TSTP Solution File: GEO192+3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GEO192+3 : TPTP v8.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 16 20:35:29 EDT 2022
% Result : Theorem 0.20s 0.43s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GEO192+3 : TPTP v8.1.0. Released v4.0.0.
% 0.12/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.34 % Computer : n006.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Wed Aug 31 06:50:39 EDT 2022
% 0.20/0.34 % CPUTime :
% 0.20/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.20/0.34 Usage: tptp [options] [-file:]file
% 0.20/0.34 -h, -? prints this message.
% 0.20/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.20/0.34 -m, -model generate model.
% 0.20/0.34 -p, -proof generate proof.
% 0.20/0.34 -c, -core generate unsat core of named formulas.
% 0.20/0.34 -st, -statistics display statistics.
% 0.20/0.34 -t:timeout set timeout (in second).
% 0.20/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.20/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.20/0.34 -<param>:<value> configuration parameter and value.
% 0.20/0.34 -o:<output-file> file to place output in.
% 0.20/0.43 % SZS status Theorem
% 0.20/0.43 % SZS output start Proof
% 0.20/0.43 tff(distinct_points_type, type, (
% 0.20/0.43 distinct_points: ( $i * $i ) > $o)).
% 0.20/0.43 tff(intersection_point_type, type, (
% 0.20/0.43 intersection_point: ( $i * $i ) > $i)).
% 0.20/0.43 tff(tptp_fun_Z_0_type, type, (
% 0.20/0.43 tptp_fun_Z_0: $i)).
% 0.20/0.43 tff(tptp_fun_Y_1_type, type, (
% 0.20/0.43 tptp_fun_Y_1: $i)).
% 0.20/0.43 tff(tptp_fun_X_2_type, type, (
% 0.20/0.43 tptp_fun_X_2: $i)).
% 0.20/0.43 tff(apart_point_and_line_type, type, (
% 0.20/0.43 apart_point_and_line: ( $i * $i ) > $o)).
% 0.20/0.43 tff(convergent_lines_type, type, (
% 0.20/0.43 convergent_lines: ( $i * $i ) > $o)).
% 0.20/0.43 tff(distinct_lines_type, type, (
% 0.20/0.43 distinct_lines: ( $i * $i ) > $o)).
% 0.20/0.43 tff(1,assumption,(~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2))), introduced(assumption)).
% 0.20/0.43 tff(2,plain,
% 0.20/0.43 ((~((~convergent_lines(X!2, Y!1)) | (distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))) <=> (~((~convergent_lines(X!2, Y!1)) | (distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(3,plain,
% 0.20/0.43 ((~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))) <=> (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(4,plain,
% 0.20/0.43 ((~![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (apart_point_and_line(intersection_point(X, Y), Z) => (distinct_lines(X, Z) & distinct_lines(Y, Z))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(5,axiom,(~![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (apart_point_and_line(intersection_point(X, Y), Z) => (distinct_lines(X, Z) & distinct_lines(Y, Z))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.20/0.43 tff(6,plain,
% 0.20/0.43 (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.20/0.43 tff(7,plain,
% 0.20/0.43 (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[6, 3])).
% 0.20/0.43 tff(8,plain,
% 0.20/0.43 (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.20/0.43 tff(9,plain,
% 0.20/0.43 (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[8, 3])).
% 0.20/0.43 tff(10,plain,
% 0.20/0.43 (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[9, 3])).
% 0.20/0.43 tff(11,plain,
% 0.20/0.43 (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[10, 3])).
% 0.20/0.43 tff(12,plain,
% 0.20/0.43 (~![X: $i, Y: $i, Z: $i] : ((~convergent_lines(X, Y)) | (distinct_lines(X, Z) & distinct_lines(Y, Z)) | (~apart_point_and_line(intersection_point(X, Y), Z)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[11, 3])).
% 0.20/0.43 tff(13,plain,(
% 0.20/0.43 ~((~convergent_lines(X!2, Y!1)) | (distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.20/0.43 inference(skolemize,[status(sab)],[12])).
% 0.20/0.43 tff(14,plain,
% 0.20/0.43 (~((~convergent_lines(X!2, Y!1)) | (distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[13, 2])).
% 0.20/0.43 tff(15,plain,
% 0.20/0.43 (convergent_lines(X!2, Y!1)),
% 0.20/0.43 inference(or_elim,[status(thm)],[14])).
% 0.20/0.43 tff(16,plain,
% 0.20/0.43 (^[X: $i, Y: $i] : refl(((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(17,plain,
% 0.20/0.43 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[16])).
% 0.20/0.43 tff(18,plain,
% 0.20/0.43 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(19,plain,
% 0.20/0.43 (^[X: $i, Y: $i] : rewrite((convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(20,plain,
% 0.20/0.43 (![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[19])).
% 0.20/0.43 tff(21,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), X)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci3')).
% 0.20/0.43 tff(22,plain,
% 0.20/0.43 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.20/0.43 tff(23,plain,
% 0.20/0.43 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[22, 18])).
% 0.20/0.43 tff(24,plain,(
% 0.20/0.43 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.43 inference(skolemize,[status(sab)],[23])).
% 0.20/0.43 tff(25,plain,
% 0.20/0.43 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[24, 17])).
% 0.20/0.43 tff(26,plain,
% 0.20/0.43 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(27,plain,
% 0.20/0.43 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(28,plain,
% 0.20/0.43 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), X!2))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.20/0.43 tff(29,plain,
% 0.20/0.43 (~apart_point_and_line(intersection_point(X!2, Y!1), X!2)),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[28, 25, 15])).
% 0.20/0.43 tff(30,plain,
% 0.20/0.43 (apart_point_and_line(intersection_point(X!2, Y!1), Z!0)),
% 0.20/0.43 inference(or_elim,[status(thm)],[14])).
% 0.20/0.43 tff(31,plain,
% 0.20/0.43 (^[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))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(32,plain,
% 0.20/0.43 (![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)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[31])).
% 0.20/0.43 tff(33,plain,
% 0.20/0.43 (![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)))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(34,plain,
% 0.20/0.43 (^[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)))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(35,plain,
% 0.20/0.43 (![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)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[34])).
% 0.20/0.43 tff(36,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')).
% 0.20/0.43 tff(37,plain,
% 0.20/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.20/0.43 tff(38,plain,
% 0.20/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[37, 33])).
% 0.20/0.43 tff(39,plain,(
% 0.20/0.43 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.43 inference(skolemize,[status(sab)],[38])).
% 0.20/0.43 tff(40,plain,
% 0.20/0.43 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[39, 32])).
% 0.20/0.43 tff(41,plain,
% 0.20/0.43 (((~![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(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(42,plain,
% 0.20/0.43 ((distinct_lines(Z!0, X!2) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0))) <=> ((~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(43,plain,
% 0.20/0.43 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Z!0, X!2) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2)))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[42])).
% 0.20/0.43 tff(44,plain,
% 0.20/0.43 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Z!0, X!2) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2))),
% 0.20/0.44 inference(transitivity,[status(thm)],[43, 41])).
% 0.20/0.44 tff(45,plain,
% 0.20/0.44 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Z!0, X!2) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(46,plain,
% 0.20/0.44 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | distinct_lines(Z!0, X!2)),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.20/0.44 tff(47,plain,
% 0.20/0.44 (distinct_lines(Z!0, X!2)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[46, 40, 30, 29])).
% 0.20/0.44 tff(48,plain,
% 0.20/0.44 (^[X: $i, Y: $i] : refl(((~distinct_lines(X, Y)) | convergent_lines(X, Y)) <=> ((~distinct_lines(X, Y)) | convergent_lines(X, Y)))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(49,plain,
% 0.20/0.44 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[48])).
% 0.20/0.44 tff(50,plain,
% 0.20/0.44 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(51,plain,
% 0.20/0.44 (^[X: $i, Y: $i] : rewrite((distinct_lines(X, Y) => convergent_lines(X, Y)) <=> ((~distinct_lines(X, Y)) | convergent_lines(X, Y)))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(52,plain,
% 0.20/0.44 (![X: $i, Y: $i] : (distinct_lines(X, Y) => convergent_lines(X, Y)) <=> ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[51])).
% 0.20/0.44 tff(53,axiom,(![X: $i, Y: $i] : (distinct_lines(X, Y) => convergent_lines(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+1.ax','p1')).
% 0.20/0.44 tff(54,plain,
% 0.20/0.44 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.20/0.44 tff(55,plain,
% 0.20/0.44 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[54, 50])).
% 0.20/0.44 tff(56,plain,(
% 0.20/0.44 ![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.20/0.44 inference(skolemize,[status(sab)],[55])).
% 0.20/0.44 tff(57,plain,
% 0.20/0.44 (![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[56, 49])).
% 0.20/0.44 tff(58,plain,
% 0.20/0.44 (((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(Z!0, X!2)) | convergent_lines(Z!0, X!2))) <=> ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(Z!0, X!2)) | convergent_lines(Z!0, X!2))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(59,plain,
% 0.20/0.44 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(Z!0, X!2)) | convergent_lines(Z!0, X!2))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(60,plain,
% 0.20/0.44 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(Z!0, X!2)) | convergent_lines(Z!0, X!2)),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.20/0.44 tff(61,plain,
% 0.20/0.44 (convergent_lines(Z!0, X!2)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[60, 57, 47])).
% 0.20/0.44 tff(62,plain,
% 0.20/0.44 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), Z!0)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), Z!0)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(63,plain,
% 0.20/0.44 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), Z!0)))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(64,plain,
% 0.20/0.44 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), Z!0))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[63, 62])).
% 0.20/0.44 tff(65,plain,
% 0.20/0.44 (~apart_point_and_line(intersection_point(Z!0, X!2), Z!0)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[64, 25, 61])).
% 0.20/0.44 tff(66,plain,
% 0.20/0.44 (^[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))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(67,plain,
% 0.20/0.44 (![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)))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[66])).
% 0.20/0.44 tff(68,plain,
% 0.20/0.44 (![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)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(69,plain,
% 0.20/0.44 (^[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)))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(70,plain,
% 0.20/0.44 (![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)))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[69])).
% 0.20/0.44 tff(71,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')).
% 0.20/0.44 tff(72,plain,
% 0.20/0.44 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.20/0.44 tff(73,plain,
% 0.20/0.44 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[72, 68])).
% 0.20/0.44 tff(74,plain,(
% 0.20/0.44 ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.44 inference(skolemize,[status(sab)],[73])).
% 0.20/0.44 tff(75,plain,
% 0.20/0.44 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[74, 67])).
% 0.20/0.44 tff(76,plain,
% 0.20/0.44 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(77,plain,
% 0.20/0.44 ((distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0))) <=> ((~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(78,plain,
% 0.20/0.44 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2))))),
% 0.20/0.44 inference(monotonicity,[status(thm)],[77])).
% 0.20/0.44 tff(79,plain,
% 0.20/0.44 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2)))),
% 0.20/0.44 inference(transitivity,[status(thm)],[78, 76])).
% 0.20/0.44 tff(80,plain,
% 0.20/0.44 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(81,plain,
% 0.20/0.44 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(X!2, Y!1), Z!0)) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[80, 79])).
% 0.20/0.44 tff(82,plain,
% 0.20/0.44 ($false),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[81, 75, 30, 65, 1])).
% 0.20/0.44 tff(83,plain,(distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2))), inference(lemma,lemma(discharge,[]))).
% 0.20/0.44 tff(84,assumption,(~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)), introduced(assumption)).
% 0.20/0.44 tff(85,plain,
% 0.20/0.44 (^[X: $i, Y: $i] : refl(((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(86,plain,
% 0.20/0.44 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[85])).
% 0.20/0.44 tff(87,plain,
% 0.20/0.44 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(88,plain,
% 0.20/0.44 (^[X: $i, Y: $i] : rewrite((convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(89,plain,
% 0.20/0.44 (![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), Y))) <=> ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[88])).
% 0.20/0.44 tff(90,axiom,(![X: $i, Y: $i] : (convergent_lines(X, Y) => (~apart_point_and_line(intersection_point(X, Y), Y)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ci4')).
% 0.20/0.44 tff(91,plain,
% 0.20/0.44 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.20/0.44 tff(92,plain,
% 0.20/0.44 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[91, 87])).
% 0.20/0.44 tff(93,plain,(
% 0.20/0.44 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.44 inference(skolemize,[status(sab)],[92])).
% 0.20/0.44 tff(94,plain,
% 0.20/0.44 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[93, 86])).
% 0.20/0.44 tff(95,plain,
% 0.20/0.44 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), X!2)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), X!2)))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(96,plain,
% 0.20/0.44 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), X!2)))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(97,plain,
% 0.20/0.44 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Z!0, X!2)) | (~apart_point_and_line(intersection_point(Z!0, X!2), X!2))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.20/0.44 tff(98,plain,
% 0.20/0.44 (~apart_point_and_line(intersection_point(Z!0, X!2), X!2)),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[97, 94, 61])).
% 0.20/0.44 tff(99,assumption,(convergent_lines(X!2, X!2)), introduced(assumption)).
% 0.20/0.44 tff(100,plain,
% 0.20/0.44 (^[X: $i] : refl((~convergent_lines(X, X)) <=> (~convergent_lines(X, X)))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(101,plain,
% 0.20/0.44 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.20/0.44 inference(quant_intro,[status(thm)],[100])).
% 0.20/0.44 tff(102,plain,
% 0.20/0.44 (![X: $i] : (~convergent_lines(X, X)) <=> ![X: $i] : (~convergent_lines(X, X))),
% 0.20/0.44 inference(rewrite,[status(thm)],[])).
% 0.20/0.44 tff(103,axiom,(![X: $i] : (~convergent_lines(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','apart3')).
% 0.20/0.44 tff(104,plain,
% 0.20/0.44 (![X: $i] : (~convergent_lines(X, X))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[103, 102])).
% 0.20/0.44 tff(105,plain,(
% 0.20/0.44 ![X: $i] : (~convergent_lines(X, X))),
% 0.20/0.44 inference(skolemize,[status(sab)],[104])).
% 0.20/0.44 tff(106,plain,
% 0.20/0.44 (![X: $i] : (~convergent_lines(X, X))),
% 0.20/0.44 inference(modus_ponens,[status(thm)],[105, 101])).
% 0.20/0.44 tff(107,plain,
% 0.20/0.44 ((~![X: $i] : (~convergent_lines(X, X))) | (~convergent_lines(X!2, X!2))),
% 0.20/0.44 inference(quant_inst,[status(thm)],[])).
% 0.20/0.44 tff(108,plain,
% 0.20/0.44 ($false),
% 0.20/0.44 inference(unit_resolution,[status(thm)],[107, 106, 99])).
% 0.20/0.44 tff(109,plain,(~convergent_lines(X!2, X!2)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.44 tff(110,assumption,(~distinct_lines(Y!1, X!2)), introduced(assumption)).
% 0.20/0.44 tff(111,plain,
% 0.20/0.44 (^[X: $i, Y: $i, Z: $i] : refl((distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))))),
% 0.20/0.44 inference(bind,[status(th)],[])).
% 0.20/0.44 tff(112,plain,
% 0.20/0.44 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[111])).
% 0.20/0.45 tff(113,plain,
% 0.20/0.45 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(114,plain,
% 0.20/0.45 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_lines(Y, Z) | convergent_lines(X, Z)) <=> (distinct_lines(Y, Z) | convergent_lines(X, Z))), ((convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z))) <=> (convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z))))), rewrite((convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z))) <=> (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))), ((convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z))) <=> (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(115,plain,
% 0.20/0.45 (![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[114])).
% 0.20/0.45 tff(116,axiom,(![X: $i, Y: $i, Z: $i] : (convergent_lines(X, Y) => (distinct_lines(Y, Z) | convergent_lines(X, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ceq3')).
% 0.20/0.45 tff(117,plain,
% 0.20/0.45 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[116, 115])).
% 0.20/0.45 tff(118,plain,
% 0.20/0.45 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[117, 113])).
% 0.20/0.45 tff(119,plain,(
% 0.20/0.45 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.45 inference(skolemize,[status(sab)],[118])).
% 0.20/0.45 tff(120,plain,
% 0.20/0.45 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[119, 112])).
% 0.20/0.45 tff(121,plain,
% 0.20/0.45 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(122,plain,
% 0.20/0.45 ((distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1))) <=> ((~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(123,plain,
% 0.20/0.45 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2)))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[122])).
% 0.20/0.45 tff(124,plain,
% 0.20/0.45 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2))),
% 0.20/0.45 inference(transitivity,[status(thm)],[123, 121])).
% 0.20/0.45 tff(125,plain,
% 0.20/0.45 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2) | (~convergent_lines(X!2, Y!1)))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(126,plain,
% 0.20/0.45 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!2, Y!1)) | distinct_lines(Y!1, X!2) | convergent_lines(X!2, X!2)),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.20/0.45 tff(127,plain,
% 0.20/0.45 ($false),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[126, 120, 15, 110, 109])).
% 0.20/0.45 tff(128,plain,(distinct_lines(Y!1, X!2)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.45 tff(129,plain,
% 0.20/0.45 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(130,plain,
% 0.20/0.45 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(131,plain,
% 0.20/0.45 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(X!2, Y!1)) | (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[130, 129])).
% 0.20/0.45 tff(132,plain,
% 0.20/0.45 (~apart_point_and_line(intersection_point(X!2, Y!1), Y!1)),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[131, 94, 15])).
% 0.20/0.45 tff(133,plain,
% 0.20/0.45 (^[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))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(134,plain,
% 0.20/0.45 (![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)))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[133])).
% 0.20/0.45 tff(135,plain,
% 0.20/0.45 (^[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)))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(136,plain,
% 0.20/0.45 (![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)))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[135])).
% 0.20/0.45 tff(137,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(138,plain,
% 0.20/0.45 (^[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))))))),
% 0.20/0.45 inference(bind,[status(th)],[])).
% 0.20/0.45 tff(139,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(quant_intro,[status(thm)],[138])).
% 0.20/0.45 tff(140,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')).
% 0.20/0.45 tff(141,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[140, 139])).
% 0.20/0.45 tff(142,plain,
% 0.20/0.45 (![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))))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[141, 137])).
% 0.20/0.45 tff(143,plain,(
% 0.20/0.45 ![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))))),
% 0.20/0.45 inference(skolemize,[status(sab)],[142])).
% 0.20/0.45 tff(144,plain,
% 0.20/0.45 (![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)))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[143, 136])).
% 0.20/0.45 tff(145,plain,
% 0.20/0.45 (![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)))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[144, 134])).
% 0.20/0.45 tff(146,plain,
% 0.20/0.45 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(147,plain,
% 0.20/0.45 ((apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2)))) <=> (apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2))))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(148,plain,
% 0.20/0.45 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2)))))),
% 0.20/0.45 inference(monotonicity,[status(thm)],[147])).
% 0.20/0.45 tff(149,plain,
% 0.20/0.45 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2))))),
% 0.20/0.45 inference(transitivity,[status(thm)],[148, 146])).
% 0.20/0.45 tff(150,plain,
% 0.20/0.45 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | (~distinct_lines(Y!1, X!2)) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2))))),
% 0.20/0.45 inference(quant_inst,[status(thm)],[])).
% 0.20/0.45 tff(151,plain,
% 0.20/0.45 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(intersection_point(X!2, Y!1), Y!1) | apart_point_and_line(intersection_point(X!2, Y!1), X!2) | (~distinct_lines(Y!1, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Y!1) | (~distinct_points(intersection_point(X!2, Y!1), intersection_point(Z!0, X!2)))),
% 0.20/0.45 inference(modus_ponens,[status(thm)],[150, 149])).
% 0.20/0.45 tff(152,plain,
% 0.20/0.45 ($false),
% 0.20/0.45 inference(unit_resolution,[status(thm)],[151, 145, 132, 29, 128, 98, 84, 83])).
% 0.20/0.45 tff(153,plain,(apart_point_and_line(intersection_point(Z!0, X!2), Y!1)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.45 tff(154,assumption,(~distinct_lines(Y!1, Z!0)), introduced(assumption)).
% 0.20/0.45 tff(155,plain,
% 0.20/0.45 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Y!1, Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(Y!1, Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)))),
% 0.20/0.45 inference(rewrite,[status(thm)],[])).
% 0.20/0.45 tff(156,plain,
% 0.20/0.45 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(Y!1, Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(157,plain,
% 0.20/0.46 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(Y!1, Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[156, 155])).
% 0.20/0.46 tff(158,plain,
% 0.20/0.46 ($false),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[157, 40, 154, 65, 153])).
% 0.20/0.46 tff(159,plain,(distinct_lines(Y!1, Z!0)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.46 tff(160,plain,
% 0.20/0.46 (((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(Y!1, Z!0)) | convergent_lines(Y!1, Z!0))) <=> ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(Y!1, Z!0)) | convergent_lines(Y!1, Z!0))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(161,plain,
% 0.20/0.46 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | ((~distinct_lines(Y!1, Z!0)) | convergent_lines(Y!1, Z!0))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(162,plain,
% 0.20/0.46 ((~![X: $i, Y: $i] : ((~distinct_lines(X, Y)) | convergent_lines(X, Y))) | (~distinct_lines(Y!1, Z!0)) | convergent_lines(Y!1, Z!0)),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[161, 160])).
% 0.20/0.46 tff(163,plain,
% 0.20/0.46 (convergent_lines(Y!1, Z!0)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[162, 57, 159])).
% 0.20/0.46 tff(164,plain,
% 0.20/0.46 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Z!0)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Z!0)))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(165,plain,
% 0.20/0.46 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Z!0)))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(166,plain,
% 0.20/0.46 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Z!0))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[165, 164])).
% 0.20/0.46 tff(167,plain,
% 0.20/0.46 (~apart_point_and_line(intersection_point(Y!1, Z!0), Z!0)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[166, 94, 163])).
% 0.20/0.46 tff(168,plain,
% 0.20/0.46 ((~(~((~distinct_lines(X!2, Z!0)) | (~distinct_lines(Y!1, Z!0))))) <=> ((~distinct_lines(X!2, Z!0)) | (~distinct_lines(Y!1, Z!0)))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(169,plain,
% 0.20/0.46 ((distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0)) <=> (~((~distinct_lines(X!2, Z!0)) | (~distinct_lines(Y!1, Z!0))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(170,plain,
% 0.20/0.46 ((~(distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0))) <=> (~(~((~distinct_lines(X!2, Z!0)) | (~distinct_lines(Y!1, Z!0)))))),
% 0.20/0.46 inference(monotonicity,[status(thm)],[169])).
% 0.20/0.46 tff(171,plain,
% 0.20/0.46 ((~(distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0))) <=> ((~distinct_lines(X!2, Z!0)) | (~distinct_lines(Y!1, Z!0)))),
% 0.20/0.46 inference(transitivity,[status(thm)],[170, 168])).
% 0.20/0.46 tff(172,plain,
% 0.20/0.46 (~(distinct_lines(X!2, Z!0) & distinct_lines(Y!1, Z!0))),
% 0.20/0.46 inference(or_elim,[status(thm)],[14])).
% 0.20/0.46 tff(173,plain,
% 0.20/0.46 ((~distinct_lines(X!2, Z!0)) | (~distinct_lines(Y!1, Z!0))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[172, 171])).
% 0.20/0.46 tff(174,plain,
% 0.20/0.46 (~distinct_lines(X!2, Z!0)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[173, 159])).
% 0.20/0.46 tff(175,plain,
% 0.20/0.46 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(X!2, Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~apart_point_and_line(intersection_point(Y!1, 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(X!2, Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2)))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(176,plain,
% 0.20/0.46 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(X!2, Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2)))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(177,plain,
% 0.20/0.46 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | distinct_lines(X!2, Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[176, 175])).
% 0.20/0.46 tff(178,plain,
% 0.20/0.46 (~apart_point_and_line(intersection_point(Y!1, Z!0), X!2)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[177, 40, 174, 167])).
% 0.20/0.46 tff(179,plain,
% 0.20/0.46 (((~![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_lines(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_lines(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(180,plain,
% 0.20/0.46 ((apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~distinct_lines(Z!0, X!2)) | (~distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0)))) <=> ((~distinct_lines(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0))))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(181,plain,
% 0.20/0.46 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~distinct_lines(Z!0, X!2)) | (~distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | ((~distinct_lines(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0)))))),
% 0.20/0.46 inference(monotonicity,[status(thm)],[180])).
% 0.20/0.46 tff(182,plain,
% 0.20/0.46 (((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~distinct_lines(Z!0, X!2)) | (~distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (~distinct_lines(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0))))),
% 0.20/0.46 inference(transitivity,[status(thm)],[181, 179])).
% 0.20/0.46 tff(183,plain,
% 0.20/0.46 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | (~distinct_lines(Z!0, X!2)) | (~distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0))))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(184,plain,
% 0.20/0.46 ((~![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_lines(Z!0, X!2)) | apart_point_and_line(intersection_point(Z!0, X!2), X!2) | apart_point_and_line(intersection_point(Z!0, X!2), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), Z!0) | apart_point_and_line(intersection_point(Y!1, Z!0), X!2) | (~distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0)))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[183, 182])).
% 0.20/0.46 tff(185,plain,
% 0.20/0.46 (~distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0))),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[184, 145, 47, 98, 65, 167, 178])).
% 0.20/0.46 tff(186,plain,
% 0.20/0.46 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Y!1)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Y!1)))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(187,plain,
% 0.20/0.46 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Y!1)))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(188,plain,
% 0.20/0.46 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(Y!1, Z!0)) | (~apart_point_and_line(intersection_point(Y!1, Z!0), Y!1))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[187, 186])).
% 0.20/0.46 tff(189,plain,
% 0.20/0.46 (~apart_point_and_line(intersection_point(Y!1, Z!0), Y!1)),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[188, 25, 163])).
% 0.20/0.46 tff(190,plain,
% 0.20/0.46 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0)))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(191,plain,
% 0.20/0.46 ((distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1))) <=> ((~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0)))),
% 0.20/0.46 inference(rewrite,[status(thm)],[])).
% 0.20/0.46 tff(192,plain,
% 0.20/0.46 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | (~apart_point_and_line(intersection_point(Z!0, 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(intersection_point(Z!0, X!2), Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0))))),
% 0.20/0.46 inference(monotonicity,[status(thm)],[191])).
% 0.20/0.46 tff(193,plain,
% 0.20/0.46 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | (~apart_point_and_line(intersection_point(Z!0, 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(intersection_point(Z!0, X!2), Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0)))),
% 0.20/0.46 inference(transitivity,[status(thm)],[192, 190])).
% 0.20/0.46 tff(194,plain,
% 0.20/0.46 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)))),
% 0.20/0.46 inference(quant_inst,[status(thm)],[])).
% 0.20/0.46 tff(195,plain,
% 0.20/0.46 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(intersection_point(Z!0, X!2), Y!1)) | apart_point_and_line(intersection_point(Y!1, Z!0), Y!1) | distinct_points(intersection_point(Z!0, X!2), intersection_point(Y!1, Z!0))),
% 0.20/0.46 inference(modus_ponens,[status(thm)],[194, 193])).
% 0.20/0.46 tff(196,plain,
% 0.20/0.46 ($false),
% 0.20/0.46 inference(unit_resolution,[status(thm)],[195, 75, 153, 189, 185])).
% 0.20/0.46 % SZS output end Proof
%------------------------------------------------------------------------------