%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO199+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:05:53 AM UTC 2025
% Result : Theorem 0.19s 0.41s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : GEO199+1 : TPTP v9.0.0. Released v3.3.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n007.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Thu Jun 19 14:42:19 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.19/0.41 % SZS status Theorem
% 0.19/0.41 % SZS output start Proof
% 0.19/0.41 tff(convergent_lines_type, type, (
% 0.19/0.41 convergent_lines: ( $i * $i ) > $o)).
% 0.19/0.41 tff(tptp_fun_X_2_type, type, (
% 0.19/0.41 tptp_fun_X_2: $i)).
% 0.19/0.41 tff(distinct_lines_type, type, (
% 0.19/0.41 distinct_lines: ( $i * $i ) > $o)).
% 0.19/0.41 tff(tptp_fun_Y_1_type, type, (
% 0.19/0.41 tptp_fun_Y_1: $i)).
% 0.19/0.41 tff(line_connecting_type, type, (
% 0.19/0.41 line_connecting: ( $i * $i ) > $i)).
% 0.19/0.41 tff(intersection_point_type, type, (
% 0.19/0.41 intersection_point: ( $i * $i ) > $i)).
% 0.19/0.41 tff(tptp_fun_Z_0_type, type, (
% 0.19/0.41 tptp_fun_Z_0: $i)).
% 0.19/0.41 tff(apart_point_and_line_type, type, (
% 0.19/0.41 apart_point_and_line: ( $i * $i ) > $o)).
% 0.19/0.41 tff(distinct_points_type, type, (
% 0.19/0.41 distinct_points: ( $i * $i ) > $o)).
% 0.19/0.41 tff(1,plain,
% 0.19/0.41 (^[X: $i] : refl((~convergent_lines(X, X)) <=> (~convergent_lines(X, X)))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(2,plain,
% 0.19/0.41 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.41 tff(3,plain,
% 0.19/0.41 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(4,axiom,(![X: $i] : (~convergent_lines(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','apart3')).
% 0.19/0.41 tff(5,plain,
% 0.19/0.41 (![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.41 tff(6,plain,(
% 0.19/0.41 ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.41 inference(skolemize,[status(sab)],[5])).
% 0.19/0.41 tff(7,plain,
% 0.19/0.41 (![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.41 tff(8,plain,
% 0.19/0.41 ((~![X: $i] : (~convergent_lines(X, X))) | (~convergent_lines(X!2, X!2))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(9,plain,
% 0.19/0.41 (~convergent_lines(X!2, X!2)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.19/0.41 tff(10,assumption,(~convergent_lines(Y!1, X!2)), introduced(assumption)).
% 0.19/0.41 tff(11,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(intersection_point(Y, X), Z)) | (~(convergent_lines(X, Y) & convergent_lines(Z, Y) & convergent_lines(X, Z) & (~apart_point_and_line(intersection_point(X, Y), Z)))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(intersection_point(Y, X), Z)) | (~(convergent_lines(X, Y) & convergent_lines(Z, Y) & convergent_lines(X, Z) & (~apart_point_and_line(intersection_point(X, Y), Z))))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(12,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : ((((convergent_lines(X, Y) & convergent_lines(Z, Y)) & convergent_lines(X, Z)) & (~apart_point_and_line(intersection_point(X, Y), Z))) => (~apart_point_and_line(intersection_point(Y, X), Z)))) <=> (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(intersection_point(Y, X), Z)) | (~(convergent_lines(X, Y) & convergent_lines(Z, Y) & convergent_lines(X, Z) & (~apart_point_and_line(intersection_point(X, Y), Z))))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(13,axiom,(~![X: $i, Y: $i, Z: $i] : ((((convergent_lines(X, Y) & convergent_lines(Z, Y)) & convergent_lines(X, Z)) & (~apart_point_and_line(intersection_point(X, Y), Z))) => (~apart_point_and_line(intersection_point(Y, X), Z)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.19/0.41 tff(14,plain,
% 0.19/0.41 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(intersection_point(Y, X), Z)) | (~(convergent_lines(X, Y) & convergent_lines(Z, Y) & convergent_lines(X, Z) & (~apart_point_and_line(intersection_point(X, Y), Z)))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.19/0.41 tff(15,plain,
% 0.19/0.41 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(intersection_point(Y, X), Z)) | (~(convergent_lines(X, Y) & convergent_lines(Z, Y) & convergent_lines(X, Z) & (~apart_point_and_line(intersection_point(X, Y), Z)))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[14, 11])).
% 0.19/0.41 tff(16,plain,
% 0.19/0.41 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(intersection_point(Y, X), Z)) | (~(convergent_lines(X, Y) & convergent_lines(Z, Y) & convergent_lines(X, Z) & (~apart_point_and_line(intersection_point(X, Y), Z)))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.19/0.41 tff(17,plain,
% 0.19/0.41 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(intersection_point(Y, X), Z)) | (~(convergent_lines(X, Y) & convergent_lines(Z, Y) & convergent_lines(X, Z) & (~apart_point_and_line(intersection_point(X, Y), Z)))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[16, 11])).
% 0.19/0.41 tff(18,plain,(
% 0.19/0.41 ~((~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | (~(convergent_lines(X!2, Y!1) & convergent_lines(Z!0, Y!1) & convergent_lines(X!2, Z!0) & (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))))),
% 0.19/0.41 inference(skolemize,[status(sab)],[17])).
% 0.19/0.41 tff(19,plain,
% 0.19/0.41 (convergent_lines(X!2, Y!1) & convergent_lines(Z!0, Y!1) & convergent_lines(X!2, Z!0) & (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0))),
% 0.19/0.41 inference(or_elim,[status(thm)],[18])).
% 0.19/0.41 tff(20,plain,
% 0.19/0.41 (convergent_lines(X!2, Y!1)),
% 0.19/0.41 inference(and_elim,[status(thm)],[19])).
% 0.19/0.41 tff(21,plain,
% 0.19/0.41 (^[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))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(22,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[21])).
% 0.19/0.41 tff(23,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(24,plain,
% 0.19/0.41 (^[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)))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(25,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[24])).
% 0.19/0.41 tff(26,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')).
% 0.19/0.41 tff(27,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[26, 25])).
% 0.19/0.41 tff(28,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[27, 23])).
% 0.19/0.41 tff(29,plain,(
% 0.19/0.41 ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.41 inference(skolemize,[status(sab)],[28])).
% 0.19/0.41 tff(30,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[29, 22])).
% 0.19/0.41 tff(31,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!2, Y!1)) | convergent_lines(Y!1, X!2) | convergent_lines(X!2, X!2))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | convergent_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(32,plain,
% 0.19/0.41 ((convergent_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1))) <=> ((~convergent_lines(X!2, Y!1)) | convergent_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(33,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!2, Y!1)) | convergent_lines(Y!1, X!2) | convergent_lines(X!2, X!2)))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[32])).
% 0.19/0.41 tff(34,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | convergent_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.19/0.41 inference(transitivity,[status(thm)],[33, 31])).
% 0.19/0.41 tff(35,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(36,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | convergent_lines(Y!1, X!2) | convergent_lines(X!2, X!2)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.19/0.41 tff(37,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[36, 30, 20, 10, 9])).
% 0.19/0.41 tff(38,plain,(convergent_lines(Y!1, X!2)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(39,plain,
% 0.19/0.41 (^[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))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(40,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[39])).
% 0.19/0.41 tff(41,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(42,plain,
% 0.19/0.41 (^[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))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(43,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[42])).
% 0.19/0.41 tff(44,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')).
% 0.19/0.41 tff(45,plain,
% 0.19/0.41 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.19/0.41 tff(46,plain,
% 0.19/0.41 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.19/0.41 tff(47,plain,(
% 0.19/0.41 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.41 inference(skolemize,[status(sab)],[46])).
% 0.19/0.41 tff(48,plain,
% 0.19/0.41 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[47, 40])).
% 0.19/0.41 tff(49,plain,
% 0.19/0.41 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Y!1, X!2)) | (~apart_point_and_line(intersection_point(Y!1, X!2), X!2)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Y!1, X!2)) | (~apart_point_and_line(intersection_point(Y!1, X!2), X!2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(50,plain,
% 0.19/0.42 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Y!1, X!2)) | (~apart_point_and_line(intersection_point(Y!1, X!2), X!2)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(51,plain,
% 0.19/0.42 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Y!1, X!2)) | (~apart_point_and_line(intersection_point(Y!1, X!2), X!2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.19/0.42 tff(52,plain,
% 0.19/0.42 ((~convergent_lines(Y!1, X!2)) | (~apart_point_and_line(intersection_point(Y!1, X!2), X!2))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[51, 48])).
% 0.19/0.42 tff(53,plain,
% 0.19/0.42 (~apart_point_and_line(intersection_point(Y!1, X!2), X!2)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[52, 38])).
% 0.19/0.42 tff(54,assumption,(~convergent_lines(Y!1, Z!0)), introduced(assumption)).
% 0.19/0.42 tff(55,plain,
% 0.19/0.42 ((~![X: $i] : (~convergent_lines(X, X))) | (~convergent_lines(Z!0, Z!0))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(56,plain,
% 0.19/0.42 (~convergent_lines(Z!0, Z!0)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[55, 7])).
% 0.19/0.42 tff(57,plain,
% 0.19/0.42 (convergent_lines(Z!0, Y!1)),
% 0.19/0.42 inference(and_elim,[status(thm)],[19])).
% 0.19/0.42 tff(58,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(Z!0, Y!1)) | convergent_lines(Z!0, Z!0) | convergent_lines(Y!1, Z!0))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(Z!0, Y!1)) | convergent_lines(Z!0, Z!0) | convergent_lines(Y!1, Z!0))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(59,plain,
% 0.19/0.42 ((convergent_lines(Y!1, Z!0) | convergent_lines(Z!0, Z!0) | (~convergent_lines(Z!0, Y!1))) <=> ((~convergent_lines(Z!0, Y!1)) | convergent_lines(Z!0, Z!0) | convergent_lines(Y!1, Z!0))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(60,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!1, Z!0) | convergent_lines(Z!0, Z!0) | (~convergent_lines(Z!0, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(Z!0, Y!1)) | convergent_lines(Z!0, Z!0) | convergent_lines(Y!1, Z!0)))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[59])).
% 0.19/0.42 tff(61,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!1, Z!0) | convergent_lines(Z!0, Z!0) | (~convergent_lines(Z!0, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(Z!0, Y!1)) | convergent_lines(Z!0, Z!0) | convergent_lines(Y!1, Z!0))),
% 0.19/0.42 inference(transitivity,[status(thm)],[60, 58])).
% 0.19/0.42 tff(62,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!1, Z!0) | convergent_lines(Z!0, Z!0) | (~convergent_lines(Z!0, Y!1)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(63,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(Z!0, Y!1)) | convergent_lines(Z!0, Z!0) | convergent_lines(Y!1, Z!0)),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.19/0.42 tff(64,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[63, 30, 57, 56, 54])).
% 0.19/0.42 tff(65,plain,(convergent_lines(Y!1, Z!0)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(66,plain,
% 0.19/0.42 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Z!0)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Z!0)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(67,plain,
% 0.19/0.42 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Z!0)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(68,plain,
% 0.19/0.42 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Z!0))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.19/0.42 tff(69,plain,
% 0.19/0.42 (~apart_point_and_line(intersection_point(Y!1, Z!0), Z!0)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[68, 48, 65])).
% 0.19/0.42 tff(70,plain,
% 0.19/0.42 (apart_point_and_line(intersection_point(Y!1, X!2), Z!0)),
% 0.19/0.42 inference(or_elim,[status(thm)],[18])).
% 0.19/0.42 tff(71,plain,
% 0.19/0.42 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(72,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[71])).
% 0.19/0.42 tff(73,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(74,plain,
% 0.19/0.42 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(X, Z) | apart_point_and_line(Z, Y)) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y))), ((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))))), rewrite((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))), ((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(75,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[74])).
% 0.19/0.42 tff(76,axiom,(![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ceq1')).
% 0.19/0.42 tff(77,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.19/0.42 tff(78,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[77, 73])).
% 0.19/0.42 tff(79,plain,(
% 0.19/0.42 ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[78])).
% 0.19/0.42 tff(80,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[79, 72])).
% 0.19/0.42 tff(81,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(82,plain,
% 0.19/0.42 ((distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0))) <=> ((~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(83,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[82])).
% 0.19/0.42 tff(84,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[83, 81])).
% 0.19/0.42 tff(85,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(86,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.19/0.42 tff(87,plain,
% 0.19/0.42 (distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[86, 80, 70, 69])).
% 0.19/0.42 tff(88,plain,
% 0.19/0.42 (^[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)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(89,plain,
% 0.19/0.42 (![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))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[88])).
% 0.19/0.42 tff(90,plain,
% 0.19/0.42 (![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))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(91,plain,
% 0.19/0.42 (^[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)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(92,plain,
% 0.19/0.42 (![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))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[91])).
% 0.19/0.42 tff(93,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')).
% 0.19/0.42 tff(94,plain,
% 0.19/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.19/0.42 tff(95,plain,
% 0.19/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[94, 90])).
% 0.19/0.42 tff(96,plain,(
% 0.19/0.42 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.42 inference(skolemize,[status(sab)],[95])).
% 0.19/0.42 tff(97,plain,
% 0.19/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[96, 89])).
% 0.19/0.42 tff(98,plain,
% 0.19/0.42 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(99,plain,
% 0.19/0.42 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(100,plain,
% 0.19/0.42 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.19/0.42 tff(101,plain,
% 0.19/0.42 (~apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[100, 97, 87])).
% 0.19/0.42 tff(102,plain,
% 0.19/0.42 (^[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)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(103,plain,
% 0.19/0.42 (![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))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[102])).
% 0.19/0.42 tff(104,plain,
% 0.19/0.42 (![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))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(105,plain,
% 0.19/0.42 (^[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)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(106,plain,
% 0.19/0.42 (![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))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[105])).
% 0.19/0.42 tff(107,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')).
% 0.19/0.42 tff(108,plain,
% 0.19/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.19/0.42 tff(109,plain,
% 0.19/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[108, 104])).
% 0.19/0.42 tff(110,plain,(
% 0.19/0.42 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.42 inference(skolemize,[status(sab)],[109])).
% 0.19/0.42 tff(111,plain,
% 0.19/0.42 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[110, 103])).
% 0.19/0.42 tff(112,plain,
% 0.19/0.42 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(113,plain,
% 0.19/0.42 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(114,plain,
% 0.19/0.42 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[113, 112])).
% 0.19/0.42 tff(115,plain,
% 0.19/0.42 (~apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[114, 111, 87])).
% 0.19/0.42 tff(116,assumption,(~distinct_lines(Z!0, Y!1)), introduced(assumption)).
% 0.19/0.42 tff(117,assumption,(~apart_point_and_line(intersection_point(Y!1, X!2), Y!1)), introduced(assumption)).
% 0.19/0.42 tff(118,plain,
% 0.19/0.42 (^[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))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(119,plain,
% 0.19/0.42 (![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)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[118])).
% 0.19/0.42 tff(120,plain,
% 0.19/0.42 (![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)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(121,plain,
% 0.19/0.42 (^[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)))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(122,plain,
% 0.19/0.43 (![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)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[121])).
% 0.19/0.43 tff(123,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/sandbox/benchmark/Axioms/GEO006+0.ax','ceq2')).
% 0.19/0.43 tff(124,plain,
% 0.19/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.19/0.43 tff(125,plain,
% 0.19/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[124, 120])).
% 0.19/0.43 tff(126,plain,(
% 0.19/0.43 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.43 inference(skolemize,[status(sab)],[125])).
% 0.19/0.43 tff(127,plain,
% 0.19/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[126, 119])).
% 0.19/0.43 tff(128,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | distinct_lines(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)))) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | distinct_lines(Z!0, Y!1))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(129,plain,
% 0.19/0.43 ((distinct_lines(Z!0, Y!1) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0))) <=> ((~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | distinct_lines(Z!0, Y!1))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(130,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Z!0, Y!1) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | distinct_lines(Z!0, Y!1)))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[129])).
% 0.19/0.43 tff(131,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Z!0, Y!1) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | distinct_lines(Z!0, Y!1))),
% 0.19/0.43 inference(transitivity,[status(thm)],[130, 128])).
% 0.19/0.43 tff(132,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Z!0, Y!1) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(133,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(Y!1, X!2), Z!0)) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | distinct_lines(Z!0, Y!1)),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[132, 131])).
% 0.19/0.43 tff(134,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[133, 127, 70, 117, 116])).
% 0.19/0.43 tff(135,plain,(apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | distinct_lines(Z!0, Y!1)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.43 tff(136,plain,
% 0.19/0.43 (apart_point_and_line(intersection_point(Y!1, X!2), Y!1)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[135, 116])).
% 0.19/0.43 tff(137,plain,
% 0.19/0.43 (^[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))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(138,plain,
% 0.19/0.43 (![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)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[137])).
% 0.19/0.43 tff(139,plain,
% 0.19/0.43 (![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)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(140,plain,
% 0.19/0.43 (^[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))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(141,plain,
% 0.19/0.43 (![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)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[140])).
% 0.19/0.43 tff(142,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')).
% 0.19/0.43 tff(143,plain,
% 0.19/0.43 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[142, 141])).
% 0.19/0.43 tff(144,plain,
% 0.19/0.43 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[143, 139])).
% 0.19/0.43 tff(145,plain,(
% 0.19/0.43 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.43 inference(skolemize,[status(sab)],[144])).
% 0.19/0.43 tff(146,plain,
% 0.19/0.43 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[145, 138])).
% 0.19/0.43 tff(147,plain,
% 0.19/0.43 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Y!1, X!2)) | (~apart_point_and_line(intersection_point(Y!1, X!2), Y!1)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Y!1, X!2)) | (~apart_point_and_line(intersection_point(Y!1, X!2), Y!1)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(148,plain,
% 0.19/0.43 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Y!1, X!2)) | (~apart_point_and_line(intersection_point(Y!1, X!2), Y!1)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(149,plain,
% 0.19/0.43 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Y!1, X!2)) | (~apart_point_and_line(intersection_point(Y!1, X!2), Y!1))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[148, 147])).
% 0.19/0.43 tff(150,plain,
% 0.19/0.43 ((~convergent_lines(Y!1, X!2)) | (~apart_point_and_line(intersection_point(Y!1, X!2), Y!1))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[149, 146])).
% 0.19/0.43 tff(151,plain,
% 0.19/0.43 (~convergent_lines(Y!1, X!2)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[150, 136])).
% 0.19/0.43 tff(152,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!1, X!2) | (~convergent_lines(Y!1, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | convergent_lines(Y!1, X!2) | (~convergent_lines(Y!1, Y!1)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(153,plain,
% 0.19/0.43 ((convergent_lines(Y!1, X!2) | convergent_lines(Y!1, X!2) | (~convergent_lines(Y!1, Y!1))) <=> (convergent_lines(Y!1, X!2) | (~convergent_lines(Y!1, Y!1)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(154,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!1, X!2) | convergent_lines(Y!1, X!2) | (~convergent_lines(Y!1, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!1, X!2) | (~convergent_lines(Y!1, Y!1))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[153])).
% 0.19/0.43 tff(155,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!1, X!2) | convergent_lines(Y!1, X!2) | (~convergent_lines(Y!1, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | convergent_lines(Y!1, X!2) | (~convergent_lines(Y!1, Y!1)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[154, 152])).
% 0.19/0.43 tff(156,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Y!1, X!2) | convergent_lines(Y!1, X!2) | (~convergent_lines(Y!1, Y!1)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(157,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | convergent_lines(Y!1, X!2) | (~convergent_lines(Y!1, Y!1))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[156, 155])).
% 0.19/0.43 tff(158,plain,
% 0.19/0.43 (~convergent_lines(Y!1, Y!1)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[157, 30, 151])).
% 0.19/0.43 tff(159,plain,
% 0.19/0.43 (^[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))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(160,plain,
% 0.19/0.43 (![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)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[159])).
% 0.19/0.43 tff(161,plain,
% 0.19/0.43 (![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)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(162,plain,
% 0.19/0.43 (^[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)))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(163,plain,
% 0.19/0.43 (![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)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[162])).
% 0.19/0.43 tff(164,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')).
% 0.19/0.43 tff(165,plain,
% 0.19/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[164, 163])).
% 0.19/0.43 tff(166,plain,
% 0.19/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[165, 161])).
% 0.19/0.43 tff(167,plain,(
% 0.19/0.43 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.43 inference(skolemize,[status(sab)],[166])).
% 0.19/0.43 tff(168,plain,
% 0.19/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[167, 160])).
% 0.19/0.43 tff(169,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Z!0, Y!1) | convergent_lines(Y!1, Y!1) | (~convergent_lines(Y!1, Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | distinct_lines(Z!0, Y!1) | convergent_lines(Y!1, Y!1) | (~convergent_lines(Y!1, Z!0)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(170,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Z!0, Y!1) | convergent_lines(Y!1, Y!1) | (~convergent_lines(Y!1, Z!0)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(171,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | distinct_lines(Z!0, Y!1) | convergent_lines(Y!1, Y!1) | (~convergent_lines(Y!1, Z!0))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[170, 169])).
% 0.19/0.43 tff(172,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[171, 168, 116, 65, 158])).
% 0.19/0.43 tff(173,plain,(distinct_lines(Z!0, Y!1)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.43 tff(174,plain,
% 0.19/0.43 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Y!1)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Y!1)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(175,plain,
% 0.19/0.43 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Y!1)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(176,plain,
% 0.19/0.43 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Y!1))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[175, 174])).
% 0.19/0.43 tff(177,plain,
% 0.19/0.43 (~apart_point_and_line(intersection_point(Y!1, Z!0), Y!1)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[176, 146, 65])).
% 0.19/0.43 tff(178,plain,
% 0.19/0.43 (((~![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)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(179,plain,
% 0.19/0.43 ((~![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)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(180,plain,
% 0.19/0.43 ((~![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))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[179, 178])).
% 0.19/0.43 tff(181,plain,
% 0.19/0.43 (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[180, 48, 20])).
% 0.19/0.43 tff(182,plain,
% 0.19/0.43 (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)),
% 0.19/0.43 inference(and_elim,[status(thm)],[19])).
% 0.19/0.43 tff(183,plain,
% 0.19/0.43 (^[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))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(184,plain,
% 0.19/0.43 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V))) <=> ![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)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[183])).
% 0.19/0.43 tff(185,plain,
% 0.19/0.43 (^[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)))))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(186,plain,
% 0.19/0.43 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![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)))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[185])).
% 0.19/0.43 tff(187,plain,
% 0.19/0.43 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![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))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(188,plain,
% 0.19/0.43 (^[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))))))),
% 0.19/0.44 inference(bind,[status(th)],[])).
% 0.19/0.44 tff(189,plain,
% 0.19/0.44 (![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))))),
% 0.19/0.44 inference(quant_intro,[status(thm)],[188])).
% 0.19/0.44 tff(190,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')).
% 0.19/0.44 tff(191,plain,
% 0.19/0.44 (![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))))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[190, 189])).
% 0.19/0.44 tff(192,plain,
% 0.19/0.44 (![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))))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[191, 187])).
% 0.19/0.44 tff(193,plain,(
% 0.19/0.44 ![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))))),
% 0.19/0.44 inference(skolemize,[status(sab)],[192])).
% 0.19/0.44 tff(194,plain,
% 0.19/0.44 (![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)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[193, 186])).
% 0.19/0.44 tff(195,plain,
% 0.19/0.44 (![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)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[194, 184])).
% 0.19/0.44 tff(196,plain,
% 0.19/0.44 (((~![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!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1))))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(197,plain,
% 0.19/0.44 ((apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1))) | (~distinct_lines(Z!0, Y!1))) <=> (apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1))))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(198,plain,
% 0.19/0.44 (((~![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!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1))) | (~distinct_lines(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_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)))))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[197])).
% 0.19/0.44 tff(199,plain,
% 0.19/0.44 (((~![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!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1))) | (~distinct_lines(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_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1))))),
% 0.19/0.44 inference(transitivity,[status(thm)],[198, 196])).
% 0.19/0.44 tff(200,plain,
% 0.19/0.44 ((~![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!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1))) | (~distinct_lines(Z!0, Y!1)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(201,plain,
% 0.19/0.44 ((~![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!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[200, 199])).
% 0.19/0.44 tff(202,plain,
% 0.19/0.44 ((~distinct_lines(Z!0, Y!1)) | (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[201, 195, 182, 181, 69, 177])).
% 0.19/0.44 tff(203,plain,
% 0.19/0.44 (~distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1))),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[202, 173])).
% 0.19/0.44 tff(204,plain,
% 0.19/0.44 (((~![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)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(205,plain,
% 0.19/0.44 ((~![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)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(206,plain,
% 0.19/0.44 ((~![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))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[205, 204])).
% 0.19/0.44 tff(207,plain,
% 0.19/0.44 (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[206, 146, 20])).
% 0.19/0.44 tff(208,plain,
% 0.19/0.44 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(209,plain,
% 0.19/0.44 ((distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2))) <=> (apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(210,plain,
% 0.19/0.44 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2))))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[209])).
% 0.19/0.44 tff(211,plain,
% 0.19/0.44 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2)))),
% 0.19/0.44 inference(transitivity,[status(thm)],[210, 208])).
% 0.19/0.44 tff(212,plain,
% 0.19/0.44 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(213,plain,
% 0.19/0.44 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_points(intersection_point(Y!1, Z!0), intersection_point(X!2, Y!1)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[212, 211])).
% 0.19/0.44 tff(214,plain,
% 0.19/0.44 (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[213, 80, 207, 203])).
% 0.19/0.44 tff(215,plain,
% 0.19/0.44 (((~![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!1, X!2), X!2) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, 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_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(Y!1, X!2), X!2) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(216,plain,
% 0.19/0.45 ((apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), X!2) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))) <=> (apart_point_and_line(intersection_point(Y!1, X!2), X!2) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(217,plain,
% 0.19/0.45 (((~![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!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), X!2) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, 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_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(Y!1, X!2), X!2) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[216])).
% 0.19/0.45 tff(218,plain,
% 0.19/0.45 (((~![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!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), X!2) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, 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_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(Y!1, X!2), X!2) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.45 inference(transitivity,[status(thm)],[217, 215])).
% 0.19/0.45 tff(219,plain,
% 0.19/0.45 ((~![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!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), X!2) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.45 inference(quant_inst,[status(thm)],[])).
% 0.19/0.45 tff(220,plain,
% 0.19/0.45 ((~![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!1, X!2), X!2) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))),
% 0.19/0.45 inference(modus_ponens,[status(thm)],[219, 218])).
% 0.19/0.45 tff(221,plain,
% 0.19/0.45 (apart_point_and_line(intersection_point(Y!1, X!2), X!2) | (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[220, 195, 214, 87, 115, 101])).
% 0.19/0.45 tff(222,plain,
% 0.19/0.45 (~distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[221, 53])).
% 0.19/0.45 tff(223,plain,
% 0.19/0.45 (~apart_point_and_line(intersection_point(Y!1, X!2), Y!1)),
% 0.19/0.45 inference(unit_resolution,[status(thm)],[150, 38])).
% 0.19/0.45 tff(224,plain,
% 0.19/0.45 (((~![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!1, X!2), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, 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_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(225,plain,
% 0.19/0.45 ((apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))) <=> (apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.45 inference(rewrite,[status(thm)],[])).
% 0.19/0.45 tff(226,plain,
% 0.19/0.45 (((~![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!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, 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_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))))),
% 0.19/0.45 inference(monotonicity,[status(thm)],[225])).
% 0.19/0.45 tff(227,plain,
% 0.19/0.45 (((~![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!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, 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_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.46 inference(transitivity,[status(thm)],[226, 224])).
% 0.19/0.46 tff(228,plain,
% 0.19/0.46 ((~![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!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(229,plain,
% 0.19/0.46 ((~![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!1, X!2), Y!1) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | (~distinct_points(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, Z!0), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | apart_point_and_line(intersection_point(Y!1, X!2), line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[228, 227])).
% 0.19/0.46 tff(230,plain,
% 0.19/0.46 (apart_point_and_line(intersection_point(Y!1, X!2), Y!1) | (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[229, 195, 177, 87, 115, 101])).
% 0.19/0.46 tff(231,plain,
% 0.19/0.46 (~distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[230, 223])).
% 0.19/0.46 tff(232,plain,
% 0.19/0.46 (^[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))))),
% 0.19/0.46 inference(bind,[status(th)],[])).
% 0.19/0.46 tff(233,plain,
% 0.19/0.46 (![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)))),
% 0.19/0.46 inference(quant_intro,[status(thm)],[232])).
% 0.19/0.46 tff(234,plain,
% 0.19/0.46 (![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)))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(235,plain,
% 0.19/0.46 (^[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)))))),
% 0.19/0.46 inference(bind,[status(th)],[])).
% 0.19/0.46 tff(236,plain,
% 0.19/0.46 (![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)))),
% 0.19/0.46 inference(quant_intro,[status(thm)],[235])).
% 0.19/0.46 tff(237,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')).
% 0.19/0.46 tff(238,plain,
% 0.19/0.46 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[237, 236])).
% 0.19/0.46 tff(239,plain,
% 0.19/0.46 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[238, 234])).
% 0.19/0.46 tff(240,plain,(
% 0.19/0.46 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.46 inference(skolemize,[status(sab)],[239])).
% 0.19/0.46 tff(241,plain,
% 0.19/0.46 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[240, 233])).
% 0.19/0.46 tff(242,plain,
% 0.19/0.46 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | ((~distinct_lines(Y!1, X!2)) | distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (~distinct_lines(Y!1, X!2)) | distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(243,plain,
% 0.19/0.46 ((distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, X!2))) <=> ((~distinct_lines(Y!1, X!2)) | distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(244,plain,
% 0.19/0.46 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, X!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | ((~distinct_lines(Y!1, X!2)) | distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[243])).
% 0.19/0.46 tff(245,plain,
% 0.19/0.46 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, X!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (~distinct_lines(Y!1, X!2)) | distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))))),
% 0.19/0.46 inference(transitivity,[status(thm)],[244, 242])).
% 0.19/0.46 tff(246,plain,
% 0.19/0.46 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | (~distinct_lines(Y!1, X!2)))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(247,plain,
% 0.19/0.46 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (~distinct_lines(Y!1, X!2)) | distinct_lines(Y!1, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0))) | distinct_lines(X!2, line_connecting(intersection_point(Y!1, X!2), intersection_point(Y!1, Z!0)))),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[246, 245])).
% 0.19/0.46 tff(248,plain,
% 0.19/0.46 (~distinct_lines(Y!1, X!2)),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[247, 241, 231, 222])).
% 0.19/0.46 tff(249,plain,
% 0.19/0.46 (((~![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))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(250,plain,
% 0.19/0.46 ((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))),
% 0.19/0.46 inference(rewrite,[status(thm)],[])).
% 0.19/0.46 tff(251,plain,
% 0.19/0.46 (((~![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)))),
% 0.19/0.46 inference(monotonicity,[status(thm)],[250])).
% 0.19/0.46 tff(252,plain,
% 0.19/0.46 (((~![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))),
% 0.19/0.46 inference(transitivity,[status(thm)],[251, 249])).
% 0.19/0.46 tff(253,plain,
% 0.19/0.46 ((~![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)))),
% 0.19/0.46 inference(quant_inst,[status(thm)],[])).
% 0.19/0.46 tff(254,plain,
% 0.19/0.46 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2)),
% 0.19/0.46 inference(modus_ponens,[status(thm)],[253, 252])).
% 0.19/0.46 tff(255,plain,
% 0.19/0.46 ($false),
% 0.19/0.46 inference(unit_resolution,[status(thm)],[254, 168, 20, 248, 9])).
% 0.19/0.46 % SZS output end Proof
% 0.19/0.46 % E exiting
%------------------------------------------------------------------------------