%------------------------------------------------------------------------------
% File     : ePrincess---1.0
% Problem  : GEO208+3 : TPTP v8.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : ePrincess-casc -timeout=%d %s

% Computer : n017.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  : 600s
% DateTime : Sat Jul 16 03:48:44 EDT 2022

% Result   : Theorem 14.39s 3.99s
% Output   : Proof 15.73s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11  % Problem  : GEO208+3 : TPTP v8.1.0. Released v4.0.0.
% 0.11/0.12  % Command  : ePrincess-casc -timeout=%d %s
% 0.11/0.32  % Computer : n017.cluster.edu
% 0.11/0.32  % Model    : x86_64 x86_64
% 0.11/0.32  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32  % Memory   : 8042.1875MB
% 0.11/0.32  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32  % CPULimit : 300
% 0.11/0.32  % WCLimit  : 600
% 0.11/0.32  % DateTime : Fri Jun 17 21:46:12 EDT 2022
% 0.11/0.32  % CPUTime  : 
% 0.18/0.57          ____       _                          
% 0.18/0.57    ___  / __ \_____(_)___  ________  __________
% 0.18/0.57   / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/
% 0.18/0.57  /  __/ ____/ /  / / / / / /__/  __(__  |__  ) 
% 0.18/0.57  \___/_/   /_/  /_/_/ /_/\___/\___/____/____/  
% 0.18/0.57  
% 0.18/0.57  A Theorem Prover for First-Order Logic
% 0.18/0.57  (ePrincess v.1.0)
% 0.18/0.57  
% 0.18/0.57  (c) Philipp Rümmer, 2009-2015
% 0.18/0.57  (c) Peter Backeman, 2014-2015
% 0.18/0.57  (contributions by Angelo Brillout, Peter Baumgartner)
% 0.18/0.57  Free software under GNU Lesser General Public License (LGPL).
% 0.18/0.57  Bug reports to peter@backeman.se
% 0.18/0.57  
% 0.18/0.57  For more information, visit http://user.uu.se/~petba168/breu/
% 0.18/0.57  
% 0.18/0.57  Loading /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 0.73/0.62  Prover 0: Options:  -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 1.69/0.97  Prover 0: Preprocessing ...
% 2.11/1.18  Prover 0: Warning: ignoring some quantifiers
% 2.39/1.20  Prover 0: Constructing countermodel ...
% 13.37/3.80  Prover 0: gave up
% 13.37/3.80  Prover 1: Options:  +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all
% 13.61/3.84  Prover 1: Preprocessing ...
% 14.05/3.95  Prover 1: Constructing countermodel ...
% 14.05/3.99  Prover 1: proved (184ms)
% 14.39/3.99  
% 14.39/3.99  No countermodel exists, formula is valid
% 14.39/3.99  % SZS status Theorem for theBenchmark
% 14.39/3.99  
% 14.39/3.99  Generating proof ... found it (size 25)
% 15.39/4.23  
% 15.39/4.23  % SZS output start Proof for theBenchmark
% 15.39/4.23  Assumed formulas after preprocessing and simplification: 
% 15.39/4.23  | (0)  ? [v0] :  ? [v1] :  ? [v2] :  ? [v3] : ( ~ (v3 = 0) & incident_point_and_line(v0, v2) = 0 & incident_point_and_line(v0, v1) = 0 & parallel_lines(v1, v2) = 0 & equal_lines(v1, v2) = v3 &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] :  ! [v8] :  ! [v9] : (v9 = 0 | v8 = 0 |  ~ (unorthogonal_lines(v5, v7) = v9) |  ~ (apart_point_and_line(v4, v5) = v8) |  ~ (distinct_lines(v5, v6) = 0) |  ? [v10] :  ? [v11] : (unorthogonal_lines(v6, v7) = v11 & apart_point_and_line(v4, v6) = v10 & (v11 = 0 | v10 = 0))) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] :  ! [v8] : (v8 = 0 | v7 = 0 |  ~ (unorthogonal_lines(v4, v6) = v8) |  ~ (unorthogonal_lines(v4, v5) = v7) |  ? [v9] : ( ~ (v9 = 0) & convergent_lines(v5, v6) = v9)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] :  ! [v8] : ( ~ (unorthogonal_lines(v4, v6) = v8) |  ~ (unorthogonal_lines(v4, v5) = v7) |  ? [v9] :  ? [v10] :  ? [v11] :  ? [v12] : (unorthogonal_lines(v5, v6) = v12 & convergent_lines(v5, v6) = v11 & convergent_lines(v4, v6) = v10 & convergent_lines(v4, v5) = v9 & ( ~ (v12 = 0) |  ~ (v11 = 0) | (v10 = 0 & v8 = 0) | (v9 = 0 & v7 = 0)))) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v7 = 0 |  ~ (apart_point_and_line(v4, v5) = v7) |  ~ (distinct_lines(v5, v6) = 0) |  ? [v8] :  ? [v9] : (apart_point_and_line(v4, v6) = v8 & convergent_lines(v5, v6) = v9 & (v9 = 0 | v8 = 0))) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v7 = 0 |  ~ (apart_point_and_line(v4, v5) = 0) |  ~ (distinct_lines(v5, v6) = v7) | apart_point_and_line(v4, v6) = 0) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v7 = 0 |  ~ (apart_point_and_line(v4, v5) = 0) |  ~ (distinct_points(v4, v6) = v7) | apart_point_and_line(v6, v5) = 0) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v7 = 0 |  ~ (convergent_lines(v4, v6) = v7) |  ~ (convergent_lines(v4, v5) = 0) | convergent_lines(v5, v6) = 0) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v7 = 0 |  ~ (convergent_lines(v4, v5) = 0) |  ~ (distinct_lines(v5, v6) = v7) | convergent_lines(v4, v6) = 0) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v7 = 0 |  ~ (distinct_lines(v4, v6) = v7) |  ~ (distinct_lines(v4, v5) = 0) | distinct_lines(v5, v6) = 0) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v7 = 0 |  ~ (distinct_points(v4, v6) = v7) |  ~ (distinct_points(v4, v5) = 0) | distinct_points(v5, v6) = 0) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (orthogonal_lines(v7, v6) = v5) |  ~ (orthogonal_lines(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (incident_point_and_line(v7, v6) = v5) |  ~ (incident_point_and_line(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (parallel_lines(v7, v6) = v5) |  ~ (parallel_lines(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (equal_lines(v7, v6) = v5) |  ~ (equal_lines(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (equal_points(v7, v6) = v5) |  ~ (equal_points(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (orthogonal_through_point(v7, v6) = v5) |  ~ (orthogonal_through_point(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (unorthogonal_lines(v7, v6) = v5) |  ~ (unorthogonal_lines(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (parallel_through_point(v7, v6) = v5) |  ~ (parallel_through_point(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (intersection_point(v7, v6) = v5) |  ~ (intersection_point(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (line_connecting(v7, v6) = v5) |  ~ (line_connecting(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (apart_point_and_line(v7, v6) = v5) |  ~ (apart_point_and_line(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (convergent_lines(v7, v6) = v5) |  ~ (convergent_lines(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (distinct_lines(v7, v6) = v5) |  ~ (distinct_lines(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : (v5 = v4 |  ~ (distinct_points(v7, v6) = v5) |  ~ (distinct_points(v7, v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : ( ~ (unorthogonal_lines(v4, v6) = v7) |  ~ (unorthogonal_lines(v4, v5) = 0) |  ? [v8] :  ? [v9] :  ? [v10] :  ? [v11] : (unorthogonal_lines(v5, v6) = v11 & convergent_lines(v5, v6) = v10 & convergent_lines(v4, v6) = v9 & convergent_lines(v4, v5) = v8 & ( ~ (v8 = 0) | (v11 = 0 & v10 = 0) | (v9 = 0 & v7 = 0)))) &  ! [v4] :  ! [v5] :  ! [v6] :  ! [v7] : ( ~ (distinct_lines(v6, v7) = 0) |  ~ (distinct_points(v4, v5) = 0) |  ? [v8] :  ? [v9] :  ? [v10] :  ? [v11] : (apart_point_and_line(v5, v7) = v11 & apart_point_and_line(v5, v6) = v10 & apart_point_and_line(v4, v7) = v9 & apart_point_and_line(v4, v6) = v8 & (v11 = 0 | v10 = 0 | v9 = 0 | v8 = 0))) &  ! [v4] :  ! [v5] :  ! [v6] : (v6 = 0 |  ~ (orthogonal_lines(v4, v5) = v6) | unorthogonal_lines(v4, v5) = 0) &  ! [v4] :  ! [v5] :  ! [v6] : (v6 = 0 |  ~ (incident_point_and_line(v4, v5) = v6) | apart_point_and_line(v4, v5) = 0) &  ! [v4] :  ! [v5] :  ! [v6] : (v6 = 0 |  ~ (parallel_lines(v4, v5) = v6) | convergent_lines(v4, v5) = 0) &  ! [v4] :  ! [v5] :  ! [v6] : (v6 = 0 |  ~ (equal_lines(v4, v5) = v6) | distinct_lines(v4, v5) = 0) &  ! [v4] :  ! [v5] :  ! [v6] : (v6 = 0 |  ~ (equal_points(v4, v5) = v6) | distinct_points(v4, v5) = 0) &  ! [v4] :  ! [v5] :  ! [v6] : (v6 = 0 |  ~ (unorthogonal_lines(v4, v5) = v6) | convergent_lines(v4, v5) = 0) &  ! [v4] :  ! [v5] :  ! [v6] : (v5 = v4 |  ~ (point(v6) = v5) |  ~ (point(v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] : (v5 = v4 |  ~ (line(v6) = v5) |  ~ (line(v6) = v4)) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (orthogonal_through_point(v5, v4) = v6) |  ~ (unorthogonal_lines(v6, v5) = 0)) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (orthogonal_through_point(v5, v4) = v6) |  ~ (apart_point_and_line(v4, v6) = 0)) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (orthogonal_through_point(v4, v5) = v6) |  ? [v7] :  ? [v8] :  ? [v9] : (point(v5) = v8 & line(v6) = v9 & line(v4) = v7 & ( ~ (v8 = 0) |  ~ (v7 = 0) | v9 = 0))) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (parallel_through_point(v5, v4) = v6) |  ~ (apart_point_and_line(v4, v6) = 0)) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (parallel_through_point(v5, v4) = v6) |  ~ (convergent_lines(v6, v5) = 0)) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (parallel_through_point(v4, v5) = v6) |  ? [v7] :  ? [v8] :  ? [v9] : (point(v5) = v8 & line(v6) = v9 & line(v4) = v7 & ( ~ (v8 = 0) |  ~ (v7 = 0) | v9 = 0))) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (intersection_point(v4, v5) = v6) |  ~ (apart_point_and_line(v6, v5) = 0) |  ? [v7] : ( ~ (v7 = 0) & convergent_lines(v4, v5) = v7)) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (intersection_point(v4, v5) = v6) |  ~ (apart_point_and_line(v6, v4) = 0) |  ? [v7] : ( ~ (v7 = 0) & convergent_lines(v4, v5) = v7)) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (intersection_point(v4, v5) = v6) |  ? [v7] :  ? [v8] :  ? [v9] :  ? [v10] : (point(v6) = v10 & line(v5) = v8 & line(v4) = v7 & convergent_lines(v4, v5) = v9 & ( ~ (v9 = 0) |  ~ (v8 = 0) |  ~ (v7 = 0) | v10 = 0))) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (line_connecting(v4, v5) = v6) |  ~ (apart_point_and_line(v5, v6) = 0) |  ? [v7] : ( ~ (v7 = 0) & distinct_points(v4, v5) = v7)) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (line_connecting(v4, v5) = v6) |  ~ (apart_point_and_line(v4, v6) = 0) |  ? [v7] : ( ~ (v7 = 0) & distinct_points(v4, v5) = v7)) &  ! [v4] :  ! [v5] :  ! [v6] : ( ~ (line_connecting(v4, v5) = v6) |  ? [v7] :  ? [v8] :  ? [v9] :  ? [v10] : (point(v5) = v8 & point(v4) = v7 & line(v6) = v10 & distinct_points(v4, v5) = v9 & ( ~ (v9 = 0) |  ~ (v8 = 0) |  ~ (v7 = 0) | v10 = 0))) &  ! [v4] :  ! [v5] : ( ~ (orthogonal_lines(v4, v5) = 0) |  ? [v6] : ( ~ (v6 = 0) & unorthogonal_lines(v4, v5) = v6)) &  ! [v4] :  ! [v5] : ( ~ (incident_point_and_line(v4, v5) = 0) |  ? [v6] : ( ~ (v6 = 0) & apart_point_and_line(v4, v5) = v6)) &  ! [v4] :  ! [v5] : ( ~ (parallel_lines(v4, v5) = 0) |  ? [v6] : ( ~ (v6 = 0) & convergent_lines(v4, v5) = v6)) &  ! [v4] :  ! [v5] : ( ~ (equal_lines(v4, v5) = 0) |  ? [v6] : ( ~ (v6 = 0) & distinct_lines(v4, v5) = v6)) &  ! [v4] :  ! [v5] : ( ~ (equal_points(v4, v5) = 0) |  ? [v6] : ( ~ (v6 = 0) & distinct_points(v4, v5) = v6)) &  ! [v4] :  ! [v5] : ( ~ (distinct_lines(v4, v5) = 0) | convergent_lines(v4, v5) = 0) &  ! [v4] :  ~ (convergent_lines(v4, v4) = 0) &  ! [v4] :  ~ (distinct_lines(v4, v4) = 0) &  ! [v4] :  ~ (distinct_points(v4, v4) = 0))
% 15.73/4.28  | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3 yields:
% 15.73/4.28  | (1)  ~ (all_0_0_0 = 0) & incident_point_and_line(all_0_3_3, all_0_1_1) = 0 & incident_point_and_line(all_0_3_3, all_0_2_2) = 0 & parallel_lines(all_0_2_2, all_0_1_1) = 0 & equal_lines(all_0_2_2, all_0_1_1) = all_0_0_0 &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] :  ! [v4] :  ! [v5] : (v5 = 0 | v4 = 0 |  ~ (unorthogonal_lines(v1, v3) = v5) |  ~ (apart_point_and_line(v0, v1) = v4) |  ~ (distinct_lines(v1, v2) = 0) |  ? [v6] :  ? [v7] : (unorthogonal_lines(v2, v3) = v7 & apart_point_and_line(v0, v2) = v6 & (v7 = 0 | v6 = 0))) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] :  ! [v4] : (v4 = 0 | v3 = 0 |  ~ (unorthogonal_lines(v0, v2) = v4) |  ~ (unorthogonal_lines(v0, v1) = v3) |  ? [v5] : ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] :  ! [v4] : ( ~ (unorthogonal_lines(v0, v2) = v4) |  ~ (unorthogonal_lines(v0, v1) = v3) |  ? [v5] :  ? [v6] :  ? [v7] :  ? [v8] : (unorthogonal_lines(v1, v2) = v8 & convergent_lines(v1, v2) = v7 & convergent_lines(v0, v2) = v6 & convergent_lines(v0, v1) = v5 & ( ~ (v8 = 0) |  ~ (v7 = 0) | (v6 = 0 & v4 = 0) | (v5 = 0 & v3 = 0)))) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (apart_point_and_line(v0, v1) = v3) |  ~ (distinct_lines(v1, v2) = 0) |  ? [v4] :  ? [v5] : (apart_point_and_line(v0, v2) = v4 & convergent_lines(v1, v2) = v5 & (v5 = 0 | v4 = 0))) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (apart_point_and_line(v0, v1) = 0) |  ~ (distinct_lines(v1, v2) = v3) | apart_point_and_line(v0, v2) = 0) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (apart_point_and_line(v0, v1) = 0) |  ~ (distinct_points(v0, v2) = v3) | apart_point_and_line(v2, v1) = 0) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (convergent_lines(v0, v2) = v3) |  ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v1, v2) = 0) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (convergent_lines(v0, v1) = 0) |  ~ (distinct_lines(v1, v2) = v3) | convergent_lines(v0, v2) = 0) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (distinct_lines(v0, v2) = v3) |  ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (distinct_points(v0, v2) = v3) |  ~ (distinct_points(v0, v1) = 0) | distinct_points(v1, v2) = 0) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (orthogonal_lines(v3, v2) = v1) |  ~ (orthogonal_lines(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (incident_point_and_line(v3, v2) = v1) |  ~ (incident_point_and_line(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (parallel_lines(v3, v2) = v1) |  ~ (parallel_lines(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (equal_lines(v3, v2) = v1) |  ~ (equal_lines(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (equal_points(v3, v2) = v1) |  ~ (equal_points(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (orthogonal_through_point(v3, v2) = v1) |  ~ (orthogonal_through_point(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (unorthogonal_lines(v3, v2) = v1) |  ~ (unorthogonal_lines(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (parallel_through_point(v3, v2) = v1) |  ~ (parallel_through_point(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (intersection_point(v3, v2) = v1) |  ~ (intersection_point(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (line_connecting(v3, v2) = v1) |  ~ (line_connecting(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (apart_point_and_line(v3, v2) = v1) |  ~ (apart_point_and_line(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (convergent_lines(v3, v2) = v1) |  ~ (convergent_lines(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (distinct_lines(v3, v2) = v1) |  ~ (distinct_lines(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (distinct_points(v3, v2) = v1) |  ~ (distinct_points(v3, v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : ( ~ (unorthogonal_lines(v0, v2) = v3) |  ~ (unorthogonal_lines(v0, v1) = 0) |  ? [v4] :  ? [v5] :  ? [v6] :  ? [v7] : (unorthogonal_lines(v1, v2) = v7 & convergent_lines(v1, v2) = v6 & convergent_lines(v0, v2) = v5 & convergent_lines(v0, v1) = v4 & ( ~ (v4 = 0) | (v7 = 0 & v6 = 0) | (v5 = 0 & v3 = 0)))) &  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : ( ~ (distinct_lines(v2, v3) = 0) |  ~ (distinct_points(v0, v1) = 0) |  ? [v4] :  ? [v5] :  ? [v6] :  ? [v7] : (apart_point_and_line(v1, v3) = v7 & apart_point_and_line(v1, v2) = v6 & apart_point_and_line(v0, v3) = v5 & apart_point_and_line(v0, v2) = v4 & (v7 = 0 | v6 = 0 | v5 = 0 | v4 = 0))) &  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (orthogonal_lines(v0, v1) = v2) | unorthogonal_lines(v0, v1) = 0) &  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (incident_point_and_line(v0, v1) = v2) | apart_point_and_line(v0, v1) = 0) &  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (parallel_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) &  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (equal_lines(v0, v1) = v2) | distinct_lines(v0, v1) = 0) &  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (equal_points(v0, v1) = v2) | distinct_points(v0, v1) = 0) &  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (unorthogonal_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0) &  ! [v0] :  ! [v1] :  ! [v2] : (v1 = v0 |  ~ (point(v2) = v1) |  ~ (point(v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] : (v1 = v0 |  ~ (line(v2) = v1) |  ~ (line(v2) = v0)) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) |  ~ (unorthogonal_lines(v2, v1) = 0)) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) |  ~ (apart_point_and_line(v0, v2) = 0)) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (orthogonal_through_point(v0, v1) = v2) |  ? [v3] :  ? [v4] :  ? [v5] : (point(v1) = v4 & line(v2) = v5 & line(v0) = v3 & ( ~ (v4 = 0) |  ~ (v3 = 0) | v5 = 0))) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) |  ~ (apart_point_and_line(v0, v2) = 0)) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) |  ~ (convergent_lines(v2, v1) = 0)) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (parallel_through_point(v0, v1) = v2) |  ? [v3] :  ? [v4] :  ? [v5] : (point(v1) = v4 & line(v2) = v5 & line(v0) = v3 & ( ~ (v4 = 0) |  ~ (v3 = 0) | v5 = 0))) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (intersection_point(v0, v1) = v2) |  ~ (apart_point_and_line(v2, v1) = 0) |  ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3)) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (intersection_point(v0, v1) = v2) |  ~ (apart_point_and_line(v2, v0) = 0) |  ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3)) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (intersection_point(v0, v1) = v2) |  ? [v3] :  ? [v4] :  ? [v5] :  ? [v6] : (point(v2) = v6 & line(v1) = v4 & line(v0) = v3 & convergent_lines(v0, v1) = v5 & ( ~ (v5 = 0) |  ~ (v4 = 0) |  ~ (v3 = 0) | v6 = 0))) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (line_connecting(v0, v1) = v2) |  ~ (apart_point_and_line(v1, v2) = 0) |  ? [v3] : ( ~ (v3 = 0) & distinct_points(v0, v1) = v3)) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (line_connecting(v0, v1) = v2) |  ~ (apart_point_and_line(v0, v2) = 0) |  ? [v3] : ( ~ (v3 = 0) & distinct_points(v0, v1) = v3)) &  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (line_connecting(v0, v1) = v2) |  ? [v3] :  ? [v4] :  ? [v5] :  ? [v6] : (point(v1) = v4 & point(v0) = v3 & line(v2) = v6 & distinct_points(v0, v1) = v5 & ( ~ (v5 = 0) |  ~ (v4 = 0) |  ~ (v3 = 0) | v6 = 0))) &  ! [v0] :  ! [v1] : ( ~ (orthogonal_lines(v0, v1) = 0) |  ? [v2] : ( ~ (v2 = 0) & unorthogonal_lines(v0, v1) = v2)) &  ! [v0] :  ! [v1] : ( ~ (incident_point_and_line(v0, v1) = 0) |  ? [v2] : ( ~ (v2 = 0) & apart_point_and_line(v0, v1) = v2)) &  ! [v0] :  ! [v1] : ( ~ (parallel_lines(v0, v1) = 0) |  ? [v2] : ( ~ (v2 = 0) & convergent_lines(v0, v1) = v2)) &  ! [v0] :  ! [v1] : ( ~ (equal_lines(v0, v1) = 0) |  ? [v2] : ( ~ (v2 = 0) & distinct_lines(v0, v1) = v2)) &  ! [v0] :  ! [v1] : ( ~ (equal_points(v0, v1) = 0) |  ? [v2] : ( ~ (v2 = 0) & distinct_points(v0, v1) = v2)) &  ! [v0] :  ! [v1] : ( ~ (distinct_lines(v0, v1) = 0) | convergent_lines(v0, v1) = 0) &  ! [v0] :  ~ (convergent_lines(v0, v0) = 0) &  ! [v0] :  ~ (distinct_lines(v0, v0) = 0) &  ! [v0] :  ~ (distinct_points(v0, v0) = 0)
% 15.73/4.29  |
% 15.73/4.29  | Applying alpha-rule on (1) yields:
% 15.73/4.29  | (2)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (intersection_point(v0, v1) = v2) |  ? [v3] :  ? [v4] :  ? [v5] :  ? [v6] : (point(v2) = v6 & line(v1) = v4 & line(v0) = v3 & convergent_lines(v0, v1) = v5 & ( ~ (v5 = 0) |  ~ (v4 = 0) |  ~ (v3 = 0) | v6 = 0)))
% 15.73/4.29  | (3)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (line_connecting(v0, v1) = v2) |  ? [v3] :  ? [v4] :  ? [v5] :  ? [v6] : (point(v1) = v4 & point(v0) = v3 & line(v2) = v6 & distinct_points(v0, v1) = v5 & ( ~ (v5 = 0) |  ~ (v4 = 0) |  ~ (v3 = 0) | v6 = 0)))
% 15.73/4.29  | (4)  ! [v0] :  ~ (distinct_lines(v0, v0) = 0)
% 15.73/4.29  | (5)  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (equal_lines(v0, v1) = v2) | distinct_lines(v0, v1) = 0)
% 15.73/4.29  | (6)  ! [v0] :  ! [v1] : ( ~ (incident_point_and_line(v0, v1) = 0) |  ? [v2] : ( ~ (v2 = 0) & apart_point_and_line(v0, v1) = v2))
% 15.73/4.29  | (7)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (intersection_point(v0, v1) = v2) |  ~ (apart_point_and_line(v2, v0) = 0) |  ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3))
% 15.73/4.29  | (8)  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (equal_points(v0, v1) = v2) | distinct_points(v0, v1) = 0)
% 15.73/4.29  | (9)  ! [v0] :  ~ (convergent_lines(v0, v0) = 0)
% 15.73/4.29  | (10)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (intersection_point(v3, v2) = v1) |  ~ (intersection_point(v3, v2) = v0))
% 15.73/4.29  | (11)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] :  ! [v4] : (v4 = 0 | v3 = 0 |  ~ (unorthogonal_lines(v0, v2) = v4) |  ~ (unorthogonal_lines(v0, v1) = v3) |  ? [v5] : ( ~ (v5 = 0) & convergent_lines(v1, v2) = v5))
% 15.73/4.29  | (12)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (distinct_lines(v3, v2) = v1) |  ~ (distinct_lines(v3, v2) = v0))
% 15.73/4.29  | (13)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (line_connecting(v0, v1) = v2) |  ~ (apart_point_and_line(v0, v2) = 0) |  ? [v3] : ( ~ (v3 = 0) & distinct_points(v0, v1) = v3))
% 15.73/4.29  | (14)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] :  ! [v4] :  ! [v5] : (v5 = 0 | v4 = 0 |  ~ (unorthogonal_lines(v1, v3) = v5) |  ~ (apart_point_and_line(v0, v1) = v4) |  ~ (distinct_lines(v1, v2) = 0) |  ? [v6] :  ? [v7] : (unorthogonal_lines(v2, v3) = v7 & apart_point_and_line(v0, v2) = v6 & (v7 = 0 | v6 = 0)))
% 15.73/4.29  | (15)  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (unorthogonal_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0)
% 15.73/4.29  | (16)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (convergent_lines(v3, v2) = v1) |  ~ (convergent_lines(v3, v2) = v0))
% 15.73/4.29  | (17)  ! [v0] :  ! [v1] : ( ~ (distinct_lines(v0, v1) = 0) | convergent_lines(v0, v1) = 0)
% 15.73/4.29  | (18)  ~ (all_0_0_0 = 0)
% 15.73/4.29  | (19) equal_lines(all_0_2_2, all_0_1_1) = all_0_0_0
% 15.73/4.29  | (20)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (line_connecting(v0, v1) = v2) |  ~ (apart_point_and_line(v1, v2) = 0) |  ? [v3] : ( ~ (v3 = 0) & distinct_points(v0, v1) = v3))
% 15.73/4.30  | (21)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : ( ~ (unorthogonal_lines(v0, v2) = v3) |  ~ (unorthogonal_lines(v0, v1) = 0) |  ? [v4] :  ? [v5] :  ? [v6] :  ? [v7] : (unorthogonal_lines(v1, v2) = v7 & convergent_lines(v1, v2) = v6 & convergent_lines(v0, v2) = v5 & convergent_lines(v0, v1) = v4 & ( ~ (v4 = 0) | (v7 = 0 & v6 = 0) | (v5 = 0 & v3 = 0))))
% 15.73/4.30  | (22)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (distinct_points(v0, v2) = v3) |  ~ (distinct_points(v0, v1) = 0) | distinct_points(v1, v2) = 0)
% 15.73/4.30  | (23)  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (orthogonal_lines(v0, v1) = v2) | unorthogonal_lines(v0, v1) = 0)
% 15.73/4.30  | (24)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) |  ~ (unorthogonal_lines(v2, v1) = 0))
% 15.73/4.30  | (25)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (orthogonal_lines(v3, v2) = v1) |  ~ (orthogonal_lines(v3, v2) = v0))
% 15.73/4.30  | (26)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (unorthogonal_lines(v3, v2) = v1) |  ~ (unorthogonal_lines(v3, v2) = v0))
% 15.73/4.30  | (27)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (orthogonal_through_point(v3, v2) = v1) |  ~ (orthogonal_through_point(v3, v2) = v0))
% 15.73/4.30  | (28) incident_point_and_line(all_0_3_3, all_0_2_2) = 0
% 15.73/4.30  | (29)  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (incident_point_and_line(v0, v1) = v2) | apart_point_and_line(v0, v1) = 0)
% 15.73/4.30  | (30)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (intersection_point(v0, v1) = v2) |  ~ (apart_point_and_line(v2, v1) = 0) |  ? [v3] : ( ~ (v3 = 0) & convergent_lines(v0, v1) = v3))
% 15.73/4.30  | (31)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (parallel_through_point(v0, v1) = v2) |  ? [v3] :  ? [v4] :  ? [v5] : (point(v1) = v4 & line(v2) = v5 & line(v0) = v3 & ( ~ (v4 = 0) |  ~ (v3 = 0) | v5 = 0)))
% 15.73/4.30  | (32)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (convergent_lines(v0, v1) = 0) |  ~ (distinct_lines(v1, v2) = v3) | convergent_lines(v0, v2) = 0)
% 15.73/4.30  | (33) incident_point_and_line(all_0_3_3, all_0_1_1) = 0
% 15.73/4.30  | (34)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (parallel_lines(v3, v2) = v1) |  ~ (parallel_lines(v3, v2) = v0))
% 15.73/4.30  | (35)  ! [v0] :  ! [v1] :  ! [v2] : (v1 = v0 |  ~ (point(v2) = v1) |  ~ (point(v2) = v0))
% 15.73/4.30  | (36)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (apart_point_and_line(v0, v1) = v3) |  ~ (distinct_lines(v1, v2) = 0) |  ? [v4] :  ? [v5] : (apart_point_and_line(v0, v2) = v4 & convergent_lines(v1, v2) = v5 & (v5 = 0 | v4 = 0)))
% 15.73/4.30  | (37)  ! [v0] :  ! [v1] :  ! [v2] : (v2 = 0 |  ~ (parallel_lines(v0, v1) = v2) | convergent_lines(v0, v1) = 0)
% 15.73/4.30  | (38)  ! [v0] :  ! [v1] :  ! [v2] : (v1 = v0 |  ~ (line(v2) = v1) |  ~ (line(v2) = v0))
% 15.73/4.30  | (39)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (parallel_through_point(v3, v2) = v1) |  ~ (parallel_through_point(v3, v2) = v0))
% 15.73/4.30  | (40)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (orthogonal_through_point(v0, v1) = v2) |  ? [v3] :  ? [v4] :  ? [v5] : (point(v1) = v4 & line(v2) = v5 & line(v0) = v3 & ( ~ (v4 = 0) |  ~ (v3 = 0) | v5 = 0)))
% 15.73/4.30  | (41)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (orthogonal_through_point(v1, v0) = v2) |  ~ (apart_point_and_line(v0, v2) = 0))
% 15.73/4.30  | (42)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (apart_point_and_line(v0, v1) = 0) |  ~ (distinct_lines(v1, v2) = v3) | apart_point_and_line(v0, v2) = 0)
% 15.73/4.30  | (43)  ! [v0] :  ! [v1] : ( ~ (orthogonal_lines(v0, v1) = 0) |  ? [v2] : ( ~ (v2 = 0) & unorthogonal_lines(v0, v1) = v2))
% 15.73/4.30  | (44)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (distinct_points(v3, v2) = v1) |  ~ (distinct_points(v3, v2) = v0))
% 15.73/4.30  | (45) parallel_lines(all_0_2_2, all_0_1_1) = 0
% 15.73/4.30  | (46)  ! [v0] :  ! [v1] : ( ~ (parallel_lines(v0, v1) = 0) |  ? [v2] : ( ~ (v2 = 0) & convergent_lines(v0, v1) = v2))
% 15.73/4.30  | (47)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) |  ~ (convergent_lines(v2, v1) = 0))
% 15.73/4.30  | (48)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (convergent_lines(v0, v2) = v3) |  ~ (convergent_lines(v0, v1) = 0) | convergent_lines(v1, v2) = 0)
% 15.73/4.30  | (49)  ! [v0] :  ! [v1] :  ! [v2] : ( ~ (parallel_through_point(v1, v0) = v2) |  ~ (apart_point_and_line(v0, v2) = 0))
% 15.73/4.30  | (50)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : ( ~ (distinct_lines(v2, v3) = 0) |  ~ (distinct_points(v0, v1) = 0) |  ? [v4] :  ? [v5] :  ? [v6] :  ? [v7] : (apart_point_and_line(v1, v3) = v7 & apart_point_and_line(v1, v2) = v6 & apart_point_and_line(v0, v3) = v5 & apart_point_and_line(v0, v2) = v4 & (v7 = 0 | v6 = 0 | v5 = 0 | v4 = 0)))
% 15.73/4.30  | (51)  ! [v0] :  ! [v1] : ( ~ (equal_points(v0, v1) = 0) |  ? [v2] : ( ~ (v2 = 0) & distinct_points(v0, v1) = v2))
% 15.73/4.30  | (52)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (distinct_lines(v0, v2) = v3) |  ~ (distinct_lines(v0, v1) = 0) | distinct_lines(v1, v2) = 0)
% 15.73/4.30  | (53)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (line_connecting(v3, v2) = v1) |  ~ (line_connecting(v3, v2) = v0))
% 15.73/4.30  | (54)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] :  ! [v4] : ( ~ (unorthogonal_lines(v0, v2) = v4) |  ~ (unorthogonal_lines(v0, v1) = v3) |  ? [v5] :  ? [v6] :  ? [v7] :  ? [v8] : (unorthogonal_lines(v1, v2) = v8 & convergent_lines(v1, v2) = v7 & convergent_lines(v0, v2) = v6 & convergent_lines(v0, v1) = v5 & ( ~ (v8 = 0) |  ~ (v7 = 0) | (v6 = 0 & v4 = 0) | (v5 = 0 & v3 = 0))))
% 15.73/4.31  | (55)  ! [v0] :  ~ (distinct_points(v0, v0) = 0)
% 15.73/4.31  | (56)  ! [v0] :  ! [v1] : ( ~ (equal_lines(v0, v1) = 0) |  ? [v2] : ( ~ (v2 = 0) & distinct_lines(v0, v1) = v2))
% 15.73/4.31  | (57)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v3 = 0 |  ~ (apart_point_and_line(v0, v1) = 0) |  ~ (distinct_points(v0, v2) = v3) | apart_point_and_line(v2, v1) = 0)
% 15.73/4.31  | (58)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (equal_points(v3, v2) = v1) |  ~ (equal_points(v3, v2) = v0))
% 15.73/4.31  | (59)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (incident_point_and_line(v3, v2) = v1) |  ~ (incident_point_and_line(v3, v2) = v0))
% 15.73/4.31  | (60)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (equal_lines(v3, v2) = v1) |  ~ (equal_lines(v3, v2) = v0))
% 15.73/4.31  | (61)  ! [v0] :  ! [v1] :  ! [v2] :  ! [v3] : (v1 = v0 |  ~ (apart_point_and_line(v3, v2) = v1) |  ~ (apart_point_and_line(v3, v2) = v0))
% 15.73/4.31  |
% 15.73/4.31  | Instantiating formula (6) with all_0_2_2, all_0_3_3 and discharging atoms incident_point_and_line(all_0_3_3, all_0_2_2) = 0, yields:
% 15.73/4.31  | (62)  ? [v0] : ( ~ (v0 = 0) & apart_point_and_line(all_0_3_3, all_0_2_2) = v0)
% 15.73/4.31  |
% 15.73/4.31  | Instantiating formula (46) with all_0_1_1, all_0_2_2 and discharging atoms parallel_lines(all_0_2_2, all_0_1_1) = 0, yields:
% 15.73/4.31  | (63)  ? [v0] : ( ~ (v0 = 0) & convergent_lines(all_0_2_2, all_0_1_1) = v0)
% 15.73/4.31  |
% 15.73/4.31  | Instantiating formula (5) with all_0_0_0, all_0_1_1, all_0_2_2 and discharging atoms equal_lines(all_0_2_2, all_0_1_1) = all_0_0_0, yields:
% 15.73/4.31  | (64) all_0_0_0 = 0 | distinct_lines(all_0_2_2, all_0_1_1) = 0
% 15.73/4.31  |
% 15.73/4.31  | Instantiating (63) with all_8_0_4 yields:
% 15.73/4.31  | (65)  ~ (all_8_0_4 = 0) & convergent_lines(all_0_2_2, all_0_1_1) = all_8_0_4
% 15.73/4.31  |
% 15.73/4.31  | Applying alpha-rule on (65) yields:
% 15.73/4.31  | (66)  ~ (all_8_0_4 = 0)
% 15.73/4.31  | (67) convergent_lines(all_0_2_2, all_0_1_1) = all_8_0_4
% 15.73/4.31  |
% 15.73/4.31  | Instantiating (62) with all_10_0_5 yields:
% 15.73/4.31  | (68)  ~ (all_10_0_5 = 0) & apart_point_and_line(all_0_3_3, all_0_2_2) = all_10_0_5
% 15.73/4.31  |
% 15.73/4.31  | Applying alpha-rule on (68) yields:
% 15.73/4.31  | (69)  ~ (all_10_0_5 = 0)
% 15.73/4.31  | (70) apart_point_and_line(all_0_3_3, all_0_2_2) = all_10_0_5
% 15.73/4.31  |
% 15.73/4.31  +-Applying beta-rule and splitting (64), into two cases.
% 15.73/4.31  |-Branch one:
% 15.73/4.31  | (71) distinct_lines(all_0_2_2, all_0_1_1) = 0
% 15.73/4.31  |
% 15.73/4.31  	| Instantiating formula (36) with all_10_0_5, all_0_1_1, all_0_2_2, all_0_3_3 and discharging atoms apart_point_and_line(all_0_3_3, all_0_2_2) = all_10_0_5, distinct_lines(all_0_2_2, all_0_1_1) = 0, yields:
% 15.73/4.31  	| (72) all_10_0_5 = 0 |  ? [v0] :  ? [v1] : (apart_point_and_line(all_0_3_3, all_0_1_1) = v0 & convergent_lines(all_0_2_2, all_0_1_1) = v1 & (v1 = 0 | v0 = 0))
% 15.73/4.31  	|
% 15.73/4.31  	| Instantiating formula (17) with all_0_1_1, all_0_2_2 and discharging atoms distinct_lines(all_0_2_2, all_0_1_1) = 0, yields:
% 15.73/4.31  	| (73) convergent_lines(all_0_2_2, all_0_1_1) = 0
% 15.73/4.31  	|
% 15.73/4.31  	+-Applying beta-rule and splitting (72), into two cases.
% 15.73/4.31  	|-Branch one:
% 15.73/4.31  	| (74) all_10_0_5 = 0
% 15.73/4.31  	|
% 15.73/4.31  		| Equations (74) can reduce 69 to:
% 15.73/4.31  		| (75) $false
% 15.73/4.31  		|
% 15.73/4.31  		|-The branch is then unsatisfiable
% 15.73/4.31  	|-Branch two:
% 15.73/4.31  	| (69)  ~ (all_10_0_5 = 0)
% 15.73/4.31  	| (77)  ? [v0] :  ? [v1] : (apart_point_and_line(all_0_3_3, all_0_1_1) = v0 & convergent_lines(all_0_2_2, all_0_1_1) = v1 & (v1 = 0 | v0 = 0))
% 15.73/4.31  	|
% 15.73/4.31  		| Instantiating (77) with all_36_0_7, all_36_1_8 yields:
% 15.73/4.31  		| (78) apart_point_and_line(all_0_3_3, all_0_1_1) = all_36_1_8 & convergent_lines(all_0_2_2, all_0_1_1) = all_36_0_7 & (all_36_0_7 = 0 | all_36_1_8 = 0)
% 15.73/4.31  		|
% 15.73/4.31  		| Applying alpha-rule on (78) yields:
% 15.73/4.31  		| (79) apart_point_and_line(all_0_3_3, all_0_1_1) = all_36_1_8
% 15.73/4.31  		| (80) convergent_lines(all_0_2_2, all_0_1_1) = all_36_0_7
% 15.73/4.31  		| (81) all_36_0_7 = 0 | all_36_1_8 = 0
% 15.73/4.31  		|
% 15.73/4.31  		| Instantiating formula (16) with all_0_2_2, all_0_1_1, all_36_0_7, all_8_0_4 and discharging atoms convergent_lines(all_0_2_2, all_0_1_1) = all_36_0_7, convergent_lines(all_0_2_2, all_0_1_1) = all_8_0_4, yields:
% 15.73/4.31  		| (82) all_36_0_7 = all_8_0_4
% 15.73/4.31  		|
% 15.73/4.31  		| Instantiating formula (16) with all_0_2_2, all_0_1_1, 0, all_36_0_7 and discharging atoms convergent_lines(all_0_2_2, all_0_1_1) = all_36_0_7, convergent_lines(all_0_2_2, all_0_1_1) = 0, yields:
% 15.73/4.31  		| (83) all_36_0_7 = 0
% 15.73/4.31  		|
% 15.73/4.31  		| Combining equations (82,83) yields a new equation:
% 15.73/4.31  		| (84) all_8_0_4 = 0
% 15.73/4.31  		|
% 15.73/4.31  		| Simplifying 84 yields:
% 15.73/4.31  		| (85) all_8_0_4 = 0
% 15.73/4.31  		|
% 15.73/4.31  		| Equations (85) can reduce 66 to:
% 15.73/4.31  		| (75) $false
% 15.73/4.31  		|
% 15.73/4.31  		|-The branch is then unsatisfiable
% 15.73/4.31  |-Branch two:
% 15.73/4.31  | (87)  ~ (distinct_lines(all_0_2_2, all_0_1_1) = 0)
% 15.73/4.31  | (88) all_0_0_0 = 0
% 15.73/4.31  |
% 15.73/4.31  	| Equations (88) can reduce 18 to:
% 15.73/4.31  	| (75) $false
% 15.73/4.31  	|
% 15.73/4.31  	|-The branch is then unsatisfiable
% 15.73/4.31  % SZS output end Proof for theBenchmark
% 15.73/4.31  
% 15.73/4.31  3732ms
%------------------------------------------------------------------------------