%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO202+2 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n004.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 05:05:55 AM UTC 2025
% Result : Theorem 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GEO202+2 : TPTP v9.0.0. Released v3.3.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n004.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 14:40:53 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 tff(distinct_points_type, type, (
% 0.20/0.40 distinct_points: ( $i * $i ) > $o)).
% 0.20/0.40 tff(tptp_fun_Y_1_type, type, (
% 0.20/0.40 tptp_fun_Y_1: $i)).
% 0.20/0.40 tff(tptp_fun_X_2_type, type, (
% 0.20/0.40 tptp_fun_X_2: $i)).
% 0.20/0.40 tff(apart_point_and_line_type, type, (
% 0.20/0.40 apart_point_and_line: ( $i * $i ) > $o)).
% 0.20/0.40 tff(line_connecting_type, type, (
% 0.20/0.40 line_connecting: ( $i * $i ) > $i)).
% 0.20/0.40 tff(tptp_fun_Z_0_type, type, (
% 0.20/0.40 tptp_fun_Z_0: $i)).
% 0.20/0.40 tff(convergent_lines_type, type, (
% 0.20/0.40 convergent_lines: ( $i * $i ) > $o)).
% 0.20/0.40 tff(intersection_point_type, type, (
% 0.20/0.40 intersection_point: ( $i * $i ) > $i)).
% 0.20/0.40 tff(distinct_lines_type, type, (
% 0.20/0.40 distinct_lines: ( $i * $i ) > $o)).
% 0.20/0.40 tff(1,plain,
% 0.20/0.40 (^[X: $i] : refl((~distinct_points(X, X)) <=> (~distinct_points(X, X)))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(2,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.40 tff(3,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(4,axiom,(![X: $i] : (~distinct_points(X, X))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO008+0.ax','apart1')).
% 0.20/0.40 tff(5,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.40 tff(6,plain,(
% 0.20/0.40 ![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(skolemize,[status(sab)],[5])).
% 0.20/0.40 tff(7,plain,
% 0.20/0.40 (![X: $i] : (~distinct_points(X, X))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.40 tff(8,plain,
% 0.20/0.40 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(X!2, X!2))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(9,plain,
% 0.20/0.40 (~distinct_points(X!2, X!2)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.20/0.40 tff(10,plain,
% 0.20/0.40 (((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1))) | distinct_points(X!2, X!2)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(11,plain,
% 0.20/0.40 ((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[10, 9])).
% 0.20/0.40 tff(12,plain,
% 0.20/0.40 (((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Z!0))) | distinct_points(X!2, X!2)),
% 0.20/0.40 inference(tautology,[status(thm)],[])).
% 0.20/0.40 tff(13,plain,
% 0.20/0.40 ((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Z!0))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[12, 9])).
% 0.20/0.40 tff(14,assumption,(apart_point_and_line(X!2, line_connecting(X!2, Z!0))), introduced(assumption)).
% 0.20/0.40 tff(15,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(intersection_point(line_connecting(X, Y), line_connecting(X, Z)), X)) | (~(distinct_points(X, Y) & distinct_points(X, Z) & convergent_lines(line_connecting(X, Y), line_connecting(X, Z)))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~distinct_points(intersection_point(line_connecting(X, Y), line_connecting(X, Z)), X)) | (~(distinct_points(X, Y) & distinct_points(X, Z) & convergent_lines(line_connecting(X, Y), line_connecting(X, Z))))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(16,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : (((distinct_points(X, Y) & distinct_points(X, Z)) & convergent_lines(line_connecting(X, Y), line_connecting(X, Z))) => (~distinct_points(intersection_point(line_connecting(X, Y), line_connecting(X, Z)), X)))) <=> (~![X: $i, Y: $i, Z: $i] : ((~distinct_points(intersection_point(line_connecting(X, Y), line_connecting(X, Z)), X)) | (~(distinct_points(X, Y) & distinct_points(X, Z) & convergent_lines(line_connecting(X, Y), line_connecting(X, Z))))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(17,axiom,(~![X: $i, Y: $i, Z: $i] : (((distinct_points(X, Y) & distinct_points(X, Z)) & convergent_lines(line_connecting(X, Y), line_connecting(X, Z))) => (~distinct_points(intersection_point(line_connecting(X, Y), line_connecting(X, Z)), X)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','con')).
% 0.20/0.40 tff(18,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~distinct_points(intersection_point(line_connecting(X, Y), line_connecting(X, Z)), X)) | (~(distinct_points(X, Y) & distinct_points(X, Z) & convergent_lines(line_connecting(X, Y), line_connecting(X, Z)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.20/0.40 tff(19,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~distinct_points(intersection_point(line_connecting(X, Y), line_connecting(X, Z)), X)) | (~(distinct_points(X, Y) & distinct_points(X, Z) & convergent_lines(line_connecting(X, Y), line_connecting(X, Z)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[18, 15])).
% 0.20/0.40 tff(20,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~distinct_points(intersection_point(line_connecting(X, Y), line_connecting(X, Z)), X)) | (~(distinct_points(X, Y) & distinct_points(X, Z) & convergent_lines(line_connecting(X, Y), line_connecting(X, Z)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.20/0.40 tff(21,plain,
% 0.20/0.40 (~![X: $i, Y: $i, Z: $i] : ((~distinct_points(intersection_point(line_connecting(X, Y), line_connecting(X, Z)), X)) | (~(distinct_points(X, Y) & distinct_points(X, Z) & convergent_lines(line_connecting(X, Y), line_connecting(X, Z)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[20, 15])).
% 0.20/0.40 tff(22,plain,(
% 0.20/0.40 ~((~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)) | (~(distinct_points(X!2, Y!1) & distinct_points(X!2, Z!0) & convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[21])).
% 0.20/0.40 tff(23,plain,
% 0.20/0.40 (distinct_points(X!2, Y!1) & distinct_points(X!2, Z!0) & convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))),
% 0.20/0.40 inference(or_elim,[status(thm)],[22])).
% 0.20/0.40 tff(24,plain,
% 0.20/0.40 (distinct_points(X!2, Z!0)),
% 0.20/0.40 inference(and_elim,[status(thm)],[23])).
% 0.20/0.40 tff(25,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(26,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))) <=> ![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[25])).
% 0.20/0.40 tff(27,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(Z, X) & distinct_points(Z, Y)) <=> (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))), (((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))))), rewrite(((~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))), (((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(28,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[27])).
% 0.20/0.40 tff(29,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(30,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y))) <=> ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))), ((distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> (distinct_points(X, Y) => ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))))), rewrite((distinct_points(X, Y) => ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))), ((distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(31,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[30])).
% 0.20/0.40 tff(32,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO008+0.ax','con1')).
% 0.20/0.40 tff(33,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.20/0.40 tff(34,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[33, 29])).
% 0.20/0.40 tff(35,plain,(
% 0.20/0.40 ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[34])).
% 0.20/0.40 tff(36,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[35, 28])).
% 0.20/0.40 tff(37,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[36, 26])).
% 0.20/0.40 tff(38,plain,
% 0.20/0.40 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Z!0)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Z!0)))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(39,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Z!0)))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(40,plain,
% 0.20/0.40 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Z!0))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[39, 38])).
% 0.20/0.40 tff(41,plain,
% 0.20/0.40 ($false),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[40, 37, 24, 14, 13])).
% 0.20/0.40 tff(42,plain,(~apart_point_and_line(X!2, line_connecting(X!2, Z!0))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.40 tff(43,assumption,(~distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1))), introduced(assumption)).
% 0.20/0.40 tff(44,plain,
% 0.20/0.40 (^[X: $i] : refl((~distinct_lines(X, X)) <=> (~distinct_lines(X, X)))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(45,plain,
% 0.20/0.40 (![X: $i] : (~distinct_lines(X, X)) <=> ![X: $i] : (~distinct_lines(X, X))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[44])).
% 0.20/0.40 tff(46,plain,
% 0.20/0.40 (![X: $i] : (~distinct_lines(X, X)) <=> ![X: $i] : (~distinct_lines(X, X))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(47,axiom,(![X: $i] : (~distinct_lines(X, X))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO008+0.ax','apart2')).
% 0.20/0.40 tff(48,plain,
% 0.20/0.40 (![X: $i] : (~distinct_lines(X, X))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.20/0.40 tff(49,plain,(
% 0.20/0.40 ![X: $i] : (~distinct_lines(X, X))),
% 0.20/0.40 inference(skolemize,[status(sab)],[48])).
% 0.20/0.40 tff(50,plain,
% 0.20/0.40 (![X: $i] : (~distinct_lines(X, X))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[49, 45])).
% 0.20/0.40 tff(51,plain,
% 0.20/0.40 ((~![X: $i] : (~distinct_lines(X, X))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(52,plain,
% 0.20/0.40 (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[51, 50])).
% 0.20/0.40 tff(53,plain,
% 0.20/0.40 (convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))),
% 0.20/0.40 inference(and_elim,[status(thm)],[23])).
% 0.20/0.40 tff(54,plain,
% 0.20/0.40 (^[X: $i, Y: $i] : refl(((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ((~convergent_lines(X, Y)) | distinct_lines(X, Y)))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(55,plain,
% 0.20/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[54])).
% 0.20/0.40 tff(56,plain,
% 0.20/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(57,plain,
% 0.20/0.40 (^[X: $i, Y: $i] : rewrite((convergent_lines(X, Y) => distinct_lines(X, Y)) <=> ((~convergent_lines(X, Y)) | distinct_lines(X, Y)))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(58,plain,
% 0.20/0.40 (![X: $i, Y: $i] : (convergent_lines(X, Y) => distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[57])).
% 0.20/0.40 tff(59,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => distinct_lines(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO008+0.ax','ceq3')).
% 0.20/0.40 tff(60,plain,
% 0.20/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.20/0.40 tff(61,plain,
% 0.20/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.20/0.40 tff(62,plain,(
% 0.20/0.40 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.20/0.40 inference(skolemize,[status(sab)],[61])).
% 0.20/0.40 tff(63,plain,
% 0.20/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[62, 55])).
% 0.20/0.40 tff(64,plain,
% 0.20/0.40 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | ((~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | (~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(65,plain,
% 0.20/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | ((~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(66,plain,
% 0.20/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | (~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.20/0.40 tff(67,plain,
% 0.20/0.40 (distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[66, 63, 53])).
% 0.20/0.40 tff(68,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : refl((distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(69,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[68])).
% 0.20/0.40 tff(70,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(71,plain,
% 0.20/0.40 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_lines(X, Z) | distinct_lines(Y, Z)) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z))), ((distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z))) <=> (distinct_lines(X, Y) => (distinct_lines(Y, Z) | distinct_lines(X, Z))))), rewrite((distinct_lines(X, Y) => (distinct_lines(Y, Z) | distinct_lines(X, Z))) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))), ((distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z))) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(72,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[71])).
% 0.20/0.40 tff(73,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z)))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO008+0.ax','apart5')).
% 0.20/0.40 tff(74,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.20/0.40 tff(75,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[74, 70])).
% 0.20/0.40 tff(76,plain,(
% 0.20/0.40 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.20/0.40 inference(skolemize,[status(sab)],[75])).
% 0.20/0.40 tff(77,plain,
% 0.20/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[76, 69])).
% 0.20/0.40 tff(78,plain,
% 0.20/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(79,plain,
% 0.20/0.41 ((distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))) <=> (distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(80,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[79])).
% 0.20/0.41 tff(81,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1)))),
% 0.20/0.41 inference(transitivity,[status(thm)],[80, 78])).
% 0.20/0.41 tff(82,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(83,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.20/0.41 tff(84,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[83, 77, 67, 52, 43])).
% 0.20/0.41 tff(85,plain,(distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.41 tff(86,plain,
% 0.20/0.41 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(87,plain,
% 0.20/0.41 (~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[86, 7])).
% 0.20/0.41 tff(88,plain,
% 0.20/0.41 (^[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)))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(89,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)))) <=> ![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))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[88])).
% 0.20/0.41 tff(90,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)))) <=> ![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))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(91,plain,
% 0.20/0.41 (^[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))))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(92,plain,
% 0.20/0.41 (![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))))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[91])).
% 0.20/0.41 tff(93,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')).
% 0.20/0.41 tff(94,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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.20/0.41 tff(95,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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[94, 90])).
% 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))))),
% 0.20/0.41 inference(skolemize,[status(sab)],[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))))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[96, 89])).
% 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))))) | ((~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)))) | distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!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(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)))) | distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(99,plain,
% 0.20/0.41 ((distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0))))) <=> ((~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)))) | distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(100,plain,
% 0.20/0.41 ((~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)))) <=> (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(101,plain,
% 0.20/0.41 ((distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1))))) <=> (distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[100])).
% 0.20/0.41 tff(102,plain,
% 0.20/0.41 ((distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1))))) <=> ((~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)))) | distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[101, 99])).
% 0.20/0.41 tff(103,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(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!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(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)))) | distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[102])).
% 0.20/0.41 tff(104,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(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!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(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)))) | distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))),
% 0.20/0.41 inference(transitivity,[status(thm)],[103, 98])).
% 0.20/0.41 tff(105,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(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~convergent_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(106,plain,
% 0.20/0.42 ((~![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(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)))) | distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.20/0.42 tff(107,plain,
% 0.20/0.42 (~(apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[106, 97, 53, 87])).
% 0.20/0.42 tff(108,plain,
% 0.20/0.42 ((apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0))) | (~apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(109,plain,
% 0.20/0.42 (~apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[108, 107])).
% 0.20/0.42 tff(110,plain,
% 0.20/0.42 ((apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0))) | (~apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)))),
% 0.20/0.42 inference(tautology,[status(thm)],[])).
% 0.20/0.42 tff(111,plain,
% 0.20/0.42 (~apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[110, 107])).
% 0.20/0.42 tff(112,plain,
% 0.20/0.42 (distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)),
% 0.20/0.42 inference(or_elim,[status(thm)],[22])).
% 0.20/0.42 tff(113,plain,
% 0.20/0.42 (^[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))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(114,plain,
% 0.20/0.42 (![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)))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[113])).
% 0.20/0.42 tff(115,plain,
% 0.20/0.42 (^[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)))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(116,plain,
% 0.20/0.42 (![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)))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[115])).
% 0.20/0.42 tff(117,plain,
% 0.20/0.42 (![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))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(118,plain,
% 0.20/0.42 (^[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))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(119,plain,
% 0.20/0.42 (![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))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[118])).
% 0.20/0.42 tff(120,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')).
% 0.20/0.42 tff(121,plain,
% 0.20/0.42 (![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))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[120, 119])).
% 0.20/0.42 tff(122,plain,
% 0.20/0.42 (![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))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[121, 117])).
% 0.20/0.42 tff(123,plain,(
% 0.20/0.42 ![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))))),
% 0.20/0.42 inference(skolemize,[status(sab)],[122])).
% 0.20/0.42 tff(124,plain,
% 0.20/0.42 (![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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[123, 116])).
% 0.20/0.42 tff(125,plain,
% 0.20/0.42 (![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)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[124, 114])).
% 0.20/0.42 tff(126,plain,
% 0.20/0.42 (((~![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)))) | ((~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!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)))) | (~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(127,plain,
% 0.20/0.42 ((apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | (~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)) | (~distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1)))) <=> ((~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(128,plain,
% 0.20/0.42 (((~![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(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | (~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)) | (~distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!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)))) | ((~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1)))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[127])).
% 0.20/0.42 tff(129,plain,
% 0.20/0.42 (((~![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(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | (~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)) | (~distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!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)))) | (~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1))))),
% 0.20/0.43 inference(transitivity,[status(thm)],[128, 126])).
% 0.20/0.43 tff(130,plain,
% 0.20/0.43 ((~![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(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | (~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)) | (~distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(131,plain,
% 0.20/0.43 ((~![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)))) | (~distinct_points(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), X!2)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Y!1)) | apart_point_and_line(intersection_point(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)), line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Z!0), line_connecting(X!2, Y!1)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[130, 129])).
% 0.20/0.43 tff(132,plain,
% 0.20/0.43 (apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[131, 125, 112, 111, 109, 85])).
% 0.20/0.43 tff(133,plain,
% 0.20/0.43 (apart_point_and_line(X!2, line_connecting(X!2, Y!1))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[132, 42])).
% 0.20/0.43 tff(134,plain,
% 0.20/0.43 (distinct_points(X!2, Y!1)),
% 0.20/0.43 inference(and_elim,[status(thm)],[23])).
% 0.20/0.43 tff(135,plain,
% 0.20/0.43 (((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(136,plain,
% 0.20/0.43 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(137,plain,
% 0.20/0.43 ((~![X: $i, Y: $i, Z: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[136, 135])).
% 0.20/0.43 tff(138,plain,
% 0.20/0.43 ($false),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[137, 37, 134, 133, 11])).
% 0.20/0.43 % SZS output end Proof
% 0.20/0.43 % E exiting
%------------------------------------------------------------------------------