SET007 Axioms: SET007+905.ax
%------------------------------------------------------------------------------
% File : SET007+905 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Lines on Planes in n-Dimensional Euclidean Spaces
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : euclidlp [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 153 ( 0 unt; 0 def)
% Number of atoms : 1177 ( 157 equ)
% Maximal formula atoms : 20 ( 7 avg)
% Number of connectives : 1137 ( 113 ~; 17 |; 315 &)
% ( 20 <=>; 672 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 13 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 19 ( 18 usr; 0 prp; 1-3 aty)
% Number of functors : 30 ( 30 usr; 4 con; 0-4 aty)
% Number of variables : 700 ( 660 !; 40 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ~ v1_xboole_0(k1_euclidlp(A)) ) ).
fof(fc2_euclidlp,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A))
& m1_subset_1(D,k1_euclid(A)) )
=> ~ v1_xboole_0(k4_euclidlp(A,B,C,D)) ) ).
fof(fc3_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ~ v1_xboole_0(k5_euclidlp(A)) ) ).
fof(t1_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(D))
=> ( k9_euclid(D,k6_real_1(A,B),k9_euclid(D,C,E)) = k9_euclid(D,k6_real_1(k4_real_1(A,C),B),E)
& k9_euclid(D,k6_real_1(np__1,B),k9_euclid(D,C,E)) = k9_euclid(D,k6_real_1(C,B),E) ) ) ) ) ) ) ).
fof(t2_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> k7_euclid(A,B,k7_euclid(A,C,D)) = k7_euclid(A,k7_euclid(A,B,C),D) ) ) ) ) ).
fof(t3_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> k8_euclid(A,B,k5_euclid(A)) = B ) ) ).
fof(t4_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> k8_euclid(A,k5_euclid(A),B) = k6_euclid(A,B) ) ) ).
fof(t5_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> k8_euclid(A,B,k7_euclid(A,C,D)) = k8_euclid(A,k8_euclid(A,B,C),D) ) ) ) ) ).
fof(t6_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> k8_euclid(A,B,C) = k7_euclid(A,B,k6_euclid(A,C)) ) ) ) ).
fof(t7_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ( k8_euclid(A,B,B) = k5_euclid(A)
& k7_euclid(A,B,k6_euclid(A,B)) = k5_euclid(A) ) ) ) ).
fof(t8_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ( k6_euclid(B,k9_euclid(B,A,C)) = k9_euclid(B,k1_real_1(A),C)
& k6_euclid(B,k9_euclid(B,A,C)) = k9_euclid(B,A,k6_euclid(B,C)) ) ) ) ) ).
fof(t9_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> k8_euclid(A,B,k8_euclid(A,C,D)) = k7_euclid(A,k8_euclid(A,B,C),D) ) ) ) ) ).
fof(t10_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> k7_euclid(A,B,k8_euclid(A,C,D)) = k8_euclid(A,k7_euclid(A,B,C),D) ) ) ) ) ).
fof(t11_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( B = k7_euclid(A,C,D)
<=> C = k8_euclid(A,B,D) ) ) ) ) ) ).
fof(t12_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ( B = k7_euclid(A,k7_euclid(A,C,D),E)
<=> k8_euclid(A,B,C) = k7_euclid(A,D,E) ) ) ) ) ) ) ).
fof(t13_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> k6_euclid(A,k7_euclid(A,k7_euclid(A,B,C),D)) = k7_euclid(A,k7_euclid(A,k6_euclid(A,B),k6_euclid(A,C)),k6_euclid(A,D)) ) ) ) ) ).
fof(t14_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ( B = C
<=> k8_euclid(A,B,C) = k5_euclid(A) ) ) ) ) ).
fof(t15_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(B))
=> ~ ( k8_euclid(B,C,D) = k9_euclid(B,A,E)
& C != D
& A = np__0 ) ) ) ) ) ) ).
fof(t16_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(C))
=> ( k9_euclid(C,k5_real_1(A,B),D) = k7_euclid(C,k9_euclid(C,A,D),k9_euclid(C,k1_real_1(B),D))
& k9_euclid(C,k5_real_1(A,B),D) = k7_euclid(C,k9_euclid(C,A,D),k6_euclid(C,k9_euclid(C,B,D)))
& k9_euclid(C,k5_real_1(A,B),D) = k8_euclid(C,k9_euclid(C,A,D),k9_euclid(C,B,D)) ) ) ) ) ) ).
fof(t17_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
=> ( k9_euclid(B,A,k8_euclid(B,C,D)) = k7_euclid(B,k9_euclid(B,A,C),k6_euclid(B,k9_euclid(B,A,D)))
& k9_euclid(B,A,k8_euclid(B,C,D)) = k7_euclid(B,k9_euclid(B,A,C),k9_euclid(B,k1_real_1(A),D))
& k9_euclid(B,A,k8_euclid(B,C,D)) = k8_euclid(B,k9_euclid(B,A,C),k9_euclid(B,A,D)) ) ) ) ) ) ).
fof(t18_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(D))
=> k9_euclid(D,k5_real_1(k5_real_1(A,B),C),E) = k8_euclid(D,k8_euclid(D,k9_euclid(D,A,E),k9_euclid(D,B,E)),k9_euclid(D,C,E)) ) ) ) ) ) ).
fof(t19_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(D))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(D))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(D))
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(D))
=> k8_euclid(D,E,k7_euclid(D,k7_euclid(D,k9_euclid(D,A,F),k9_euclid(D,B,G)),k9_euclid(D,C,H))) = k7_euclid(D,E,k7_euclid(D,k7_euclid(D,k9_euclid(D,k1_real_1(A),F),k9_euclid(D,k1_real_1(B),G)),k9_euclid(D,k1_real_1(C),H))) ) ) ) ) ) ) ) ) ).
fof(t20_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(D))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(D))
=> k8_euclid(D,E,k9_euclid(D,k3_real_1(k3_real_1(A,B),C),F)) = k7_euclid(D,k7_euclid(D,k7_euclid(D,E,k9_euclid(D,k1_real_1(A),F)),k9_euclid(D,k1_real_1(B),F)),k9_euclid(D,k1_real_1(C),F)) ) ) ) ) ) ) ).
fof(t21_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> k7_euclid(A,k7_euclid(A,B,C),k7_euclid(A,D,E)) = k7_euclid(A,k7_euclid(A,B,D),k7_euclid(A,C,E)) ) ) ) ) ) ).
fof(t22_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(A))
=> k7_euclid(A,k7_euclid(A,k7_euclid(A,B,C),D),k7_euclid(A,k7_euclid(A,E,F),G)) = k7_euclid(A,k7_euclid(A,k7_euclid(A,B,E),k7_euclid(A,C,F)),k7_euclid(A,D,G)) ) ) ) ) ) ) ) ).
fof(t23_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> k8_euclid(A,k7_euclid(A,B,C),k7_euclid(A,D,E)) = k7_euclid(A,k8_euclid(A,B,D),k8_euclid(A,C,E)) ) ) ) ) ) ).
fof(t24_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(A))
=> k8_euclid(A,k7_euclid(A,k7_euclid(A,B,C),D),k7_euclid(A,k7_euclid(A,E,F),G)) = k7_euclid(A,k7_euclid(A,k8_euclid(A,B,E),k8_euclid(A,C,F)),k8_euclid(A,D,G)) ) ) ) ) ) ) ) ).
fof(t25_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(B))
=> k9_euclid(B,A,k7_euclid(B,k7_euclid(B,C,D),E)) = k7_euclid(B,k7_euclid(B,k9_euclid(B,A,C),k9_euclid(B,A,D)),k9_euclid(B,A,E)) ) ) ) ) ) ).
fof(t26_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(D))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(D))
=> k9_euclid(D,A,k7_euclid(D,k9_euclid(D,B,E),k9_euclid(D,C,F))) = k7_euclid(D,k9_euclid(D,k4_real_1(A,B),E),k9_euclid(D,k4_real_1(A,C),F)) ) ) ) ) ) ) ).
fof(t27_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(E))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(E))
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(E))
=> k9_euclid(E,A,k7_euclid(E,k7_euclid(E,k9_euclid(E,B,F),k9_euclid(E,C,G)),k9_euclid(E,D,H))) = k7_euclid(E,k7_euclid(E,k9_euclid(E,k4_real_1(A,B),F),k9_euclid(E,k4_real_1(A,C),G)),k9_euclid(E,k4_real_1(A,D),H)) ) ) ) ) ) ) ) ) ).
fof(t28_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(E))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(E))
=> k7_euclid(E,k7_euclid(E,k9_euclid(E,A,F),k9_euclid(E,B,G)),k7_euclid(E,k9_euclid(E,C,F),k9_euclid(E,D,G))) = k7_euclid(E,k9_euclid(E,k3_real_1(A,C),F),k9_euclid(E,k3_real_1(B,D),G)) ) ) ) ) ) ) ) ).
fof(t29_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(G))
=> ! [I] :
( m2_finseq_2(I,k1_numbers,k1_euclid(G))
=> ! [J] :
( m2_finseq_2(J,k1_numbers,k1_euclid(G))
=> k7_euclid(G,k7_euclid(G,k7_euclid(G,k9_euclid(G,A,H),k9_euclid(G,B,I)),k9_euclid(G,C,J)),k7_euclid(G,k7_euclid(G,k9_euclid(G,D,H),k9_euclid(G,E,I)),k9_euclid(G,F,J))) = k7_euclid(G,k7_euclid(G,k9_euclid(G,k3_real_1(A,D),H),k9_euclid(G,k3_real_1(B,E),I)),k9_euclid(G,k3_real_1(C,F),J)) ) ) ) ) ) ) ) ) ) ) ).
fof(t30_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(E))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(E))
=> k8_euclid(E,k7_euclid(E,k9_euclid(E,A,F),k9_euclid(E,B,G)),k7_euclid(E,k9_euclid(E,C,F),k9_euclid(E,D,G))) = k7_euclid(E,k9_euclid(E,k5_real_1(A,C),F),k9_euclid(E,k5_real_1(B,D),G)) ) ) ) ) ) ) ) ).
fof(t31_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(G))
=> ! [I] :
( m2_finseq_2(I,k1_numbers,k1_euclid(G))
=> ! [J] :
( m2_finseq_2(J,k1_numbers,k1_euclid(G))
=> k8_euclid(G,k7_euclid(G,k7_euclid(G,k9_euclid(G,A,H),k9_euclid(G,B,I)),k9_euclid(G,C,J)),k7_euclid(G,k7_euclid(G,k9_euclid(G,D,H),k9_euclid(G,E,I)),k9_euclid(G,F,J))) = k7_euclid(G,k7_euclid(G,k9_euclid(G,k5_real_1(A,D),H),k9_euclid(G,k5_real_1(B,E),I)),k9_euclid(G,k5_real_1(C,F),J)) ) ) ) ) ) ) ) ) ) ) ).
fof(t32_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(D))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(D))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(D))
=> ( k3_real_1(k3_real_1(A,B),C) = np__1
=> k7_euclid(D,k7_euclid(D,k9_euclid(D,A,E),k9_euclid(D,B,F)),k9_euclid(D,C,G)) = k7_euclid(D,k7_euclid(D,E,k9_euclid(D,B,k8_euclid(D,F,E))),k9_euclid(D,C,k8_euclid(D,G,E))) ) ) ) ) ) ) ) ) ).
fof(t33_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(C))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(C))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(C))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(C))
=> ~ ( D = k7_euclid(C,k7_euclid(C,E,k9_euclid(C,A,k8_euclid(C,F,E))),k9_euclid(C,B,k8_euclid(C,G,E)))
& ! [H] :
( m1_subset_1(H,k1_numbers)
=> ~ ( D = k7_euclid(C,k7_euclid(C,k9_euclid(C,H,E),k9_euclid(C,A,F)),k9_euclid(C,B,G))
& k3_real_1(k3_real_1(H,A),B) = np__1 ) ) ) ) ) ) ) ) ) ) ).
fof(t34_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,A)
& k4_jordan2c(A) = k5_euclid(A) ) ) ).
fof(t35_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_euclid(A)))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( ( v1_euclid_4(B,A)
& r2_hidden(C,B)
& r2_hidden(D,B) )
=> ( C = D
| B = k2_euclid_4(A,C,D) ) ) ) ) ) ) ).
fof(t36_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ~ ( r2_hidden(B,k2_euclid_4(A,D,E))
& r2_hidden(C,k2_euclid_4(A,D,E))
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> k8_euclid(A,C,B) != k9_euclid(A,F,k8_euclid(A,E,D)) ) ) ) ) ) ) ) ).
fof(d1_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ( r1_euclidlp(A,B,C)
<=> ( B != k5_euclid(A)
& C != k5_euclid(A)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& B = k9_euclid(A,D,C) ) ) ) ) ) ) ).
fof(t37_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ~ ( r1_euclidlp(A,B,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( D != np__0
& B = k9_euclid(A,D,C) ) ) ) ) ) ) ).
fof(t38_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( ( r2_euclidlp(A,B,C)
& r2_euclidlp(A,C,D) )
=> r2_euclidlp(A,B,D) ) ) ) ) ) ).
fof(d2_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ( r3_euclidlp(A,B,C)
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( k7_euclid(A,k9_euclid(A,D,B),k9_euclid(A,E,C)) = k5_euclid(A)
=> ( D = np__0
& E = np__0 ) ) ) ) ) ) ) ) ).
fof(t39_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ( r3_euclidlp(A,B,C)
=> ( B != k5_euclid(A)
& C != k5_euclid(A) ) ) ) ) ) ).
fof(t40_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(E))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(E))
=> ( ( r3_euclidlp(E,F,G)
& k7_euclid(E,k9_euclid(E,A,F),k9_euclid(E,B,G)) = k7_euclid(E,k9_euclid(E,C,F),k9_euclid(E,D,G)) )
=> ( A = C
& B = D ) ) ) ) ) ) ) ) ) ).
fof(t41_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(E))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(E))
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(E))
=> ! [I] :
( m2_finseq_2(I,k1_numbers,k1_euclid(E))
=> ! [J] :
( m2_finseq_2(J,k1_numbers,k1_euclid(E))
=> ~ ( r3_euclidlp(E,J,F)
& J = k7_euclid(E,k9_euclid(E,A,G),k9_euclid(E,B,H))
& F = k7_euclid(E,k9_euclid(E,C,G),k9_euclid(E,D,H))
& ! [K] :
( m1_subset_1(K,k1_numbers)
=> ! [L] :
( m1_subset_1(L,k1_numbers)
=> ! [M] :
( m1_subset_1(M,k1_numbers)
=> ! [N] :
( m1_subset_1(N,k1_numbers)
=> ~ ( G = k7_euclid(E,k9_euclid(E,K,J),k9_euclid(E,L,F))
& H = k7_euclid(E,k9_euclid(E,M,J),k9_euclid(E,N,F)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t42_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ~ ( r3_euclidlp(A,B,C)
& B = C ) ) ) ) ).
fof(t43_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ~ ( r3_euclidlp(A,k8_euclid(A,B,C),k8_euclid(A,D,C))
& B = D ) ) ) ) ) ).
fof(t44_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
=> ( r3_euclidlp(B,C,D)
=> r3_euclidlp(B,k7_euclid(B,C,k9_euclid(B,A,D)),D) ) ) ) ) ) ).
fof(t45_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(A))
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(A))
=> ! [I] :
( m2_finseq_2(I,k1_numbers,k1_euclid(A))
=> ( ( r3_euclidlp(A,k8_euclid(A,B,C),k8_euclid(A,D,E))
& r2_hidden(F,k2_euclid_4(A,C,B))
& r2_hidden(G,k2_euclid_4(A,C,B))
& r2_hidden(H,k2_euclid_4(A,E,D))
& r2_hidden(I,k2_euclid_4(A,E,D)) )
=> ( F = G
| H = I
| r3_euclidlp(A,k8_euclid(A,G,F),k8_euclid(A,I,H)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t46_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ( r2_euclidlp(A,B,C)
=> ( ~ r3_euclidlp(A,B,C)
& B != k5_euclid(A)
& C != k5_euclid(A) ) ) ) ) ) ).
fof(t47_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ~ ( ~ r3_euclidlp(A,B,C)
& B != k5_euclid(A)
& C != k5_euclid(A)
& ~ r2_euclidlp(A,B,C) ) ) ) ) ).
fof(t48_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
& r2_hidden(E,k2_euclid_4(A,B,C))
& r1_euclid_4(A,k8_euclid(A,B,C),k8_euclid(A,D,E)) ) ) ) ) ) ).
fof(d3_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ( r4_euclidlp(A,B,C)
<=> ( B != k5_euclid(A)
& C != k5_euclid(A)
& r1_euclid_4(A,B,C) ) ) ) ) ) ).
fof(t49_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( ( r4_euclidlp(A,B,C)
& r2_euclidlp(A,C,D) )
=> r4_euclidlp(A,B,D) ) ) ) ) ) ).
fof(t50_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ( r4_euclidlp(A,B,C)
=> r3_euclidlp(A,B,C) ) ) ) ) ).
fof(t51_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ~ ( r2_euclidlp(A,B,C)
& r4_euclidlp(A,B,C) ) ) ) ) ).
fof(t52_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ~ ( r4_euclidlp(A,B,C)
& r2_euclidlp(A,B,C) ) ) ) ) ).
fof(t53_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> r2_hidden(k2_euclid_4(A,B,C),k1_euclidlp(A)) ) ) ) ).
fof(t54_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( ( r2_hidden(B,D)
& r2_hidden(C,D) )
=> r1_tarski(k2_euclid_4(A,B,C),D) ) ) ) ) ) ).
fof(t55_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( ~ ( ~ r1_xboole_0(B,C)
& ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ~ ( r2_hidden(D,B)
& r2_hidden(D,C) ) ) )
& ~ ( ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
& r2_hidden(D,B)
& r2_hidden(D,C) )
& r1_xboole_0(B,C) ) ) ) ) ) ).
fof(t56_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r1_xboole_0(C,D)
& r2_hidden(B,C)
& r2_hidden(B,D) ) ) ) ) ) ).
fof(t57_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ? [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
& ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
& B = k2_euclid_4(A,C,D) ) ) ) ) ).
fof(t58_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ? [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
& r2_hidden(C,B) ) ) ) ).
fof(t59_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r2_hidden(B,C)
& v1_euclid_4(C,A)
& ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ~ ( D != B
& r2_hidden(D,C) ) ) ) ) ) ) ).
fof(t60_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( ~ r2_hidden(B,C)
& v1_euclid_4(C,A)
& ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ~ ( C = k2_euclid_4(A,D,E)
& r4_euclidlp(A,k8_euclid(A,B,D),k8_euclid(A,E,D)) ) ) ) ) ) ) ) ).
fof(t61_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( ~ r2_hidden(B,C)
& v1_euclid_4(C,A)
& ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ~ ( C = k2_euclid_4(A,D,E)
& r3_euclidlp(A,k8_euclid(A,B,D),k8_euclid(A,E,D)) ) ) ) ) ) ) ) ).
fof(d6_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> k3_euclidlp(A,B,C) = k5_seq_4(k2_euclidlp(A,B,C)) ) ) ) ).
fof(t63_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
& r2_hidden(D,C)
& k4_euclid_4(A,k8_euclid(A,B,D)) = k3_euclidlp(A,B,C) ) ) ) ) ).
fof(t64_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> r1_xreal_0(np__0,k3_euclidlp(A,B,C)) ) ) ) ).
fof(t65_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( r2_hidden(B,C)
<=> k3_euclidlp(A,B,C) = np__0 ) ) ) ) ).
fof(d7_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( r5_euclidlp(A,B,C)
<=> ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
& ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
& ? [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
& ? [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(A))
& B = k2_euclid_4(A,D,E)
& C = k2_euclid_4(A,F,G)
& r2_euclidlp(A,k8_euclid(A,E,D),k8_euclid(A,G,F)) ) ) ) ) ) ) ) ) ).
fof(t66_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( ( r5_euclidlp(A,B,C)
& r5_euclidlp(A,C,D) )
=> r5_euclidlp(A,B,D) ) ) ) ) ) ).
fof(d8_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( r6_euclidlp(A,B,C)
<=> ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
& ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
& ? [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
& ? [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(A))
& B = k2_euclid_4(A,D,E)
& C = k2_euclid_4(A,F,G)
& r4_euclidlp(A,k8_euclid(A,E,D),k8_euclid(A,G,F)) ) ) ) ) ) ) ) ) ).
fof(t67_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( ( r6_euclidlp(A,B,C)
& r5_euclidlp(A,C,D) )
=> r6_euclidlp(A,B,D) ) ) ) ) ) ).
fof(t68_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( ~ r2_hidden(B,C)
& v1_euclid_4(C,A)
& ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r2_hidden(B,D)
& r6_euclidlp(A,D,C)
& ~ r1_xboole_0(D,C) ) ) ) ) ) ) ).
fof(t69_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r1_xboole_0(B,C)
& ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ~ ( r2_hidden(D,B)
& ~ r2_hidden(D,C) ) ) ) ) ) ) ).
fof(t70_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( ( r2_hidden(B,D)
& r2_hidden(C,D) )
=> ( B = C
| ( k2_euclid_4(A,B,C) = D
& v1_euclid_4(D,A) ) ) ) ) ) ) ) ).
fof(t71_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( ( v1_euclid_4(B,A)
& v1_euclid_4(C,A)
& B = C )
=> r5_euclidlp(A,B,C) ) ) ) ) ).
fof(t72_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( r5_euclidlp(A,B,C)
=> ( v1_euclid_4(B,A)
& v1_euclid_4(C,A) ) ) ) ) ) ).
fof(t73_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( r6_euclidlp(A,B,C)
=> ( v1_euclid_4(B,A)
& v1_euclid_4(C,A) ) ) ) ) ) ).
fof(t74_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k1_euclidlp(B))
=> ( ( r2_hidden(C,D)
& r2_hidden(k9_euclid(B,A,C),D) )
=> ( A = np__1
| r2_hidden(k5_euclid(B),D) ) ) ) ) ) ) ).
fof(t75_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(B))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k1_euclidlp(B))
=> ~ ( r2_hidden(C,E)
& r2_hidden(D,E)
& ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(B))
=> ~ ( r2_hidden(F,E)
& k8_euclid(B,F,C) = k9_euclid(B,A,k8_euclid(B,D,C)) ) ) ) ) ) ) ) ) ).
fof(t76_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r2_hidden(B,E)
& r2_hidden(C,E)
& r2_hidden(D,E)
& B != C
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> k8_euclid(A,D,B) != k9_euclid(A,F,k8_euclid(A,C,B)) ) ) ) ) ) ) ) ).
fof(t77_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( r5_euclidlp(A,B,C)
=> ( B = C
| r1_xboole_0(B,C) ) ) ) ) ) ).
fof(t78_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r5_euclidlp(A,B,C)
& B != C
& ~ r1_xboole_0(B,C) ) ) ) ) ).
fof(t79_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( r5_euclidlp(A,B,C)
=> ( r1_xboole_0(B,C)
| B = C ) ) ) ) ) ).
fof(t80_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( v1_euclid_4(C,A)
& ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r2_hidden(B,D)
& r5_euclidlp(A,D,C) ) ) ) ) ) ) ).
fof(t81_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( ~ r2_hidden(B,C)
& v1_euclid_4(C,A)
& ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r2_hidden(B,D)
& r5_euclidlp(A,D,C)
& D != C ) ) ) ) ) ) ).
fof(t82_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_subset_1(F,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [G] :
( m2_subset_1(G,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( ( r2_hidden(B,F)
& r2_hidden(C,F)
& r2_hidden(D,G)
& r2_hidden(E,G)
& r6_euclidlp(A,F,G) )
=> ( B = C
| D = E
| r4_euclidlp(A,k8_euclid(A,C,B),k8_euclid(A,E,D)) ) ) ) ) ) ) ) ) ) ).
fof(t83_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r6_euclidlp(A,B,C)
& B = C ) ) ) ) ).
fof(t84_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( v1_euclid_4(D,A)
& D = k2_euclid_4(A,B,C)
& B = C ) ) ) ) ) ).
fof(t85_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_subset_1(F,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [G] :
( m2_subset_1(G,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( ( r2_hidden(B,F)
& r2_hidden(C,F)
& r2_hidden(D,G)
& r2_hidden(E,G)
& r5_euclidlp(A,F,G) )
=> ( B = C
| D = E
| r2_euclidlp(A,k8_euclid(A,C,B),k8_euclid(A,E,D)) ) ) ) ) ) ) ) ) ) ).
fof(t86_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
=> ! [G] :
( m2_subset_1(G,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [H] :
( m2_subset_1(H,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( ( r3_euclidlp(A,k8_euclid(A,B,C),k8_euclid(A,D,C))
& r2_hidden(E,k2_euclid_4(A,C,B))
& r2_hidden(F,k2_euclid_4(A,C,D))
& G = k2_euclid_4(A,B,D)
& H = k2_euclid_4(A,E,F) )
=> ( r5_euclidlp(A,G,H)
<=> ? [I] :
( m1_subset_1(I,k1_numbers)
& I != np__0
& k8_euclid(A,E,C) = k9_euclid(A,I,k8_euclid(A,B,C))
& k8_euclid(A,F,C) = k9_euclid(A,I,k8_euclid(A,D,C)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t87_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( v1_euclid_4(B,A)
& v1_euclid_4(C,A)
& B != C
& ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ~ ( r2_hidden(D,B)
& ~ r2_hidden(D,C) ) ) ) ) ) ) ).
fof(t88_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r6_euclidlp(A,C,D)
& r2_hidden(B,D)
& ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r2_hidden(B,E)
& r6_euclidlp(A,E,D)
& r5_euclidlp(A,E,C) ) ) ) ) ) ) ) ).
fof(t89_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r2_hidden(B,C)
& r2_hidden(B,D)
& r6_euclidlp(A,C,D)
& ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ~ ( B != E
& r2_hidden(E,C)
& ~ r2_hidden(E,D) ) ) ) ) ) ) ) ).
fof(d10_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_euclid(A)))
=> ( v1_euclidlp(B,A)
<=> ? [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
& ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
& ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
& r3_euclidlp(A,k8_euclid(A,D,C),k8_euclid(A,E,C))
& B = k4_euclidlp(A,C,D,E) ) ) ) ) ) ) ).
fof(t90_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( r2_hidden(B,k4_euclidlp(A,B,C,D))
& r2_hidden(C,k4_euclidlp(A,B,C,D))
& r2_hidden(D,k4_euclidlp(A,B,C,D)) ) ) ) ) ) ).
fof(t91_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(A))
=> ( ( r2_hidden(B,k4_euclidlp(A,C,D,E))
& r2_hidden(F,k4_euclidlp(A,C,D,E))
& r2_hidden(G,k4_euclidlp(A,C,D,E)) )
=> r1_tarski(k4_euclidlp(A,B,F,G),k4_euclidlp(A,C,D,E)) ) ) ) ) ) ) ) ) ).
fof(t92_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_euclid(A)))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
=> ( r2_hidden(C,k4_euclidlp(A,D,E,F))
=> ( ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( m1_subset_1(H,k1_numbers)
=> ! [I] :
( m1_subset_1(I,k1_numbers)
=> ~ ( k3_real_1(k3_real_1(G,H),I) = np__0
& C = k7_euclid(A,k7_euclid(A,k9_euclid(A,G,D),k9_euclid(A,H,E)),k9_euclid(A,I,F)) ) ) ) )
| r2_hidden(k5_euclid(A),k4_euclidlp(A,D,E,F)) ) ) ) ) ) ) ) ) ).
fof(t93_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
=> ( ( r2_hidden(B,k4_euclidlp(A,C,D,E))
& r2_hidden(F,k4_euclidlp(A,C,D,E)) )
=> r1_tarski(k2_euclid_4(A,B,F),k4_euclidlp(A,C,D,E)) ) ) ) ) ) ) ) ).
fof(t94_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_euclid(A)))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ( ( v1_euclidlp(B,A)
& r2_hidden(C,B) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( D != np__1
& r2_hidden(k9_euclid(A,D,C),B) ) )
| r2_hidden(k5_euclid(A),B) ) ) ) ) ) ).
fof(t95_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(D))
=> ! [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(D))
=> ! [G] :
( m2_finseq_2(G,k1_numbers,k1_euclid(D))
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(D))
=> ~ ( r3_euclidlp(D,k8_euclid(D,E,E),k8_euclid(D,F,E))
& r2_hidden(G,k4_euclidlp(D,E,H,F))
& G = k7_euclid(D,k7_euclid(D,k9_euclid(D,A,E),k9_euclid(D,B,H)),k9_euclid(D,C,F))
& k3_real_1(k3_real_1(A,B),C) != np__1
& ~ r2_hidden(k5_euclid(D),k4_euclidlp(D,E,H,F)) ) ) ) ) ) ) ) ) ) ).
fof(t96_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ( r2_hidden(B,k4_euclidlp(A,C,D,E))
<=> ? [F] :
( m1_subset_1(F,k1_numbers)
& ? [G] :
( m1_subset_1(G,k1_numbers)
& ? [H] :
( m1_subset_1(H,k1_numbers)
& k3_real_1(k3_real_1(F,G),H) = np__1
& B = k7_euclid(A,k7_euclid(A,k9_euclid(A,F,C),k9_euclid(A,G,D)),k9_euclid(A,H,E)) ) ) ) ) ) ) ) ) ) ).
fof(t97_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( m2_finseq_2(H,k1_numbers,k1_euclid(G))
=> ! [I] :
( m2_finseq_2(I,k1_numbers,k1_euclid(G))
=> ! [J] :
( m2_finseq_2(J,k1_numbers,k1_euclid(G))
=> ! [K] :
( m2_finseq_2(K,k1_numbers,k1_euclid(G))
=> ( ( r3_euclidlp(G,k8_euclid(G,H,I),k8_euclid(G,J,I))
& r2_hidden(K,k4_euclidlp(G,I,H,J))
& k3_real_1(k3_real_1(A,B),C) = np__1
& K = k7_euclid(G,k7_euclid(G,k9_euclid(G,A,I),k9_euclid(G,B,H)),k9_euclid(G,C,J))
& k3_real_1(k3_real_1(D,E),F) = np__1
& K = k7_euclid(G,k7_euclid(G,k9_euclid(G,D,I),k9_euclid(G,E,H)),k9_euclid(G,F,J)) )
=> ( A = D
& B = E
& C = F ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t98_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> r2_hidden(k4_euclidlp(A,B,C,D),k5_euclidlp(A)) ) ) ) ) ).
fof(t99_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ( ( r2_hidden(B,E)
& r2_hidden(C,E)
& r2_hidden(D,E) )
=> r1_tarski(k4_euclidlp(A,B,C,D),E) ) ) ) ) ) ) ).
fof(t100_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ( ( r2_hidden(B,E)
& r2_hidden(C,E)
& r2_hidden(D,E)
& r3_euclidlp(A,k8_euclid(A,C,B),k8_euclid(A,D,B)) )
=> E = k4_euclidlp(A,B,C,D) ) ) ) ) ) ) ).
fof(t101_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ( ( v1_euclidlp(B,A)
& r1_tarski(B,C) )
=> B = C ) ) ) ) ).
fof(t102_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ( r1_tarski(k2_euclid_4(A,B,C),k4_euclidlp(A,B,C,D))
& r1_tarski(k2_euclid_4(A,C,D),k4_euclidlp(A,B,C,D))
& r1_tarski(k2_euclid_4(A,D,B),k4_euclidlp(A,B,C,D)) ) ) ) ) ) ).
fof(t103_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ( ( r2_hidden(B,D)
& r2_hidden(C,D) )
=> r1_tarski(k2_euclid_4(A,B,C),D) ) ) ) ) ) ).
fof(d12_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( r7_euclidlp(A,B,C)
<=> ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
& ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
& ? [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(A))
& r1_tarski(B,k4_euclidlp(A,D,E,F))
& r1_tarski(C,k4_euclidlp(A,D,E,F)) ) ) ) ) ) ) ) ).
fof(t104_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( r7_euclidlp(A,B,C)
<=> ? [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
& r1_tarski(B,D)
& r1_tarski(C,D) ) ) ) ) ) ).
fof(t105_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( r5_euclidlp(A,B,C)
=> r7_euclidlp(A,B,C) ) ) ) ) ).
fof(t106_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( v1_euclid_4(B,A)
& v1_euclid_4(C,A)
& r7_euclidlp(A,B,C)
& r1_xboole_0(B,C)
& ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ~ ( r1_tarski(B,D)
& r1_tarski(C,D)
& v1_euclidlp(D,A) ) ) ) ) ) ) ).
fof(t107_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ? [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
& r2_hidden(B,D)
& r1_tarski(C,D) ) ) ) ) ).
fof(t108_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( ~ r2_hidden(B,C)
& v1_euclid_4(C,A)
& ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ~ ( r2_hidden(B,D)
& r1_tarski(C,D)
& v1_euclidlp(D,A) ) ) ) ) ) ) ).
fof(t109_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ( ( r2_hidden(B,D)
& r1_tarski(C,D)
& v1_euclid_4(C,A) )
=> ( r2_hidden(B,C)
| v1_euclidlp(D,A) ) ) ) ) ) ) ).
fof(t110_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ( ( v1_euclid_4(C,A)
& r2_hidden(B,D)
& r1_tarski(C,D)
& r2_hidden(B,E)
& r1_tarski(C,E) )
=> ( r2_hidden(B,C)
| D = E ) ) ) ) ) ) ) ).
fof(t111_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( v1_euclid_4(B,A)
& v1_euclid_4(C,A)
& r7_euclidlp(A,B,C)
& B != C
& ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ~ ( r1_tarski(B,D)
& r1_tarski(C,D)
& v1_euclidlp(D,A) ) ) ) ) ) ) ).
fof(t112_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( v1_euclid_4(B,A)
& v1_euclid_4(C,A)
& B != C
& ~ r1_xboole_0(B,C)
& ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ~ ( r1_tarski(B,D)
& r1_tarski(C,D)
& v1_euclidlp(D,A) ) ) ) ) ) ) ).
fof(t113_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ( ( v1_euclid_4(B,A)
& v1_euclid_4(C,A)
& r1_tarski(B,D)
& r1_tarski(C,D)
& r1_tarski(B,E)
& r1_tarski(C,E) )
=> ( B = C
| r1_xboole_0(B,C)
| D = E ) ) ) ) ) ) ) ).
fof(t114_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r5_euclidlp(A,B,C)
& B != C
& ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ~ ( r1_tarski(B,D)
& r1_tarski(C,D)
& v1_euclidlp(D,A) ) ) ) ) ) ) ).
fof(t115_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r6_euclidlp(A,B,C)
& ~ r1_xboole_0(B,C)
& ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ~ ( v1_euclidlp(D,A)
& r1_tarski(B,D)
& r1_tarski(C,D) ) ) ) ) ) ) ).
fof(t116_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [F] :
( m2_subset_1(F,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ( ( r1_tarski(C,F)
& r1_tarski(D,F)
& r1_tarski(E,F)
& r2_hidden(B,C)
& r2_hidden(B,D)
& r2_hidden(B,E)
& r6_euclidlp(A,C,E)
& r6_euclidlp(A,D,E) )
=> C = D ) ) ) ) ) ) ) ).
fof(t117_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ~ ( r7_euclidlp(A,B,C)
& r6_euclidlp(A,B,C)
& r1_xboole_0(B,C) ) ) ) ) ).
fof(t118_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [F] :
( m2_subset_1(F,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ( ( r1_tarski(C,F)
& r1_tarski(D,F)
& r6_euclidlp(A,C,D)
& r2_hidden(B,F)
& r5_euclidlp(A,E,D)
& r2_hidden(B,E) )
=> ( r1_tarski(E,F)
& r6_euclidlp(A,E,C) ) ) ) ) ) ) ) ) ).
fof(t119_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ( ( r1_tarski(B,E)
& r1_tarski(C,E)
& r1_tarski(D,E)
& r6_euclidlp(A,B,C)
& r6_euclidlp(A,B,D) )
=> r5_euclidlp(A,C,D) ) ) ) ) ) ) ).
fof(t120_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [F] :
( m2_subset_1(F,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ~ ( r1_tarski(B,F)
& r1_tarski(C,F)
& r1_tarski(D,F)
& r5_euclidlp(A,B,C)
& r5_euclidlp(A,C,D)
& B != C
& C != D
& D != B
& ~ r1_xboole_0(E,B)
& ~ r1_xboole_0(E,C)
& r1_xboole_0(E,D) ) ) ) ) ) ) ) ).
fof(t121_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( ( r7_euclidlp(A,B,C)
& v1_euclid_4(B,A)
& v1_euclid_4(C,A)
& r1_xboole_0(B,C) )
=> r5_euclidlp(A,B,C) ) ) ) ) ).
fof(t122_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(A))
=> ! [F] :
( m2_subset_1(F,k1_zfmisc_1(k1_euclid(A)),k5_euclidlp(A))
=> ~ ( r2_hidden(B,F)
& r2_hidden(C,F)
& r2_hidden(D,F)
& r2_hidden(E,F)
& r3_euclidlp(A,k8_euclid(A,C,B),k8_euclid(A,E,D))
& r1_xboole_0(k2_euclid_4(A,B,C),k2_euclid_4(A,D,E)) ) ) ) ) ) ) ) ).
fof(symmetry_r2_euclidlp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> ( r2_euclidlp(A,B,C)
=> r2_euclidlp(A,C,B) ) ) ).
fof(redefinition_r2_euclidlp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> ( r2_euclidlp(A,B,C)
<=> r1_euclidlp(A,B,C) ) ) ).
fof(symmetry_r3_euclidlp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> ( r3_euclidlp(A,B,C)
=> r3_euclidlp(A,C,B) ) ) ).
fof(symmetry_r4_euclidlp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A)) )
=> ( r4_euclidlp(A,B,C)
=> r4_euclidlp(A,C,B) ) ) ).
fof(symmetry_r5_euclidlp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclidlp(A))
& m1_subset_1(C,k1_euclidlp(A)) )
=> ( r5_euclidlp(A,B,C)
=> r5_euclidlp(A,C,B) ) ) ).
fof(symmetry_r6_euclidlp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclidlp(A))
& m1_subset_1(C,k1_euclidlp(A)) )
=> ( r6_euclidlp(A,B,C)
=> r6_euclidlp(A,C,B) ) ) ).
fof(dt_k1_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_subset_1(k1_euclidlp(A),k1_zfmisc_1(k1_zfmisc_1(k1_euclid(A)))) ) ).
fof(dt_k2_euclidlp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclidlp(A)) )
=> m1_subset_1(k2_euclidlp(A,B,C),k1_zfmisc_1(k1_numbers)) ) ).
fof(dt_k3_euclidlp,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclidlp(A)) )
=> m1_subset_1(k3_euclidlp(A,B,C),k1_numbers) ) ).
fof(dt_k4_euclidlp,axiom,
! [A,B,C,D] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k1_euclid(A))
& m1_subset_1(C,k1_euclid(A))
& m1_subset_1(D,k1_euclid(A)) )
=> m1_subset_1(k4_euclidlp(A,B,C,D),k1_zfmisc_1(k1_euclid(A))) ) ).
fof(dt_k5_euclidlp,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_subset_1(k5_euclidlp(A),k1_zfmisc_1(k1_zfmisc_1(k1_euclid(A)))) ) ).
fof(d4_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_euclidlp(A) = a_1_0_euclidlp(A) ) ).
fof(d5_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> k2_euclidlp(A,B,C) = a_3_0_euclidlp(A,B,C) ) ) ) ).
fof(t62_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(k1_euclid(A)),k1_euclidlp(A))
=> ( E = k2_euclid_4(A,B,C)
=> a_4_0_euclidlp(A,B,C,D) = k2_euclidlp(A,D,E) ) ) ) ) ) ) ).
fof(d9_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k1_numbers,k1_euclid(A))
=> ! [C] :
( m2_finseq_2(C,k1_numbers,k1_euclid(A))
=> ! [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(A))
=> k4_euclidlp(A,B,C,D) = a_4_1_euclidlp(A,B,C,D) ) ) ) ) ).
fof(d11_euclidlp,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k5_euclidlp(A) = a_1_1_euclidlp(A) ) ).
fof(fraenkel_a_1_0_euclidlp,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_0_euclidlp(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_euclid(B)))
& A = C
& ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
& ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(B))
& C = k2_euclid_4(B,D,E) ) ) ) ) ) ).
fof(fraenkel_a_3_0_euclidlp,axiom,
! [A,B,C,D] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_finseq_2(C,k1_numbers,k1_euclid(B))
& m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k1_euclidlp(B)) )
=> ( r2_hidden(A,a_3_0_euclidlp(B,C,D))
<=> ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(B))
& A = k4_euclid_4(B,k8_euclid(B,C,E))
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_4_0_euclidlp,axiom,
! [A,B,C,D,E] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_finseq_2(C,k1_numbers,k1_euclid(B))
& m2_finseq_2(D,k1_numbers,k1_euclid(B))
& m2_finseq_2(E,k1_numbers,k1_euclid(B)) )
=> ( r2_hidden(A,a_4_0_euclidlp(B,C,D,E))
<=> ? [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(B))
& A = k4_euclid_4(B,k8_euclid(B,E,F))
& r2_hidden(F,k2_euclid_4(B,C,D)) ) ) ) ).
fof(fraenkel_a_4_1_euclidlp,axiom,
! [A,B,C,D,E] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_finseq_2(C,k1_numbers,k1_euclid(B))
& m2_finseq_2(D,k1_numbers,k1_euclid(B))
& m2_finseq_2(E,k1_numbers,k1_euclid(B)) )
=> ( r2_hidden(A,a_4_1_euclidlp(B,C,D,E))
<=> ? [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(B))
& A = F
& ? [G] :
( m1_subset_1(G,k1_numbers)
& ? [H] :
( m1_subset_1(H,k1_numbers)
& ? [I] :
( m1_subset_1(I,k1_numbers)
& k3_real_1(k3_real_1(G,H),I) = np__1
& F = k7_euclid(B,k7_euclid(B,k9_euclid(B,G,C),k9_euclid(B,H,D)),k9_euclid(B,I,E)) ) ) ) ) ) ) ).
fof(fraenkel_a_1_1_euclidlp,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_1_euclidlp(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_euclid(B)))
& A = C
& ? [D] :
( m2_finseq_2(D,k1_numbers,k1_euclid(B))
& ? [E] :
( m2_finseq_2(E,k1_numbers,k1_euclid(B))
& ? [F] :
( m2_finseq_2(F,k1_numbers,k1_euclid(B))
& C = k4_euclidlp(B,D,E,F) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------