%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO181+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 05:05:47 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
% 0.06/0.12 % Problem : GEO181+3 : TPTP v9.0.0. Released v4.0.0.
% 0.06/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n014.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 14:40:09 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.39 % SZS status Theorem
% 0.19/0.39 % SZS output start Proof
% 0.19/0.39 tff(distinct_lines_type, type, (
% 0.19/0.39 distinct_lines: ( $i * $i ) > $o)).
% 0.19/0.39 tff(parallel_through_point_type, type, (
% 0.19/0.39 parallel_through_point: ( $i * $i ) > $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(line_connecting_type, type, (
% 0.19/0.39 line_connecting: ( $i * $i ) > $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(tptp_fun_X_2_type, type, (
% 0.19/0.39 tptp_fun_X_2: $i)).
% 0.19/0.39 tff(convergent_lines_type, type, (
% 0.19/0.39 convergent_lines: ( $i * $i ) > $o)).
% 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(distinct_points_type, type, (
% 0.19/0.39 distinct_points: ( $i * $i ) > $o)).
% 0.19/0.39 tff(1,plain,
% 0.19/0.39 (^[X: $i, Y: $i] : refl((~convergent_lines(parallel_through_point(Y, X), Y)) <=> (~convergent_lines(parallel_through_point(Y, X), Y)))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(2,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y)) <=> ![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.39 tff(3,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y)) <=> ![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(4,axiom,(![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+2.ax','cp1')).
% 0.19/0.39 tff(5,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.39 tff(6,plain,(
% 0.19/0.39 ![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))),
% 0.19/0.39 inference(skolemize,[status(sab)],[5])).
% 0.19/0.39 tff(7,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.39 tff(8,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : (~convergent_lines(parallel_through_point(Y, X), Y))) | (~convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(9,plain,
% 0.19/0.39 (~convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.19/0.39 tff(10,plain,
% 0.19/0.39 (^[X: $i, Y: $i] : refl((~apart_point_and_line(X, parallel_through_point(Y, X))) <=> (~apart_point_and_line(X, parallel_through_point(Y, X))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(11,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X))) <=> ![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[10])).
% 0.19/0.39 tff(12,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X))) <=> ![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(13,axiom,(![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+2.ax','cp2')).
% 0.19/0.39 tff(14,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.19/0.39 tff(15,plain,(
% 0.19/0.39 ![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))),
% 0.19/0.39 inference(skolemize,[status(sab)],[14])).
% 0.19/0.39 tff(16,plain,
% 0.19/0.39 (![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[15, 11])).
% 0.19/0.39 tff(17,plain,
% 0.19/0.39 ((~![X: $i, Y: $i] : (~apart_point_and_line(X, parallel_through_point(Y, X)))) | (~apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(18,plain,
% 0.19/0.39 (~apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[17, 16])).
% 0.19/0.39 tff(19,assumption,(~apart_point_and_line(Z!0, line_connecting(X!2, Z!0))), introduced(assumption)).
% 0.19/0.39 tff(20,assumption,(~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))), introduced(assumption)).
% 0.19/0.39 tff(21,plain,
% 0.19/0.39 ((~(apart_point_and_line(Y!1, line_connecting(X!2, Z!0)) | (~distinct_points(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, Z!0)) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(22,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Y, line_connecting(X, Z)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))) <=> (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Y, line_connecting(X, Z)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(23,plain,
% 0.19/0.39 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => apart_point_and_line(Y, line_connecting(X, Z))))) <=> (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Y, line_connecting(X, Z)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y)))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(24,axiom,(~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => apart_point_and_line(Y, line_connecting(X, Z))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.19/0.39 tff(25,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Y, line_connecting(X, Z)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.19/0.39 tff(26,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Y, line_connecting(X, Z)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[25, 22])).
% 0.19/0.39 tff(27,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Y, line_connecting(X, Z)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[26, 22])).
% 0.19/0.39 tff(28,plain,
% 0.19/0.39 (~![X: $i, Y: $i, Z: $i] : (apart_point_and_line(Y, line_connecting(X, Z)) | (~distinct_points(X, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[27, 22])).
% 0.19/0.39 tff(29,plain,(
% 0.19/0.39 ~(apart_point_and_line(Y!1, line_connecting(X!2, Z!0)) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.19/0.39 inference(skolemize,[status(sab)],[28])).
% 0.19/0.39 tff(30,plain,
% 0.19/0.39 (~(apart_point_and_line(Y!1, line_connecting(X!2, Z!0)) | (~distinct_points(X!2, Y!1)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[29, 21])).
% 0.19/0.39 tff(31,plain,
% 0.19/0.39 (apart_point_and_line(Z!0, line_connecting(X!2, Y!1))),
% 0.19/0.39 inference(or_elim,[status(thm)],[30])).
% 0.19/0.39 tff(32,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(33,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)],[32])).
% 0.19/0.39 tff(34,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(35,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(36,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)],[35])).
% 0.19/0.40 tff(37,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(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)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.19/0.40 tff(39,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)],[38, 34])).
% 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)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[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)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[40, 33])).
% 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)))) | ((~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(43,plain,
% 0.19/0.40 ((distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) <=> ((~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(44,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(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | (~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)))) | ((~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[43])).
% 0.19/0.40 tff(45,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(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | (~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)))) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))),
% 0.19/0.40 inference(transitivity,[status(thm)],[44, 42])).
% 0.19/0.40 tff(46,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(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(47,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.19/0.40 tff(48,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[47, 41, 31, 19, 20])).
% 0.19/0.40 tff(49,plain,(distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.40 tff(50,plain,
% 0.19/0.40 (distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[49, 19])).
% 0.19/0.40 tff(51,assumption,(distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))), introduced(assumption)).
% 0.19/0.40 tff(52,assumption,(~apart_point_and_line(X!2, line_connecting(X!2, Z!0))), introduced(assumption)).
% 0.19/0.40 tff(53,plain,
% 0.19/0.40 (distinct_points(X!2, Y!1)),
% 0.19/0.40 inference(or_elim,[status(thm)],[30])).
% 0.19/0.40 tff(54,plain,
% 0.19/0.40 (^[X: $i, Y: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(55,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[54])).
% 0.19/0.40 tff(56,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(57,plain,
% 0.19/0.40 (^[X: $i, Y: $i] : rewrite((distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(58,plain,
% 0.19/0.40 (![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[57])).
% 0.19/0.40 tff(59,axiom,(![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(X, line_connecting(X, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci1')).
% 0.19/0.40 tff(60,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.19/0.40 tff(61,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[60, 56])).
% 0.19/0.40 tff(62,plain,(
% 0.19/0.40 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.40 inference(skolemize,[status(sab)],[61])).
% 0.19/0.40 tff(63,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[62, 55])).
% 0.19/0.40 tff(64,plain,
% 0.19/0.40 (((~![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))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(65,plain,
% 0.19/0.40 ((~![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))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(66,plain,
% 0.19/0.40 ((~![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)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.19/0.40 tff(67,plain,
% 0.19/0.40 (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[66, 63, 53])).
% 0.19/0.40 tff(68,plain,
% 0.19/0.40 (^[X: $i, Y: $i] : refl(((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(69,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[68])).
% 0.19/0.40 tff(70,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(71,plain,
% 0.19/0.40 (^[X: $i, Y: $i] : rewrite((distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(72,plain,
% 0.19/0.40 (![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y)))) <=> ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[71])).
% 0.19/0.40 tff(73,axiom,(![X: $i, Y: $i] : (distinct_points(X, Y) => (~apart_point_and_line(Y, line_connecting(X, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci2')).
% 0.19/0.40 tff(74,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[73, 72])).
% 0.19/0.40 tff(75,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[74, 70])).
% 0.19/0.40 tff(76,plain,(
% 0.19/0.40 ![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.40 inference(skolemize,[status(sab)],[75])).
% 0.19/0.40 tff(77,plain,
% 0.19/0.40 (![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[76, 69])).
% 0.19/0.40 tff(78,plain,
% 0.19/0.40 (((~![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))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(79,plain,
% 0.19/0.40 ((~![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))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(80,plain,
% 0.19/0.40 ((~![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)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.19/0.40 tff(81,plain,
% 0.19/0.40 (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[80, 77, 53])).
% 0.19/0.40 tff(82,plain,
% 0.19/0.40 (~apart_point_and_line(Y!1, line_connecting(X!2, Z!0))),
% 0.19/0.40 inference(or_elim,[status(thm)],[30])).
% 0.19/0.40 tff(83,plain,
% 0.19/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : refl((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(84,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[83])).
% 0.19/0.40 tff(85,plain,
% 0.19/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(rewrite((distinct_points(X, Y) & distinct_lines(U, V)) <=> (~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> (~(~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))))), rewrite((~(~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))) <=> ((~distinct_lines(U, V)) | (~distinct_points(X, Y)))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> ((~distinct_lines(U, V)) | (~distinct_points(X, Y))))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_lines(U, V)) | (~distinct_points(X, Y)))))), rewrite((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_lines(U, V)) | (~distinct_points(X, Y)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(86,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[85])).
% 0.19/0.40 tff(87,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(88,plain,
% 0.19/0.40 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((apart_point_and_line(X, U) | apart_point_and_line(X, V)) <=> (apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> ((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)))), rewrite(((apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) <=> (apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> ((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)))), rewrite(((apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))), ((((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U)))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> ((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))))), rewrite(((distinct_points(X, Y) & distinct_lines(U, V)) => (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))), (((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(89,plain,
% 0.19/0.40 (![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))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[88])).
% 0.19/0.40 tff(90,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')).
% 0.19/0.40 tff(91,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.19/0.40 tff(92,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[91, 87])).
% 0.19/0.40 tff(93,plain,(
% 0.19/0.40 ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.40 inference(skolemize,[status(sab)],[92])).
% 0.19/0.40 tff(94,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[93, 86])).
% 0.19/0.40 tff(95,plain,
% 0.19/0.40 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[94, 84])).
% 0.19/0.40 tff(96,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(Y!1, line_connecting(X!2, Z!0)) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(Y!1, line_connecting(X!2, Z!0)) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(97,plain,
% 0.19/0.41 ((apart_point_and_line(Y!1, line_connecting(X!2, Z!0)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~distinct_points(X!2, Y!1))) <=> (apart_point_and_line(Y!1, line_connecting(X!2, Z!0)) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(98,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(Y!1, line_connecting(X!2, Z!0)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~distinct_points(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(Y!1, line_connecting(X!2, Z!0)) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[97])).
% 0.19/0.41 tff(99,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(Y!1, line_connecting(X!2, Z!0)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~distinct_points(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(Y!1, line_connecting(X!2, Z!0)) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[98, 96])).
% 0.19/0.41 tff(100,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(Y!1, line_connecting(X!2, Z!0)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0))) | (~distinct_points(X!2, Y!1)))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(101,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(Y!1, line_connecting(X!2, Z!0)) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[100, 99])).
% 0.19/0.41 tff(102,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[101, 95, 82, 53, 81, 67, 52, 51])).
% 0.19/0.41 tff(103,plain,(apart_point_and_line(X!2, line_connecting(X!2, Z!0)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Z!0)))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(104,plain,
% 0.19/0.41 (apart_point_and_line(X!2, line_connecting(X!2, Z!0))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[103, 50])).
% 0.19/0.41 tff(105,plain,
% 0.19/0.41 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(106,plain,
% 0.19/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | ((~distinct_points(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(107,plain,
% 0.19/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(X, line_connecting(X, Y))))) | (~distinct_points(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[106, 105])).
% 0.19/0.41 tff(108,plain,
% 0.19/0.41 ((~distinct_points(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0)))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[107, 63])).
% 0.19/0.41 tff(109,plain,
% 0.19/0.41 (~distinct_points(X!2, Z!0)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[108, 104])).
% 0.19/0.41 tff(110,plain,
% 0.19/0.41 (^[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))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(111,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))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[110])).
% 0.19/0.41 tff(112,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))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(113,plain,
% 0.19/0.41 (^[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)))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(114,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[113])).
% 0.19/0.41 tff(115,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')).
% 0.19/0.41 tff(116,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)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.19/0.41 tff(117,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)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[116, 112])).
% 0.19/0.41 tff(118,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)))),
% 0.19/0.41 inference(skolemize,[status(sab)],[117])).
% 0.19/0.41 tff(119,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)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[118, 111])).
% 0.19/0.41 tff(120,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(X!2, Z!0) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | distinct_points(X!2, Z!0) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(121,plain,
% 0.19/0.41 ((distinct_points(X!2, Z!0) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0)))) <=> (distinct_points(X!2, Z!0) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(122,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(X!2, Z!0) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(X!2, Z!0) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[121])).
% 0.19/0.41 tff(123,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(X!2, Z!0) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | distinct_points(X!2, Z!0) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)))),
% 0.19/0.41 inference(transitivity,[status(thm)],[122, 120])).
% 0.19/0.41 tff(124,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(X!2, Z!0) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0)) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(125,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | distinct_points(X!2, Z!0) | (~apart_point_and_line(X!2, line_connecting(X!2, Z!0))) | apart_point_and_line(Z!0, line_connecting(X!2, Z!0))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[124, 123])).
% 0.19/0.41 tff(126,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[125, 119, 109, 104, 19])).
% 0.19/0.41 tff(127,plain,(apart_point_and_line(Z!0, line_connecting(X!2, Z!0))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(128,plain,
% 0.19/0.41 (((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(X!2, Z!0)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Z!0))))) <=> ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(X!2, Z!0)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Z!0))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(129,plain,
% 0.19/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | ((~distinct_points(X!2, Z!0)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Z!0))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(130,plain,
% 0.19/0.41 ((~![X: $i, Y: $i] : ((~distinct_points(X, Y)) | (~apart_point_and_line(Y, line_connecting(X, Y))))) | (~distinct_points(X!2, Z!0)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Z!0)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[129, 128])).
% 0.19/0.41 tff(131,plain,
% 0.19/0.41 ((~distinct_points(X!2, Z!0)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Z!0)))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[130, 77])).
% 0.19/0.41 tff(132,plain,
% 0.19/0.41 (~distinct_points(X!2, Z!0)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[131, 127])).
% 0.19/0.41 tff(133,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(X!2, Z!0) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~apart_point_and_line(X!2, parallel_through_point(line_connecting(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)))) | distinct_points(X!2, Z!0) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(134,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(X!2, Z!0) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(135,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(X!2, Z!0) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[134, 133])).
% 0.19/0.41 tff(136,plain,
% 0.19/0.41 (~apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[135, 119, 132, 18])).
% 0.19/0.41 tff(137,plain,
% 0.19/0.41 (^[X: $i, Y: $i, Z: $i] : refl((convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z))) <=> (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(138,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[137])).
% 0.19/0.41 tff(139,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))),
% 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] : trans(monotonicity(trans(monotonicity(rewrite((apart_point_and_line(X, Y) | apart_point_and_line(X, Z)) <=> (apart_point_and_line(X, Z) | apart_point_and_line(X, Y))), (((apart_point_and_line(X, Y) | apart_point_and_line(X, Z)) | convergent_lines(Y, Z)) <=> ((apart_point_and_line(X, Z) | apart_point_and_line(X, Y)) | convergent_lines(Y, Z)))), rewrite(((apart_point_and_line(X, Z) | apart_point_and_line(X, Y)) | convergent_lines(Y, Z)) <=> (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y))), (((apart_point_and_line(X, Y) | apart_point_and_line(X, Z)) | convergent_lines(Y, Z)) <=> (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y)))), ((distinct_lines(Y, Z) => ((apart_point_and_line(X, Y) | apart_point_and_line(X, Z)) | convergent_lines(Y, Z))) <=> (distinct_lines(Y, Z) => (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y))))), rewrite((distinct_lines(Y, Z) => (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y))) <=> (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))), ((distinct_lines(Y, Z) => ((apart_point_and_line(X, Y) | apart_point_and_line(X, Z)) | convergent_lines(Y, Z))) <=> (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(141,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) => ((apart_point_and_line(X, Y) | apart_point_and_line(X, Z)) | convergent_lines(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[140])).
% 0.19/0.41 tff(142,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) => ((apart_point_and_line(X, Y) | apart_point_and_line(X, Z)) | convergent_lines(Y, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+2.ax','cup1')).
% 0.19/0.41 tff(143,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[142, 141])).
% 0.19/0.41 tff(144,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[143, 139])).
% 0.19/0.41 tff(145,plain,(
% 0.19/0.41 ![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))),
% 0.19/0.41 inference(skolemize,[status(sab)],[144])).
% 0.19/0.41 tff(146,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[145, 138])).
% 0.19/0.41 tff(147,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))) | (apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(148,plain,
% 0.19/0.42 ((convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)))) <=> (apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(149,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))) | (convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))) | (apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[148])).
% 0.19/0.42 tff(150,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))) | (convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1))))),
% 0.19/0.42 inference(transitivity,[status(thm)],[149, 147])).
% 0.19/0.42 tff(151,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))) | (convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(152,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(Y, Z) | apart_point_and_line(X, Z) | apart_point_and_line(X, Y) | (~distinct_lines(Y, Z)))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | convergent_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[151, 150])).
% 0.19/0.42 tff(153,plain,
% 0.19/0.42 (~distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[152, 146, 67, 136, 9])).
% 0.19/0.42 tff(154,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)))) | ((~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(155,plain,
% 0.19/0.42 ((apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1))) <=> ((~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(156,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(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | ((~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[155])).
% 0.19/0.42 tff(157,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(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1))))),
% 0.19/0.42 inference(transitivity,[status(thm)],[156, 154])).
% 0.19/0.42 tff(158,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(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1)))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(159,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)))) | (~distinct_points(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[158, 157])).
% 0.19/0.42 tff(160,plain,
% 0.19/0.42 (~distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[159, 95, 53, 81, 67])).
% 0.19/0.42 tff(161,plain,
% 0.19/0.42 (^[X: $i, Y: $i, Z: $i] : refl((distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(162,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[161])).
% 0.19/0.42 tff(163,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(164,plain,
% 0.19/0.42 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_lines(X, Z) | distinct_lines(Y, Z)) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z))), ((distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z))) <=> (distinct_lines(X, Y) => (distinct_lines(Y, Z) | distinct_lines(X, Z))))), rewrite((distinct_lines(X, Y) => (distinct_lines(Y, Z) | distinct_lines(X, Z))) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))), ((distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z))) <=> (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(165,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[164])).
% 0.19/0.42 tff(166,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_lines(X, Y) => (distinct_lines(X, Z) | distinct_lines(Y, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','apart5')).
% 0.19/0.42 tff(167,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[166, 165])).
% 0.19/0.42 tff(168,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[167, 163])).
% 0.19/0.42 tff(169,plain,(
% 0.19/0.42 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.42 inference(skolemize,[status(sab)],[168])).
% 0.19/0.42 tff(170,plain,
% 0.19/0.42 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[169, 162])).
% 0.19/0.42 tff(171,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(172,plain,
% 0.19/0.43 ((distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0)))) <=> (distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(173,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0)))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[172])).
% 0.19/0.43 tff(174,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0))))),
% 0.19/0.43 inference(transitivity,[status(thm)],[173, 171])).
% 0.19/0.43 tff(175,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | (distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0))))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(176,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | distinct_lines(X, Z) | (~distinct_lines(X, Y)))) | distinct_lines(line_connecting(X!2, Y!1), line_connecting(X!2, Y!1)) | distinct_lines(parallel_through_point(line_connecting(X!2, Y!1), Z!0), line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[175, 174])).
% 0.19/0.43 tff(177,plain,
% 0.19/0.43 (~distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[176, 170, 160, 153])).
% 0.19/0.43 tff(178,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(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(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(179,plain,
% 0.19/0.43 ((distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))) <=> ((~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0)))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(180,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~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)))) | ((~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[179])).
% 0.19/0.43 tff(181,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~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)))) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0)))),
% 0.19/0.43 inference(transitivity,[status(thm)],[180, 178])).
% 0.19/0.43 tff(182,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(183,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))) | apart_point_and_line(Z!0, parallel_through_point(line_connecting(X!2, Y!1), Z!0)) | distinct_lines(line_connecting(X!2, Y!1), parallel_through_point(line_connecting(X!2, Y!1), Z!0))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[182, 181])).
% 0.19/0.43 tff(184,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[183, 41, 31, 18, 177])).
% 0.19/0.43 % SZS output end Proof
% 0.19/0.43 % E exiting
%------------------------------------------------------------------------------