%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO201+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n016.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 05:05:54 AM UTC 2025
% Result : Theorem 0.12s 0.38s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : GEO201+1 : TPTP v9.0.0. Released v3.3.0.
% 0.12/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n016.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 14:41:32 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.38 % SZS status Theorem
% 0.12/0.38 % SZS output start Proof
% 0.12/0.38 tff(distinct_lines_type, type, (
% 0.12/0.38 distinct_lines: ( $i * $i ) > $o)).
% 0.12/0.38 tff(tptp_fun_X_1_type, type, (
% 0.12/0.38 tptp_fun_X_1: $i)).
% 0.12/0.38 tff(tptp_fun_Y_0_type, type, (
% 0.12/0.38 tptp_fun_Y_0: $i)).
% 0.12/0.38 tff(convergent_lines_type, type, (
% 0.12/0.38 convergent_lines: ( $i * $i ) > $o)).
% 0.12/0.38 tff(distinct_points_type, type, (
% 0.12/0.38 distinct_points: ( $i * $i ) > $o)).
% 0.12/0.38 tff(intersection_point_type, type, (
% 0.12/0.38 intersection_point: ( $i * $i ) > $i)).
% 0.12/0.38 tff(line_connecting_type, type, (
% 0.12/0.38 line_connecting: ( $i * $i ) > $i)).
% 0.12/0.38 tff(apart_point_and_line_type, type, (
% 0.12/0.38 apart_point_and_line: ( $i * $i ) > $o)).
% 0.12/0.38 tff(1,assumption,(convergent_lines(X!1, X!1)), introduced(assumption)).
% 0.12/0.38 tff(2,plain,
% 0.12/0.38 (^[X: $i] : refl((~convergent_lines(X, X)) <=> (~convergent_lines(X, X)))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(3,plain,
% 0.12/0.38 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[2])).
% 0.12/0.38 tff(4,plain,
% 0.12/0.38 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(5,axiom,(![X: $i] : (~convergent_lines(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','apart3')).
% 0.12/0.38 tff(6,plain,
% 0.12/0.38 (![X: $i] : (~convergent_lines(X, X))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.12/0.38 tff(7,plain,(
% 0.12/0.38 ![X: $i] : (~convergent_lines(X, X))),
% 0.12/0.38 inference(skolemize,[status(sab)],[6])).
% 0.12/0.38 tff(8,plain,
% 0.12/0.38 (![X: $i] : (~convergent_lines(X, X))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.12/0.38 tff(9,plain,
% 0.12/0.38 ((~![X: $i] : (~convergent_lines(X, X))) | (~convergent_lines(X!1, X!1))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(10,plain,
% 0.12/0.38 ($false),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 0.12/0.38 tff(11,plain,(~convergent_lines(X!1, X!1)), inference(lemma,lemma(discharge,[]))).
% 0.12/0.38 tff(12,plain,
% 0.12/0.38 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))) <=> (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(13,plain,
% 0.12/0.38 ((~![X: $i, Y: $i] : (convergent_lines(X, Y) => (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))) <=> (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(14,axiom,(~![X: $i, Y: $i] : (convergent_lines(X, Y) => (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.12/0.38 tff(15,plain,
% 0.12/0.38 (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.12/0.38 tff(16,plain,
% 0.12/0.38 (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[15, 12])).
% 0.12/0.38 tff(17,plain,
% 0.12/0.38 (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.12/0.38 tff(18,plain,
% 0.12/0.38 (~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~distinct_points(intersection_point(X, Y), intersection_point(Y, X))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[17, 12])).
% 0.12/0.38 tff(19,plain,(
% 0.12/0.38 ~((~convergent_lines(X!1, Y!0)) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))),
% 0.12/0.38 inference(skolemize,[status(sab)],[18])).
% 0.12/0.38 tff(20,plain,
% 0.12/0.38 (convergent_lines(X!1, Y!0)),
% 0.12/0.38 inference(or_elim,[status(thm)],[19])).
% 0.12/0.38 tff(21,plain,
% 0.12/0.38 (^[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))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(22,plain,
% 0.12/0.38 (![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)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[21])).
% 0.12/0.39 tff(23,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(24,plain,
% 0.12/0.39 (^[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)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(25,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[24])).
% 0.12/0.39 tff(26,axiom,(![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ceq3')).
% 0.12/0.39 tff(27,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.12/0.39 tff(28,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[27, 23])).
% 0.12/0.39 tff(29,plain,(
% 0.12/0.39 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.12/0.39 inference(skolemize,[status(sab)],[28])).
% 0.12/0.39 tff(30,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[29, 22])).
% 0.12/0.39 tff(31,plain,
% 0.12/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!1, Y!0)) | distinct_lines(Y!0, X!1) | convergent_lines(X!1, X!1))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!1, Y!0)) | distinct_lines(Y!0, X!1) | convergent_lines(X!1, X!1))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(32,plain,
% 0.12/0.39 ((distinct_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0))) <=> ((~convergent_lines(X!1, Y!0)) | distinct_lines(Y!0, X!1) | convergent_lines(X!1, X!1))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(33,plain,
% 0.12/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!1, Y!0)) | distinct_lines(Y!0, X!1) | convergent_lines(X!1, X!1)))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[32])).
% 0.12/0.39 tff(34,plain,
% 0.12/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!1, Y!0)) | distinct_lines(Y!0, X!1) | convergent_lines(X!1, X!1))),
% 0.12/0.39 inference(transitivity,[status(thm)],[33, 31])).
% 0.12/0.39 tff(35,plain,
% 0.12/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(36,plain,
% 0.12/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!1, Y!0)) | distinct_lines(Y!0, X!1) | convergent_lines(X!1, X!1)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.12/0.39 tff(37,plain,
% 0.12/0.39 (distinct_lines(Y!0, X!1) | convergent_lines(X!1, X!1)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[36, 30, 20])).
% 0.12/0.39 tff(38,plain,
% 0.12/0.39 (distinct_lines(Y!0, X!1)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[37, 11])).
% 0.12/0.39 tff(39,plain,
% 0.12/0.39 (^[X: $i, Y: $i, Z: $i] : refl((convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(40,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[39])).
% 0.12/0.39 tff(41,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(42,plain,
% 0.12/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((convergent_lines(X, Z) | convergent_lines(Y, Z)) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z))), ((convergent_lines(X, Y) => (convergent_lines(X, Z) | convergent_lines(Y, Z))) <=> (convergent_lines(X, Y) => (convergent_lines(Y, Z) | convergent_lines(X, Z))))), rewrite((convergent_lines(X, Y) => (convergent_lines(Y, Z) | convergent_lines(X, Z))) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))), ((convergent_lines(X, Y) => (convergent_lines(X, Z) | convergent_lines(Y, Z))) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(43,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (convergent_lines(X, Z) | convergent_lines(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[42])).
% 0.12/0.39 tff(44,axiom,(![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (convergent_lines(X, Z) | convergent_lines(Y, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ax6')).
% 0.12/0.39 tff(45,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.12/0.39 tff(46,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.12/0.39 tff(47,plain,(
% 0.12/0.39 ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.12/0.39 inference(skolemize,[status(sab)],[46])).
% 0.12/0.39 tff(48,plain,
% 0.12/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[47, 40])).
% 0.12/0.39 tff(49,plain,
% 0.12/0.39 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!1, Y!0)) | convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!1, Y!0)) | convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(50,plain,
% 0.12/0.39 ((convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0))) <=> ((~convergent_lines(X!1, Y!0)) | convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(51,plain,
% 0.12/0.39 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!1, Y!0)) | convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1)))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[50])).
% 0.12/0.39 tff(52,plain,
% 0.12/0.39 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!1, Y!0)) | convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1))),
% 0.12/0.39 inference(transitivity,[status(thm)],[51, 49])).
% 0.12/0.39 tff(53,plain,
% 0.12/0.39 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1) | (~convergent_lines(X!1, Y!0)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(54,plain,
% 0.12/0.39 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!1, Y!0)) | convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1)),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.12/0.39 tff(55,plain,
% 0.12/0.39 (convergent_lines(Y!0, X!1) | convergent_lines(X!1, X!1)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[54, 48, 20])).
% 0.12/0.39 tff(56,plain,
% 0.12/0.39 (convergent_lines(Y!0, X!1)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[55, 11])).
% 0.12/0.39 tff(57,plain,
% 0.12/0.39 (^[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))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(58,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[57])).
% 0.12/0.39 tff(59,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(60,plain,
% 0.12/0.39 (^[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))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(61,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[60])).
% 0.12/0.39 tff(62,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), Y)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci4')).
% 0.12/0.39 tff(63,plain,
% 0.12/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.12/0.39 tff(64,plain,
% 0.12/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[63, 59])).
% 0.12/0.39 tff(65,plain,(
% 0.12/0.39 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.12/0.39 inference(skolemize,[status(sab)],[64])).
% 0.12/0.39 tff(66,plain,
% 0.12/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[65, 58])).
% 0.12/0.39 tff(67,plain,
% 0.12/0.39 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Y!0, X!1)) | (~apart_point_and_line(intersection_point(Y!0, X!1), X!1)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Y!0, X!1)) | (~apart_point_and_line(intersection_point(Y!0, X!1), X!1)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(68,plain,
% 0.12/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Y!0, X!1)) | (~apart_point_and_line(intersection_point(Y!0, X!1), X!1)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(69,plain,
% 0.12/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Y!0, X!1)) | (~apart_point_and_line(intersection_point(Y!0, X!1), X!1))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.12/0.39 tff(70,plain,
% 0.12/0.39 ((~convergent_lines(Y!0, X!1)) | (~apart_point_and_line(intersection_point(Y!0, X!1), X!1))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[69, 66])).
% 0.12/0.39 tff(71,plain,
% 0.12/0.39 (~apart_point_and_line(intersection_point(Y!0, X!1), X!1)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[70, 56])).
% 0.12/0.39 tff(72,plain,
% 0.12/0.39 (^[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))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(73,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[72])).
% 0.12/0.39 tff(74,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(75,plain,
% 0.12/0.39 (^[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))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(76,plain,
% 0.12/0.39 (![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)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[75])).
% 0.12/0.39 tff(77,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci3')).
% 0.12/0.39 tff(78,plain,
% 0.12/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.12/0.39 tff(79,plain,
% 0.12/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[78, 74])).
% 0.12/0.39 tff(80,plain,(
% 0.12/0.39 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.12/0.39 inference(skolemize,[status(sab)],[79])).
% 0.12/0.39 tff(81,plain,
% 0.12/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[80, 73])).
% 0.12/0.39 tff(82,plain,
% 0.12/0.39 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!1, Y!0)) | (~apart_point_and_line(intersection_point(X!1, Y!0), X!1)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!1, Y!0)) | (~apart_point_and_line(intersection_point(X!1, Y!0), X!1)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(83,plain,
% 0.12/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!1, Y!0)) | (~apart_point_and_line(intersection_point(X!1, Y!0), X!1)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(84,plain,
% 0.12/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!1, Y!0)) | (~apart_point_and_line(intersection_point(X!1, Y!0), X!1))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.12/0.39 tff(85,plain,
% 0.12/0.39 (~apart_point_and_line(intersection_point(X!1, Y!0), X!1)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[84, 81, 20])).
% 0.12/0.39 tff(86,plain,
% 0.12/0.39 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!1, Y!0)) | (~apart_point_and_line(intersection_point(X!1, Y!0), Y!0)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(X!1, Y!0)) | (~apart_point_and_line(intersection_point(X!1, Y!0), Y!0)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(87,plain,
% 0.12/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!1, Y!0)) | (~apart_point_and_line(intersection_point(X!1, Y!0), Y!0)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(88,plain,
% 0.12/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(X!1, Y!0)) | (~apart_point_and_line(intersection_point(X!1, Y!0), Y!0))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[87, 86])).
% 0.12/0.39 tff(89,plain,
% 0.12/0.39 (~apart_point_and_line(intersection_point(X!1, Y!0), Y!0)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[88, 66, 20])).
% 0.12/0.39 tff(90,plain,
% 0.12/0.39 (^[X: $i, Y: $i, U: $i, V: $i] : refl((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(91,plain,
% 0.12/0.39 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[90])).
% 0.12/0.39 tff(92,plain,
% 0.12/0.39 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(rewrite((distinct_points(X, Y) & distinct_lines(U, V)) <=> (~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> (~(~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))))), rewrite((~(~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))) <=> ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> ((~distinct_points(X, Y)) | (~distinct_lines(U, V))))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))))), rewrite((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(93,plain,
% 0.12/0.39 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[92])).
% 0.12/0.40 tff(94,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(95,plain,
% 0.12/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((apart_point_and_line(X, U) | apart_point_and_line(X, V)) <=> (apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> ((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)))), rewrite(((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> ((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)))), rewrite(((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> ((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))))), rewrite(((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(96,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : ((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[95])).
% 0.12/0.40 tff(97,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/sandbox/benchmark/Axioms/GEO006+0.ax','cu1')).
% 0.12/0.40 tff(98,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[97, 96])).
% 0.12/0.40 tff(99,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[98, 94])).
% 0.12/0.40 tff(100,plain,(
% 0.12/0.40 ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.12/0.40 inference(skolemize,[status(sab)],[99])).
% 0.12/0.40 tff(101,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[100, 93])).
% 0.12/0.40 tff(102,plain,
% 0.12/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[101, 91])).
% 0.12/0.40 tff(103,plain,
% 0.12/0.40 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, X!1)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, X!1)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(104,plain,
% 0.12/0.40 ((apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, X!1))) <=> (apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, X!1)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(105,plain,
% 0.12/0.40 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, X!1)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, X!1))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[104])).
% 0.12/0.40 tff(106,plain,
% 0.12/0.40 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, X!1)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, X!1)))),
% 0.12/0.40 inference(transitivity,[status(thm)],[105, 103])).
% 0.12/0.40 tff(107,plain,
% 0.12/0.40 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, X!1)))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(108,plain,
% 0.12/0.40 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, X!1))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.12/0.40 tff(109,plain,
% 0.12/0.40 ((~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, X!1))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[108, 102, 89, 85])).
% 0.12/0.40 tff(110,plain,
% 0.12/0.40 (~distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[109, 38])).
% 0.12/0.40 tff(111,plain,
% 0.12/0.40 (distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))),
% 0.12/0.40 inference(or_elim,[status(thm)],[19])).
% 0.12/0.40 tff(112,plain,
% 0.12/0.40 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(113,plain,
% 0.12/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[112])).
% 0.12/0.40 tff(114,plain,
% 0.12/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(115,plain,
% 0.12/0.40 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(X, Z) | distinct_points(Y, Z)) <=> (distinct_points(Y, Z) | distinct_points(X, Z))), ((distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> (distinct_points(X, Y) => (distinct_points(Y, Z) | distinct_points(X, Z))))), rewrite((distinct_points(X, Y) => (distinct_points(Y, Z) | distinct_points(X, Z))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))), ((distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(116,plain,
% 0.12/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[115])).
% 0.12/0.40 tff(117,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','apart4')).
% 0.12/0.40 tff(118,plain,
% 0.12/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[117, 116])).
% 0.12/0.40 tff(119,plain,
% 0.12/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[118, 114])).
% 0.12/0.40 tff(120,plain,(
% 0.12/0.40 ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.12/0.40 inference(skolemize,[status(sab)],[119])).
% 0.12/0.40 tff(121,plain,
% 0.12/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[120, 113])).
% 0.12/0.40 tff(122,plain,
% 0.12/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | ((~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(123,plain,
% 0.12/0.40 ((distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))) <=> ((~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(124,plain,
% 0.12/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | ((~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[123])).
% 0.12/0.40 tff(125,plain,
% 0.12/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)))),
% 0.12/0.40 inference(transitivity,[status(thm)],[124, 122])).
% 0.12/0.40 tff(126,plain,
% 0.12/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0)) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(127,plain,
% 0.12/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[126, 125])).
% 0.12/0.41 tff(128,plain,
% 0.12/0.41 (distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0)) | distinct_points(intersection_point(X!1, Y!0), intersection_point(X!1, Y!0))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[127, 121, 111])).
% 0.12/0.41 tff(129,plain,
% 0.12/0.41 (distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[128, 110])).
% 0.12/0.41 tff(130,plain,
% 0.12/0.41 (^[X: $i, Y: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))))),
% 0.12/0.41 inference(bind,[status(th)],[])).
% 0.12/0.41 tff(131,plain,
% 0.12/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.12/0.41 inference(quant_intro,[status(thm)],[130])).
% 0.12/0.41 tff(132,plain,
% 0.12/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(133,plain,
% 0.12/0.41 (^[X: $i, Y: $i] : rewrite((distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))))),
% 0.12/0.41 inference(bind,[status(th)],[])).
% 0.12/0.41 tff(134,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.12/0.41 inference(quant_intro,[status(thm)],[133])).
% 0.12/0.41 tff(135,axiom,(![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci1')).
% 0.12/0.41 tff(136,plain,
% 0.12/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[135, 134])).
% 0.12/0.41 tff(137,plain,
% 0.12/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[136, 132])).
% 0.12/0.41 tff(138,plain,(
% 0.12/0.41 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.12/0.41 inference(skolemize,[status(sab)],[137])).
% 0.12/0.41 tff(139,plain,
% 0.12/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[138, 131])).
% 0.12/0.41 tff(140,plain,
% 0.12/0.41 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(141,plain,
% 0.12/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(142,plain,
% 0.12/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[141, 140])).
% 0.12/0.41 tff(143,plain,
% 0.12/0.41 (~apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[142, 139, 111])).
% 0.12/0.41 tff(144,plain,
% 0.12/0.41 (^[X: $i, Y: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))))),
% 0.12/0.41 inference(bind,[status(th)],[])).
% 0.12/0.41 tff(145,plain,
% 0.12/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.12/0.41 inference(quant_intro,[status(thm)],[144])).
% 0.12/0.41 tff(146,plain,
% 0.12/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(147,plain,
% 0.12/0.41 (^[X: $i, Y: $i] : rewrite((distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))))),
% 0.12/0.41 inference(bind,[status(th)],[])).
% 0.12/0.41 tff(148,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.12/0.41 inference(quant_intro,[status(thm)],[147])).
% 0.12/0.41 tff(149,axiom,(![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci2')).
% 0.12/0.41 tff(150,plain,
% 0.12/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[149, 148])).
% 0.12/0.41 tff(151,plain,
% 0.12/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[150, 146])).
% 0.12/0.41 tff(152,plain,(
% 0.12/0.41 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.12/0.41 inference(skolemize,[status(sab)],[151])).
% 0.12/0.41 tff(153,plain,
% 0.12/0.41 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[152, 145])).
% 0.12/0.41 tff(154,plain,
% 0.12/0.41 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(155,plain,
% 0.12/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(156,plain,
% 0.12/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[155, 154])).
% 0.12/0.41 tff(157,plain,
% 0.12/0.41 (~apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[156, 153, 111])).
% 0.12/0.41 tff(158,plain,
% 0.12/0.41 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(159,plain,
% 0.12/0.41 ((apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))) <=> (apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(160,plain,
% 0.12/0.41 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[159])).
% 0.12/0.41 tff(161,plain,
% 0.12/0.41 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.12/0.41 inference(transitivity,[status(thm)],[160, 158])).
% 0.12/0.41 tff(162,plain,
% 0.12/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(163,plain,
% 0.12/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), X!1) | apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))),
% 0.12/0.42 inference(modus_ponens,[status(thm)],[162, 161])).
% 0.12/0.42 tff(164,plain,
% 0.12/0.42 (apart_point_and_line(intersection_point(Y!0, X!1), X!1) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))),
% 0.12/0.42 inference(unit_resolution,[status(thm)],[163, 102, 157, 143, 85])).
% 0.12/0.42 tff(165,plain,
% 0.12/0.42 (~distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))),
% 0.12/0.42 inference(unit_resolution,[status(thm)],[164, 129, 71])).
% 0.12/0.42 tff(166,plain,
% 0.12/0.42 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Y!0, X!1)) | (~apart_point_and_line(intersection_point(Y!0, X!1), Y!0)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Y!0, X!1)) | (~apart_point_and_line(intersection_point(Y!0, X!1), Y!0)))),
% 0.12/0.42 inference(rewrite,[status(thm)],[])).
% 0.12/0.42 tff(167,plain,
% 0.12/0.42 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Y!0, X!1)) | (~apart_point_and_line(intersection_point(Y!0, X!1), Y!0)))),
% 0.12/0.42 inference(quant_inst,[status(thm)],[])).
% 0.12/0.42 tff(168,plain,
% 0.12/0.42 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Y!0, X!1)) | (~apart_point_and_line(intersection_point(Y!0, X!1), Y!0))),
% 0.12/0.42 inference(modus_ponens,[status(thm)],[167, 166])).
% 0.12/0.42 tff(169,plain,
% 0.12/0.42 ((~convergent_lines(Y!0, X!1)) | (~apart_point_and_line(intersection_point(Y!0, X!1), Y!0))),
% 0.12/0.42 inference(unit_resolution,[status(thm)],[168, 81])).
% 0.12/0.42 tff(170,plain,
% 0.12/0.42 (~apart_point_and_line(intersection_point(Y!0, X!1), Y!0)),
% 0.12/0.42 inference(unit_resolution,[status(thm)],[169, 56])).
% 0.12/0.42 tff(171,plain,
% 0.12/0.42 (((~![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(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.12/0.42 inference(rewrite,[status(thm)],[])).
% 0.12/0.42 tff(172,plain,
% 0.12/0.42 ((apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))) <=> (apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(173,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(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[172])).
% 0.20/0.42 tff(174,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(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[173, 171])).
% 0.20/0.42 tff(175,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(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(176,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(intersection_point(Y!0, X!1), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | apart_point_and_line(intersection_point(X!1, Y!0), Y!0) | apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[175, 174])).
% 0.20/0.42 tff(177,plain,
% 0.20/0.42 (apart_point_and_line(intersection_point(Y!0, X!1), Y!0) | (~distinct_points(intersection_point(Y!0, X!1), intersection_point(X!1, Y!0))) | (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[176, 102, 157, 143, 89])).
% 0.20/0.42 tff(178,plain,
% 0.20/0.42 (~distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1)))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[177, 129, 170])).
% 0.20/0.42 tff(179,plain,
% 0.20/0.42 (^[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))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(180,plain,
% 0.20/0.42 (![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)))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[179])).
% 0.20/0.42 tff(181,plain,
% 0.20/0.42 (![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)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(182,plain,
% 0.20/0.42 (^[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)))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(183,plain,
% 0.20/0.42 (![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)))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[182])).
% 0.20/0.42 tff(184,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','apart5')).
% 0.20/0.42 tff(185,plain,
% 0.20/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[184, 183])).
% 0.20/0.42 tff(186,plain,
% 0.20/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[185, 181])).
% 0.20/0.42 tff(187,plain,(
% 0.20/0.42 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.20/0.42 inference(skolemize,[status(sab)],[186])).
% 0.20/0.42 tff(188,plain,
% 0.20/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[187, 180])).
% 0.20/0.42 tff(189,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~distinct_lines(Y!0, X!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~distinct_lines(Y!0, X!1)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(190,plain,
% 0.20/0.42 ((distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~distinct_lines(Y!0, X!1))) <=> (distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~distinct_lines(Y!0, X!1)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(191,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~distinct_lines(Y!0, X!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~distinct_lines(Y!0, X!1))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[190])).
% 0.20/0.42 tff(192,plain,
% 0.20/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~distinct_lines(Y!0, X!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~distinct_lines(Y!0, X!1)))),
% 0.20/0.42 inference(transitivity,[status(thm)],[191, 189])).
% 0.20/0.42 tff(193,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~distinct_lines(Y!0, X!1)))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(194,plain,
% 0.20/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(Y!0, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | distinct_lines(X!1, line_connecting(intersection_point(X!1, Y!0), intersection_point(Y!0, X!1))) | (~distinct_lines(Y!0, X!1))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[193, 192])).
% 0.20/0.42 tff(195,plain,
% 0.20/0.42 ($false),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[194, 188, 178, 165, 38])).
% 0.20/0.43 % SZS output end Proof
% 0.20/0.43 % E exiting
%------------------------------------------------------------------------------