TSTP Solution File: GEO191+3 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GEO191+3 : TPTP v8.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% 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 : Fri Sep 16 20:35:28 EDT 2022
% Result : Theorem 0.19s 0.50s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : GEO191+3 : TPTP v8.1.0. Released v4.0.0.
% 0.04/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n014.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Wed Aug 31 06:51:17 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34 Usage: tptp [options] [-file:]file
% 0.13/0.34 -h, -? prints this message.
% 0.13/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.34 -m, -model generate model.
% 0.13/0.34 -p, -proof generate proof.
% 0.13/0.34 -c, -core generate unsat core of named formulas.
% 0.13/0.34 -st, -statistics display statistics.
% 0.13/0.34 -t:timeout set timeout (in second).
% 0.13/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34 -<param>:<value> configuration parameter and value.
% 0.13/0.34 -o:<output-file> file to place output in.
% 0.19/0.50 % SZS status Theorem
% 0.19/0.50 % SZS output start Proof
% 0.19/0.50 tff(apart_point_and_line_type, type, (
% 0.19/0.50 apart_point_and_line: ( $i * $i ) > $o)).
% 0.19/0.50 tff(tptp_fun_V_0_type, type, (
% 0.19/0.50 tptp_fun_V_0: $i)).
% 0.19/0.50 tff(intersection_point_type, type, (
% 0.19/0.50 intersection_point: ( $i * $i ) > $i)).
% 0.19/0.50 tff(tptp_fun_Y_2_type, type, (
% 0.19/0.50 tptp_fun_Y_2: $i)).
% 0.19/0.50 tff(tptp_fun_X_3_type, type, (
% 0.19/0.50 tptp_fun_X_3: $i)).
% 0.19/0.50 tff(distinct_points_type, type, (
% 0.19/0.50 distinct_points: ( $i * $i ) > $o)).
% 0.19/0.50 tff(tptp_fun_U_1_type, type, (
% 0.19/0.50 tptp_fun_U_1: $i)).
% 0.19/0.50 tff(distinct_lines_type, type, (
% 0.19/0.50 distinct_lines: ( $i * $i ) > $o)).
% 0.19/0.50 tff(convergent_lines_type, type, (
% 0.19/0.50 convergent_lines: ( $i * $i ) > $o)).
% 0.19/0.50 tff(unorthogonal_lines_type, type, (
% 0.19/0.50 unorthogonal_lines: ( $i * $i ) > $o)).
% 0.19/0.50 tff(orthogonal_through_point_type, type, (
% 0.19/0.50 orthogonal_through_point: ( $i * $i ) > $i)).
% 0.19/0.50 tff(line_connecting_type, type, (
% 0.19/0.50 line_connecting: ( $i * $i ) > $i)).
% 0.19/0.50 tff(1,plain,
% 0.19/0.50 (^[A: $i, L: $i] : refl((~unorthogonal_lines(orthogonal_through_point(L, A), L)) <=> (~unorthogonal_lines(orthogonal_through_point(L, A), L)))),
% 0.19/0.50 inference(bind,[status(th)],[])).
% 0.19/0.50 tff(2,plain,
% 0.19/0.50 (![A: $i, L: $i] : (~unorthogonal_lines(orthogonal_through_point(L, A), L)) <=> ![A: $i, L: $i] : (~unorthogonal_lines(orthogonal_through_point(L, A), L))),
% 0.19/0.50 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.50 tff(3,plain,
% 0.19/0.50 (![A: $i, L: $i] : (~unorthogonal_lines(orthogonal_through_point(L, A), L)) <=> ![A: $i, L: $i] : (~unorthogonal_lines(orthogonal_through_point(L, A), L))),
% 0.19/0.50 inference(rewrite,[status(thm)],[])).
% 0.19/0.50 tff(4,axiom,(![A: $i, L: $i] : (~unorthogonal_lines(orthogonal_through_point(L, A), L))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+3.ax','ooc1')).
% 0.19/0.50 tff(5,plain,
% 0.19/0.50 (![A: $i, L: $i] : (~unorthogonal_lines(orthogonal_through_point(L, A), L))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.19/0.50 tff(6,plain,(
% 0.19/0.50 ![A: $i, L: $i] : (~unorthogonal_lines(orthogonal_through_point(L, A), L))),
% 0.19/0.50 inference(skolemize,[status(sab)],[5])).
% 0.19/0.50 tff(7,plain,
% 0.19/0.50 (![A: $i, L: $i] : (~unorthogonal_lines(orthogonal_through_point(L, A), L))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.19/0.50 tff(8,plain,
% 0.19/0.50 ((~![A: $i, L: $i] : (~unorthogonal_lines(orthogonal_through_point(L, A), L))) | (~unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3))),
% 0.19/0.50 inference(quant_inst,[status(thm)],[])).
% 0.19/0.50 tff(9,plain,
% 0.19/0.50 (~unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3)),
% 0.19/0.50 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.19/0.50 tff(10,plain,
% 0.19/0.50 (^[L: $i, M: $i, N: $i] : refl((unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N))) <=> (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N))))),
% 0.19/0.50 inference(bind,[status(th)],[])).
% 0.19/0.50 tff(11,plain,
% 0.19/0.50 (![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N))) <=> ![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))),
% 0.19/0.50 inference(quant_intro,[status(thm)],[10])).
% 0.19/0.50 tff(12,plain,
% 0.19/0.50 (^[L: $i, M: $i, N: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N))) <=> (~(unorthogonal_lines(L, N) | unorthogonal_lines(L, M)))), ((~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N)))) <=> (~(~(unorthogonal_lines(L, N) | unorthogonal_lines(L, M)))))), rewrite((~(~(unorthogonal_lines(L, N) | unorthogonal_lines(L, M)))) <=> (unorthogonal_lines(L, N) | unorthogonal_lines(L, M))), ((~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N)))) <=> (unorthogonal_lines(L, N) | unorthogonal_lines(L, M)))), (((~convergent_lines(M, N)) | (~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N))))) <=> ((~convergent_lines(M, N)) | (unorthogonal_lines(L, N) | unorthogonal_lines(L, M))))), rewrite(((~convergent_lines(M, N)) | (unorthogonal_lines(L, N) | unorthogonal_lines(L, M))) <=> (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))), (((~convergent_lines(M, N)) | (~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N))))) <=> (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))))),
% 0.19/0.50 inference(bind,[status(th)],[])).
% 0.19/0.50 tff(13,plain,
% 0.19/0.50 (![L: $i, M: $i, N: $i] : ((~convergent_lines(M, N)) | (~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N))))) <=> ![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))),
% 0.19/0.50 inference(quant_intro,[status(thm)],[12])).
% 0.19/0.50 tff(14,plain,
% 0.19/0.50 (![L: $i, M: $i, N: $i] : ((~convergent_lines(M, N)) | (~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N))))) <=> ![L: $i, M: $i, N: $i] : ((~convergent_lines(M, N)) | (~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N)))))),
% 0.19/0.50 inference(rewrite,[status(thm)],[])).
% 0.19/0.50 tff(15,plain,
% 0.19/0.50 (^[L: $i, M: $i, N: $i] : rewrite((((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N))) => (~convergent_lines(M, N))) <=> ((~convergent_lines(M, N)) | (~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N))))))),
% 0.19/0.50 inference(bind,[status(th)],[])).
% 0.19/0.50 tff(16,plain,
% 0.19/0.50 (![L: $i, M: $i, N: $i] : (((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N))) => (~convergent_lines(M, N))) <=> ![L: $i, M: $i, N: $i] : ((~convergent_lines(M, N)) | (~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N)))))),
% 0.19/0.50 inference(quant_intro,[status(thm)],[15])).
% 0.19/0.50 tff(17,axiom,(![L: $i, M: $i, N: $i] : (((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N))) => (~convergent_lines(M, N)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+4.ax','couo1')).
% 0.19/0.50 tff(18,plain,
% 0.19/0.50 (![L: $i, M: $i, N: $i] : ((~convergent_lines(M, N)) | (~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N)))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[17, 16])).
% 0.19/0.50 tff(19,plain,
% 0.19/0.50 (![L: $i, M: $i, N: $i] : ((~convergent_lines(M, N)) | (~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N)))))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.19/0.50 tff(20,plain,(
% 0.19/0.50 ![L: $i, M: $i, N: $i] : ((~convergent_lines(M, N)) | (~((~unorthogonal_lines(L, M)) & (~unorthogonal_lines(L, N)))))),
% 0.19/0.50 inference(skolemize,[status(sab)],[19])).
% 0.19/0.50 tff(21,plain,
% 0.19/0.50 (![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[20, 13])).
% 0.19/0.50 tff(22,plain,
% 0.19/0.50 (![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[21, 11])).
% 0.19/0.50 tff(23,plain,
% 0.19/0.50 (((~![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))) | (unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | (~convergent_lines(X!3, X!3)))) <=> ((~![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))) | unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | (~convergent_lines(X!3, X!3)))),
% 0.19/0.50 inference(rewrite,[status(thm)],[])).
% 0.19/0.50 tff(24,plain,
% 0.19/0.50 ((unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | (~convergent_lines(X!3, X!3))) <=> (unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | (~convergent_lines(X!3, X!3)))),
% 0.19/0.50 inference(rewrite,[status(thm)],[])).
% 0.19/0.50 tff(25,plain,
% 0.19/0.50 (((~![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))) | (unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | (~convergent_lines(X!3, X!3)))) <=> ((~![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))) | (unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | (~convergent_lines(X!3, X!3))))),
% 0.19/0.51 inference(monotonicity,[status(thm)],[24])).
% 0.19/0.51 tff(26,plain,
% 0.19/0.51 (((~![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))) | (unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | (~convergent_lines(X!3, X!3)))) <=> ((~![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))) | unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | (~convergent_lines(X!3, X!3)))),
% 0.19/0.51 inference(transitivity,[status(thm)],[25, 23])).
% 0.19/0.51 tff(27,plain,
% 0.19/0.51 ((~![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))) | (unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | (~convergent_lines(X!3, X!3)))),
% 0.19/0.51 inference(quant_inst,[status(thm)],[])).
% 0.19/0.51 tff(28,plain,
% 0.19/0.51 ((~![L: $i, M: $i, N: $i] : (unorthogonal_lines(L, N) | unorthogonal_lines(L, M) | (~convergent_lines(M, N)))) | unorthogonal_lines(orthogonal_through_point(X!3, intersection_point(V!0, line_connecting(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))), X!3) | (~convergent_lines(X!3, X!3))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.19/0.51 tff(29,plain,
% 0.19/0.51 (~convergent_lines(X!3, X!3)),
% 0.19/0.51 inference(unit_resolution,[status(thm)],[28, 22, 9])).
% 0.19/0.51 tff(30,assumption,(~distinct_lines(Y!2, X!3)), introduced(assumption)).
% 0.19/0.51 tff(31,plain,
% 0.19/0.51 ((~(apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(U!1, V!0), X!3) | (~(convergent_lines(X!3, Y!2) & convergent_lines(U!1, V!0) & (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1)))))) <=> (~(apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(U!1, V!0), X!3) | (~(convergent_lines(X!3, Y!2) & convergent_lines(U!1, V!0) & (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1))))))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(32,plain,
% 0.19/0.51 ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(intersection_point(U, V), Y) | apart_point_and_line(intersection_point(U, V), X) | (~(convergent_lines(X, Y) & convergent_lines(U, V) & (apart_point_and_line(intersection_point(X, Y), V) | apart_point_and_line(intersection_point(X, Y), U)))))) <=> (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(intersection_point(U, V), Y) | apart_point_and_line(intersection_point(U, V), X) | (~(convergent_lines(X, Y) & convergent_lines(U, V) & (apart_point_and_line(intersection_point(X, Y), V) | apart_point_and_line(intersection_point(X, Y), U))))))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(33,plain,
% 0.19/0.51 ((~![X: $i, Y: $i, U: $i, V: $i] : (((convergent_lines(X, Y) & convergent_lines(U, V)) & (apart_point_and_line(intersection_point(X, Y), U) | apart_point_and_line(intersection_point(X, Y), V))) => (apart_point_and_line(intersection_point(U, V), X) | apart_point_and_line(intersection_point(U, V), Y)))) <=> (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(intersection_point(U, V), Y) | apart_point_and_line(intersection_point(U, V), X) | (~(convergent_lines(X, Y) & convergent_lines(U, V) & (apart_point_and_line(intersection_point(X, Y), V) | apart_point_and_line(intersection_point(X, Y), U))))))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(34,axiom,(~![X: $i, Y: $i, U: $i, V: $i] : (((convergent_lines(X, Y) & convergent_lines(U, V)) & (apart_point_and_line(intersection_point(X, Y), U) | apart_point_and_line(intersection_point(X, Y), V))) => (apart_point_and_line(intersection_point(U, V), X) | apart_point_and_line(intersection_point(U, V), Y)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','con')).
% 0.19/0.51 tff(35,plain,
% 0.19/0.51 (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(intersection_point(U, V), Y) | apart_point_and_line(intersection_point(U, V), X) | (~(convergent_lines(X, Y) & convergent_lines(U, V) & (apart_point_and_line(intersection_point(X, Y), V) | apart_point_and_line(intersection_point(X, Y), U)))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.19/0.51 tff(36,plain,
% 0.19/0.51 (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(intersection_point(U, V), Y) | apart_point_and_line(intersection_point(U, V), X) | (~(convergent_lines(X, Y) & convergent_lines(U, V) & (apart_point_and_line(intersection_point(X, Y), V) | apart_point_and_line(intersection_point(X, Y), U)))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[35, 32])).
% 0.19/0.51 tff(37,plain,
% 0.19/0.51 (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(intersection_point(U, V), Y) | apart_point_and_line(intersection_point(U, V), X) | (~(convergent_lines(X, Y) & convergent_lines(U, V) & (apart_point_and_line(intersection_point(X, Y), V) | apart_point_and_line(intersection_point(X, Y), U)))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.19/0.51 tff(38,plain,
% 0.19/0.51 (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(intersection_point(U, V), Y) | apart_point_and_line(intersection_point(U, V), X) | (~(convergent_lines(X, Y) & convergent_lines(U, V) & (apart_point_and_line(intersection_point(X, Y), V) | apart_point_and_line(intersection_point(X, Y), U)))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[37, 32])).
% 0.19/0.51 tff(39,plain,
% 0.19/0.51 (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(intersection_point(U, V), Y) | apart_point_and_line(intersection_point(U, V), X) | (~(convergent_lines(X, Y) & convergent_lines(U, V) & (apart_point_and_line(intersection_point(X, Y), V) | apart_point_and_line(intersection_point(X, Y), U)))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[38, 32])).
% 0.19/0.51 tff(40,plain,
% 0.19/0.51 (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(intersection_point(U, V), Y) | apart_point_and_line(intersection_point(U, V), X) | (~(convergent_lines(X, Y) & convergent_lines(U, V) & (apart_point_and_line(intersection_point(X, Y), V) | apart_point_and_line(intersection_point(X, Y), U)))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[39, 32])).
% 0.19/0.51 tff(41,plain,
% 0.19/0.51 (~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(intersection_point(U, V), Y) | apart_point_and_line(intersection_point(U, V), X) | (~(convergent_lines(X, Y) & convergent_lines(U, V) & (apart_point_and_line(intersection_point(X, Y), V) | apart_point_and_line(intersection_point(X, Y), U)))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[40, 32])).
% 0.19/0.51 tff(42,plain,(
% 0.19/0.51 ~(apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(U!1, V!0), X!3) | (~(convergent_lines(X!3, Y!2) & convergent_lines(U!1, V!0) & (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1)))))),
% 0.19/0.51 inference(skolemize,[status(sab)],[41])).
% 0.19/0.51 tff(43,plain,
% 0.19/0.51 (~(apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(U!1, V!0), X!3) | (~(convergent_lines(X!3, Y!2) & convergent_lines(U!1, V!0) & (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1)))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[42, 31])).
% 0.19/0.51 tff(44,plain,
% 0.19/0.51 (convergent_lines(X!3, Y!2) & convergent_lines(U!1, V!0) & (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1))),
% 0.19/0.51 inference(or_elim,[status(thm)],[43])).
% 0.19/0.51 tff(45,plain,
% 0.19/0.51 (convergent_lines(X!3, Y!2)),
% 0.19/0.51 inference(and_elim,[status(thm)],[44])).
% 0.19/0.51 tff(46,plain,
% 0.19/0.51 (^[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))))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(47,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[46])).
% 0.19/0.51 tff(48,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(49,plain,
% 0.19/0.51 (^[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)))))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(50,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[49])).
% 0.19/0.51 tff(51,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')).
% 0.19/0.51 tff(52,plain,
% 0.19/0.51 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[51, 50])).
% 0.19/0.51 tff(53,plain,
% 0.19/0.51 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[52, 48])).
% 0.19/0.51 tff(54,plain,(
% 0.19/0.51 ![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.51 inference(skolemize,[status(sab)],[53])).
% 0.19/0.51 tff(55,plain,
% 0.19/0.51 (![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[54, 47])).
% 0.19/0.51 tff(56,plain,
% 0.19/0.51 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!3, Y!2)) | distinct_lines(Y!2, X!3) | convergent_lines(X!3, X!3))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!3, Y!2)) | distinct_lines(Y!2, X!3) | convergent_lines(X!3, X!3))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(57,plain,
% 0.19/0.51 ((distinct_lines(Y!2, X!3) | convergent_lines(X!3, X!3) | (~convergent_lines(X!3, Y!2))) <=> ((~convergent_lines(X!3, Y!2)) | distinct_lines(Y!2, X!3) | convergent_lines(X!3, X!3))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(58,plain,
% 0.19/0.51 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!2, X!3) | convergent_lines(X!3, X!3) | (~convergent_lines(X!3, Y!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | ((~convergent_lines(X!3, Y!2)) | distinct_lines(Y!2, X!3) | convergent_lines(X!3, X!3)))),
% 0.19/0.51 inference(monotonicity,[status(thm)],[57])).
% 0.19/0.51 tff(59,plain,
% 0.19/0.51 (((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!2, X!3) | convergent_lines(X!3, X!3) | (~convergent_lines(X!3, Y!2)))) <=> ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!3, Y!2)) | distinct_lines(Y!2, X!3) | convergent_lines(X!3, X!3))),
% 0.19/0.51 inference(transitivity,[status(thm)],[58, 56])).
% 0.19/0.51 tff(60,plain,
% 0.19/0.51 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (distinct_lines(Y!2, X!3) | convergent_lines(X!3, X!3) | (~convergent_lines(X!3, Y!2)))),
% 0.19/0.51 inference(quant_inst,[status(thm)],[])).
% 0.19/0.51 tff(61,plain,
% 0.19/0.51 ((~![X: $i, Y: $i, Z: $i] : (distinct_lines(Y, Z) | convergent_lines(X, Z) | (~convergent_lines(X, Y)))) | (~convergent_lines(X!3, Y!2)) | distinct_lines(Y!2, X!3) | convergent_lines(X!3, X!3)),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[60, 59])).
% 0.19/0.51 tff(62,plain,
% 0.19/0.51 ($false),
% 0.19/0.51 inference(unit_resolution,[status(thm)],[61, 55, 45, 30, 29])).
% 0.19/0.51 tff(63,plain,(distinct_lines(Y!2, X!3)), inference(lemma,lemma(discharge,[]))).
% 0.19/0.51 tff(64,plain,
% 0.19/0.51 (^[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))))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(65,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[64])).
% 0.19/0.51 tff(66,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(67,plain,
% 0.19/0.51 (^[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))))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(68,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[67])).
% 0.19/0.51 tff(69,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')).
% 0.19/0.51 tff(70,plain,
% 0.19/0.51 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.19/0.51 tff(71,plain,
% 0.19/0.51 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[70, 66])).
% 0.19/0.51 tff(72,plain,(
% 0.19/0.51 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.51 inference(skolemize,[status(sab)],[71])).
% 0.19/0.51 tff(73,plain,
% 0.19/0.51 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[72, 65])).
% 0.19/0.51 tff(74,plain,
% 0.19/0.51 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!3, Y!2)) | (~apart_point_and_line(intersection_point(X!3, Y!2), X!3)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!3, Y!2)) | (~apart_point_and_line(intersection_point(X!3, Y!2), X!3)))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(75,plain,
% 0.19/0.51 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(X!3, Y!2)) | (~apart_point_and_line(intersection_point(X!3, Y!2), X!3)))),
% 0.19/0.51 inference(quant_inst,[status(thm)],[])).
% 0.19/0.51 tff(76,plain,
% 0.19/0.51 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(X!3, Y!2)) | (~apart_point_and_line(intersection_point(X!3, Y!2), X!3))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.19/0.51 tff(77,plain,
% 0.19/0.51 (~apart_point_and_line(intersection_point(X!3, Y!2), X!3)),
% 0.19/0.51 inference(unit_resolution,[status(thm)],[76, 73, 45])).
% 0.19/0.51 tff(78,plain,
% 0.19/0.51 (^[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))))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(79,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[78])).
% 0.19/0.51 tff(80,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(81,plain,
% 0.19/0.51 (^[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))))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(82,plain,
% 0.19/0.51 (![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)))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[81])).
% 0.19/0.51 tff(83,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')).
% 0.19/0.51 tff(84,plain,
% 0.19/0.51 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.19/0.51 tff(85,plain,
% 0.19/0.51 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[84, 80])).
% 0.19/0.51 tff(86,plain,(
% 0.19/0.51 ![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.51 inference(skolemize,[status(sab)],[85])).
% 0.19/0.51 tff(87,plain,
% 0.19/0.51 (![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[86, 79])).
% 0.19/0.51 tff(88,plain,
% 0.19/0.51 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!3, Y!2)) | (~apart_point_and_line(intersection_point(X!3, Y!2), Y!2)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(X!3, Y!2)) | (~apart_point_and_line(intersection_point(X!3, Y!2), Y!2)))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(89,plain,
% 0.19/0.51 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(X!3, Y!2)) | (~apart_point_and_line(intersection_point(X!3, Y!2), Y!2)))),
% 0.19/0.51 inference(quant_inst,[status(thm)],[])).
% 0.19/0.51 tff(90,plain,
% 0.19/0.51 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(X!3, Y!2)) | (~apart_point_and_line(intersection_point(X!3, Y!2), Y!2))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.19/0.51 tff(91,plain,
% 0.19/0.51 (~apart_point_and_line(intersection_point(X!3, Y!2), Y!2)),
% 0.19/0.51 inference(unit_resolution,[status(thm)],[90, 87, 45])).
% 0.19/0.51 tff(92,plain,
% 0.19/0.51 (~apart_point_and_line(intersection_point(U!1, V!0), X!3)),
% 0.19/0.51 inference(or_elim,[status(thm)],[43])).
% 0.19/0.51 tff(93,plain,
% 0.19/0.51 (~apart_point_and_line(intersection_point(U!1, V!0), Y!2)),
% 0.19/0.51 inference(or_elim,[status(thm)],[43])).
% 0.19/0.51 tff(94,plain,
% 0.19/0.51 (^[X: $i, Y: $i, U: $i, V: $i] : refl((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y))))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(95,plain,
% 0.19/0.51 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[94])).
% 0.19/0.51 tff(96,plain,
% 0.19/0.51 (^[X: $i, Y: $i, U: $i, V: $i] : trans(monotonicity(trans(monotonicity(rewrite((distinct_points(X, Y) & distinct_lines(U, V)) <=> (~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> (~(~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))))), rewrite((~(~((~distinct_lines(U, V)) | (~distinct_points(X, Y))))) <=> ((~distinct_lines(U, V)) | (~distinct_points(X, Y)))), ((~(distinct_points(X, Y) & distinct_lines(U, V))) <=> ((~distinct_lines(U, V)) | (~distinct_points(X, Y))))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_lines(U, V)) | (~distinct_points(X, Y)))))), rewrite((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | ((~distinct_lines(U, V)) | (~distinct_points(X, Y)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))), ((apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(97,plain,
% 0.19/0.51 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~(distinct_points(X, Y) & distinct_lines(U, V)))) <=> ![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[96])).
% 0.19/0.51 tff(98,plain,
% 0.19/0.51 (![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))))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(99,plain,
% 0.19/0.51 (^[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))))))),
% 0.19/0.51 inference(bind,[status(th)],[])).
% 0.19/0.51 tff(100,plain,
% 0.19/0.51 (![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))))),
% 0.19/0.51 inference(quant_intro,[status(thm)],[99])).
% 0.19/0.51 tff(101,axiom,(![X: $i, Y: $i, U: $i, V: $i] : ((distinct_points(X, Y) & distinct_lines(U, V)) => (((apart_point_and_line(X, U) | apart_point_and_line(X, V)) | apart_point_and_line(Y, U)) | apart_point_and_line(Y, V)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','cu1')).
% 0.19/0.51 tff(102,plain,
% 0.19/0.51 (![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))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.19/0.51 tff(103,plain,
% 0.19/0.51 (![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))))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[102, 98])).
% 0.19/0.51 tff(104,plain,(
% 0.19/0.51 ![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))))),
% 0.19/0.51 inference(skolemize,[status(sab)],[103])).
% 0.19/0.51 tff(105,plain,
% 0.19/0.51 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[104, 97])).
% 0.19/0.51 tff(106,plain,
% 0.19/0.51 (![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))),
% 0.19/0.51 inference(modus_ponens,[status(thm)],[105, 95])).
% 0.19/0.51 tff(107,plain,
% 0.19/0.51 (((~![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(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | (~distinct_lines(Y!2, X!3)) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | (~distinct_lines(Y!2, X!3)) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0))))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(108,plain,
% 0.19/0.51 ((apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | (~distinct_lines(Y!2, X!3)) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))) <=> (apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | (~distinct_lines(Y!2, X!3)) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0))))),
% 0.19/0.51 inference(rewrite,[status(thm)],[])).
% 0.19/0.51 tff(109,plain,
% 0.19/0.51 (((~![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(U!1, V!0), X!3) | apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | (~distinct_lines(Y!2, X!3)) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | (apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | (~distinct_lines(Y!2, X!3)) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))))),
% 0.19/0.51 inference(monotonicity,[status(thm)],[108])).
% 0.19/0.51 tff(110,plain,
% 0.19/0.51 (((~![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(U!1, V!0), X!3) | apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | (~distinct_lines(Y!2, X!3)) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0))))) <=> ((~![X: $i, Y: $i, U: $i, V: $i] : (apart_point_and_line(Y, V) | apart_point_and_line(Y, U) | apart_point_and_line(X, V) | apart_point_and_line(X, U) | (~distinct_lines(U, V)) | (~distinct_points(X, Y)))) | apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | (~distinct_lines(Y!2, X!3)) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0))))),
% 0.19/0.52 inference(transitivity,[status(thm)],[109, 107])).
% 0.19/0.52 tff(111,plain,
% 0.19/0.52 ((~![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(U!1, V!0), X!3) | apart_point_and_line(intersection_point(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | (~distinct_lines(Y!2, X!3)) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0))))),
% 0.19/0.52 inference(quant_inst,[status(thm)],[])).
% 0.19/0.52 tff(112,plain,
% 0.19/0.52 ((~![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(U!1, V!0), Y!2) | apart_point_and_line(intersection_point(U!1, V!0), X!3) | apart_point_and_line(intersection_point(X!3, Y!2), Y!2) | apart_point_and_line(intersection_point(X!3, Y!2), X!3) | (~distinct_lines(Y!2, X!3)) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.19/0.52 tff(113,plain,
% 0.19/0.52 ((~distinct_lines(Y!2, X!3)) | (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)))),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[112, 106, 93, 92, 91, 77])).
% 0.19/0.52 tff(114,plain,
% 0.19/0.52 (~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0))),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[113, 63])).
% 0.19/0.52 tff(115,assumption,(~distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0))), introduced(assumption)).
% 0.19/0.52 tff(116,plain,
% 0.19/0.52 (convergent_lines(U!1, V!0)),
% 0.19/0.52 inference(and_elim,[status(thm)],[44])).
% 0.19/0.52 tff(117,plain,
% 0.19/0.52 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), V!0)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), V!0)))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(118,plain,
% 0.19/0.52 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | ((~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), V!0)))),
% 0.19/0.52 inference(quant_inst,[status(thm)],[])).
% 0.19/0.52 tff(119,plain,
% 0.19/0.52 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), Y)))) | (~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), V!0))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[118, 117])).
% 0.19/0.52 tff(120,plain,
% 0.19/0.52 (~apart_point_and_line(intersection_point(U!1, V!0), V!0)),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[119, 87, 116])).
% 0.19/0.52 tff(121,assumption,(apart_point_and_line(intersection_point(X!3, Y!2), V!0)), introduced(assumption)).
% 0.19/0.52 tff(122,plain,
% 0.19/0.52 (^[X: $i, Y: $i, Z: $i] : refl((distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))))),
% 0.19/0.52 inference(bind,[status(th)],[])).
% 0.19/0.52 tff(123,plain,
% 0.19/0.52 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.52 inference(quant_intro,[status(thm)],[122])).
% 0.19/0.52 tff(124,plain,
% 0.19/0.52 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(125,plain,
% 0.19/0.52 (^[X: $i, Y: $i, Z: $i] : trans(monotonicity(rewrite((distinct_points(X, Z) | apart_point_and_line(Z, Y)) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y))), ((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))))), rewrite((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))), ((apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))))),
% 0.19/0.52 inference(bind,[status(th)],[])).
% 0.19/0.52 tff(126,plain,
% 0.19/0.52 (![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y))) <=> ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.52 inference(quant_intro,[status(thm)],[125])).
% 0.19/0.52 tff(127,axiom,(![X: $i, Y: $i, Z: $i] : (apart_point_and_line(X, Y) => (distinct_points(X, Z) | apart_point_and_line(Z, Y)))), file('/export/starexec/sandbox/benchmark/Axioms/GEO006+0.ax','ceq1')).
% 0.19/0.52 tff(128,plain,
% 0.19/0.52 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[127, 126])).
% 0.19/0.52 tff(129,plain,
% 0.19/0.52 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[128, 124])).
% 0.19/0.52 tff(130,plain,(
% 0.19/0.52 ![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.52 inference(skolemize,[status(sab)],[129])).
% 0.19/0.52 tff(131,plain,
% 0.19/0.52 (![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[130, 123])).
% 0.19/0.52 tff(132,plain,
% 0.19/0.52 (((~![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(U!1, V!0), V!0) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), V!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(U!1, V!0), V!0) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), V!0)))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(133,plain,
% 0.19/0.52 ((distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~apart_point_and_line(intersection_point(X!3, Y!2), V!0))) <=> (apart_point_and_line(intersection_point(U!1, V!0), V!0) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), V!0)))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(134,plain,
% 0.19/0.52 (((~![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!3, Y!2), intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~apart_point_and_line(intersection_point(X!3, Y!2), V!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(U!1, V!0), V!0) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), V!0))))),
% 0.19/0.52 inference(monotonicity,[status(thm)],[133])).
% 0.19/0.52 tff(135,plain,
% 0.19/0.52 (((~![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!3, Y!2), intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~apart_point_and_line(intersection_point(X!3, Y!2), V!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(U!1, V!0), V!0) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), V!0)))),
% 0.19/0.52 inference(transitivity,[status(thm)],[134, 132])).
% 0.19/0.52 tff(136,plain,
% 0.19/0.52 ((~![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!3, Y!2), intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | (~apart_point_and_line(intersection_point(X!3, Y!2), V!0)))),
% 0.19/0.52 inference(quant_inst,[status(thm)],[])).
% 0.19/0.52 tff(137,plain,
% 0.19/0.52 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(intersection_point(U!1, V!0), V!0) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), V!0))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[136, 135])).
% 0.19/0.52 tff(138,plain,
% 0.19/0.52 ($false),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[137, 131, 121, 120, 115])).
% 0.19/0.52 tff(139,plain,(distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), V!0))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.52 tff(140,plain,
% 0.19/0.52 (~apart_point_and_line(intersection_point(X!3, Y!2), V!0)),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[139, 114])).
% 0.19/0.52 tff(141,plain,
% 0.19/0.52 (((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), U!1)))) <=> ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), U!1)))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(142,plain,
% 0.19/0.52 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | ((~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), U!1)))),
% 0.19/0.52 inference(quant_inst,[status(thm)],[])).
% 0.19/0.52 tff(143,plain,
% 0.19/0.52 ((~![X: $i, Y: $i] : ((~convergent_lines(X, Y)) | (~apart_point_and_line(intersection_point(X, Y), X)))) | (~convergent_lines(U!1, V!0)) | (~apart_point_and_line(intersection_point(U!1, V!0), U!1))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[142, 141])).
% 0.19/0.52 tff(144,plain,
% 0.19/0.52 (~apart_point_and_line(intersection_point(U!1, V!0), U!1)),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[143, 73, 116])).
% 0.19/0.52 tff(145,assumption,(apart_point_and_line(intersection_point(X!3, Y!2), U!1)), introduced(assumption)).
% 0.19/0.52 tff(146,plain,
% 0.19/0.52 (((~![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(U!1, V!0), U!1) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), U!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(U!1, V!0), U!1) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), U!1)))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(147,plain,
% 0.19/0.52 ((distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~apart_point_and_line(intersection_point(X!3, Y!2), U!1))) <=> (apart_point_and_line(intersection_point(U!1, V!0), U!1) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), U!1)))),
% 0.19/0.52 inference(rewrite,[status(thm)],[])).
% 0.19/0.52 tff(148,plain,
% 0.19/0.52 (((~![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!3, Y!2), intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~apart_point_and_line(intersection_point(X!3, Y!2), U!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(U!1, V!0), U!1) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), U!1))))),
% 0.19/0.52 inference(monotonicity,[status(thm)],[147])).
% 0.19/0.52 tff(149,plain,
% 0.19/0.52 (((~![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!3, Y!2), intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~apart_point_and_line(intersection_point(X!3, Y!2), U!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(U!1, V!0), U!1) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), U!1)))),
% 0.19/0.52 inference(transitivity,[status(thm)],[148, 146])).
% 0.19/0.52 tff(150,plain,
% 0.19/0.52 ((~![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!3, Y!2), intersection_point(U!1, V!0)) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | (~apart_point_and_line(intersection_point(X!3, Y!2), U!1)))),
% 0.19/0.52 inference(quant_inst,[status(thm)],[])).
% 0.19/0.52 tff(151,plain,
% 0.19/0.52 ((~![X: $i, Y: $i, Z: $i] : (distinct_points(X, Z) | apart_point_and_line(Z, Y) | (~apart_point_and_line(X, Y)))) | apart_point_and_line(intersection_point(U!1, V!0), U!1) | distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), U!1))),
% 0.19/0.52 inference(modus_ponens,[status(thm)],[150, 149])).
% 0.19/0.52 tff(152,plain,
% 0.19/0.52 ($false),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[151, 131, 145, 144, 115])).
% 0.19/0.52 tff(153,plain,(distinct_points(intersection_point(X!3, Y!2), intersection_point(U!1, V!0)) | (~apart_point_and_line(intersection_point(X!3, Y!2), U!1))), inference(lemma,lemma(discharge,[]))).
% 0.19/0.52 tff(154,plain,
% 0.19/0.52 (~apart_point_and_line(intersection_point(X!3, Y!2), U!1)),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[153, 114])).
% 0.19/0.52 tff(155,plain,
% 0.19/0.52 (apart_point_and_line(intersection_point(X!3, Y!2), V!0) | apart_point_and_line(intersection_point(X!3, Y!2), U!1)),
% 0.19/0.52 inference(and_elim,[status(thm)],[44])).
% 0.19/0.52 tff(156,plain,
% 0.19/0.52 ($false),
% 0.19/0.52 inference(unit_resolution,[status(thm)],[155, 154, 140])).
% 0.19/0.52 % SZS output end Proof
%------------------------------------------------------------------------------