%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO179+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:05:46 AM UTC 2025
% Result : Theorem 0.20s 0.38s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.01/0.12 % Problem : GEO179+1 : TPTP v9.0.0. Released v3.3.0.
% 0.01/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n003.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 14:40:09 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.38 % SZS status Theorem
% 0.20/0.38 % SZS output start Proof
% 0.20/0.38 tff(apart_point_and_line_type, type, (
% 0.20/0.38 apart_point_and_line: ( $i * $i ) > $o)).
% 0.20/0.38 tff(line_connecting_type, type, (
% 0.20/0.38 line_connecting: ( $i * $i ) > $i)).
% 0.20/0.38 tff(tptp_fun_X_2_type, type, (
% 0.20/0.38 tptp_fun_X_2: $i)).
% 0.20/0.38 tff(tptp_fun_Z_0_type, type, (
% 0.20/0.38 tptp_fun_Z_0: $i)).
% 0.20/0.38 tff(distinct_points_type, type, (
% 0.20/0.38 distinct_points: ( $i * $i ) > $o)).
% 0.20/0.38 tff(tptp_fun_Y_1_type, type, (
% 0.20/0.38 tptp_fun_Y_1: $i)).
% 0.20/0.38 tff(distinct_lines_type, type, (
% 0.20/0.38 distinct_lines: ( $i * $i ) > $o)).
% 0.20/0.38 tff(1,assumption,(~distinct_points(Z!0, X!2)), introduced(assumption)).
% 0.20/0.38 tff(2,plain,
% 0.20/0.38 ((~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y)))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(3,plain,
% 0.20/0.38 ((~![X: $i, Y: $i, Z: $i] : ((distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y))) => (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y)))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(4,axiom,(~![X: $i, Y: $i, Z: $i] : ((distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y))) => (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.20/0.38 tff(5,plain,
% 0.20/0.38 (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.20/0.38 tff(6,plain,
% 0.20/0.38 (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.20/0.38 tff(7,plain,
% 0.20/0.38 (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.20/0.38 tff(8,plain,
% 0.20/0.38 (~![X: $i, Y: $i, Z: $i] : ((~(distinct_points(X, Y) & apart_point_and_line(Z, line_connecting(X, Y)))) | (distinct_lines(line_connecting(X, Y), line_connecting(Z, X)) & distinct_lines(line_connecting(X, Y), line_connecting(Z, Y))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.20/0.38 tff(9,plain,(
% 0.20/0.38 ~((~(distinct_points(X!2, Y!1) & apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) & distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))))),
% 0.20/0.38 inference(skolemize,[status(sab)],[8])).
% 0.20/0.38 tff(10,plain,
% 0.20/0.38 (distinct_points(X!2, Y!1) & apart_point_and_line(Z!0, line_connecting(X!2, Y!1))),
% 0.20/0.38 inference(or_elim,[status(thm)],[9])).
% 0.20/0.38 tff(11,plain,
% 0.20/0.38 (distinct_points(X!2, Y!1)),
% 0.20/0.38 inference(and_elim,[status(thm)],[10])).
% 0.20/0.38 tff(12,plain,
% 0.20/0.38 (^[X: $i, Y: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(13,plain,
% 0.20/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[12])).
% 0.20/0.38 tff(14,plain,
% 0.20/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(15,plain,
% 0.20/0.38 (^[X: $i, Y: $i] : rewrite((distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(16,plain,
% 0.20/0.38 (![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[15])).
% 0.20/0.38 tff(17,axiom,(![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci1')).
% 0.20/0.38 tff(18,plain,
% 0.20/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.20/0.38 tff(19,plain,
% 0.20/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.20/0.38 tff(20,plain,(
% 0.20/0.38 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.38 inference(skolemize,[status(sab)],[19])).
% 0.20/0.38 tff(21,plain,
% 0.20/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[20, 13])).
% 0.20/0.38 tff(22,plain,
% 0.20/0.38 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(23,plain,
% 0.20/0.38 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(24,plain,
% 0.20/0.38 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.20/0.38 tff(25,plain,
% 0.20/0.38 (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[24, 21, 11])).
% 0.20/0.38 tff(26,plain,
% 0.20/0.38 (apart_point_and_line(Z!0, line_connecting(X!2, Y!1))),
% 0.20/0.38 inference(and_elim,[status(thm)],[10])).
% 0.20/0.38 tff(27,plain,
% 0.20/0.38 (^[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))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(28,plain,
% 0.20/0.38 (![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)))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[27])).
% 0.20/0.38 tff(29,plain,
% 0.20/0.38 (![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)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(30,plain,
% 0.20/0.38 (^[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)))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(31,plain,
% 0.20/0.38 (![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)))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[30])).
% 0.20/0.38 tff(32,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')).
% 0.20/0.38 tff(33,plain,
% 0.20/0.38 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.20/0.38 tff(34,plain,
% 0.20/0.38 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[33, 29])).
% 0.20/0.38 tff(35,plain,(
% 0.20/0.38 ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.38 inference(skolemize,[status(sab)],[34])).
% 0.20/0.38 tff(36,plain,
% 0.20/0.38 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[35, 28])).
% 0.20/0.38 tff(37,plain,
% 0.20/0.38 (((~![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(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(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(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(38,plain,
% 0.20/0.38 ((distinct_points(Z!0, X!2) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) <=> (apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(39,plain,
% 0.20/0.38 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, X!2) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, 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(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2)))),
% 0.20/0.38 inference(monotonicity,[status(thm)],[38])).
% 0.20/0.38 tff(40,plain,
% 0.20/0.38 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, X!2) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, 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(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2))),
% 0.20/0.38 inference(transitivity,[status(thm)],[39, 37])).
% 0.20/0.38 tff(41,plain,
% 0.20/0.38 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, X!2) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(42,plain,
% 0.20/0.38 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | distinct_points(Z!0, X!2)),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.20/0.38 tff(43,plain,
% 0.20/0.38 ($false),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[42, 36, 26, 25, 1])).
% 0.20/0.38 tff(44,plain,(distinct_points(Z!0, X!2)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.38 tff(45,plain,
% 0.20/0.38 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(46,plain,
% 0.20/0.38 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2))))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(47,plain,
% 0.20/0.38 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.20/0.38 tff(48,plain,
% 0.20/0.38 ((~distinct_points(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2)))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[47, 21])).
% 0.20/0.38 tff(49,plain,
% 0.20/0.38 (~apart_point_and_line(Z!0, line_connecting(Z!0, X!2))),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[48, 44])).
% 0.20/0.38 tff(50,assumption,(~distinct_points(Z!0, Y!1)), introduced(assumption)).
% 0.20/0.38 tff(51,plain,
% 0.20/0.38 (^[X: $i, Y: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(52,plain,
% 0.20/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[51])).
% 0.20/0.38 tff(53,plain,
% 0.20/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(54,plain,
% 0.20/0.38 (^[X: $i, Y: $i] : rewrite((distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))))),
% 0.20/0.38 inference(bind,[status(th)],[])).
% 0.20/0.38 tff(55,plain,
% 0.20/0.38 (![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.38 inference(quant_intro,[status(thm)],[54])).
% 0.20/0.38 tff(56,axiom,(![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci2')).
% 0.20/0.38 tff(57,plain,
% 0.20/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.20/0.38 tff(58,plain,
% 0.20/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[57, 53])).
% 0.20/0.38 tff(59,plain,(
% 0.20/0.38 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.38 inference(skolemize,[status(sab)],[58])).
% 0.20/0.38 tff(60,plain,
% 0.20/0.38 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[59, 52])).
% 0.20/0.38 tff(61,plain,
% 0.20/0.38 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(62,plain,
% 0.20/0.39 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(63,plain,
% 0.20/0.39 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.20/0.39 tff(64,plain,
% 0.20/0.39 (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[63, 60, 11])).
% 0.20/0.39 tff(65,plain,
% 0.20/0.39 (((~![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(Y!1, line_connecting(X!2, Y!1)) | distinct_points(Z!0, Y!1) | (~apart_point_and_line(Z!0, line_connecting(X!2, 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(Y!1, line_connecting(X!2, Y!1)) | distinct_points(Z!0, Y!1) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(66,plain,
% 0.20/0.39 ((distinct_points(Z!0, Y!1) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) <=> (apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | distinct_points(Z!0, Y!1) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(67,plain,
% 0.20/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, Y!1) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, 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(Y!1, line_connecting(X!2, Y!1)) | distinct_points(Z!0, Y!1) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))))),
% 0.20/0.39 inference(monotonicity,[status(thm)],[66])).
% 0.20/0.39 tff(68,plain,
% 0.20/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, Y!1) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, 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(Y!1, line_connecting(X!2, Y!1)) | distinct_points(Z!0, Y!1) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.39 inference(transitivity,[status(thm)],[67, 65])).
% 0.20/0.39 tff(69,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(Z!0, Y!1) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(70,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | distinct_points(Z!0, Y!1) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.20/0.39 tff(71,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[70, 36, 26, 64, 50])).
% 0.20/0.39 tff(72,plain,(distinct_points(Z!0, Y!1)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.39 tff(73,plain,
% 0.20/0.39 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(74,plain,
% 0.20/0.39 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(75,plain,
% 0.20/0.39 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.20/0.39 tff(76,plain,
% 0.20/0.39 (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[75, 21, 72])).
% 0.20/0.39 tff(77,assumption,(~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))), introduced(assumption)).
% 0.20/0.39 tff(78,assumption,(~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))), introduced(assumption)).
% 0.20/0.39 tff(79,plain,
% 0.20/0.39 (^[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))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(80,plain,
% 0.20/0.39 (![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)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[79])).
% 0.20/0.39 tff(81,plain,
% 0.20/0.39 (![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)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(82,plain,
% 0.20/0.39 (^[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)))))),
% 0.20/0.39 inference(bind,[status(th)],[])).
% 0.20/0.39 tff(83,plain,
% 0.20/0.39 (![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)))),
% 0.20/0.39 inference(quant_intro,[status(thm)],[82])).
% 0.20/0.39 tff(84,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')).
% 0.20/0.39 tff(85,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.20/0.39 tff(86,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[85, 81])).
% 0.20/0.39 tff(87,plain,(
% 0.20/0.39 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.39 inference(skolemize,[status(sab)],[86])).
% 0.20/0.39 tff(88,plain,
% 0.20/0.39 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[87, 80])).
% 0.20/0.39 tff(89,plain,
% 0.20/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(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(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(90,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(91,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.20/0.39 tff(92,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[91, 88, 26, 78, 77])).
% 0.20/0.39 tff(93,plain,(apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.39 tff(94,plain,
% 0.20/0.39 (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[93, 76])).
% 0.20/0.39 tff(95,plain,
% 0.20/0.39 ((~(~((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))))) <=> ((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(96,plain,
% 0.20/0.39 ((distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) & distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))) <=> (~((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(97,plain,
% 0.20/0.39 ((~(distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) & distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))) <=> (~(~((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))))))),
% 0.20/0.39 inference(monotonicity,[status(thm)],[96])).
% 0.20/0.39 tff(98,plain,
% 0.20/0.39 ((~(distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) & distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))) <=> ((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1))))),
% 0.20/0.39 inference(transitivity,[status(thm)],[97, 95])).
% 0.20/0.39 tff(99,plain,
% 0.20/0.39 (~(distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) & distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))),
% 0.20/0.39 inference(or_elim,[status(thm)],[9])).
% 0.20/0.39 tff(100,plain,
% 0.20/0.39 ((~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, Y!1)))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.20/0.39 tff(101,plain,
% 0.20/0.39 (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2))),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[100, 94])).
% 0.20/0.39 tff(102,plain,
% 0.20/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(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)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(103,plain,
% 0.20/0.39 ((distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) <=> (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)))),
% 0.20/0.39 inference(rewrite,[status(thm)],[])).
% 0.20/0.39 tff(104,plain,
% 0.20/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(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(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2))))),
% 0.20/0.39 inference(monotonicity,[status(thm)],[103])).
% 0.20/0.39 tff(105,plain,
% 0.20/0.39 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(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(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)))),
% 0.20/0.39 inference(transitivity,[status(thm)],[104, 102])).
% 0.20/0.39 tff(106,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.20/0.39 inference(quant_inst,[status(thm)],[])).
% 0.20/0.39 tff(107,plain,
% 0.20/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(Z!0, X!2)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(Z!0, X!2))),
% 0.20/0.39 inference(modus_ponens,[status(thm)],[106, 105])).
% 0.20/0.39 tff(108,plain,
% 0.20/0.39 ($false),
% 0.20/0.39 inference(unit_resolution,[status(thm)],[107, 88, 26, 101, 49])).
% 0.20/0.39 % SZS output end Proof
% 0.20/0.40 % E exiting
%------------------------------------------------------------------------------