%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO192+3 : TPTP v9.0.0. Released v4.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n014.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 05:05:51 AM UTC 2025
% Result : Theorem 0.19s 0.39s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : GEO192+3 : TPTP v9.0.0. Released v4.0.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n014.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Thu Jun 19 14:40:54 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.19/0.39 % SZS status Theorem
% 0.19/0.39 % SZS output start Proof
% 0.19/0.39 tff(distinct_points_type, type, (
% 0.19/0.39 distinct_points: ( $i * $i ) > $o)).
% 0.19/0.39 tff(intersection_point_type, type, (
% 0.19/0.39 intersection_point: ( $i * $i ) > $i)).
% 0.19/0.39 tff(tptp_fun_X_2_type, type, (
% 0.19/0.39 tptp_fun_X_2: $i)).
% 0.19/0.39 tff(tptp_fun_Z_0_type, type, (
% 0.19/0.39 tptp_fun_Z_0: $i)).
% 0.19/0.39 tff(tptp_fun_Y_1_type, type, (
% 0.19/0.39 tptp_fun_Y_1: $i)).
% 0.19/0.39 tff(apart_point_and_line_type, type, (
% 0.19/0.39 apart_point_and_line: ( $i * $i ) > $o)).
% 0.19/0.39 tff(convergent_lines_type, type, (
% 0.19/0.39 convergent_lines: ( $i * $i ) > $o)).
% 0.19/0.39 tff(distinct_lines_type, type, (
% 0.19/0.39 distinct_lines: ( $i * $i ) > $o)).
% 0.19/0.39 tff(1,assumption,(~distinct_lines(Z!0, X!2)), introduced(assumption)).
% 0.19/0.39 tff(2,plain,
% 0.19/0.39 ((~((~convergent_lines(X!2, Y!1)) | (distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))) <=> (~((~convergent_lines(X!2, Y!1)) | (distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(3,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))) <=> (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(4,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (apart_point_and_line(intersection_point(X, Y), Z) => (distinct_lines(X, Z) & distinct_lines(Y, Z))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(5,axiom,(~![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (apart_point_and_line(intersection_point(X, Y), Z) => (distinct_lines(X, Z) & distinct_lines(Y, Z))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.19/0.39 tff(6,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.19/0.39 tff(7,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[6, 3])).
% 0.19/0.39 tff(8,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.19/0.39 tff(9,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[8, 3])).
% 0.19/0.39 tff(10,plain,(
% 0.19/0.39 ~((~convergent_lines(X!2, Y!1)) | (distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[9])).
% 0.19/0.39 tff(11,plain,
% 0.19/0.39 (~((~convergent_lines(X!2, Y!1)) | (distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[10, 2])).
% 0.19/0.39 tff(12,plain,
% 0.19/0.39 (convergent_lines(X!2, Y!1)),
% 0.19/0.39 inference(or_elim,[status(thm)],[11])).
% 0.19/0.39 tff(13,plain,
% 0.19/0.39 (^[X: $i, Y: $i] : refl(((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(14,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[13])).
% 0.19/0.39 tff(15,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(16,plain,
% 0.19/0.39 (^[X: $i, Y: $i] : rewrite((convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(17,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[16])).
% 0.19/0.39 tff(18,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')).
% 0.19/0.39 tff(19,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.19/0.39 tff(20,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.19/0.39 tff(21,plain,(
% 0.19/0.39 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[20])).
% 0.19/0.39 tff(22,plain,
% 0.19/0.39 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[21, 14])).
% 0.19/0.39 tff(23,plain,
% 0.19/0.39 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(24,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(25,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.19/0.39 tff(26,plain,
% 0.19/0.39 (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[25, 22, 12])).
% 0.19/0.39 tff(27,plain,
% 0.19/0.39 (apart_point_and_line(intersection_point(X!2, Y!1), Z!0)),
% 0.19/0.39 inference(or_elim,[status(thm)],[11])).
% 0.19/0.39 tff(28,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : refl((distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(29,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[28])).
% 0.19/0.39 tff(30,plain,
% 0.19/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(31,plain,
% 0.19/0.39 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_lines(Y, Z) | apart_point_and_line(X, Z)) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z))), ((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))))), rewrite((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))), ((apart_point_and_line(X, Y) => (distinct_lines(Y, Z) | apart_point_and_line(X, Z))) <=> (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(32,plain,
% 0.19/0.40 (![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)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[31])).
% 0.19/0.40 tff(33,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')).
% 0.19/0.40 tff(34,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[33, 32])).
% 0.19/0.40 tff(35,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[34, 30])).
% 0.19/0.40 tff(36,plain,(
% 0.19/0.40 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[35])).
% 0.19/0.40 tff(37,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[36, 29])).
% 0.19/0.40 tff(38,plain,
% 0.19/0.40 (((~![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), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2))) <=> ((~![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), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(39,plain,
% 0.19/0.40 ((distinct_lines(Z!0, X!2) | 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, Y!1), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(40,plain,
% 0.19/0.40 (((~![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, X!2) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~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), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2)))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[39])).
% 0.19/0.40 tff(41,plain,
% 0.19/0.40 (((~![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, X!2) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~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), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2))),
% 0.19/0.40 inference(transitivity,[status(thm)],[40, 38])).
% 0.19/0.40 tff(42,plain,
% 0.19/0.40 ((~![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, X!2) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(43,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[42, 41])).
% 0.19/0.40 tff(44,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[43, 37, 27, 26, 1])).
% 0.19/0.40 tff(45,plain,(distinct_lines(Z!0, X!2)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.40 tff(46,plain,
% 0.19/0.40 (^[X: $i, Y: $i] : refl(((~distinct_lines(X, Y)) | convergent_lines(X, Y)) <=> ((~distinct_lines(X, Y)) | convergent_lines(X, Y)))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(47,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[46])).
% 0.19/0.40 tff(48,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(49,plain,
% 0.19/0.40 (^[X: $i, Y: $i] : rewrite((distinct_lines(X, Y) => convergent_lines(X, Y)) <=> ((~distinct_lines(X, Y)) | convergent_lines(X, Y)))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(50,plain,
% 0.19/0.40 (![X: $i, Y: $i] : (distinct_lines(X, Y) => convergent_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[49])).
% 0.19/0.40 tff(51,axiom,(![X: $i, Y: $i] : (distinct_lines(X, Y) => convergent_lines(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+1.ax','p1')).
% 0.19/0.40 tff(52,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.19/0.40 tff(53,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[52, 48])).
% 0.19/0.40 tff(54,plain,(
% 0.19/0.40 ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.40 inference(skolemize,[status(sab)],[53])).
% 0.19/0.40 tff(55,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[54, 47])).
% 0.19/0.40 tff(56,plain,
% 0.19/0.40 (((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(Z!0, X!2)) | convergent_lines(Z!0, X!2))) <=> ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(Z!0, X!2)) | convergent_lines(Z!0, X!2))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(57,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(Z!0, X!2)) | convergent_lines(Z!0, X!2))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(58,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(Z!0, X!2)) | convergent_lines(Z!0, X!2)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.19/0.40 tff(59,plain,
% 0.19/0.40 (convergent_lines(Z!0, X!2)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[58, 55, 45])).
% 0.19/0.40 tff(60,plain,
% 0.19/0.40 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), Z!0)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), Z!0)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(61,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), Z!0)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(62,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), Z!0))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.19/0.40 tff(63,plain,
% 0.19/0.40 ((~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), Z!0))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[62, 22])).
% 0.19/0.40 tff(64,plain,
% 0.19/0.40 (~apart_point_and_line(intersection_point(Z!0, X!2), Z!0)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[63, 59])).
% 0.19/0.40 tff(65,assumption,(~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1))), introduced(assumption)).
% 0.19/0.40 tff(66,plain,
% 0.19/0.40 (^[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))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(67,plain,
% 0.19/0.40 (![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)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[66])).
% 0.19/0.40 tff(68,plain,
% 0.19/0.40 (![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)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(69,plain,
% 0.19/0.40 (^[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))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(70,plain,
% 0.19/0.40 (![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)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[69])).
% 0.19/0.40 tff(71,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')).
% 0.19/0.40 tff(72,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.19/0.40 tff(73,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[72, 68])).
% 0.19/0.40 tff(74,plain,(
% 0.19/0.40 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[73])).
% 0.19/0.40 tff(75,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[74, 67])).
% 0.19/0.40 tff(76,plain,
% 0.19/0.40 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))) <=> ((~![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)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(77,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(78,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.19/0.40 tff(79,plain,
% 0.19/0.40 (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[78, 75, 12])).
% 0.19/0.40 tff(80,plain,
% 0.19/0.40 (((~![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), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), 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(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | distinct_lines(Z!0, Y!1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(81,plain,
% 0.19/0.40 ((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), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | distinct_lines(Z!0, Y!1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(82,plain,
% 0.19/0.40 (((~![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), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | distinct_lines(Z!0, Y!1)))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[81])).
% 0.19/0.40 tff(83,plain,
% 0.19/0.40 (((~![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), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | distinct_lines(Z!0, Y!1))),
% 0.19/0.40 inference(transitivity,[status(thm)],[82, 80])).
% 0.19/0.40 tff(84,plain,
% 0.19/0.40 ((~![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)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(85,plain,
% 0.19/0.40 ((~![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), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | distinct_lines(Z!0, Y!1)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.19/0.40 tff(86,plain,
% 0.19/0.40 (distinct_lines(Z!0, Y!1)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[85, 37, 27, 79])).
% 0.19/0.40 tff(87,plain,
% 0.19/0.40 (((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(Z!0, Y!1)) | convergent_lines(Z!0, Y!1))) <=> ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(Z!0, Y!1)) | convergent_lines(Z!0, Y!1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(88,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(Z!0, Y!1)) | convergent_lines(Z!0, Y!1))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(89,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(Z!0, Y!1)) | convergent_lines(Z!0, Y!1)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.19/0.40 tff(90,plain,
% 0.19/0.40 (convergent_lines(Z!0, Y!1)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[89, 55, 86])).
% 0.19/0.40 tff(91,plain,
% 0.19/0.40 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Z!0, Y!1)) | (~apart_point_and_line(intersection_point(Z!0, Y!1), Z!0)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Z!0, Y!1)) | (~apart_point_and_line(intersection_point(Z!0, Y!1), Z!0)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(92,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Z!0, Y!1)) | (~apart_point_and_line(intersection_point(Z!0, Y!1), Z!0)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(93,plain,
% 0.19/0.40 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Z!0, Y!1)) | (~apart_point_and_line(intersection_point(Z!0, Y!1), Z!0))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[92, 91])).
% 0.19/0.40 tff(94,plain,
% 0.19/0.40 (~apart_point_and_line(intersection_point(Z!0, Y!1), Z!0)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[93, 22, 90])).
% 0.19/0.40 tff(95,plain,
% 0.19/0.40 (^[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))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(96,plain,
% 0.19/0.40 (![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)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[95])).
% 0.19/0.40 tff(97,plain,
% 0.19/0.40 (![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)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(98,plain,
% 0.19/0.40 (^[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)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(99,plain,
% 0.19/0.40 (![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)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[98])).
% 0.19/0.40 tff(100,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')).
% 0.19/0.40 tff(101,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.19/0.40 tff(102,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[101, 97])).
% 0.19/0.40 tff(103,plain,(
% 0.19/0.40 ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[102])).
% 0.19/0.40 tff(104,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[103, 96])).
% 0.19/0.40 tff(105,plain,
% 0.19/0.40 (((~![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), Z!0)) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1)))) <=> ((~![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), Z!0)) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(106,plain,
% 0.19/0.40 ((distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1)) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0))) <=> ((~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(107,plain,
% 0.19/0.41 (((~![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(X!2, Y!1), intersection_point(Z!0, Y!1)) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~apart_point_and_line(intersection_point(X!2, 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(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[106])).
% 0.19/0.41 tff(108,plain,
% 0.19/0.41 (((~![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(X!2, Y!1), intersection_point(Z!0, Y!1)) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~apart_point_and_line(intersection_point(X!2, 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(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1)))),
% 0.19/0.41 inference(transitivity,[status(thm)],[107, 105])).
% 0.19/0.41 tff(109,plain,
% 0.19/0.41 ((~![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(X!2, Y!1), intersection_point(Z!0, Y!1)) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(110,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[109, 108])).
% 0.19/0.41 tff(111,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[110, 104, 27, 94, 65])).
% 0.19/0.41 tff(112,plain,(distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(113,assumption,(~apart_point_and_line(intersection_point(Z!0, Y!1), X!2)), introduced(assumption)).
% 0.19/0.41 tff(114,plain,
% 0.19/0.41 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Z!0, Y!1)) | (~apart_point_and_line(intersection_point(Z!0, Y!1), Y!1)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Z!0, Y!1)) | (~apart_point_and_line(intersection_point(Z!0, Y!1), Y!1)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(115,plain,
% 0.19/0.41 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Z!0, Y!1)) | (~apart_point_and_line(intersection_point(Z!0, Y!1), Y!1)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(116,plain,
% 0.19/0.41 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Z!0, Y!1)) | (~apart_point_and_line(intersection_point(Z!0, Y!1), Y!1))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.19/0.41 tff(117,plain,
% 0.19/0.41 (~apart_point_and_line(intersection_point(Z!0, Y!1), Y!1)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[116, 75, 90])).
% 0.19/0.41 tff(118,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(119,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)],[118])).
% 0.19/0.41 tff(120,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(121,axiom,(![X: $i] : (~convergent_lines(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','apart3')).
% 0.19/0.41 tff(122,plain,
% 0.19/0.41 (![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[121, 120])).
% 0.19/0.41 tff(123,plain,(
% 0.19/0.41 ![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.41 inference(skolemize,[status(sab)],[122])).
% 0.19/0.41 tff(124,plain,
% 0.19/0.41 (![X: $i] : (~convergent_lines(X, X))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[123, 119])).
% 0.19/0.41 tff(125,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(126,plain,
% 0.19/0.41 (~convergent_lines(X!2, X!2)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[125, 124])).
% 0.19/0.41 tff(127,assumption,(~distinct_lines(Y!1, X!2)), introduced(assumption)).
% 0.19/0.41 tff(128,plain,
% 0.19/0.41 (^[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))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(129,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[128])).
% 0.19/0.41 tff(130,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(131,plain,
% 0.19/0.41 (^[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)))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(132,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[131])).
% 0.19/0.41 tff(133,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')).
% 0.19/0.41 tff(134,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[133, 132])).
% 0.19/0.41 tff(135,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[134, 130])).
% 0.19/0.41 tff(136,plain,(
% 0.19/0.41 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.41 inference(skolemize,[status(sab)],[135])).
% 0.19/0.41 tff(137,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[136, 129])).
% 0.19/0.41 tff(138,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(139,plain,
% 0.19/0.41 ((distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1))) <=> ((~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(140,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2)))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[139])).
% 0.19/0.41 tff(141,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.19/0.41 inference(transitivity,[status(thm)],[140, 138])).
% 0.19/0.41 tff(142,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(143,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2)),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[142, 141])).
% 0.19/0.41 tff(144,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[143, 137, 12, 127, 126])).
% 0.19/0.41 tff(145,plain,(distinct_lines(Y!1, X!2)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(146,plain,
% 0.19/0.41 (^[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))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(147,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))) <=> ![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(quant_intro,[status(thm)],[146])).
% 0.19/0.41 tff(148,plain,
% 0.19/0.41 (^[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)))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(149,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)))) <=> ![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(quant_intro,[status(thm)],[148])).
% 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)))) <=> ![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(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(151,plain,
% 0.19/0.41 (^[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(152,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)],[151])).
% 0.19/0.41 tff(153,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')).
% 0.19/0.41 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_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[153, 152])).
% 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_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[154, 150])).
% 0.19/0.41 tff(156,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)],[155])).
% 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)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[156, 149])).
% 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)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[157, 147])).
% 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, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(160,plain,
% 0.19/0.41 ((apart_point_and_line(intersection_point(Z!0, Y!1), X!2) | apart_point_and_line(intersection_point(Z!0, 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), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(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), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(161,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(Z!0, Y!1), X!2) | apart_point_and_line(intersection_point(Z!0, 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), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1)))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[160])).
% 0.19/0.42 tff(162,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Z!0, Y!1), X!2) | apart_point_and_line(intersection_point(Z!0, 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), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1))))),
% 0.19/0.42 inference(transitivity,[status(thm)],[161, 159])).
% 0.19/0.42 tff(163,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Z!0, Y!1), X!2) | apart_point_and_line(intersection_point(Z!0, 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), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(164,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), X!2) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, Y!1)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[163, 162])).
% 0.19/0.42 tff(165,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[164, 158, 79, 26, 145, 117, 113, 112])).
% 0.19/0.42 tff(166,plain,(apart_point_and_line(intersection_point(Z!0, Y!1), X!2)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(167,assumption,(~distinct_lines(X!2, Z!0)), introduced(assumption)).
% 0.19/0.42 tff(168,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(X!2, Z!0) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~apart_point_and_line(intersection_point(Z!0, Y!1), X!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(X!2, Z!0) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~apart_point_and_line(intersection_point(Z!0, Y!1), X!2)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(169,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(X!2, Z!0) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~apart_point_and_line(intersection_point(Z!0, Y!1), X!2)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(170,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(X!2, Z!0) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~apart_point_and_line(intersection_point(Z!0, Y!1), X!2))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[169, 168])).
% 0.19/0.42 tff(171,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[170, 37, 167, 94, 166])).
% 0.19/0.42 tff(172,plain,(distinct_lines(X!2, Z!0)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(173,plain,
% 0.19/0.42 ((~(~((~distinct_lines(X!2, Z!0)) | (~distinct_lines(Y!1, Z!0))))) <=> ((~distinct_lines(X!2, Z!0)) | (~distinct_lines(Y!1, Z!0)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(174,plain,
% 0.19/0.42 ((distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0)) <=> (~((~distinct_lines(X!2, Z!0)) | (~distinct_lines(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(X!2, Z!0) & distinct_lines(Y!1, Z!0))) <=> (~(~((~distinct_lines(X!2, Z!0)) | (~distinct_lines(Y!1, Z!0)))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[174])).
% 0.19/0.42 tff(176,plain,
% 0.19/0.42 ((~(distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0))) <=> ((~distinct_lines(X!2, Z!0)) | (~distinct_lines(Y!1, Z!0)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[175, 173])).
% 0.19/0.42 tff(177,plain,
% 0.19/0.42 (~(distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0))),
% 0.19/0.42 inference(or_elim,[status(thm)],[11])).
% 0.19/0.42 tff(178,plain,
% 0.19/0.42 ((~distinct_lines(X!2, Z!0)) | (~distinct_lines(Y!1, Z!0))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[177, 176])).
% 0.19/0.42 tff(179,plain,
% 0.19/0.42 (~distinct_lines(Y!1, Z!0)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[178, 172])).
% 0.19/0.42 tff(180,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(Y!1, Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(Y!1, Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(181,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(Y!1, Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(182,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(Y!1, Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[181, 180])).
% 0.19/0.42 tff(183,plain,
% 0.19/0.42 (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[182, 37, 179, 64])).
% 0.19/0.42 tff(184,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(Z!0, Y!1), intersection_point(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)))) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(185,plain,
% 0.19/0.42 ((apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~distinct_lines(Z!0, Y!1)) | (~distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2)))) <=> (apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(186,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~distinct_lines(Z!0, Y!1)) | (~distinct_points(intersection_point(Z!0, Y!1), intersection_point(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)))) | (apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2)))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[185])).
% 0.19/0.42 tff(187,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~distinct_lines(Z!0, Y!1)) | (~distinct_points(intersection_point(Z!0, Y!1), intersection_point(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)))) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2))))),
% 0.19/0.42 inference(transitivity,[status(thm)],[186, 184])).
% 0.19/0.42 tff(188,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | (~distinct_lines(Z!0, Y!1)) | (~distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2))))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(189,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_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~distinct_lines(Z!0, Y!1)) | apart_point_and_line(intersection_point(Z!0, Y!1), Y!1) | apart_point_and_line(intersection_point(Z!0, Y!1), Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[188, 187])).
% 0.19/0.43 tff(190,plain,
% 0.19/0.43 (~distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[189, 158, 64, 86, 117, 94, 183])).
% 0.19/0.43 tff(191,plain,
% 0.19/0.43 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), X!2)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), X!2)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(192,plain,
% 0.19/0.43 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), X!2)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(193,plain,
% 0.19/0.43 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), X!2))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[192, 191])).
% 0.19/0.43 tff(194,plain,
% 0.19/0.43 (~apart_point_and_line(intersection_point(Z!0, X!2), X!2)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[193, 75, 59])).
% 0.19/0.43 tff(195,plain,
% 0.19/0.43 (((~![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(Z!0, Y!1), X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | distinct_points(intersection_point(Z!0, Y!1), intersection_point(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(Z!0, Y!1), X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(196,plain,
% 0.19/0.43 ((distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | (~apart_point_and_line(intersection_point(Z!0, Y!1), X!2))) <=> ((~apart_point_and_line(intersection_point(Z!0, Y!1), X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(197,plain,
% 0.19/0.43 (((~![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(Z!0, Y!1), intersection_point(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | (~apart_point_and_line(intersection_point(Z!0, Y!1), 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(Z!0, Y!1), X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[196])).
% 0.19/0.43 tff(198,plain,
% 0.19/0.43 (((~![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(Z!0, Y!1), intersection_point(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | (~apart_point_and_line(intersection_point(Z!0, Y!1), 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(Z!0, Y!1), X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[197, 195])).
% 0.19/0.43 tff(199,plain,
% 0.19/0.43 ((~![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(Z!0, Y!1), intersection_point(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | (~apart_point_and_line(intersection_point(Z!0, Y!1), X!2)))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(200,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(Z!0, Y!1), X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | distinct_points(intersection_point(Z!0, Y!1), intersection_point(Z!0, X!2))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[199, 198])).
% 0.19/0.43 tff(201,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[200, 104, 166, 194, 190])).
% 0.19/0.43 % SZS output end Proof
% 0.19/0.43 % E exiting
%------------------------------------------------------------------------------