%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO205+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.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 05:05:56 AM UTC 2025
% Result : Theorem 0.20s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : GEO205+1 : TPTP v9.0.0. Released v3.3.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n008.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 14:41:18 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.39 % SZS status Theorem
% 0.20/0.39 % SZS output start Proof
% 0.20/0.39 tff(convergent_lines_type, type, (
% 0.20/0.39 convergent_lines: ( $i * $i ) > $o)).
% 0.20/0.39 tff(tptp_fun_X_2_type, type, (
% 0.20/0.39 tptp_fun_X_2: $i)).
% 0.20/0.39 tff(distinct_lines_type, type, (
% 0.20/0.39 distinct_lines: ( $i * $i ) > $o)).
% 0.20/0.39 tff(tptp_fun_Y_1_type, type, (
% 0.20/0.39 tptp_fun_Y_1: $i)).
% 0.20/0.39 tff(apart_point_and_line_type, type, (
% 0.20/0.39 apart_point_and_line: ( $i * $i ) > $o)).
% 0.20/0.39 tff(intersection_point_type, type, (
% 0.20/0.39 intersection_point: ( $i * $i ) > $i)).
% 0.20/0.39 tff(tptp_fun_Z_0_type, type, (
% 0.20/0.39 tptp_fun_Z_0: $i)).
% 0.20/0.39 tff(distinct_points_type, type, (
% 0.20/0.39 distinct_points: ( $i * $i ) > $o)).
% 0.20/0.39 tff(1,assumption,(convergent_lines(X!2, X!2)), introduced(assumption)).
% 0.20/0.39 tff(2,plain,
% 0.20/0.39 (^[X: $i] : refl((~convergent_lines(X, X)) <=> (~convergent_lines(X, X)))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(3,plain,
% 0.20/0.39 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[2])).
% 0.20/0.39 tff(4,plain,
% 0.20/0.39 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(5,axiom,(![X: $i] : (~convergent_lines(X, X))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO006+0.ax','apart3')).
% 0.20/0.39 tff(6,plain,
% 0.20/0.39 (![X: $i] : (~convergent_lines(X, X))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.20/0.39 tff(7,plain,(
% 0.20/0.39 ![X: $i] : (~convergent_lines(X, X))),
% 0.20/0.39 inference(skolemize,[status(sab)],[6])).
% 0.20/0.39 tff(8,plain,
% 0.20/0.39 (![X: $i] : (~convergent_lines(X, X))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.20/0.39 tff(9,plain,
% 0.20/0.39 ((~![X: $i] : (~convergent_lines(X, X))) | (~convergent_lines(X!2, X!2))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(10,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 0.20/0.39 tff(11,plain,(~convergent_lines(X!2, X!2)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.39 tff(12,assumption,(~convergent_lines(X!2, Z!0)), introduced(assumption)).
% 0.20/0.39 tff(13,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z))))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(14,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : ((convergent_lines(X, Y) & (~distinct_lines(Y, Z))) => (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z))))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(15,axiom,(~![X: $i, Y: $i, Z: $i] : ((convergent_lines(X, Y) & (~distinct_lines(Y, Z))) => (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','con')).
% 0.20/0.39 tff(16,plain,
% 0.20/0.39 (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.20/0.39 tff(17,plain,
% 0.20/0.39 (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[16, 13])).
% 0.20/0.39 tff(18,plain,
% 0.20/0.39 (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.20/0.39 tff(19,plain,
% 0.20/0.39 (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[18, 13])).
% 0.20/0.39 tff(20,plain,(
% 0.20/0.39 ~((~(convergent_lines(X!2, Y!1) & (~distinct_lines(Y!1, Z!0)))) | (convergent_lines(X!2, Z!0) & (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))))),
% 0.20/0.39 inference(skolemize,[status(sab)],[19])).
% 0.20/0.39 tff(21,plain,
% 0.20/0.39 (convergent_lines(X!2, Y!1) & (~distinct_lines(Y!1, Z!0))),
% 0.20/0.39 inference(or_elim,[status(thm)],[20])).
% 0.20/0.39 tff(22,plain,
% 0.20/0.39 (~distinct_lines(Y!1, Z!0)),
% 0.20/0.39 inference(and_elim,[status(thm)],[21])).
% 0.20/0.39 tff(23,plain,
% 0.20/0.39 (convergent_lines(X!2, Y!1)),
% 0.20/0.39 inference(and_elim,[status(thm)],[21])).
% 0.20/0.39 tff(24,plain,
% 0.20/0.39 (^[X: $i, Y: $i, Z: $i] : refl((distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(25,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[24])).
% 0.20/0.39 tff(26,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(27,plain,
% 0.20/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_lines(Y, Z) | convergent_lines(X, Z)) <=> (distinct_lines(Y, Z) | convergent_lines(X, Z))), ((convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z))) <=> (convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z))))), rewrite((convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z))) <=> (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))), ((convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z))) <=> (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(28,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[27])).
% 0.20/0.39 tff(29,axiom,(![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z)))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO006+0.ax','ceq3')).
% 0.20/0.39 tff(30,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.20/0.39 tff(31,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[30, 26])).
% 0.20/0.39 tff(32,plain,(
% 0.20/0.39 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.39 inference(skolemize,[status(sab)],[31])).
% 0.20/0.39 tff(33,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[32, 25])).
% 0.20/0.39 tff(34,plain,
% 0.20/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!1, Z!0) | convergent_lines(X!2, Z!0) | (~convergent_lines(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | distinct_lines(Y!1, Z!0) | convergent_lines(X!2, Z!0) | (~convergent_lines(X!2, Y!1)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(35,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!1, Z!0) | convergent_lines(X!2, Z!0) | (~convergent_lines(X!2, Y!1)))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(36,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | distinct_lines(Y!1, Z!0) | convergent_lines(X!2, Z!0) | (~convergent_lines(X!2, Y!1))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.20/0.39 tff(37,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[36, 33, 23, 22, 12])).
% 0.20/0.39 tff(38,plain,(convergent_lines(X!2, Z!0)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.39 tff(39,plain,
% 0.20/0.39 (^[X: $i, Y: $i] : refl(((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(40,plain,
% 0.20/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[39])).
% 0.20/0.39 tff(41,plain,
% 0.20/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(42,plain,
% 0.20/0.39 (^[X: $i, Y: $i] : rewrite((convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(43,plain,
% 0.20/0.39 (![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[42])).
% 0.20/0.39 tff(44,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO006+0.ax','ci4')).
% 0.20/0.39 tff(45,plain,
% 0.20/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.20/0.39 tff(46,plain,
% 0.20/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.20/0.39 tff(47,plain,(
% 0.20/0.39 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.39 inference(skolemize,[status(sab)],[46])).
% 0.20/0.39 tff(48,plain,
% 0.20/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[47, 40])).
% 0.20/0.39 tff(49,plain,
% 0.20/0.39 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!2, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Z!0), Z!0)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(X!2, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Z!0), Z!0)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(50,plain,
% 0.20/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!2, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Z!0), Z!0)))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(51,plain,
% 0.20/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(X!2, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Z!0), Z!0))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.20/0.39 tff(52,plain,
% 0.20/0.39 (~apart_point_and_line(intersection_point(X!2, Z!0), Z!0)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[51, 48, 38])).
% 0.20/0.39 tff(53,plain,
% 0.20/0.39 (^[X: $i, Y: $i, Z: $i] : refl((distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(54,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[53])).
% 0.20/0.39 tff(55,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(56,plain,
% 0.20/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_lines(Y, Z) | apart_point_and_line(X, Z)) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z))), ((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))))), rewrite((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))), ((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(57,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[56])).
% 0.20/0.39 tff(58,axiom,(![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z)))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO006+0.ax','ceq2')).
% 0.20/0.39 tff(59,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.20/0.39 tff(60,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[59, 55])).
% 0.20/0.39 tff(61,plain,(
% 0.20/0.39 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.39 inference(skolemize,[status(sab)],[60])).
% 0.20/0.39 tff(62,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[61, 54])).
% 0.20/0.39 tff(63,plain,
% 0.20/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Y!1, Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | (~apart_point_and_line(intersection_point(X!2, Z!0), Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(Y!1, Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | (~apart_point_and_line(intersection_point(X!2, Z!0), Y!1)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(64,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Y!1, Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | (~apart_point_and_line(intersection_point(X!2, Z!0), Y!1)))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(65,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(Y!1, Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | (~apart_point_and_line(intersection_point(X!2, Z!0), Y!1))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.20/0.39 tff(66,plain,
% 0.20/0.39 (~apart_point_and_line(intersection_point(X!2, Z!0), Y!1)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[65, 62, 22, 52])).
% 0.20/0.39 tff(67,plain,
% 0.20/0.39 (^[X: $i, Y: $i] : refl(((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(68,plain,
% 0.20/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[67])).
% 0.20/0.39 tff(69,plain,
% 0.20/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(70,plain,
% 0.20/0.39 (^[X: $i, Y: $i] : rewrite((convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(71,plain,
% 0.20/0.39 (![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[70])).
% 0.20/0.39 tff(72,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X)))), file('/export/starexec/sandbox2/benchmark/Axioms/GEO006+0.ax','ci3')).
% 0.20/0.39 tff(73,plain,
% 0.20/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[72, 71])).
% 0.20/0.39 tff(74,plain,
% 0.20/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[73, 69])).
% 0.20/0.39 tff(75,plain,(
% 0.20/0.39 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.39 inference(skolemize,[status(sab)],[74])).
% 0.20/0.39 tff(76,plain,
% 0.20/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[75, 68])).
% 0.20/0.39 tff(77,plain,
% 0.20/0.39 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!2, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Z!0), X!2)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!2, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Z!0), X!2)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(78,plain,
% 0.20/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!2, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Z!0), X!2)))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(79,plain,
% 0.20/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!2, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Z!0), X!2))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[78, 77])).
% 0.20/0.39 tff(80,plain,
% 0.20/0.39 (~apart_point_and_line(intersection_point(X!2, Z!0), X!2)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[79, 76, 38])).
% 0.20/0.39 tff(81,plain,
% 0.20/0.39 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(82,plain,
% 0.20/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(83,plain,
% 0.20/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.20/0.39 tff(84,plain,
% 0.20/0.39 (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[83, 76, 23])).
% 0.20/0.39 tff(85,plain,
% 0.20/0.39 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(86,plain,
% 0.20/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(87,plain,
% 0.20/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[86, 85])).
% 0.20/0.40 tff(88,plain,
% 0.20/0.40 (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[87, 48, 23])).
% 0.20/0.40 tff(89,plain,
% 0.20/0.40 ((~(~((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))) <=> ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(90,plain,
% 0.20/0.40 ((convergent_lines(X!2, Z!0) & (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) <=> (~((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(91,plain,
% 0.20/0.40 ((~(convergent_lines(X!2, Z!0) & (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))) <=> (~(~((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[90])).
% 0.20/0.40 tff(92,plain,
% 0.20/0.40 ((~(convergent_lines(X!2, Z!0) & (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))) <=> ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))),
% 0.20/0.40 inference(transitivity,[status(thm)],[91, 89])).
% 0.20/0.40 tff(93,plain,
% 0.20/0.40 (~(convergent_lines(X!2, Z!0) & (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))),
% 0.20/0.40 inference(or_elim,[status(thm)],[20])).
% 0.20/0.40 tff(94,plain,
% 0.20/0.40 ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.20/0.40 tff(95,plain,
% 0.20/0.40 (distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))),
% 0.20/0.40 inference(unit_resolution,[status(thm)],[94, 38])).
% 0.20/0.40 tff(96,plain,
% 0.20/0.40 (^[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_lines(U, V)) | (~distinct_points(X, Y))) <=> (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_lines(U, V)) | (~distinct_points(X, Y))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(97,plain,
% 0.20/0.40 (![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_lines(U, V)) | (~distinct_points(X, Y))) <=> ![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_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[96])).
% 0.20/0.40 tff(98,plain,
% 0.20/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(rewrite((distinct_points(X, Y) & distinct_lines(U, V)) <=> (~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> (~(~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))))), rewrite((~(~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))) <=> ((~distinct_lines(U, V)) | (~distinct_points(X, Y)))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> ((~distinct_lines(U, V)) | (~distinct_points(X, Y))))), ((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_lines(U, V)) | (~distinct_points(X, Y)))))), 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_lines(U, V)) | (~distinct_points(X, Y)))) <=> (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_lines(U, V)) | (~distinct_points(X, Y)))), ((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_lines(U, V)) | (~distinct_points(X, Y)))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(99,plain,
% 0.20/0.40 (![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_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[98])).
% 0.20/0.40 tff(100,plain,
% 0.20/0.40 (![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))))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(101,plain,
% 0.20/0.40 (^[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))))))),
% 0.20/0.40 inference(bind,[status(th)],[])).
% 0.20/0.40 tff(102,plain,
% 0.20/0.40 (![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))))),
% 0.20/0.40 inference(quant_intro,[status(thm)],[101])).
% 0.20/0.40 tff(103,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/GEO006+0.ax','cu1')).
% 0.20/0.40 tff(104,plain,
% 0.20/0.40 (![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))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[103, 102])).
% 0.20/0.40 tff(105,plain,
% 0.20/0.40 (![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))))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[104, 100])).
% 0.20/0.40 tff(106,plain,(
% 0.20/0.40 ![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))))),
% 0.20/0.40 inference(skolemize,[status(sab)],[105])).
% 0.20/0.40 tff(107,plain,
% 0.20/0.40 (![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_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[106, 99])).
% 0.20/0.40 tff(108,plain,
% 0.20/0.40 (![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_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.20/0.40 inference(modus_ponens,[status(thm)],[107, 97])).
% 0.20/0.40 tff(109,plain,
% 0.20/0.40 (((~![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_lines(U, V)) | (~distinct_points(X, Y)))) | ((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), 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_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Y!1))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(110,plain,
% 0.20/0.40 ((apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) <=> ((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Y!1))),
% 0.20/0.40 inference(rewrite,[status(thm)],[])).
% 0.20/0.40 tff(111,plain,
% 0.20/0.40 (((~![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_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!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_lines(U, V)) | (~distinct_points(X, Y)))) | ((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Y!1)))),
% 0.20/0.40 inference(monotonicity,[status(thm)],[110])).
% 0.20/0.40 tff(112,plain,
% 0.20/0.40 (((~![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_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!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_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Y!1))),
% 0.20/0.40 inference(transitivity,[status(thm)],[111, 109])).
% 0.20/0.40 tff(113,plain,
% 0.20/0.40 ((~![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_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))),
% 0.20/0.40 inference(quant_inst,[status(thm)],[])).
% 0.20/0.40 tff(114,plain,
% 0.20/0.40 ((~![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_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Y!1)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.20/0.41 tff(115,plain,
% 0.20/0.41 (~distinct_lines(Y!1, X!2)),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[114, 108, 95, 88, 84, 80, 66])).
% 0.20/0.41 tff(116,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(117,plain,
% 0.20/0.41 ((distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1))) <=> ((~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(118,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2)))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[117])).
% 0.20/0.41 tff(119,plain,
% 0.20/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.20/0.41 inference(transitivity,[status(thm)],[118, 116])).
% 0.20/0.41 tff(120,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(121,plain,
% 0.20/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2)),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[120, 119])).
% 0.20/0.41 tff(122,plain,
% 0.20/0.41 ($false),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[121, 33, 23, 115, 11])).
% 0.20/0.41 % SZS output end Proof
% 0.20/0.41 % E exiting
%------------------------------------------------------------------------------