 13 (d3_xboole_0_3): fresh9(X, X, Y, Z, W) = true2.
% 2.81/0.72 Axiom 14 (d2_zfmisc_1_5): fresh5(X, X, Y, Z, W) = W.
% 2.81/0.72 Axiom 15 (t33_zfmisc_1_1): fresh(ordered_pair(X, Y), ordered_pair(Z, W), Y, W) = Y.
% 2.81/0.72 Axiom 16 (d2_zfmisc_1_1): fresh21(X, X, Y, Z, W, V) = equiv2(Y, Z, V).
% 2.81/0.72 Axiom 17 (d3_xboole_0_1): fresh14(X, X, Y, Z, W, V) = equiv(Y, Z, V).
% 2.81/0.72 Axiom 18 (d3_xboole_0_2): fresh12(X, X, Y, Z, W, V) = in(V, W).
% 2.81/0.72 Axiom 19 (d3_xboole_0_3): fresh10(in(X, Y), true2, Z, Y, X) = fresh9(in(X, Z), true2, Z, Y, X).
% 2.81/0.72 Axiom 20 (d3_xboole_0_5): fresh7(equiv(X, Y, Z), true2, X, Z) = in(Z, X).
% 2.81/0.72 Axiom 21 (d3_xboole_0_6): fresh6(equiv(X, Y, Z), true2, Y, Z) = in(Z, Y).
% 2.81/0.72 Axiom 22 (t33_zfmisc_1): fresh2(ordered_pair(X, Y), ordered_pair(Z, W), X, Z) = X.
% 2.81/0.72 Axiom 23 (d2_zfmisc_1_3): fresh23(X, X, Y, Z, W, V, U) = fresh24(W, ordered_pair(V, U), Y, Z, W).
% 2.81/0.72 Axiom 24 (d2_zfmisc_1_1): fresh21(in(X, Y), true2, Z, W, Y, X) = fresh22(Y, cartesian_product2(Z, W), Z, W, X).
% 2.81/0.72 Axiom 25 (d2_zfmisc_1_3): fresh18(X, X, Y, Z, W, V, U) = equiv2(Y, Z, W).
% 2.81/0.72 Axiom 26 (d2_zfmisc_1_6): fresh16(equiv2(X, Y, Z), true2, X, Y, Z) = in(f(X, Y, Z), Y).
% 2.81/0.72 Axiom 27 (d2_zfmisc_1_7): fresh15(equiv2(X, Y, Z), true2, X, Y, Z) = in(e2(X, Y, Z), X).
% 2.81/0.72 Axiom 28 (d3_xboole_0_1): fresh14(in(X, Y), true2, Z, W, Y, X) = fresh13(Y, set_intersection2(Z, W), Z, W, X).
% 2.81/0.72 Axiom 29 (d2_zfmisc_1_5): fresh5(equiv2(X, Y, Z), true2, X, Y, Z) = ordered_pair(e2(X, Y, Z), f(X, Y, Z)).
% 2.81/0.72 Axiom 30 (d2_zfmisc_1_3): fresh23(in(X, Y), true2, Z, Y, W, V, X) = fresh18(in(V, Z), true2, Z, Y, W, V, X).
% 2.81/0.72 Axiom 31 (d3_xboole_0_2): fresh12(equiv(X, Y, Z), true2, X, Y, W, Z) = fresh11(W, set_intersection2(X, Y), W, Z).
% 2.81/0.72
% 2.81/0.72 Lemma 32: equiv(cartesian_product2(b, c), cartesian_product2(d, e), a) = true2.
% 2.81/0.72 Proof:
% 2.81/0.72 equiv(cartesian_product2(b, c), cartesian_product2(d, e), a)
% 2.81/0.72 = { by axiom 17 (d3_xboole_0_1) R->L }
% 2.81/0.72 fresh14(true2, true2, cartesian_product2(b, c), cartesian_product2(d, e), set_intersection2(cartesian_product2(b, c), cartesian_product2(d, e)), a)
% 2.81/0.72 = { by axiom 6 (t104_zfmisc_1) R->L }
% 2.81/0.72 fresh14(in(a, set_intersection2(cartesian_product2(b, c), cartesian_product2(d, e))), true2, cartesian_product2(b, c), cartesian_product2(d, e), set_intersection2(cartesian_product2(b, c), cartesian_product2(d, e)), a)
% 2.81/0.72 = { by axiom 28 (d3_xboole_0_1) }
% 2.81/0.72 fresh13(set_intersection2(cartesian_product2(b, c), cartesian_product2(d, e)), set_intersection2(cartesian_product2(b, c), cartesian_product2(d, e)), cartesian_product2(b, c), cartesian_product2(d, e), a)
% 2.81/0.72 = { by axiom 11 (d3_xboole_0_1) }
% 2.81/0.72 true2
% 2.81/0.72
% 2.81/0.72 Lemma 33: equiv2(b, c, a) = true2.
% 2.81/0.72 Proof:
% 2.81/0.72 equiv2(b, c, a)
% 2.81/0.72 = { by axiom 16 (d2_zfmisc_1_1) R->L }
% 2.81/0.72 fresh21(true2, true2, b, c, cartesian_product2(b, c), a)
% 2.81/0.72 = { by axiom 3 (d3_xboole_0_5) R->L }
% 2.81/0.72 fresh21(fresh7(true2, true2, cartesian_product2(b, c), a), true2, b, c, cartesian_product2(b, c), a)
% 2.81/0.72 = { by lemma 32 R->L }
% 2.81/0.72 fresh21(fresh7(equiv(cartesian_product2(b, c), cartesian_product2(d, e), a), true2, cartesian_product2(b, c), a), true2, b, c, cartesian_product2(b, c), a)
% 2.81/0.72 = { by axiom 20 (d3_xboole_0_5) }
% 2.81/0.72 fresh21(in(a, cartesian_product2(b, c)), true2, b, c, cartesian_product2(b, c), a)
% 2.81/0.72 = { by axiom 24 (d2_zfmisc_1_1) }
% 2.81/0.72 fresh22(cartesian_product2(b, c), cartesian_product2(b, c), b, c, a)
% 2.81/0.72 = { by axiom 8 (d2_zfmisc_1_1) }
% 2.81/0.72 true2
% 2.81/0.72
% 2.81/0.72 Lemma 34: equiv2(d, e, a) = true2.
% 2.81/0.72 Proof:
% 2.81/0.72 equiv2(d, e, a)
% 2.81/0.72 = { by axiom 16 (d2_zfmisc_1_1) R->L }
% 2.81/0.72 fresh21(true2, true2, d, e, cartesian_product2(d, e), a)
% 2.81/0.72 = { by axiom 4 (d3_xboole_0_6) R->L }
% 2.81/0.72 fresh21(fresh6(true2, true2, cartesian_product2(d, e), a), true2, d, e, cartesian_product2(d, e), a)
% 2.81/0.72 = { by lemma 32 R->L }
% 2.81/0.72 fresh21(fresh6(equiv(cartesian_product2(b, c), cartesian_product2(d, e), a), true2, cartesian_product2(d, e), a), true2, d, e, cartesian_product2(d, e), a)
% 2.81/0.72 = { by axiom 21 (d3_xboole_0_6) }
% 2.81/0.72 fresh21(in(a, cartesian_product2(d, e)), true2, d, e, cartesian_product2(d, e), a)
% 2.81/0.72 = { by axiom 24 (d2_zfmisc_1_1) }
% 2.81/0.72 fresh22(cartesian_product2(d, e), cartesian_product2(d, e), d, e, a)
% 2.81/0.72 = { by axiom 8 (d2_zfmisc_1_1) }
% 2.81/0.72 true2
% 2.81/0.72
% 2.81/0.72 Lemma 35: ordered_pair(e2(b, c, a), f(b, c, a)) = a.
% 2.81/0.72 Proof:
% 2.81/0.72 ordered_pair(e2(b, c, a), f(b, c, a))
% 2.81/0.72 = { by axiom 29 (d2_zfmisc_1_5) R->L }
% 2.81/0.72 fresh5(equiv2(b, c, a), true2, b, c, a)
% 2.81/0.72 = { by lemma 33 }
% 2.81/0.72 fresh5(true2, true2, b, c, a)
% 2.81/0.72 = { by axiom 14 (d2_zfmisc_1_5) }
% 2.81/0.72 a
% 2.81/0.72
% 2.81/0.72 Lemma 36: ordered_pair(e2(d, e, a), f(d, e, a)) = a.
% 2.81/0.72 Proof:
% 2.81/0.72 ordered_pair(e2(d, e, a), f(d, e, a))
% 2.81/0.72 = { by axiom 29 (d2_zfmisc_1_5) R->L }
% 2.81/0.72 fresh5(equiv2(d, e, a), true2, d, e, a)
% 2.81/0.72 = { by lemma 34 }
% 2.81/0.72 fresh5(true2, true2, d, e, a)
% 2.81/0.72 = { by axiom 14 (d2_zfmisc_1_5) }
% 2.81/0.73 a
% 2.81/0.73
% 2.81/0.73 Lemma 37: equiv2(set_intersection2(b, d), set_intersection2(c, e), a) = true2.
% 2.81/0.73 Proof:
% 2.81/0.73 equiv2(set_intersection2(b, d), set_intersection2(c, e), a)
% 2.81/0.73 = { by lemma 35 R->L }
% 2.81/0.73 equiv2(set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)))
% 2.81/0.73 = { by axiom 25 (d2_zfmisc_1_3) R->L }
% 2.81/0.73 fresh18(true2, true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 2 (d3_xboole_0_2) R->L }
% 2.81/0.73 fresh18(fresh11(set_intersection2(b, d), set_intersection2(b, d), set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 31 (d3_xboole_0_2) R->L }
% 2.81/0.73 fresh18(fresh12(equiv(b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 12 (d3_xboole_0_3) R->L }
% 2.81/0.73 fresh18(fresh12(fresh10(true2, true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 10 (d2_zfmisc_1_7) R->L }
% 2.81/0.73 fresh18(fresh12(fresh10(fresh15(true2, true2, d, e, a), true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by lemma 34 R->L }
% 2.81/0.73 fresh18(fresh12(fresh10(fresh15(equiv2(d, e, a), true2, d, e, a), true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 27 (d2_zfmisc_1_7) }
% 2.81/0.73 fresh18(fresh12(fresh10(in(e2(d, e, a), d), true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 22 (t33_zfmisc_1) R->L }
% 2.81/0.73 fresh18(fresh12(fresh10(in(fresh2(ordered_pair(e2(d, e, a), f(d, e, a)), ordered_pair(e2(b, c, a), f(b, c, a)), e2(d, e, a), e2(b, c, a)), d), true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by lemma 35 }
% 2.81/0.73 fresh18(fresh12(fresh10(in(fresh2(ordered_pair(e2(d, e, a), f(d, e, a)), a, e2(d, e, a), e2(b, c, a)), d), true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by lemma 36 }
% 2.81/0.73 fresh18(fresh12(fresh10(in(fresh2(a, a, e2(d, e, a), e2(b, c, a)), d), true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 5 (t33_zfmisc_1) }
% 2.81/0.73 fresh18(fresh12(fresh10(in(e2(b, c, a), d), true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 19 (d3_xboole_0_3) }
% 2.81/0.73 fresh18(fresh12(fresh9(in(e2(b, c, a), b), true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 27 (d2_zfmisc_1_7) R->L }
% 2.81/0.73 fresh18(fresh12(fresh9(fresh15(equiv2(b, c, a), true2, b, c, a), true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by lemma 33 }
% 2.81/0.73 fresh18(fresh12(fresh9(fresh15(true2, true2, b, c, a), true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 10 (d2_zfmisc_1_7) }
% 2.81/0.73 fresh18(fresh12(fresh9(true2, true2, b, d, e2(b, c, a)), true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 13 (d3_xboole_0_3) }
% 2.81/0.73 fresh18(fresh12(true2, true2, b, d, set_intersection2(b, d), e2(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 18 (d3_xboole_0_2) }
% 2.81/0.73 fresh18(in(e2(b, c, a), set_intersection2(b, d)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 30 (d2_zfmisc_1_3) R->L }
% 2.81/0.73 fresh23(in(f(b, c, a), set_intersection2(c, e)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 18 (d3_xboole_0_2) R->L }
% 2.81/0.73 fresh23(fresh12(true2, true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 13 (d3_xboole_0_3) R->L }
% 2.81/0.73 fresh23(fresh12(fresh9(true2, true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 9 (d2_zfmisc_1_6) R->L }
% 2.81/0.73 fresh23(fresh12(fresh9(fresh16(true2, true2, b, c, a), true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by lemma 33 R->L }
% 2.81/0.73 fresh23(fresh12(fresh9(fresh16(equiv2(b, c, a), true2, b, c, a), true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 26 (d2_zfmisc_1_6) }
% 2.81/0.73 fresh23(fresh12(fresh9(in(f(b, c, a), c), true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 19 (d3_xboole_0_3) R->L }
% 2.81/0.73 fresh23(fresh12(fresh10(in(f(b, c, a), e), true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 1 (t33_zfmisc_1_1) R->L }
% 2.81/0.73 fresh23(fresh12(fresh10(in(fresh(a, a, f(d, e, a), f(b, c, a)), e), true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by lemma 36 R->L }
% 2.81/0.73 fresh23(fresh12(fresh10(in(fresh(ordered_pair(e2(d, e, a), f(d, e, a)), a, f(d, e, a), f(b, c, a)), e), true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by lemma 35 R->L }
% 2.81/0.73 fresh23(fresh12(fresh10(in(fresh(ordered_pair(e2(d, e, a), f(d, e, a)), ordered_pair(e2(b, c, a), f(b, c, a)), f(d, e, a), f(b, c, a)), e), true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 15 (t33_zfmisc_1_1) }
% 2.81/0.73 fresh23(fresh12(fresh10(in(f(d, e, a), e), true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 26 (d2_zfmisc_1_6) R->L }
% 2.81/0.73 fresh23(fresh12(fresh10(fresh16(equiv2(d, e, a), true2, d, e, a), true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by lemma 34 }
% 2.81/0.73 fresh23(fresh12(fresh10(fresh16(true2, true2, d, e, a), true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 9 (d2_zfmisc_1_6) }
% 2.81/0.73 fresh23(fresh12(fresh10(true2, true2, c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 12 (d3_xboole_0_3) }
% 2.81/0.73 fresh23(fresh12(equiv(c, e, f(b, c, a)), true2, c, e, set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 31 (d3_xboole_0_2) }
% 2.81/0.73 fresh23(fresh11(set_intersection2(c, e), set_intersection2(c, e), set_intersection2(c, e), f(b, c, a)), true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 2 (d3_xboole_0_2) }
% 2.81/0.73 fresh23(true2, true2, set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)), e2(b, c, a), f(b, c, a))
% 2.81/0.73 = { by axiom 23 (d2_zfmisc_1_3) }
% 2.81/0.73 fresh24(ordered_pair(e2(b, c, a), f(b, c, a)), ordered_pair(e2(b, c, a), f(b, c, a)), set_intersection2(b, d), set_intersection2(c, e), ordered_pair(e2(b, c, a), f(b, c, a)))
% 2.81/0.73 = { by axiom 7 (d2_zfmisc_1_3) }
% 2.81/0.73 true2
% 2.81/0.73
% 2.81/0.73 Goal 1 (t104_zfmisc_1_1): tuple2(a, in(X, set_intersection2(b, d)), in(Y, set_intersection2(c, e))) = tuple2(ordered_pair(X, Y), true2, true2).
% 2.81/0.73 The goal is true when:
% 2.81/0.73 X = e2(set_intersection2(b, d), set_intersection2(c, e), a)
% 2.81/0.73 Y = f(set_intersection2(b, d), set_intersection2(c, e), a)
% 2.81/0.73
% 2.81/0.73 Proof:
% 2.81/0.73 tuple2(a, in(e2(set_intersection2(b, d), set_intersection2(c, e), a), set_intersection2(b, d)), in(f(set_intersection2(b, d), set_intersection2(c, e), a), set_intersection2(c, e)))
% 2.81/0.73 = { by axiom 27 (d2_zfmisc_1_7) R->L }
% 2.81/0.73 tuple2(a, fresh15(equiv2(set_intersection2(b, d), set_intersection2(c, e), a), true2, set_intersection2(b, d), set_intersection2(c, e), a), in(f(set_intersection2(b, d), set_intersection2(c, e), a), set_intersection2(c, e)))
% 2.81/0.73 = { by lemma 37 }
% 2.81/0.73 tuple2(a, fresh15(true2, true2, set_intersection2(b, d), set_intersection2(c, e), a), in(f(set_intersection2(b, d), set_intersection2(c, e), a), set_intersection2(c, e)))
% 2.81/0.73 = { by axiom 10 (d2_zfmisc_1_7) }
% 2.81/0.73 tuple2(a, true2, in(f(set_intersection2(b, d), set_intersection2(c, e), a), set_intersection2(c, e)))
% 2.81/0.73 = { by axiom 26 (d2_zfmisc_1_6) R->L }
% 2.81/0.73 tuple2(a, true2, fresh16(equiv2(set_intersection2(b, d), set_intersection2(c, e), a), true2, set_intersection2(b, d), set_intersection2(c, e), a))
% 2.81/0.73 = { by lemma 37 }
% 2.81/0.73 tuple2(a, true2, fresh16(true2, true2, set_intersection2(b, d), set_intersection2(c, e), a))
% 2.81/0.73 = { by axiom 9 (d2_zfmisc_1_6) }
% 2.81/0.73 tuple2(a, true2, true2)
% 2.81/0.73 = { by axiom 14 (d2_zfmisc_1_5) R->L }
% 2.81/0.73 tuple2(fresh5(true2, true2, set_intersection2(b, d), set_intersection2(c, e), a), true2, true2)
% 2.81/0.73 = { by lemma 37 R->L }
% 2.81/0.73 tuple2(fresh5(equiv2(set_intersection2(b, d), set_intersection2(c, e), a), true2, set_intersection2(b, d), set_intersection2(c, e), a), true2, true2)
% 2.81/0.73 = { by axiom 29 (d2_zfmisc_1_5) }
% 2.81/0.73 tuple2(ordered_pair(e2(set_intersection2(b, d), set_intersection2(c, e), a), f(set_intersection2(b, d), set_intersection2(c, e), a)), true2, true2)
% 2.81/0.73 % SZS output end Proof
% 2.81/0.73
% 2.81/0.73 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------