%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO205+2 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:05:56 AM UTC 2025
% Result : Theorem 0.19s 0.38s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GEO205+2 : TPTP v9.0.0. Released v3.3.0.
% 0.00/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n006.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 14:40:50 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.38 % SZS status Theorem
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 tff(distinct_lines_type, type, (
% 0.19/0.38 distinct_lines: ( $i * $i ) > $o)).
% 0.19/0.38 tff(tptp_fun_Y_1_type, type, (
% 0.19/0.38 tptp_fun_Y_1: $i)).
% 0.19/0.38 tff(tptp_fun_Z_0_type, type, (
% 0.19/0.38 tptp_fun_Z_0: $i)).
% 0.19/0.38 tff(convergent_lines_type, type, (
% 0.19/0.38 convergent_lines: ( $i * $i ) > $o)).
% 0.19/0.38 tff(tptp_fun_X_2_type, type, (
% 0.19/0.38 tptp_fun_X_2: $i)).
% 0.19/0.38 tff(distinct_points_type, type, (
% 0.19/0.38 distinct_points: ( $i * $i ) > $o)).
% 0.19/0.38 tff(intersection_point_type, type, (
% 0.19/0.38 intersection_point: ( $i * $i ) > $i)).
% 0.19/0.38 tff(apart_point_and_line_type, type, (
% 0.19/0.38 apart_point_and_line: ( $i * $i ) > $o)).
% 0.19/0.38 tff(1,plain,
% 0.19/0.38 (^[X: $i] : refl((~distinct_lines(X, X)) <=> (~distinct_lines(X, X)))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(2,plain,
% 0.19/0.38 (![X: $i] : (~distinct_lines(X, X)) <=> ![X: $i] : (~distinct_lines(X, X))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.38 tff(3,plain,
% 0.19/0.38 (![X: $i] : (~distinct_lines(X, X)) <=> ![X: $i] : (~distinct_lines(X, X))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(4,axiom,(![X: $i] : (~distinct_lines(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart2')).
% 0.19/0.38 tff(5,plain,
% 0.19/0.38 (![X: $i] : (~distinct_lines(X, X))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.38 tff(6,plain,(
% 0.19/0.38 ![X: $i] : (~distinct_lines(X, X))),
% 0.19/0.38 inference(skolemize,[status(sab)],[5])).
% 0.19/0.38 tff(7,plain,
% 0.19/0.38 (![X: $i] : (~distinct_lines(X, X))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.38 tff(8,plain,
% 0.19/0.38 ((~![X: $i] : (~distinct_lines(X, X))) | (~distinct_lines(Z!0, Z!0))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(9,plain,
% 0.19/0.38 (~distinct_lines(Z!0, Z!0)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.19/0.38 tff(10,plain,
% 0.19/0.38 ((~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z))))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(11,plain,
% 0.19/0.38 ((~![X: $i, Y: $i, Z: $i] : ((convergent_lines(X, Y) & (~distinct_lines(Y, Z))) => (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z))))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(12,axiom,(~![X: $i, Y: $i, Z: $i] : ((convergent_lines(X, Y) & (~distinct_lines(Y, Z))) => (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.19/0.38 tff(13,plain,
% 0.19/0.38 (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[12, 11])).
% 0.19/0.38 tff(14,plain,
% 0.19/0.38 (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[13, 10])).
% 0.19/0.38 tff(15,plain,
% 0.19/0.38 (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[14, 10])).
% 0.19/0.38 tff(16,plain,
% 0.19/0.38 (~![X: $i, Y: $i, Z: $i] : ((~(convergent_lines(X, Y) & (~distinct_lines(Y, Z)))) | (convergent_lines(X, Z) & (~distinct_points(intersection_point(X, Y), intersection_point(X, Z)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[15, 10])).
% 0.19/0.38 tff(17,plain,(
% 0.19/0.38 ~((~(convergent_lines(X!2, Y!1) & (~distinct_lines(Y!1, Z!0)))) | (convergent_lines(X!2, Z!0) & (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))))),
% 0.19/0.38 inference(skolemize,[status(sab)],[16])).
% 0.19/0.38 tff(18,plain,
% 0.19/0.38 (convergent_lines(X!2, Y!1) & (~distinct_lines(Y!1, Z!0))),
% 0.19/0.38 inference(or_elim,[status(thm)],[17])).
% 0.19/0.38 tff(19,plain,
% 0.19/0.38 (~distinct_lines(Y!1, Z!0)),
% 0.19/0.38 inference(and_elim,[status(thm)],[18])).
% 0.19/0.38 tff(20,plain,
% 0.19/0.38 (^[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))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(21,plain,
% 0.19/0.38 (![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)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[20])).
% 0.19/0.38 tff(22,plain,
% 0.19/0.38 (![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)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(23,plain,
% 0.19/0.38 (^[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)))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(24,plain,
% 0.19/0.38 (![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)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[23])).
% 0.19/0.38 tff(25,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/GEO008+0.ax','apart5')).
% 0.19/0.38 tff(26,plain,
% 0.19/0.38 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[25, 24])).
% 0.19/0.38 tff(27,plain,
% 0.19/0.38 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[26, 22])).
% 0.19/0.38 tff(28,plain,(
% 0.19/0.38 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[27])).
% 0.19/0.38 tff(29,plain,
% 0.19/0.38 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[28, 21])).
% 0.19/0.38 tff(30,plain,
% 0.19/0.38 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(Y!1, Z!0) | distinct_lines(Z!0, Z!0) | (~distinct_lines(Z!0, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(Y!1, Z!0) | distinct_lines(Z!0, Z!0) | (~distinct_lines(Z!0, Y!1)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(31,plain,
% 0.19/0.38 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(Y!1, Z!0) | distinct_lines(Z!0, Z!0) | (~distinct_lines(Z!0, Y!1)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(32,plain,
% 0.19/0.38 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(Y!1, Z!0) | distinct_lines(Z!0, Z!0) | (~distinct_lines(Z!0, Y!1))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.19/0.38 tff(33,plain,
% 0.19/0.38 (~distinct_lines(Z!0, Y!1)),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[32, 29, 19, 9])).
% 0.19/0.38 tff(34,plain,
% 0.19/0.38 (^[X: $i] : refl((~convergent_lines(X, X)) <=> (~convergent_lines(X, X)))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(35,plain,
% 0.19/0.38 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[34])).
% 0.19/0.38 tff(36,plain,
% 0.19/0.39 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(37,axiom,(![X: $i] : (~convergent_lines(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart3')).
% 0.19/0.39 tff(38,plain,
% 0.19/0.39 (![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.19/0.39 tff(39,plain,(
% 0.19/0.39 ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.39 inference(skolemize,[status(sab)],[38])).
% 0.19/0.39 tff(40,plain,
% 0.19/0.39 (![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[39, 35])).
% 0.19/0.39 tff(41,plain,
% 0.19/0.39 ((~![X: $i] : (~convergent_lines(X, X))) | (~convergent_lines(X!2, X!2))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(42,plain,
% 0.19/0.39 (~convergent_lines(X!2, X!2)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[41, 40])).
% 0.19/0.39 tff(43,plain,
% 0.19/0.39 (^[X: $i, Y: $i] : refl(((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ((~convergent_lines(X, Y)) | distinct_lines(X, Y)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(44,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[43])).
% 0.19/0.39 tff(45,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(46,plain,
% 0.19/0.39 (^[X: $i, Y: $i] : rewrite((convergent_lines(X, Y) => distinct_lines(X, Y)) <=> ((~convergent_lines(X, Y)) | distinct_lines(X, Y)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(47,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (convergent_lines(X, Y) => distinct_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[46])).
% 0.19/0.39 tff(48,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => distinct_lines(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','ceq3')).
% 0.19/0.39 tff(49,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[48, 47])).
% 0.19/0.39 tff(50,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[49, 45])).
% 0.19/0.39 tff(51,plain,(
% 0.19/0.39 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.39 inference(skolemize,[status(sab)],[50])).
% 0.19/0.39 tff(52,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[51, 44])).
% 0.19/0.39 tff(53,plain,
% 0.19/0.39 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | ((~convergent_lines(Y!1, Z!0)) | distinct_lines(Y!1, Z!0))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | (~convergent_lines(Y!1, Z!0)) | distinct_lines(Y!1, Z!0))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(54,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | ((~convergent_lines(Y!1, Z!0)) | distinct_lines(Y!1, Z!0))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(55,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | (~convergent_lines(Y!1, Z!0)) | distinct_lines(Y!1, Z!0)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.19/0.39 tff(56,plain,
% 0.19/0.39 (~convergent_lines(Y!1, Z!0)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[55, 52, 19])).
% 0.19/0.39 tff(57,assumption,(~convergent_lines(X!2, Z!0)), introduced(assumption)).
% 0.19/0.39 tff(58,plain,
% 0.19/0.39 (convergent_lines(X!2, Y!1)),
% 0.19/0.39 inference(and_elim,[status(thm)],[18])).
% 0.19/0.39 tff(59,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : refl((convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(60,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[59])).
% 0.19/0.39 tff(61,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(62,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((convergent_lines(X, Z) | convergent_lines(Y, Z)) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z))), ((convergent_lines(X, Y) => (convergent_lines(X, Z) | convergent_lines(Y, Z))) <=> (convergent_lines(X, Y) => (convergent_lines(Y, Z) | convergent_lines(X, Z))))), rewrite((convergent_lines(X, Y) => (convergent_lines(Y, Z) | convergent_lines(X, Z))) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))), ((convergent_lines(X, Y) => (convergent_lines(X, Z) | convergent_lines(Y, Z))) <=> (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(63,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (convergent_lines(X, Z) | convergent_lines(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[62])).
% 0.19/0.39 tff(64,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/GEO008+0.ax','apart6')).
% 0.19/0.39 tff(65,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.19/0.39 tff(66,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[65, 61])).
% 0.19/0.39 tff(67,plain,(
% 0.19/0.39 ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[66])).
% 0.19/0.39 tff(68,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[67, 60])).
% 0.19/0.39 tff(69,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(X!2, Z!0) | convergent_lines(Y!1, Z!0) | (~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, Z!0) | convergent_lines(Y!1, Z!0) | (~convergent_lines(X!2, Y!1)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(70,plain,
% 0.19/0.39 ((convergent_lines(Y!1, Z!0) | convergent_lines(X!2, Z!0) | (~convergent_lines(X!2, Y!1))) <=> (convergent_lines(X!2, Z!0) | convergent_lines(Y!1, Z!0) | (~convergent_lines(X!2, Y!1)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(71,plain,
% 0.19/0.39 (((~![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(X!2, Z!0) | (~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, Z!0) | convergent_lines(Y!1, Z!0) | (~convergent_lines(X!2, Y!1))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[70])).
% 0.19/0.39 tff(72,plain,
% 0.19/0.39 (((~![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(X!2, Z!0) | (~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, Z!0) | convergent_lines(Y!1, Z!0) | (~convergent_lines(X!2, Y!1)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[71, 69])).
% 0.19/0.39 tff(73,plain,
% 0.19/0.39 ((~![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(X!2, Z!0) | (~convergent_lines(X!2, Y!1)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(74,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | convergent_lines(X!2, Z!0) | convergent_lines(Y!1, Z!0) | (~convergent_lines(X!2, Y!1))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.19/0.39 tff(75,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[74, 68, 58, 57, 56])).
% 0.19/0.39 tff(76,plain,(convergent_lines(X!2, Z!0)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.39 tff(77,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!2, Z!0)) | convergent_lines(X!2, X!2) | convergent_lines(Z!0, X!2))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Z!0)) | convergent_lines(X!2, X!2) | convergent_lines(Z!0, X!2))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(78,plain,
% 0.19/0.39 ((convergent_lines(Z!0, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Z!0))) <=> ((~convergent_lines(X!2, Z!0)) | convergent_lines(X!2, X!2) | convergent_lines(Z!0, X!2))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(79,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Z!0, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!2, Z!0)) | convergent_lines(X!2, X!2) | convergent_lines(Z!0, X!2)))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[78])).
% 0.19/0.39 tff(80,plain,
% 0.19/0.39 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Z!0, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Z!0)) | convergent_lines(X!2, X!2) | convergent_lines(Z!0, X!2))),
% 0.19/0.39 inference(transitivity,[status(thm)],[79, 77])).
% 0.19/0.39 tff(81,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (convergent_lines(Z!0, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Z!0)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(82,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Z!0)) | convergent_lines(X!2, X!2) | convergent_lines(Z!0, X!2)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.19/0.39 tff(83,plain,
% 0.19/0.39 (convergent_lines(Z!0, X!2)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[82, 68, 76, 42])).
% 0.19/0.39 tff(84,plain,
% 0.19/0.39 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | ((~convergent_lines(Z!0, X!2)) | distinct_lines(Z!0, X!2))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | (~convergent_lines(Z!0, X!2)) | distinct_lines(Z!0, X!2))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(85,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | ((~convergent_lines(Z!0, X!2)) | distinct_lines(Z!0, X!2))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(86,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | distinct_lines(X, Y))) | (~convergent_lines(Z!0, X!2)) | distinct_lines(Z!0, X!2)),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.19/0.39 tff(87,plain,
% 0.19/0.39 (distinct_lines(Z!0, X!2)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[86, 52, 83])).
% 0.19/0.39 tff(88,plain,
% 0.19/0.39 (^[X: $i] : refl((~distinct_points(X, X)) <=> (~distinct_points(X, X)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(89,plain,
% 0.19/0.39 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[88])).
% 0.19/0.39 tff(90,plain,
% 0.19/0.39 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(91,axiom,(![X: $i] : (~distinct_points(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart1')).
% 0.19/0.39 tff(92,plain,
% 0.19/0.39 (![X: $i] : (~distinct_points(X, X))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[91, 90])).
% 0.19/0.39 tff(93,plain,(
% 0.19/0.39 ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.39 inference(skolemize,[status(sab)],[92])).
% 0.19/0.39 tff(94,plain,
% 0.19/0.39 (![X: $i] : (~distinct_points(X, X))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[93, 89])).
% 0.19/0.39 tff(95,plain,
% 0.19/0.39 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(96,plain,
% 0.19/0.39 (~distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[95, 94])).
% 0.19/0.39 tff(97,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))) <=> (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(98,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[97])).
% 0.19/0.39 tff(99,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(100,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(trans(monotonicity(rewrite((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) <=> (apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))), (((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y))) <=> ((apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)) => distinct_points(Z, intersection_point(X, Y))))), rewrite(((apart_point_and_line(Z, Y) | apart_point_and_line(Z, X)) => distinct_points(Z, intersection_point(X, Y))) <=> ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y)))), (((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y))) <=> ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y))))), ((convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y)))) <=> (convergent_lines(X, Y) => ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y)))))), rewrite((convergent_lines(X, Y) => ((~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))) | distinct_points(Z, intersection_point(X, Y)))) <=> (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))), ((convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y)))) <=> (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(101,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y)))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[100])).
% 0.19/0.39 tff(102,axiom,(![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => ((apart_point_and_line(Z, X) | apart_point_and_line(Z, Y)) => distinct_points(Z, intersection_point(X, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','con2')).
% 0.19/0.40 tff(103,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.19/0.40 tff(104,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[103, 99])).
% 0.19/0.40 tff(105,plain,(
% 0.19/0.40 ![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.40 inference(skolemize,[status(sab)],[104])).
% 0.19/0.40 tff(106,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[105, 98])).
% 0.19/0.40 tff(107,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(108,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2)))) <=> ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(109,plain,
% 0.19/0.40 ((~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))) <=> (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(110,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2)))) <=> (distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[109])).
% 0.19/0.40 tff(111,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2)))) <=> ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[110, 108])).
% 0.19/0.40 tff(112,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2)))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[111])).
% 0.19/0.40 tff(113,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[112, 107])).
% 0.19/0.40 tff(114,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~convergent_lines(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(115,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Z!0), intersection_point(X!2, Z!0)) | (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[114, 113])).
% 0.19/0.40 tff(116,plain,
% 0.19/0.40 (~(apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[115, 106, 76, 96])).
% 0.19/0.40 tff(117,plain,
% 0.19/0.40 ((apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2)) | (~apart_point_and_line(intersection_point(X!2, Z!0), X!2))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(118,plain,
% 0.19/0.40 (~apart_point_and_line(intersection_point(X!2, Z!0), X!2)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[117, 116])).
% 0.19/0.40 tff(119,plain,
% 0.19/0.40 ((apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2)) | (~apart_point_and_line(intersection_point(X!2, Z!0), Z!0))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(120,plain,
% 0.19/0.40 (~apart_point_and_line(intersection_point(X!2, Z!0), Z!0)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[119, 116])).
% 0.19/0.40 tff(121,plain,
% 0.19/0.40 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(122,plain,
% 0.19/0.40 (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[121, 94])).
% 0.19/0.40 tff(123,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | ((~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(124,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))) <=> ((~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(125,plain,
% 0.19/0.40 ((~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2))) <=> (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(126,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2)))) <=> (distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[125])).
% 0.19/0.40 tff(127,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2)))) <=> ((~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)))),
% 0.19/0.40 inference(transitivity,[status(thm)],[126, 124])).
% 0.19/0.40 tff(128,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | ((~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[127])).
% 0.19/0.40 tff(129,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)))),
% 0.19/0.40 inference(transitivity,[status(thm)],[128, 123])).
% 0.19/0.40 tff(130,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1)) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(131,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Z, intersection_point(X, Y)) | (~convergent_lines(X, Y)) | (~(apart_point_and_line(Z, Y) | apart_point_and_line(Z, X))))) | (~convergent_lines(X!2, Y!1)) | (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Y!1))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[130, 129])).
% 0.19/0.40 tff(132,plain,
% 0.19/0.40 (~(apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[131, 106, 58, 122])).
% 0.19/0.40 tff(133,plain,
% 0.19/0.40 ((apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(134,plain,
% 0.19/0.40 (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[133, 132])).
% 0.19/0.40 tff(135,plain,
% 0.19/0.40 ((~(~((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))) <=> ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(136,plain,
% 0.19/0.40 ((convergent_lines(X!2, Z!0) & (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) <=> (~((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(137,plain,
% 0.19/0.40 ((~(convergent_lines(X!2, Z!0) & (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))) <=> (~(~((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[136])).
% 0.19/0.40 tff(138,plain,
% 0.19/0.40 ((~(convergent_lines(X!2, Z!0) & (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))) <=> ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))),
% 0.19/0.40 inference(transitivity,[status(thm)],[137, 135])).
% 0.19/0.40 tff(139,plain,
% 0.19/0.40 (~(convergent_lines(X!2, Z!0) & (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))),
% 0.19/0.40 inference(or_elim,[status(thm)],[17])).
% 0.19/0.40 tff(140,plain,
% 0.19/0.40 ((~convergent_lines(X!2, Z!0)) | distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[139, 138])).
% 0.19/0.40 tff(141,plain,
% 0.19/0.40 (distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[140, 76])).
% 0.19/0.40 tff(142,plain,
% 0.19/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : refl((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(143,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[142])).
% 0.19/0.40 tff(144,plain,
% 0.19/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(rewrite((distinct_points(X, Y) & distinct_lines(U, V)) <=> (~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> (~(~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))))), rewrite((~(~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))) <=> ((~distinct_lines(U, V)) | (~distinct_points(X, Y)))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> ((~distinct_lines(U, V)) | (~distinct_points(X, Y))))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_lines(U, V)) | (~distinct_points(X, Y)))))), rewrite((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_lines(U, V)) | (~distinct_points(X, Y)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(145,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[144])).
% 0.19/0.40 tff(146,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(147,plain,
% 0.19/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((apart_point_and_line(X, U) | apart_point_and_line(X, V)) <=> (apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> ((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)))), rewrite(((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> ((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)))), rewrite(((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> ((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))))), rewrite(((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(148,plain,
% 0.19/0.41 (![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))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[147])).
% 0.19/0.41 tff(149,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/GEO008+0.ax','cu1')).
% 0.19/0.41 tff(150,plain,
% 0.19/0.41 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[149, 148])).
% 0.19/0.41 tff(151,plain,
% 0.19/0.41 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[150, 146])).
% 0.19/0.41 tff(152,plain,(
% 0.19/0.41 ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.41 inference(skolemize,[status(sab)],[151])).
% 0.19/0.41 tff(153,plain,
% 0.19/0.41 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[152, 145])).
% 0.19/0.41 tff(154,plain,
% 0.19/0.41 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[153, 143])).
% 0.19/0.41 tff(155,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | ((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | (~distinct_lines(Z!0, X!2)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | (~distinct_lines(Z!0, X!2)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(156,plain,
% 0.19/0.41 ((apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | (~distinct_lines(Z!0, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0)))) <=> ((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | (~distinct_lines(Z!0, X!2)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(157,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | (~distinct_lines(Z!0, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | ((~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | (~distinct_lines(Z!0, X!2))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[156])).
% 0.19/0.41 tff(158,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | (~distinct_lines(Z!0, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | (~distinct_lines(Z!0, X!2)))),
% 0.19/0.41 inference(transitivity,[status(thm)],[157, 155])).
% 0.19/0.41 tff(159,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(X!2, Z!0), X!2) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | (~distinct_lines(Z!0, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(160,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(X!2, Z!0))) | apart_point_and_line(intersection_point(X!2, Z!0), Z!0) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Z!0) | apart_point_and_line(intersection_point(X!2, Z!0), X!2) | (~distinct_lines(Z!0, X!2))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[159, 158])).
% 0.19/0.41 tff(161,plain,
% 0.19/0.41 (apart_point_and_line(intersection_point(X!2, Y!1), Z!0)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[160, 154, 141, 134, 120, 118, 87])).
% 0.19/0.41 tff(162,plain,
% 0.19/0.41 ((apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1))),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(163,plain,
% 0.19/0.41 (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[162, 132])).
% 0.19/0.41 tff(164,plain,
% 0.19/0.41 (^[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))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(165,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[164])).
% 0.19/0.41 tff(166,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(167,plain,
% 0.19/0.41 (^[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)))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(168,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[167])).
% 0.19/0.41 tff(169,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/GEO008+0.ax','ceq2')).
% 0.19/0.41 tff(170,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[169, 168])).
% 0.19/0.41 tff(171,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[170, 166])).
% 0.19/0.41 tff(172,plain,(
% 0.19/0.41 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.41 inference(skolemize,[status(sab)],[171])).
% 0.19/0.41 tff(173,plain,
% 0.19/0.41 (![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(modus_ponens,[status(thm)],[172, 165])).
% 0.19/0.42 tff(174,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)))) | (apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | distinct_lines(Z!0, Y!1) | (~apart_point_and_line(intersection_point(X!2, Y!1), 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(X!2, Y!1), Y!1) | distinct_lines(Z!0, Y!1) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(175,plain,
% 0.19/0.42 ((distinct_lines(Z!0, Y!1) | 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(X!2, Y!1), Y!1) | distinct_lines(Z!0, Y!1) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(176,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)))) | (distinct_lines(Z!0, Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~apart_point_and_line(intersection_point(X!2, Y!1), 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(X!2, Y!1), Y!1) | distinct_lines(Z!0, Y!1) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[175])).
% 0.19/0.42 tff(177,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)))) | (distinct_lines(Z!0, Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~apart_point_and_line(intersection_point(X!2, Y!1), 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(X!2, Y!1), Y!1) | distinct_lines(Z!0, Y!1) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[176, 174])).
% 0.19/0.42 tff(178,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)))) | (distinct_lines(Z!0, Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(179,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | distinct_lines(Z!0, Y!1) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[178, 177])).
% 0.19/0.42 tff(180,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[179, 173, 163, 161, 33])).
% 0.19/0.42 % SZS output end Proof
% 0.19/0.42 % E exiting
%------------------------------------------------------------------------------