%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO185+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n009.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 05:05:48 AM UTC 2025
% Result : Theorem 0.12s 0.36s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.11 % Problem : GEO185+1 : TPTP v9.0.0. Released v3.3.0.
% 0.07/0.11 % Command : run_E %s %d THM
% 0.12/0.32 % Computer : n009.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Thu Jun 19 14:40:03 EDT 2025
% 0.12/0.32 % CPUTime :
% 0.12/0.36 % SZS status Theorem
% 0.12/0.36 % SZS output start Proof
% 0.12/0.36 tff(apart_point_and_line_type, type, (
% 0.12/0.36 apart_point_and_line: ( $i * $i ) > $o)).
% 0.12/0.36 tff(tptp_fun_V_0_type, type, (
% 0.12/0.36 tptp_fun_V_0: $i)).
% 0.12/0.36 tff(tptp_fun_X_3_type, type, (
% 0.12/0.36 tptp_fun_X_3: $i)).
% 0.12/0.36 tff(intersection_point_type, type, (
% 0.12/0.36 intersection_point: ( $i * $i ) > $i)).
% 0.12/0.36 tff(tptp_fun_U_1_type, type, (
% 0.12/0.36 tptp_fun_U_1: $i)).
% 0.12/0.36 tff(convergent_lines_type, type, (
% 0.12/0.36 convergent_lines: ( $i * $i ) > $o)).
% 0.12/0.36 tff(distinct_points_type, type, (
% 0.12/0.36 distinct_points: ( $i * $i ) > $o)).
% 0.12/0.36 tff(tptp_fun_Y_2_type, type, (
% 0.12/0.36 tptp_fun_Y_2: $i)).
% 0.12/0.36 tff(1,plain,
% 0.12/0.36 ((~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & convergent_lines(U, V) & (~distinct_points(X, intersection_point(U, V))))) | ((~apart_point_and_line(X, U)) & (~apart_point_and_line(X, V))))) <=> (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & convergent_lines(U, V) & (~distinct_points(X, intersection_point(U, V))))) | ((~apart_point_and_line(X, U)) & (~apart_point_and_line(X, V)))))),
% 0.12/0.36 inference(rewrite,[status(thm)],[])).
% 0.12/0.36 tff(2,plain,
% 0.12/0.36 ((~![X: $i, Y: $i, U: $i, V: $i] : (((distinct_points(X, Y) & convergent_lines(U, V)) & (~distinct_points(X, intersection_point(U, V)))) => ((~apart_point_and_line(X, U)) & (~apart_point_and_line(X, V))))) <=> (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & convergent_lines(U, V) & (~distinct_points(X, intersection_point(U, V))))) | ((~apart_point_and_line(X, U)) & (~apart_point_and_line(X, V)))))),
% 0.12/0.36 inference(rewrite,[status(thm)],[])).
% 0.12/0.36 tff(3,axiom,(~![X: $i, Y: $i, U: $i, V: $i] : (((distinct_points(X, Y) & convergent_lines(U, V)) & (~distinct_points(X, intersection_point(U, V)))) => ((~apart_point_and_line(X, U)) & (~apart_point_and_line(X, V))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.12/0.36 tff(4,plain,
% 0.12/0.36 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & convergent_lines(U, V) & (~distinct_points(X, intersection_point(U, V))))) | ((~apart_point_and_line(X, U)) & (~apart_point_and_line(X, V))))),
% 0.12/0.36 inference(modus_ponens,[status(thm)],[3, 2])).
% 0.12/0.36 tff(5,plain,
% 0.12/0.36 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & convergent_lines(U, V) & (~distinct_points(X, intersection_point(U, V))))) | ((~apart_point_and_line(X, U)) & (~apart_point_and_line(X, V))))),
% 0.12/0.36 inference(modus_ponens,[status(thm)],[4, 1])).
% 0.12/0.36 tff(6,plain,
% 0.12/0.36 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & convergent_lines(U, V) & (~distinct_points(X, intersection_point(U, V))))) | ((~apart_point_and_line(X, U)) & (~apart_point_and_line(X, V))))),
% 0.12/0.36 inference(modus_ponens,[status(thm)],[5, 1])).
% 0.12/0.36 tff(7,plain,
% 0.12/0.36 (~![X: $i, Y: $i, U: $i, V: $i] : ((~(distinct_points(X, Y) & convergent_lines(U, V) & (~distinct_points(X, intersection_point(U, V))))) | ((~apart_point_and_line(X, U)) & (~apart_point_and_line(X, V))))),
% 0.12/0.36 inference(modus_ponens,[status(thm)],[6, 1])).
% 0.12/0.36 tff(8,plain,(
% 0.12/0.36 ~((~(distinct_points(X!3, Y!2) & convergent_lines(U!1, V!0) & (~distinct_points(X!3, intersection_point(U!1, V!0))))) | ((~apart_point_and_line(X!3, U!1)) & (~apart_point_and_line(X!3, V!0))))),
% 0.12/0.36 inference(skolemize,[status(sab)],[7])).
% 0.12/0.36 tff(9,plain,
% 0.12/0.36 (distinct_points(X!3, Y!2) & convergent_lines(U!1, V!0) & (~distinct_points(X!3, intersection_point(U!1, V!0)))),
% 0.12/0.36 inference(or_elim,[status(thm)],[8])).
% 0.12/0.36 tff(10,plain,
% 0.12/0.36 (convergent_lines(U!1, V!0)),
% 0.12/0.36 inference(and_elim,[status(thm)],[9])).
% 0.12/0.36 tff(11,plain,
% 0.12/0.36 (^[X: $i, Y: $i] : refl(((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))))),
% 0.12/0.36 inference(bind,[status(th)],[])).
% 0.12/0.36 tff(12,plain,
% 0.12/0.36 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.12/0.36 inference(quant_intro,[status(thm)],[11])).
% 0.12/0.36 tff(13,plain,
% 0.12/0.36 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.12/0.36 inference(rewrite,[status(thm)],[])).
% 0.12/0.36 tff(14,plain,
% 0.12/0.36 (^[X: $i, Y: $i] : rewrite((convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))))),
% 0.12/0.36 inference(bind,[status(th)],[])).
% 0.12/0.36 tff(15,plain,
% 0.12/0.36 (![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.12/0.36 inference(quant_intro,[status(thm)],[14])).
% 0.12/0.36 tff(16,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), Y)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci4')).
% 0.12/0.36 tff(17,plain,
% 0.12/0.36 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.12/0.36 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.12/0.36 tff(18,plain,
% 0.12/0.36 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.12/0.36 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.12/0.36 tff(19,plain,(
% 0.12/0.36 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.12/0.36 inference(skolemize,[status(sab)],[18])).
% 0.12/0.36 tff(20,plain,
% 0.12/0.36 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.12/0.36 inference(modus_ponens,[status(thm)],[19, 12])).
% 0.12/0.36 tff(21,plain,
% 0.12/0.36 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), V!0)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), V!0)))),
% 0.12/0.36 inference(rewrite,[status(thm)],[])).
% 0.12/0.36 tff(22,plain,
% 0.12/0.36 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), V!0)))),
% 0.12/0.36 inference(quant_inst,[status(thm)],[])).
% 0.12/0.36 tff(23,plain,
% 0.12/0.36 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), V!0))),
% 0.12/0.36 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.12/0.36 tff(24,plain,
% 0.12/0.36 (~apart_point_and_line(intersection_point(U!1, V!0), V!0)),
% 0.12/0.36 inference(unit_resolution,[status(thm)],[23, 20, 10])).
% 0.12/0.36 tff(25,plain,
% 0.12/0.36 (~distinct_points(X!3, intersection_point(U!1, V!0))),
% 0.12/0.36 inference(and_elim,[status(thm)],[9])).
% 0.12/0.36 tff(26,plain,
% 0.12/0.36 (^[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))))),
% 0.12/0.36 inference(bind,[status(th)],[])).
% 0.12/0.36 tff(27,plain,
% 0.12/0.36 (![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)))),
% 0.12/0.36 inference(quant_intro,[status(thm)],[26])).
% 0.12/0.36 tff(28,plain,
% 0.12/0.36 (![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)))),
% 0.12/0.36 inference(rewrite,[status(thm)],[])).
% 0.12/0.36 tff(29,plain,
% 0.12/0.36 (^[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)))))),
% 0.12/0.36 inference(bind,[status(th)],[])).
% 0.12/0.36 tff(30,plain,
% 0.12/0.36 (![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)))),
% 0.12/0.36 inference(quant_intro,[status(thm)],[29])).
% 0.12/0.36 tff(31,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')).
% 0.12/0.36 tff(32,plain,
% 0.12/0.36 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.12/0.36 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.12/0.36 tff(33,plain,
% 0.12/0.36 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.12/0.36 inference(modus_ponens,[status(thm)],[32, 28])).
% 0.12/0.36 tff(34,plain,(
% 0.12/0.36 ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.12/0.36 inference(skolemize,[status(sab)],[33])).
% 0.12/0.36 tff(35,plain,
% 0.12/0.36 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.12/0.36 inference(modus_ponens,[status(thm)],[34, 27])).
% 0.12/0.36 tff(36,plain,
% 0.12/0.36 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0))),
% 0.12/0.36 inference(rewrite,[status(thm)],[])).
% 0.12/0.36 tff(37,plain,
% 0.12/0.36 ((distinct_points(X!3, intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~apart_point_and_line(X!3, V!0))) <=> (distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0))),
% 0.12/0.36 inference(rewrite,[status(thm)],[])).
% 0.12/0.36 tff(38,plain,
% 0.12/0.36 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(X!3, intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~apart_point_and_line(X!3, V!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0)))),
% 0.12/0.36 inference(monotonicity,[status(thm)],[37])).
% 0.12/0.36 tff(39,plain,
% 0.12/0.36 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(X!3, intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~apart_point_and_line(X!3, V!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0))),
% 0.12/0.36 inference(transitivity,[status(thm)],[38, 36])).
% 0.12/0.36 tff(40,plain,
% 0.12/0.36 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(X!3, intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~apart_point_and_line(X!3, V!0)))),
% 0.12/0.36 inference(quant_inst,[status(thm)],[])).
% 0.12/0.36 tff(41,plain,
% 0.12/0.36 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0)),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.12/0.37 tff(42,plain,
% 0.12/0.37 (~apart_point_and_line(X!3, V!0)),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[41, 35, 25, 24])).
% 0.12/0.37 tff(43,plain,
% 0.12/0.37 (^[X: $i, Y: $i] : refl(((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(44,plain,
% 0.12/0.37 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[43])).
% 0.12/0.37 tff(45,plain,
% 0.12/0.37 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(46,plain,
% 0.12/0.37 (^[X: $i, Y: $i] : rewrite((convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(47,plain,
% 0.12/0.37 (![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[46])).
% 0.12/0.37 tff(48,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci3')).
% 0.12/0.37 tff(49,plain,
% 0.12/0.37 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[48, 47])).
% 0.12/0.37 tff(50,plain,
% 0.12/0.37 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[49, 45])).
% 0.12/0.37 tff(51,plain,(
% 0.12/0.37 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.12/0.37 inference(skolemize,[status(sab)],[50])).
% 0.12/0.37 tff(52,plain,
% 0.12/0.37 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[51, 44])).
% 0.12/0.37 tff(53,plain,
% 0.12/0.37 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), U!1)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), U!1)))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(54,plain,
% 0.12/0.37 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), U!1)))),
% 0.12/0.37 inference(quant_inst,[status(thm)],[])).
% 0.12/0.37 tff(55,plain,
% 0.12/0.37 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), U!1))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.12/0.37 tff(56,plain,
% 0.12/0.37 (~apart_point_and_line(intersection_point(U!1, V!0), U!1)),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[55, 52, 10])).
% 0.12/0.37 tff(57,plain,
% 0.12/0.37 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, U!1)) | apart_point_and_line(intersection_point(U!1, V!0), U!1))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, U!1)) | apart_point_and_line(intersection_point(U!1, V!0), U!1))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(58,plain,
% 0.12/0.37 ((distinct_points(X!3, intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~apart_point_and_line(X!3, U!1))) <=> (distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, U!1)) | apart_point_and_line(intersection_point(U!1, V!0), U!1))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(59,plain,
% 0.12/0.37 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(X!3, intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~apart_point_and_line(X!3, U!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, U!1)) | apart_point_and_line(intersection_point(U!1, V!0), U!1)))),
% 0.12/0.37 inference(monotonicity,[status(thm)],[58])).
% 0.12/0.37 tff(60,plain,
% 0.12/0.37 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(X!3, intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~apart_point_and_line(X!3, U!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, U!1)) | apart_point_and_line(intersection_point(U!1, V!0), U!1))),
% 0.12/0.37 inference(transitivity,[status(thm)],[59, 57])).
% 0.12/0.37 tff(61,plain,
% 0.12/0.37 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(X!3, intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~apart_point_and_line(X!3, U!1)))),
% 0.12/0.37 inference(quant_inst,[status(thm)],[])).
% 0.12/0.37 tff(62,plain,
% 0.12/0.37 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | distinct_points(X!3, intersection_point(U!1, V!0)) | (~apart_point_and_line(X!3, U!1)) | apart_point_and_line(intersection_point(U!1, V!0), U!1)),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.12/0.37 tff(63,plain,
% 0.12/0.37 (~apart_point_and_line(X!3, U!1)),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[62, 35, 25, 56])).
% 0.12/0.37 tff(64,plain,
% 0.12/0.37 ((~(~(apart_point_and_line(X!3, U!1) | apart_point_and_line(X!3, V!0)))) <=> (apart_point_and_line(X!3, U!1) | apart_point_and_line(X!3, V!0))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(65,plain,
% 0.12/0.37 (((~apart_point_and_line(X!3, U!1)) & (~apart_point_and_line(X!3, V!0))) <=> (~(apart_point_and_line(X!3, U!1) | apart_point_and_line(X!3, V!0)))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(66,plain,
% 0.12/0.37 ((~((~apart_point_and_line(X!3, U!1)) & (~apart_point_and_line(X!3, V!0)))) <=> (~(~(apart_point_and_line(X!3, U!1) | apart_point_and_line(X!3, V!0))))),
% 0.12/0.37 inference(monotonicity,[status(thm)],[65])).
% 0.12/0.37 tff(67,plain,
% 0.12/0.37 ((~((~apart_point_and_line(X!3, U!1)) & (~apart_point_and_line(X!3, V!0)))) <=> (apart_point_and_line(X!3, U!1) | apart_point_and_line(X!3, V!0))),
% 0.12/0.37 inference(transitivity,[status(thm)],[66, 64])).
% 0.12/0.37 tff(68,plain,
% 0.12/0.37 (~((~apart_point_and_line(X!3, U!1)) & (~apart_point_and_line(X!3, V!0)))),
% 0.12/0.37 inference(or_elim,[status(thm)],[8])).
% 0.12/0.37 tff(69,plain,
% 0.12/0.37 (apart_point_and_line(X!3, U!1) | apart_point_and_line(X!3, V!0)),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[68, 67])).
% 0.12/0.37 tff(70,plain,
% 0.12/0.37 ($false),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[69, 63, 42])).
% 0.12/0.37 % SZS output end Proof
% 0.12/0.37 % E exiting
%------------------------------------------------------------------------------