%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO196+2 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.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 : Sat Jun 21 05:05:53 AM UTC 2025
% Result : Theorem 0.12s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : GEO196+2 : TPTP v9.0.0. Released v3.3.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n005.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 14:40:50 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(apart_point_and_line_type, type, (
% 0.12/0.39 apart_point_and_line: ( $i * $i ) > $o)).
% 0.12/0.39 tff(tptp_fun_X_3_type, type, (
% 0.12/0.39 tptp_fun_X_3: $i)).
% 0.12/0.39 tff(intersection_point_type, type, (
% 0.12/0.39 intersection_point: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_V_0_type, type, (
% 0.12/0.39 tptp_fun_V_0: $i)).
% 0.12/0.39 tff(tptp_fun_U_1_type, type, (
% 0.12/0.39 tptp_fun_U_1: $i)).
% 0.12/0.39 tff(distinct_points_type, type, (
% 0.12/0.39 distinct_points: ( $i * $i ) > $o)).
% 0.12/0.39 tff(tptp_fun_Y_2_type, type, (
% 0.12/0.39 tptp_fun_Y_2: $i)).
% 0.12/0.39 tff(convergent_lines_type, type, (
% 0.12/0.39 convergent_lines: ( $i * $i ) > $o)).
% 0.12/0.39 tff(distinct_lines_type, type, (
% 0.12/0.39 distinct_lines: ( $i * $i ) > $o)).
% 0.12/0.39 tff(1,assumption,(distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0))), introduced(assumption)).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (^[X: $i] : refl((~distinct_points(X, X)) <=> (~distinct_points(X, X)))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[2])).
% 0.12/0.39 tff(4,plain,
% 0.12/0.39 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(5,axiom,(![X: $i] : (~distinct_points(X, X))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO008+0.ax','apart1')).
% 0.12/0.39 tff(6,plain,
% 0.12/0.39 (![X: $i] : (~distinct_points(X, X))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.12/0.39 tff(7,plain,(
% 0.12/0.39 ![X: $i] : (~distinct_points(X, X))),
% 0.12/0.39 inference(skolemize,[status(sab)],[6])).
% 0.12/0.39 tff(8,plain,
% 0.12/0.39 (![X: $i] : (~distinct_points(X, X))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.12/0.39 tff(9,plain,
% 0.12/0.39 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(10,plain,
% 0.12/0.39 ($false),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 0.12/0.39 tff(11,plain,(~distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0))), inference(lemma,lemma(discharge,[]))).
% 0.12/0.39 tff(12,plain,
% 0.12/0.39 ((~![X: $i, Y: $i, U: $i, V: $i] : ((~(convergent_lines(X, Y) & convergent_lines(U, V) & (~apart_point_and_line(intersection_point(X, Y), U)) & (~apart_point_and_line(intersection_point(X, Y), V)))) | ((~apart_point_and_line(intersection_point(U, V), X)) & (~apart_point_and_line(intersection_point(U, V), Y))))) <=> (~![X: $i, Y: $i, U: $i, V: $i] : ((~(convergent_lines(X, Y) & convergent_lines(U, V) & (~apart_point_and_line(intersection_point(X, Y), U)) & (~apart_point_and_line(intersection_point(X, Y), V)))) | ((~apart_point_and_line(intersection_point(U, V), X)) & (~apart_point_and_line(intersection_point(U, V), Y)))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(13,plain,
% 0.12/0.39 ((~![X: $i, Y: $i, U: $i, V: $i] : ((((convergent_lines(X, Y) & convergent_lines(U, V)) & (~apart_point_and_line(intersection_point(X, Y), U))) & (~apart_point_and_line(intersection_point(X, Y), V))) => ((~apart_point_and_line(intersection_point(U, V), X)) & (~apart_point_and_line(intersection_point(U, V), Y))))) <=> (~![X: $i, Y: $i, U: $i, V: $i] : ((~(convergent_lines(X, Y) & convergent_lines(U, V) & (~apart_point_and_line(intersection_point(X, Y), U)) & (~apart_point_and_line(intersection_point(X, Y), V)))) | ((~apart_point_and_line(intersection_point(U, V), X)) & (~apart_point_and_line(intersection_point(U, V), Y)))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(14,axiom,(~![X: $i, Y: $i, U: $i, V: $i] : ((((convergent_lines(X, Y) & convergent_lines(U, V)) & (~apart_point_and_line(intersection_point(X, Y), U))) & (~apart_point_and_line(intersection_point(X, Y), V))) => ((~apart_point_and_line(intersection_point(U, V), X)) & (~apart_point_and_line(intersection_point(U, V), Y))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','con')).
% 0.12/0.39 tff(15,plain,
% 0.12/0.39 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(convergent_lines(X, Y) & convergent_lines(U, V) & (~apart_point_and_line(intersection_point(X, Y), U)) & (~apart_point_and_line(intersection_point(X, Y), V)))) | ((~apart_point_and_line(intersection_point(U, V), X)) & (~apart_point_and_line(intersection_point(U, V), Y))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.12/0.39 tff(16,plain,
% 0.12/0.39 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(convergent_lines(X, Y) & convergent_lines(U, V) & (~apart_point_and_line(intersection_point(X, Y), U)) & (~apart_point_and_line(intersection_point(X, Y), V)))) | ((~apart_point_and_line(intersection_point(U, V), X)) & (~apart_point_and_line(intersection_point(U, V), Y))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[15, 12])).
% 0.12/0.39 tff(17,plain,
% 0.12/0.39 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(convergent_lines(X, Y) & convergent_lines(U, V) & (~apart_point_and_line(intersection_point(X, Y), U)) & (~apart_point_and_line(intersection_point(X, Y), V)))) | ((~apart_point_and_line(intersection_point(U, V), X)) & (~apart_point_and_line(intersection_point(U, V), Y))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.12/0.39 tff(18,plain,
% 0.12/0.39 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(convergent_lines(X, Y) & convergent_lines(U, V) & (~apart_point_and_line(intersection_point(X, Y), U)) & (~apart_point_and_line(intersection_point(X, Y), V)))) | ((~apart_point_and_line(intersection_point(U, V), X)) & (~apart_point_and_line(intersection_point(U, V), Y))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[17, 12])).
% 0.12/0.39 tff(19,plain,(
% 0.12/0.39 ~((~(convergent_lines(X!3, Y!2) & convergent_lines(U!1, V!0) & (~apart_point_and_line(intersection_point(X!3, Y!2), U!1)) & (~apart_point_and_line(intersection_point(X!3, Y!2), V!0)))) | ((~apart_point_and_line(intersection_point(U!1, V!0), X!3)) & (~apart_point_and_line(intersection_point(U!1, V!0), Y!2))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[18])).
% 0.12/0.39 tff(20,plain,
% 0.12/0.39 (convergent_lines(X!3, Y!2) & convergent_lines(U!1, V!0) & (~apart_point_and_line(intersection_point(X!3, Y!2), U!1)) & (~apart_point_and_line(intersection_point(X!3, Y!2), V!0))),
% 0.12/0.39 inference(or_elim,[status(thm)],[19])).
% 0.12/0.39 tff(21,plain,
% 0.12/0.39 (convergent_lines(U!1, V!0)),
% 0.12/0.39 inference(and_elim,[status(thm)],[20])).
% 0.12/0.39 tff(22,plain,
% 0.12/0.39 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))) <=> (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(23,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[22])).
% 0.12/0.39 tff(24,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(25,plain,
% 0.12/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(trans(monotonicity(rewrite((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) <=> (apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))), (((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y))) <=> ((apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)) => distinct_points(Z, intersection_point(X, Y))))), rewrite(((apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)) => distinct_points(Z, intersection_point(X, Y))) <=> ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y)))), (((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y))) <=> ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y))))), ((convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y)))) <=> (convergent_lines(X, Y) => ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y)))))), rewrite((convergent_lines(X, Y) => ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y)))) <=> (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))), ((convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y)))) <=> (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(26,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y)))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[25])).
% 0.12/0.39 tff(27,axiom,(![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO008+0.ax','con2')).
% 0.12/0.39 tff(28,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.12/0.39 tff(29,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[28, 24])).
% 0.12/0.39 tff(30,plain,(
% 0.12/0.39 ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[29])).
% 0.12/0.39 tff(31,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[30, 23])).
% 0.12/0.39 tff(32,plain,
% 0.12/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | ((~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))) | distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))) | distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(33,plain,
% 0.12/0.39 ((distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)) | (~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1)))) <=> ((~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))) | distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(34,plain,
% 0.12/0.39 ((~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))) <=> (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(35,plain,
% 0.12/0.39 ((distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)) | (~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1)))) <=> (distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)) | (~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[34])).
% 0.12/0.39 tff(36,plain,
% 0.12/0.39 ((distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)) | (~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1)))) <=> ((~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))) | distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)))),
% 0.12/0.39 inference(transitivity,[status(thm)],[35, 33])).
% 0.12/0.39 tff(37,plain,
% 0.12/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)) | (~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | ((~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))) | distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0))))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[36])).
% 0.12/0.39 tff(38,plain,
% 0.12/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)) | (~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))) | distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)))),
% 0.12/0.39 inference(transitivity,[status(thm)],[37, 32])).
% 0.12/0.39 tff(39,plain,
% 0.12/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0)) | (~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(40,plain,
% 0.12/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(U!1, V!0)) | (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))) | distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[39, 38])).
% 0.12/0.39 tff(41,plain,
% 0.12/0.39 ((~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))) | distinct_points(intersection_point(U!1, V!0), intersection_point(U!1, V!0))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[40, 31, 21])).
% 0.12/0.39 tff(42,plain,
% 0.12/0.39 (~(apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[41, 11])).
% 0.12/0.40 tff(43,plain,
% 0.12/0.40 ((apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1)) | (~apart_point_and_line(intersection_point(U!1, V!0), U!1))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(44,plain,
% 0.12/0.40 (~apart_point_and_line(intersection_point(U!1, V!0), U!1)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[43, 42])).
% 0.12/0.40 tff(45,plain,
% 0.12/0.40 ((apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1)) | (~apart_point_and_line(intersection_point(U!1, V!0), V!0))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(46,plain,
% 0.12/0.40 (~apart_point_and_line(intersection_point(U!1, V!0), V!0)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[45, 42])).
% 0.12/0.40 tff(47,plain,
% 0.12/0.40 (^[X: $i, Y: $i] : refl(((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ((~convergent_lines(X, Y)) | distinct_lines(X, Y)))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(48,plain,
% 0.12/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[47])).
% 0.12/0.40 tff(49,plain,
% 0.12/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(50,plain,
% 0.12/0.40 (^[X: $i, Y: $i] : rewrite((convergent_lines(X, Y) => distinct_lines(X, Y)) <=> ((~convergent_lines(X, Y)) | distinct_lines(X, Y)))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(51,plain,
% 0.12/0.40 (![X: $i, Y: $i] : (convergent_lines(X, Y) => distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[50])).
% 0.12/0.40 tff(52,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => distinct_lines(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO008+0.ax','ceq3')).
% 0.12/0.40 tff(53,plain,
% 0.12/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[52, 51])).
% 0.12/0.40 tff(54,plain,
% 0.12/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[53, 49])).
% 0.12/0.40 tff(55,plain,(
% 0.12/0.40 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.12/0.40 inference(skolemize,[status(sab)],[54])).
% 0.12/0.40 tff(56,plain,
% 0.12/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[55, 48])).
% 0.12/0.40 tff(57,plain,
% 0.12/0.40 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | ((~convergent_lines(U!1, V!0)) | distinct_lines(U!1, V!0))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | (~convergent_lines(U!1, V!0)) | distinct_lines(U!1, V!0))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(58,plain,
% 0.12/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | ((~convergent_lines(U!1, V!0)) | distinct_lines(U!1, V!0))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(59,plain,
% 0.12/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | (~convergent_lines(U!1, V!0)) | distinct_lines(U!1, V!0)),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.12/0.40 tff(60,plain,
% 0.12/0.40 (distinct_lines(U!1, V!0)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[59, 56, 21])).
% 0.12/0.40 tff(61,plain,
% 0.12/0.40 (~apart_point_and_line(intersection_point(X!3, Y!2), V!0)),
% 0.12/0.40 inference(and_elim,[status(thm)],[20])).
% 0.12/0.40 tff(62,plain,
% 0.12/0.40 (~apart_point_and_line(intersection_point(X!3, Y!2), U!1)),
% 0.12/0.40 inference(and_elim,[status(thm)],[20])).
% 0.12/0.40 tff(63,plain,
% 0.12/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : refl((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(64,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[63])).
% 0.12/0.40 tff(65,plain,
% 0.12/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(rewrite((distinct_points(X, Y) & distinct_lines(U, V)) <=> (~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> (~(~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))))), rewrite((~(~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))) <=> ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> ((~distinct_points(X, Y)) | (~distinct_lines(U, V))))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))))), rewrite((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(66,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[65])).
% 0.12/0.40 tff(67,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(68,plain,
% 0.12/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((apart_point_and_line(X, U) | apart_point_and_line(X, V)) <=> (apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> ((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)))), rewrite(((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> ((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)))), rewrite(((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> ((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))))), rewrite(((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(69,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : ((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[68])).
% 0.12/0.40 tff(70,axiom,(![X: $i, Y: $i, U: $i, V: $i] : ((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO008+0.ax','cu1')).
% 0.12/0.40 tff(71,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[70, 69])).
% 0.12/0.40 tff(72,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[71, 67])).
% 0.12/0.40 tff(73,plain,(
% 0.12/0.40 ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.12/0.40 inference(skolemize,[status(sab)],[72])).
% 0.12/0.40 tff(74,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[73, 66])).
% 0.12/0.40 tff(75,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[74, 64])).
% 0.12/0.40 tff(76,plain,
% 0.12/0.40 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_lines(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))) | apart_point_and_line(intersection_point(U!1, V!0), U!1))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_lines(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))) | apart_point_and_line(intersection_point(U!1, V!0), U!1))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(77,plain,
% 0.12/0.40 ((apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0))) <=> (apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_lines(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))) | apart_point_and_line(intersection_point(U!1, V!0), U!1))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(78,plain,
% 0.12/0.40 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_lines(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))) | apart_point_and_line(intersection_point(U!1, V!0), U!1)))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[77])).
% 0.12/0.40 tff(79,plain,
% 0.12/0.40 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_lines(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))) | apart_point_and_line(intersection_point(U!1, V!0), U!1))),
% 0.12/0.40 inference(transitivity,[status(thm)],[78, 76])).
% 0.12/0.40 tff(80,plain,
% 0.12/0.40 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0)))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(81,plain,
% 0.12/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_lines(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))) | apart_point_and_line(intersection_point(U!1, V!0), U!1)),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[80, 79])).
% 0.12/0.41 tff(82,plain,
% 0.12/0.41 (~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[81, 75, 62, 61, 60, 46, 44])).
% 0.12/0.41 tff(83,plain,
% 0.12/0.41 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(84,plain,
% 0.12/0.41 ((apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0))) <=> (apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(85,plain,
% 0.12/0.41 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0))))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[84])).
% 0.12/0.41 tff(86,plain,
% 0.12/0.41 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0)))),
% 0.20/0.41 inference(transitivity,[status(thm)],[85, 83])).
% 0.20/0.41 tff(87,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0)))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(88,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(X!3, Y!2), U!1) | apart_point_and_line(intersection_point(X!3, Y!2), V!0) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2))) | (~distinct_lines(U!1, V!0))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[87, 86])).
% 0.20/0.41 tff(89,plain,
% 0.20/0.41 (~distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[88, 75, 62, 61, 60])).
% 0.20/0.41 tff(90,plain,
% 0.20/0.41 (convergent_lines(X!3, Y!2)),
% 0.20/0.41 inference(and_elim,[status(thm)],[20])).
% 0.20/0.41 tff(91,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | ((~convergent_lines(X!3, Y!2)) | distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!3, Y!2)) | distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(92,plain,
% 0.20/0.41 ((distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~convergent_lines(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3)))) <=> ((~convergent_lines(X!3, Y!2)) | distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(93,plain,
% 0.20/0.41 ((~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3))) <=> (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(94,plain,
% 0.20/0.41 ((distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~convergent_lines(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3)))) <=> (distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~convergent_lines(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[93])).
% 0.20/0.41 tff(95,plain,
% 0.20/0.41 ((distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~convergent_lines(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3)))) <=> ((~convergent_lines(X!3, Y!2)) | distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[94, 92])).
% 0.20/0.41 tff(96,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~convergent_lines(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | ((~convergent_lines(X!3, Y!2)) | distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3)))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[95])).
% 0.20/0.41 tff(97,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~convergent_lines(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!3, Y!2)) | distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[96, 91])).
% 0.20/0.41 tff(98,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~convergent_lines(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(99,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!3, Y!2)) | distinct_points(intersection_point(X!3, Y!2), intersection_point(X!3, Y!2)) | (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.20/0.41 tff(100,plain,
% 0.20/0.41 (~(apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[99, 31, 90, 89])).
% 0.20/0.41 tff(101,plain,
% 0.20/0.41 ((apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3)) | (~apart_point_and_line(intersection_point(X!3, Y!2), X!3))),
% 0.20/0.41 inference(tautology,[status(thm)],[])).
% 0.20/0.41 tff(102,plain,
% 0.20/0.41 (~apart_point_and_line(intersection_point(X!3, Y!2), X!3)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[101, 100])).
% 0.20/0.41 tff(103,plain,
% 0.20/0.41 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(104,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[103])).
% 0.20/0.41 tff(105,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(106,plain,
% 0.20/0.41 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(X, Z) | apart_point_and_line(Z, Y)) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y))), ((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))))), rewrite((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))), ((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(107,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[106])).
% 0.20/0.41 tff(108,axiom,(![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO008+0.ax','ceq1')).
% 0.20/0.41 tff(109,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[108, 107])).
% 0.20/0.41 tff(110,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[109, 105])).
% 0.20/0.41 tff(111,plain,(
% 0.20/0.41 ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(skolemize,[status(sab)],[110])).
% 0.20/0.41 tff(112,plain,
% 0.20/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[111, 104])).
% 0.20/0.41 tff(113,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(U!1, V!0), X!3)) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(U!1, V!0), X!3)) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(114,plain,
% 0.20/0.41 ((distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | (~apart_point_and_line(intersection_point(U!1, V!0), X!3))) <=> ((~apart_point_and_line(intersection_point(U!1, V!0), X!3)) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(115,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | (~apart_point_and_line(intersection_point(U!1, V!0), X!3)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(U!1, V!0), X!3)) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[114])).
% 0.20/0.42 tff(116,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | (~apart_point_and_line(intersection_point(U!1, V!0), X!3)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(U!1, V!0), X!3)) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)))),
% 0.20/0.42 inference(transitivity,[status(thm)],[115, 113])).
% 0.20/0.42 tff(117,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | (~apart_point_and_line(intersection_point(U!1, V!0), X!3)))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(118,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(U!1, V!0), X!3)) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[117, 116])).
% 0.20/0.42 tff(119,plain,
% 0.20/0.42 ((~apart_point_and_line(intersection_point(U!1, V!0), X!3)) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[118, 112, 102])).
% 0.20/0.42 tff(120,plain,
% 0.20/0.42 (~apart_point_and_line(intersection_point(U!1, V!0), X!3)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[119, 82])).
% 0.20/0.42 tff(121,assumption,(~distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))), introduced(assumption)).
% 0.20/0.42 tff(122,plain,
% 0.20/0.42 ((apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3)) | (~apart_point_and_line(intersection_point(X!3, Y!2), Y!2))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(123,plain,
% 0.20/0.42 (~apart_point_and_line(intersection_point(X!3, Y!2), Y!2)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[122, 100])).
% 0.20/0.42 tff(124,assumption,(apart_point_and_line(intersection_point(U!1, V!0), Y!2)), introduced(assumption)).
% 0.20/0.42 tff(125,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(U!1, V!0), Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(U!1, V!0), Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(126,plain,
% 0.20/0.42 ((distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | (~apart_point_and_line(intersection_point(U!1, V!0), Y!2))) <=> ((~apart_point_and_line(intersection_point(U!1, V!0), Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(127,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | (~apart_point_and_line(intersection_point(U!1, V!0), Y!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(U!1, V!0), Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[126])).
% 0.20/0.42 tff(128,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | (~apart_point_and_line(intersection_point(U!1, V!0), Y!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(U!1, V!0), Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)))),
% 0.20/0.42 inference(transitivity,[status(thm)],[127, 125])).
% 0.20/0.42 tff(129,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | (~apart_point_and_line(intersection_point(U!1, V!0), Y!2)))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(130,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(U!1, V!0), Y!2)) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[129, 128])).
% 0.20/0.42 tff(131,plain,
% 0.20/0.42 ($false),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[130, 112, 124, 123, 121])).
% 0.20/0.42 tff(132,plain,(distinct_points(intersection_point(U!1, V!0), intersection_point(X!3, Y!2)) | (~apart_point_and_line(intersection_point(U!1, V!0), Y!2))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.42 tff(133,plain,
% 0.20/0.42 (~apart_point_and_line(intersection_point(U!1, V!0), Y!2)),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[132, 82])).
% 0.20/0.42 tff(134,plain,
% 0.20/0.42 ((~(~(apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(U!1, V!0), Y!2)))) <=> (apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(U!1, V!0), Y!2))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(135,plain,
% 0.20/0.42 (((~apart_point_and_line(intersection_point(U!1, V!0), X!3)) & (~apart_point_and_line(intersection_point(U!1, V!0), Y!2))) <=> (~(apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(U!1, V!0), Y!2)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(136,plain,
% 0.20/0.42 ((~((~apart_point_and_line(intersection_point(U!1, V!0), X!3)) & (~apart_point_and_line(intersection_point(U!1, V!0), Y!2)))) <=> (~(~(apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(U!1, V!0), Y!2))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[135])).
% 0.20/0.42 tff(137,plain,
% 0.20/0.42 ((~((~apart_point_and_line(intersection_point(U!1, V!0), X!3)) & (~apart_point_and_line(intersection_point(U!1, V!0), Y!2)))) <=> (apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(U!1, V!0), Y!2))),
% 0.20/0.42 inference(transitivity,[status(thm)],[136, 134])).
% 0.20/0.42 tff(138,plain,
% 0.20/0.42 (~((~apart_point_and_line(intersection_point(U!1, V!0), X!3)) & (~apart_point_and_line(intersection_point(U!1, V!0), Y!2)))),
% 0.20/0.42 inference(or_elim,[status(thm)],[19])).
% 0.20/0.42 tff(139,plain,
% 0.20/0.42 (apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(U!1, V!0), Y!2)),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[138, 137])).
% 0.20/0.42 tff(140,plain,
% 0.20/0.42 ($false),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[139, 133, 120])).
% 0.20/0.42 % SZS output end Proof
% 0.20/0.42 % E exiting
%------------------------------------------------------------------------------