%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GEO188+2 : TPTP v9.0.0. Released v3.3.0.
% 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 05:05:50 AM UTC 2025
% Result : Theorem 0.19s 0.40s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GEO188+2 : TPTP v9.0.0. Released v3.3.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n014.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Thu Jun 19 14:40:39 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.19/0.40 % SZS status Theorem
% 0.19/0.40 % SZS output start Proof
% 0.19/0.40 tff(distinct_points_type, type, (
% 0.19/0.40 distinct_points: ( $i * $i ) > $o)).
% 0.19/0.40 tff(tptp_fun_Z_0_type, type, (
% 0.19/0.40 tptp_fun_Z_0: $i)).
% 0.19/0.40 tff(apart_point_and_line_type, type, (
% 0.19/0.40 apart_point_and_line: ( $i * $i ) > $o)).
% 0.19/0.40 tff(line_connecting_type, type, (
% 0.19/0.40 line_connecting: ( $i * $i ) > $i)).
% 0.19/0.40 tff(tptp_fun_Y_1_type, type, (
% 0.19/0.40 tptp_fun_Y_1: $i)).
% 0.19/0.40 tff(tptp_fun_X_2_type, type, (
% 0.19/0.40 tptp_fun_X_2: $i)).
% 0.19/0.40 tff(distinct_lines_type, type, (
% 0.19/0.40 distinct_lines: ( $i * $i ) > $o)).
% 0.19/0.40 tff(1,plain,
% 0.19/0.40 (^[X: $i] : refl((~distinct_points(X, X)) <=> (~distinct_points(X, X)))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(2,plain,
% 0.19/0.40 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.40 tff(3,plain,
% 0.19/0.40 (![X: $i] : (~distinct_points(X, X)) <=> ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(4,axiom,(![X: $i] : (~distinct_points(X, X))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart1')).
% 0.19/0.40 tff(5,plain,
% 0.19/0.40 (![X: $i] : (~distinct_points(X, X))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.40 tff(6,plain,(
% 0.19/0.40 ![X: $i] : (~distinct_points(X, X))),
% 0.19/0.40 inference(skolemize,[status(sab)],[5])).
% 0.19/0.40 tff(7,plain,
% 0.19/0.40 (![X: $i] : (~distinct_points(X, X))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.40 tff(8,plain,
% 0.19/0.40 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(Y!1, Y!1))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(9,plain,
% 0.19/0.40 (~distinct_points(Y!1, Y!1)),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.19/0.40 tff(10,plain,
% 0.19/0.40 (((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0))) | distinct_points(Y!1, Y!1)),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(11,plain,
% 0.19/0.40 ((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[10, 9])).
% 0.19/0.40 tff(12,assumption,(~distinct_points(Z!0, Y!1)), introduced(assumption)).
% 0.19/0.40 tff(13,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(X, line_connecting(Z, Y))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y))))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(X, line_connecting(Z, Y))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y)))))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(14,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : ((((distinct_points(X, Y) & distinct_points(X, Z)) & distinct_points(Y, Z)) & (~apart_point_and_line(Z, line_connecting(X, Y)))) => (~apart_point_and_line(X, line_connecting(Z, Y))))) <=> (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(X, line_connecting(Z, Y))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y)))))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(15,axiom,(~![X: $i, Y: $i, Z: $i] : ((((distinct_points(X, Y) & distinct_points(X, Z)) & distinct_points(Y, Z)) & (~apart_point_and_line(Z, line_connecting(X, Y)))) => (~apart_point_and_line(X, line_connecting(Z, Y))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.19/0.40 tff(16,plain,
% 0.19/0.40 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(X, line_connecting(Z, Y))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.19/0.40 tff(17,plain,
% 0.19/0.40 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(X, line_connecting(Z, Y))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[16, 13])).
% 0.19/0.40 tff(18,plain,
% 0.19/0.40 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(X, line_connecting(Z, Y))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.19/0.40 tff(19,plain,
% 0.19/0.40 (~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(X, line_connecting(Z, Y))) | (~(distinct_points(X, Y) & distinct_points(X, Z) & distinct_points(Y, Z) & (~apart_point_and_line(Z, line_connecting(X, Y))))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[18, 13])).
% 0.19/0.40 tff(20,plain,(
% 0.19/0.40 ~((~apart_point_and_line(X!2, line_connecting(Z!0, Y!1))) | (~(distinct_points(X!2, Y!1) & distinct_points(X!2, Z!0) & distinct_points(Y!1, Z!0) & (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))))))),
% 0.19/0.40 inference(skolemize,[status(sab)],[19])).
% 0.19/0.40 tff(21,plain,
% 0.19/0.40 (distinct_points(X!2, Y!1) & distinct_points(X!2, Z!0) & distinct_points(Y!1, Z!0) & (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1)))),
% 0.19/0.40 inference(or_elim,[status(thm)],[20])).
% 0.19/0.40 tff(22,plain,
% 0.19/0.40 (distinct_points(Y!1, Z!0)),
% 0.19/0.40 inference(and_elim,[status(thm)],[21])).
% 0.19/0.40 tff(23,plain,
% 0.19/0.40 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(24,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[23])).
% 0.19/0.40 tff(25,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(26,plain,
% 0.19/0.40 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(X, Z) | distinct_points(Y, Z)) <=> (distinct_points(Y, Z) | distinct_points(X, Z))), ((distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> (distinct_points(X, Y) => (distinct_points(Y, Z) | distinct_points(X, Z))))), rewrite((distinct_points(X, Y) => (distinct_points(Y, Z) | distinct_points(X, Z))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))), ((distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(27,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[26])).
% 0.19/0.40 tff(28,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (distinct_points(X, Z) | distinct_points(Y, Z)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','apart4')).
% 0.19/0.40 tff(29,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.19/0.40 tff(30,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.19/0.40 tff(31,plain,(
% 0.19/0.40 ![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[30])).
% 0.19/0.40 tff(32,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[31, 24])).
% 0.19/0.40 tff(33,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(Y!1, Y!1) | (~distinct_points(Y!1, Z!0)) | distinct_points(Z!0, Y!1))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | distinct_points(Y!1, Y!1) | (~distinct_points(Y!1, Z!0)) | distinct_points(Z!0, Y!1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(34,plain,
% 0.19/0.40 ((distinct_points(Z!0, Y!1) | distinct_points(Y!1, Y!1) | (~distinct_points(Y!1, Z!0))) <=> (distinct_points(Y!1, Y!1) | (~distinct_points(Y!1, Z!0)) | distinct_points(Z!0, Y!1))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(35,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(Z!0, Y!1) | distinct_points(Y!1, Y!1) | (~distinct_points(Y!1, Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(Y!1, Y!1) | (~distinct_points(Y!1, Z!0)) | distinct_points(Z!0, Y!1)))),
% 0.19/0.40 inference(monotonicity,[status(thm)],[34])).
% 0.19/0.40 tff(36,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(Z!0, Y!1) | distinct_points(Y!1, Y!1) | (~distinct_points(Y!1, Z!0)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | distinct_points(Y!1, Y!1) | (~distinct_points(Y!1, Z!0)) | distinct_points(Z!0, Y!1))),
% 0.19/0.40 inference(transitivity,[status(thm)],[35, 33])).
% 0.19/0.40 tff(37,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | (distinct_points(Z!0, Y!1) | distinct_points(Y!1, Y!1) | (~distinct_points(Y!1, Z!0)))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(38,plain,
% 0.19/0.40 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(Y, Z) | distinct_points(X, Z) | (~distinct_points(X, Y)))) | distinct_points(Y!1, Y!1) | (~distinct_points(Y!1, Z!0)) | distinct_points(Z!0, Y!1)),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[37, 36])).
% 0.19/0.40 tff(39,plain,
% 0.19/0.40 ($false),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[38, 32, 22, 9, 12])).
% 0.19/0.40 tff(40,plain,(distinct_points(Z!0, Y!1)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.40 tff(41,assumption,(apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))), introduced(assumption)).
% 0.19/0.40 tff(42,plain,
% 0.19/0.40 (^[X: $i, Y: $i, Z: $i] : refl(((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))) <=> ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(43,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))) <=> ![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[42])).
% 0.19/0.40 tff(44,plain,
% 0.19/0.40 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(Z, X) & distinct_points(Z, Y)) <=> (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y))))), (((distinct_points(Z, X) & distinct_points(Z, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y))) <=> ((~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y))))), rewrite(((~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y))) <=> ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))), (((distinct_points(Z, X) & distinct_points(Z, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y))) <=> ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(45,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[44])).
% 0.19/0.40 tff(46,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(47,plain,
% 0.19/0.40 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y))) <=> ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))), ((distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> (distinct_points(X, Y) => ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))))), rewrite((distinct_points(X, Y) => ((~apart_point_and_line(Z, line_connecting(X, Y))) | (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)))), ((distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(48,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y)))) <=> ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[47])).
% 0.19/0.40 tff(49,axiom,(![X: $i, Y: $i, Z: $i] : (distinct_points(X, Y) => (apart_point_and_line(Z, line_connecting(X, Y)) => (distinct_points(Z, X) & distinct_points(Z, Y))))), file('/export/starexec/sandbox/benchmark/Axioms/GEO008+0.ax','con1')).
% 0.19/0.40 tff(50,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.19/0.40 tff(51,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[50, 46])).
% 0.19/0.40 tff(52,plain,(
% 0.19/0.40 ![X: $i, Y: $i, Z: $i] : ((distinct_points(Z, X) & distinct_points(Z, Y)) | (~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)))),
% 0.19/0.40 inference(skolemize,[status(sab)],[51])).
% 0.19/0.40 tff(53,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[52, 45])).
% 0.19/0.40 tff(54,plain,
% 0.19/0.40 (![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[53, 43])).
% 0.19/0.40 tff(55,plain,
% 0.19/0.40 (((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0)))) | (~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0)))) | (~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))))),
% 0.19/0.40 inference(rewrite,[status(thm)],[])).
% 0.19/0.40 tff(56,plain,
% 0.19/0.40 (((~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0))))) <=> ((~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0)))) | (~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(57,plain,
% 0.19/0.41 ((~((~distinct_points(Y!1, Z!0)) | (~distinct_points(Y!1, Y!1)))) <=> (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(58,plain,
% 0.19/0.41 (((~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Z!0)) | (~distinct_points(Y!1, Y!1))))) <=> ((~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0)))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[57])).
% 0.19/0.41 tff(59,plain,
% 0.19/0.41 (((~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Z!0)) | (~distinct_points(Y!1, Y!1))))) <=> ((~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0)))) | (~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[58, 56])).
% 0.19/0.41 tff(60,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Z!0)) | (~distinct_points(Y!1, Y!1)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0)))) | (~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[59])).
% 0.19/0.41 tff(61,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Z!0)) | (~distinct_points(Y!1, Y!1)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0)))) | (~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[60, 55])).
% 0.19/0.41 tff(62,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Z!0)) | (~distinct_points(Y!1, Y!1)))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(63,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Y!1, Y!1)) | (~distinct_points(Y!1, Z!0)))) | (~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.19/0.41 tff(64,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[63, 54, 41, 40, 11])).
% 0.19/0.41 tff(65,plain,(~apart_point_and_line(Y!1, line_connecting(Z!0, Y!1))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(66,assumption,(apart_point_and_line(X!2, line_connecting(X!2, Y!1))), introduced(assumption)).
% 0.19/0.41 tff(67,plain,
% 0.19/0.41 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(X!2, X!2))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(68,plain,
% 0.19/0.41 (~distinct_points(X!2, X!2)),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[67, 7])).
% 0.19/0.41 tff(69,plain,
% 0.19/0.41 (((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1))) | distinct_points(X!2, X!2)),
% 0.19/0.41 inference(tautology,[status(thm)],[])).
% 0.19/0.41 tff(70,plain,
% 0.19/0.41 ((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1))),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[69, 68])).
% 0.19/0.41 tff(71,plain,
% 0.19/0.41 (distinct_points(X!2, Y!1)),
% 0.19/0.41 inference(and_elim,[status(thm)],[21])).
% 0.19/0.41 tff(72,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!2, Y!1)) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!2, Y!1)) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(73,plain,
% 0.19/0.41 (((~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1)) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1))))) <=> ((~distinct_points(X!2, Y!1)) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(74,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1)) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(X!2, Y!1)) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1)))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[73])).
% 0.19/0.41 tff(75,plain,
% 0.19/0.41 (((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1)) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!2, Y!1)) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[74, 72])).
% 0.19/0.41 tff(76,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~apart_point_and_line(X!2, line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1)) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(77,plain,
% 0.19/0.41 ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(X!2, Y!1)) | (~((~distinct_points(X!2, X!2)) | (~distinct_points(X!2, Y!1)))) | (~apart_point_and_line(X!2, line_connecting(X!2, Y!1)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.19/0.41 tff(78,plain,
% 0.19/0.41 ($false),
% 0.19/0.41 inference(unit_resolution,[status(thm)],[77, 54, 71, 70, 66])).
% 0.19/0.41 tff(79,plain,(~apart_point_and_line(X!2, line_connecting(X!2, Y!1))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.41 tff(80,plain,
% 0.19/0.41 (apart_point_and_line(X!2, line_connecting(Z!0, Y!1))),
% 0.19/0.41 inference(or_elim,[status(thm)],[20])).
% 0.19/0.41 tff(81,plain,
% 0.19/0.41 (^[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))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(82,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[81])).
% 0.19/0.41 tff(83,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(84,plain,
% 0.19/0.41 (^[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)))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(85,plain,
% 0.19/0.41 (![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)))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[84])).
% 0.19/0.41 tff(86,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/GEO008+0.ax','ceq2')).
% 0.19/0.41 tff(87,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[86, 85])).
% 0.19/0.41 tff(88,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[87, 83])).
% 0.19/0.41 tff(89,plain,(
% 0.19/0.41 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.41 inference(skolemize,[status(sab)],[88])).
% 0.19/0.41 tff(90,plain,
% 0.19/0.41 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[89, 82])).
% 0.19/0.41 tff(91,plain,
% 0.19/0.41 (((~![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(X!2, line_connecting(Z!0, Y!1))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Z!0, Y!1), 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(X!2, line_connecting(Z!0, Y!1))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(92,plain,
% 0.19/0.41 ((distinct_lines(line_connecting(Z!0, 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(Z!0, Y!1)))) <=> ((~apart_point_and_line(X!2, line_connecting(Z!0, Y!1))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1)))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(93,plain,
% 0.19/0.41 (((~![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(Z!0, 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(Z!0, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | ((~apart_point_and_line(X!2, line_connecting(Z!0, Y!1))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[92])).
% 0.19/0.42 tff(94,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(Z!0, 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(Z!0, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(X!2, line_connecting(Z!0, Y!1))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[93, 91])).
% 0.19/0.42 tff(95,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (distinct_lines(line_connecting(Z!0, 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(Z!0, Y!1))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(96,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | apart_point_and_line(X, Z) | (~apart_point_and_line(X, Y)))) | (~apart_point_and_line(X!2, line_connecting(Z!0, Y!1))) | apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.19/0.42 tff(97,plain,
% 0.19/0.42 (apart_point_and_line(X!2, line_connecting(X!2, Y!1)) | distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[96, 90, 80])).
% 0.19/0.42 tff(98,plain,
% 0.19/0.42 (distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[97, 79])).
% 0.19/0.42 tff(99,plain,
% 0.19/0.42 (((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1))) | distinct_points(Y!1, Y!1)),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(100,plain,
% 0.19/0.42 ((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[99, 9])).
% 0.19/0.42 tff(101,assumption,(apart_point_and_line(Y!1, line_connecting(X!2, Y!1))), introduced(assumption)).
% 0.19/0.42 tff(102,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1)) | (~((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1)) | (~((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1)))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(103,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1)) | (~((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1)))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(104,plain,
% 0.19/0.42 ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))) | (~distinct_points(X!2, Y!1)) | (~((~distinct_points(Y!1, X!2)) | (~distinct_points(Y!1, Y!1))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[103, 102])).
% 0.19/0.42 tff(105,plain,
% 0.19/0.42 ($false),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[104, 54, 71, 101, 100])).
% 0.19/0.42 tff(106,plain,(~apart_point_and_line(Y!1, line_connecting(X!2, Y!1))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.42 tff(107,plain,
% 0.19/0.42 (~apart_point_and_line(Z!0, line_connecting(X!2, Y!1))),
% 0.19/0.42 inference(and_elim,[status(thm)],[21])).
% 0.19/0.42 tff(108,plain,
% 0.19/0.42 (^[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_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_points(X, Y)) | (~distinct_lines(U, V))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(109,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_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)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[108])).
% 0.19/0.42 tff(110,plain,
% 0.19/0.42 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(rewrite((distinct_points(X, Y) & distinct_lines(U, V)) <=> (~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> (~(~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))))), rewrite((~(~((~distinct_points(X, Y)) | (~distinct_lines(U, V))))) <=> ((~distinct_points(X, Y)) | (~distinct_lines(U, V)))), ((~(distinct_points(X, Y) & distinct_lines(U, 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) | (~(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_points(X, Y)) | (~distinct_lines(U, V)))))), 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_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_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_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_points(X, Y)) | (~distinct_lines(U, V)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(111,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_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)))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[110])).
% 0.19/0.42 tff(112,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_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))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(113,plain,
% 0.19/0.42 (^[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))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(114,plain,
% 0.19/0.42 (![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))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[113])).
% 0.19/0.42 tff(115,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/GEO008+0.ax','cu1')).
% 0.19/0.42 tff(116,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_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.19/0.42 tff(117,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_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[116, 112])).
% 0.19/0.42 tff(118,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_points(X, Y) & distinct_lines(U, V))))),
% 0.19/0.42 inference(skolemize,[status(sab)],[117])).
% 0.19/0.42 tff(119,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_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[118, 111])).
% 0.19/0.42 tff(120,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_points(X, Y)) | (~distinct_lines(U, V)))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[119, 109])).
% 0.19/0.42 tff(121,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_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))) | apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~distinct_points(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_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))) | apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~distinct_points(Y!1, Z!0)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(122,plain,
% 0.19/0.42 ((apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | (~distinct_points(Y!1, Z!0)) | (~distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1)))) <=> (apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))) | apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~distinct_points(Y!1, Z!0)))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(123,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_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | (~distinct_points(Y!1, Z!0)) | (~distinct_lines(line_connecting(Z!0, 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_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))) | apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~distinct_points(Y!1, Z!0))))),
% 0.19/0.42 inference(monotonicity,[status(thm)],[122])).
% 0.19/0.42 tff(124,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_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | (~distinct_points(Y!1, Z!0)) | (~distinct_lines(line_connecting(Z!0, 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_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))) | apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~distinct_points(Y!1, Z!0)))),
% 0.19/0.42 inference(transitivity,[status(thm)],[123, 121])).
% 0.19/0.42 tff(125,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_points(X, Y)) | (~distinct_lines(U, V)))) | (apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | (~distinct_points(Y!1, Z!0)) | (~distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))))),
% 0.19/0.42 inference(quant_inst,[status(thm)],[])).
% 0.19/0.42 tff(126,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_points(X, Y)) | (~distinct_lines(U, V)))) | apart_point_and_line(Z!0, line_connecting(X!2, Y!1)) | apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))) | apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)) | (~distinct_points(Y!1, Z!0))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.19/0.42 tff(127,plain,
% 0.19/0.42 (apart_point_and_line(Y!1, line_connecting(X!2, Y!1)) | (~distinct_lines(line_connecting(Z!0, Y!1), line_connecting(X!2, Y!1))) | apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[126, 120, 22, 107])).
% 0.19/0.42 tff(128,plain,
% 0.19/0.42 (apart_point_and_line(Y!1, line_connecting(Z!0, Y!1)) | apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[127, 106, 98])).
% 0.19/0.42 tff(129,plain,
% 0.19/0.42 (apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[128, 65])).
% 0.19/0.42 tff(130,assumption,(~distinct_points(Z!0, Z!0)), introduced(assumption)).
% 0.19/0.42 tff(131,plain,
% 0.19/0.42 (((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1))) | distinct_points(Z!0, Z!0)),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(132,plain,
% 0.19/0.42 ((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[131, 130])).
% 0.19/0.42 tff(133,assumption,(apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))), introduced(assumption)).
% 0.19/0.42 tff(134,plain,
% 0.19/0.42 (((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(135,plain,
% 0.19/0.42 (((~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1))))) <=> ((~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(136,plain,
% 0.19/0.42 ((~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))) <=> (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(137,plain,
% 0.19/0.42 (((~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1))))) <=> ((~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[136])).
% 0.19/0.43 tff(138,plain,
% 0.19/0.43 (((~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1))))) <=> ((~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))),
% 0.19/0.43 inference(transitivity,[status(thm)],[137, 135])).
% 0.19/0.43 tff(139,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[138])).
% 0.19/0.43 tff(140,plain,
% 0.19/0.43 (((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))))) <=> ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))))),
% 0.19/0.43 inference(transitivity,[status(thm)],[139, 134])).
% 0.19/0.43 tff(141,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | ((~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(142,plain,
% 0.19/0.43 ((~![X: $i, Y: $i, Z: $i] : ((~apart_point_and_line(Z, line_connecting(X, Y))) | (~distinct_points(X, Y)) | (~((~distinct_points(Z, X)) | (~distinct_points(Z, Y)))))) | (~distinct_points(Z!0, Y!1)) | (~((~distinct_points(Z!0, Z!0)) | (~distinct_points(Z!0, Y!1)))) | (~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[141, 140])).
% 0.19/0.43 tff(143,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[142, 54, 133, 40, 132])).
% 0.19/0.43 tff(144,plain,((~apart_point_and_line(Z!0, line_connecting(Z!0, Y!1))) | distinct_points(Z!0, Z!0)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.43 tff(145,plain,
% 0.19/0.43 (distinct_points(Z!0, Z!0)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[144, 129])).
% 0.19/0.43 tff(146,plain,
% 0.19/0.43 ((~![X: $i] : (~distinct_points(X, X))) | (~distinct_points(Z!0, Z!0))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(147,plain,
% 0.19/0.43 ($false),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[146, 7, 145])).
% 0.19/0.43 % SZS output end Proof
% 0.19/0.43 % E exiting
%------------------------------------------------------------------------------