SET007 Axioms: SET007+895.ax
%------------------------------------------------------------------------------
% File : SET007+895 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of Connected Subsets of the Real Line
% 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 : rcomp_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 115 ( 0 unt; 0 def)
% Number of atoms : 890 ( 99 equ)
% Maximal formula atoms : 41 ( 7 avg)
% Number of connectives : 874 ( 99 ~; 31 |; 367 &)
% ( 10 <=>; 367 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 46 ( 45 usr; 0 prp; 1-5 aty)
% Number of functors : 52 ( 52 usr; 6 con; 0-4 aty)
% Number of variables : 336 ( 320 !; 16 ?)
% 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_rcomp_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ v1_xboole_0(k2_subset_1(A)) ) ).
fof(cc1_rcomp_3,axiom,
! [A] :
( m1_topmetr(A,k8_metric_1)
=> v1_borsuk_6(A) ) ).
fof(fc2_rcomp_3,axiom,
! [A] :
( ( v1_borsuk_6(A)
& l1_struct_0(A) )
=> ( v1_membered(u1_struct_0(A))
& v2_membered(u1_struct_0(A)) ) ) ).
fof(rc1_rcomp_3,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v1_membered(A)
& v2_membered(A)
& v1_seq_4(A)
& v2_seq_4(A) ) ).
fof(fc3_rcomp_3,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_membered(k1_rcomp_2(A,B))
& v2_membered(k1_rcomp_2(A,B))
& v1_seq_4(k1_rcomp_2(A,B))
& v2_seq_4(k1_rcomp_2(A,B))
& v3_seq_4(k1_rcomp_2(A,B)) ) ) ).
fof(fc4_rcomp_3,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_membered(k2_rcomp_2(A,B))
& v2_membered(k2_rcomp_2(A,B))
& v1_seq_4(k2_rcomp_2(A,B))
& v2_seq_4(k2_rcomp_2(A,B))
& v3_seq_4(k2_rcomp_2(A,B)) ) ) ).
fof(fc5_rcomp_3,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_membered(k2_rcomp_1(A,B))
& v2_membered(k2_rcomp_1(A,B))
& v1_seq_4(k2_rcomp_1(A,B))
& v2_seq_4(k2_rcomp_1(A,B))
& v3_seq_4(k2_rcomp_1(A,B))
& v3_rcomp_1(k2_rcomp_1(A,B)) ) ) ).
fof(fc6_rcomp_3,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_membered(k1_rcomp_1(A,B))
& v2_membered(k1_rcomp_1(A,B))
& v1_seq_4(k1_rcomp_1(A,B))
& v2_seq_4(k1_rcomp_1(A,B))
& v3_seq_4(k1_rcomp_1(A,B))
& v1_rcomp_1(k1_rcomp_1(A,B))
& v2_rcomp_1(k1_rcomp_1(A,B))
& v1_jct_misc(k1_rcomp_1(A,B)) ) ) ).
fof(fc7_rcomp_3,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_membered(k1_rcomp_2(A,B))
& v2_membered(k1_rcomp_2(A,B))
& v1_seq_4(k1_rcomp_2(A,B))
& v2_seq_4(k1_rcomp_2(A,B))
& v3_seq_4(k1_rcomp_2(A,B))
& v1_jct_misc(k1_rcomp_2(A,B)) ) ) ).
fof(fc8_rcomp_3,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_membered(k2_rcomp_2(A,B))
& v2_membered(k2_rcomp_2(A,B))
& v1_seq_4(k2_rcomp_2(A,B))
& v2_seq_4(k2_rcomp_2(A,B))
& v3_seq_4(k2_rcomp_2(A,B))
& v1_jct_misc(k2_rcomp_2(A,B)) ) ) ).
fof(fc9_rcomp_3,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_membered(k2_rcomp_1(A,B))
& v2_membered(k2_rcomp_1(A,B))
& v1_seq_4(k2_rcomp_1(A,B))
& v2_seq_4(k2_rcomp_1(A,B))
& v3_seq_4(k2_rcomp_1(A,B))
& v3_rcomp_1(k2_rcomp_1(A,B))
& v1_jct_misc(k2_rcomp_1(A,B)) ) ) ).
fof(rc2_rcomp_3,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& ~ v1_xboole_0(A)
& v1_membered(A)
& v2_membered(A)
& v1_seq_4(A)
& v2_seq_4(A)
& v3_seq_4(A)
& v3_rcomp_1(A)
& v1_jct_misc(A) ) ).
fof(fc10_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ v1_xboole_0(k2_limfunc1(A))
& v1_membered(k2_limfunc1(A))
& v2_membered(k2_limfunc1(A))
& v1_seq_4(k2_limfunc1(A))
& ~ v2_seq_4(k2_limfunc1(A))
& v2_rcomp_1(k2_limfunc1(A))
& v1_jct_misc(k2_limfunc1(A)) ) ) ).
fof(fc11_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ v1_xboole_0(k12_prob_1(A))
& v1_membered(k12_prob_1(A))
& v2_membered(k12_prob_1(A))
& v1_seq_4(k12_prob_1(A))
& ~ v2_seq_4(k12_prob_1(A))
& v3_rcomp_1(k12_prob_1(A))
& v1_jct_misc(k12_prob_1(A)) ) ) ).
fof(fc12_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ v1_xboole_0(k3_limfunc1(A))
& v1_membered(k3_limfunc1(A))
& v2_membered(k3_limfunc1(A))
& ~ v1_seq_4(k3_limfunc1(A))
& v2_seq_4(k3_limfunc1(A))
& v2_rcomp_1(k3_limfunc1(A))
& v1_jct_misc(k3_limfunc1(A)) ) ) ).
fof(fc13_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ v1_xboole_0(k4_limfunc1(A))
& v1_membered(k4_limfunc1(A))
& v2_membered(k4_limfunc1(A))
& ~ v1_seq_4(k4_limfunc1(A))
& v2_seq_4(k4_limfunc1(A))
& v3_rcomp_1(k4_limfunc1(A))
& v1_jct_misc(k4_limfunc1(A)) ) ) ).
fof(fc14_rcomp_3,axiom,
( ~ v1_xboole_0(k2_subset_1(k1_numbers))
& v1_membered(k2_subset_1(k1_numbers))
& v2_membered(k2_subset_1(k1_numbers))
& ~ v1_seq_4(k2_subset_1(k1_numbers))
& ~ v2_seq_4(k2_subset_1(k1_numbers))
& v2_rcomp_1(k2_subset_1(k1_numbers))
& v3_rcomp_1(k2_subset_1(k1_numbers))
& v1_jct_misc(k2_subset_1(k1_numbers)) ) ).
fof(rc3_rcomp_3,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& ~ v1_xboole_0(A)
& v1_membered(A)
& v2_membered(A)
& ~ v3_seq_4(A)
& v2_rcomp_1(A)
& v3_rcomp_1(A)
& v1_jct_misc(A) ) ).
fof(fc15_rcomp_3,axiom,
! [A] :
( ( v2_topalg_2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topalg_2))) )
=> ( v1_pre_topc(k3_pre_topc(k3_topalg_2,A))
& v2_pre_topc(k3_pre_topc(k3_topalg_2,A))
& v1_borsuk_6(k3_pre_topc(k3_topalg_2,A))
& v3_topalg_2(k3_pre_topc(k3_topalg_2,A)) ) ) ).
fof(fc16_rcomp_3,axiom,
! [A] :
( ( v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( v1_membered(k5_toprealb(A))
& v2_membered(k5_toprealb(A))
& v2_topalg_2(k5_toprealb(A)) ) ) ).
fof(fc17_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ v3_struct_0(k4_topmetr(A,A))
& v1_pre_topc(k4_topmetr(A,A))
& v2_pre_topc(k4_topmetr(A,A))
& v2_compts_1(k4_topmetr(A,A))
& v6_group_1(k4_topmetr(A,A))
& v3_realset2(k4_topmetr(A,A))
& v10_pscomp_1(k4_topmetr(A,A))
& v1_borsuk_6(k4_topmetr(A,A)) ) ) ).
fof(rc4_rcomp_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v3_pre_topc(B,A)
& v4_pre_topc(B,A)
& v2_connsp_1(B,A) ) ) ).
fof(rc5_rcomp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_connsp_1(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B)
& v3_pre_topc(B,A)
& v4_pre_topc(B,A)
& v2_connsp_1(B,A) ) ) ).
fof(rc6_rcomp_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
& ~ v1_xboole_0(B)
& v1_tops_2(B,A)
& v2_tops_2(B,A)
& v1_rcomp_3(B,A) ) ) ).
fof(t1_rcomp_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_membered(A)
& v2_seq_4(A) )
=> ! [B] :
( ( v2_rcomp_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( r1_tarski(A,B)
=> r2_hidden(k3_seq_4(A),B) ) ) ) ).
fof(t2_rcomp_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v2_membered(A)
& v1_seq_4(A) )
=> ! [B] :
( ( v2_rcomp_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( r1_tarski(A,B)
=> r2_hidden(k2_seq_4(A),B) ) ) ) ).
fof(t3_rcomp_3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> k7_pscomp_1(k4_subset_1(k1_numbers,A,B)) = k4_subset_1(k1_numbers,k7_pscomp_1(A),k7_pscomp_1(B)) ) ) ).
fof(t4_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> k4_pscomp_1(k1_rcomp_2(A,B)) = A ) ) ) ).
fof(t5_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> k3_pscomp_1(k1_rcomp_2(A,B)) = B ) ) ) ).
fof(t6_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> k4_pscomp_1(k2_rcomp_2(A,B)) = A ) ) ) ).
fof(t7_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> k3_pscomp_1(k2_rcomp_2(A,B)) = B ) ) ) ).
fof(t8_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( k1_rcomp_1(A,B) = k1_rcomp_1(C,D)
=> ( ( ~ r1_xreal_0(A,B)
& ~ r1_xreal_0(C,D) )
| ( A = C
& B = D ) ) ) ) ) ) ) ).
fof(t9_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( k2_rcomp_1(A,B) = k2_rcomp_1(C,D)
=> ( ( r1_xreal_0(B,A)
& r1_xreal_0(D,C) )
| ( A = C
& B = D ) ) ) ) ) ) ) ).
fof(t10_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( k2_rcomp_2(A,B) = k2_rcomp_2(C,D)
=> ( ( r1_xreal_0(B,A)
& r1_xreal_0(D,C) )
| ( A = C
& B = D ) ) ) ) ) ) ) ).
fof(t11_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( k1_rcomp_2(A,B) = k1_rcomp_2(C,D)
=> ( ( r1_xreal_0(B,A)
& r1_xreal_0(D,C) )
| ( A = C
& B = D ) ) ) ) ) ) ) ).
fof(t12_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r1_tarski(k1_rcomp_2(A,B),k1_rcomp_1(C,D))
=> ( r1_xreal_0(B,A)
| ( r1_xreal_0(C,A)
& r1_xreal_0(B,D) ) ) ) ) ) ) ) ).
fof(t13_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r1_tarski(k1_rcomp_2(A,B),k1_rcomp_2(C,D))
=> ( r1_xreal_0(B,A)
| ( r1_xreal_0(C,A)
& r1_xreal_0(B,D) ) ) ) ) ) ) ) ).
fof(t14_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r1_tarski(k2_rcomp_2(A,B),k1_rcomp_1(C,D))
=> ( r1_xreal_0(B,A)
| ( r1_xreal_0(C,A)
& r1_xreal_0(B,D) ) ) ) ) ) ) ) ).
fof(t15_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r1_tarski(k2_rcomp_2(A,B),k2_rcomp_2(C,D))
=> ( r1_xreal_0(B,A)
| ( r1_xreal_0(C,A)
& r1_xreal_0(B,D) ) ) ) ) ) ) ) ).
fof(t16_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_subset_1(k1_numbers,k1_rcomp_1(A,B)) = k4_subset_1(k1_numbers,k12_prob_1(A),k4_limfunc1(B)) ) ) ).
fof(t17_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_subset_1(k1_numbers,k2_rcomp_1(A,B)) = k4_subset_1(k1_numbers,k2_limfunc1(A),k3_limfunc1(B)) ) ) ).
fof(t18_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_subset_1(k1_numbers,k1_rcomp_2(A,B)) = k4_subset_1(k1_numbers,k12_prob_1(A),k3_limfunc1(B)) ) ) ).
fof(t19_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k3_subset_1(k1_numbers,k2_rcomp_2(A,B)) = k4_subset_1(k1_numbers,k2_limfunc1(A),k4_limfunc1(B)) ) ) ).
fof(t20_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> k5_subset_1(k1_numbers,k1_rcomp_1(A,B),k4_subset_1(k1_numbers,k2_limfunc1(A),k3_limfunc1(B))) = k2_tarski(A,B) ) ) ) ).
fof(t21_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_pscomp_1(k2_limfunc1(A)) = A ) ).
fof(t22_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_pscomp_1(k12_prob_1(A)) = A ) ).
fof(t23_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> k4_pscomp_1(k3_limfunc1(A)) = A ) ).
fof(t24_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> k4_pscomp_1(k4_limfunc1(A)) = A ) ).
fof(t25_rcomp_3,axiom,
! [A] :
( ( v3_seq_4(A)
& v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( ( r2_hidden(k4_pscomp_1(A),A)
& r2_hidden(k3_pscomp_1(A),A) )
=> A = k1_rcomp_1(k4_pscomp_1(A),k3_pscomp_1(A)) ) ) ).
fof(t26_rcomp_3,axiom,
! [A] :
( ( v3_seq_4(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( ~ r2_hidden(k4_pscomp_1(A),A)
=> r1_tarski(A,k2_rcomp_2(k4_pscomp_1(A),k3_pscomp_1(A))) ) ) ).
fof(t27_rcomp_3,axiom,
! [A] :
( ( v3_seq_4(A)
& v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( r2_hidden(k3_pscomp_1(A),A)
=> ( r2_hidden(k4_pscomp_1(A),A)
| A = k2_rcomp_2(k4_pscomp_1(A),k3_pscomp_1(A)) ) ) ) ).
fof(t28_rcomp_3,axiom,
! [A] :
( ( v3_seq_4(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( ~ r2_hidden(k3_pscomp_1(A),A)
=> r1_tarski(A,k1_rcomp_2(k4_pscomp_1(A),k3_pscomp_1(A))) ) ) ).
fof(t29_rcomp_3,axiom,
! [A] :
( ( v3_seq_4(A)
& v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( r2_hidden(k4_pscomp_1(A),A)
=> ( r2_hidden(k3_pscomp_1(A),A)
| A = k1_rcomp_2(k4_pscomp_1(A),k3_pscomp_1(A)) ) ) ) ).
fof(t30_rcomp_3,axiom,
! [A] :
( ( v3_seq_4(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ~ ( ~ r2_hidden(k4_pscomp_1(A),A)
& ~ r2_hidden(k3_pscomp_1(A),A)
& ~ r1_tarski(A,k2_rcomp_1(k4_pscomp_1(A),k3_pscomp_1(A))) ) ) ).
fof(t31_rcomp_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_seq_4(A)
& v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ~ ( ~ r2_hidden(k4_pscomp_1(A),A)
& ~ r2_hidden(k3_pscomp_1(A),A)
& A != k2_rcomp_1(k4_pscomp_1(A),k3_pscomp_1(A)) ) ) ).
fof(t32_rcomp_3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v1_seq_4(A)
=> r1_tarski(A,k2_limfunc1(k3_pscomp_1(A))) ) ) ).
fof(t33_rcomp_3,axiom,
! [A] :
( ( v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( ( v1_seq_4(A)
& r2_hidden(k3_pscomp_1(A),A) )
=> ( v2_seq_4(A)
| A = k2_limfunc1(k3_pscomp_1(A)) ) ) ) ).
fof(t34_rcomp_3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v1_seq_4(A)
=> ( r2_hidden(k3_pscomp_1(A),A)
| r1_tarski(A,k12_prob_1(k3_pscomp_1(A))) ) ) ) ).
fof(t35_rcomp_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( v1_seq_4(A)
=> ( v2_seq_4(A)
| r2_hidden(k3_pscomp_1(A),A)
| A = k12_prob_1(k3_pscomp_1(A)) ) ) ) ).
fof(t36_rcomp_3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v2_seq_4(A)
=> r1_tarski(A,k3_limfunc1(k4_pscomp_1(A))) ) ) ).
fof(t37_rcomp_3,axiom,
! [A] :
( ( v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( ( v2_seq_4(A)
& r2_hidden(k4_pscomp_1(A),A) )
=> ( v1_seq_4(A)
| A = k3_limfunc1(k4_pscomp_1(A)) ) ) ) ).
fof(t38_rcomp_3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v2_seq_4(A)
=> ( r2_hidden(k4_pscomp_1(A),A)
| r1_tarski(A,k4_limfunc1(k4_pscomp_1(A))) ) ) ) ).
fof(t39_rcomp_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( v2_seq_4(A)
=> ( v1_seq_4(A)
| r2_hidden(k4_pscomp_1(A),A)
| A = k4_limfunc1(k4_pscomp_1(A)) ) ) ) ).
fof(t40_rcomp_3,axiom,
! [A] :
( ( v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ~ ( ~ v1_seq_4(A)
& ~ v2_seq_4(A)
& A != k1_numbers ) ) ).
fof(t41_rcomp_3,axiom,
! [A] :
( ( v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ~ ( ~ v1_xboole_0(A)
& A != k1_numbers
& ! [B] :
( v1_xreal_0(B)
=> A != k2_limfunc1(B) )
& ! [B] :
( v1_xreal_0(B)
=> A != k12_prob_1(B) )
& ! [B] :
( v1_xreal_0(B)
=> A != k3_limfunc1(B) )
& ! [B] :
( v1_xreal_0(B)
=> A != k4_limfunc1(B) )
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( r1_xreal_0(B,C)
& A = k1_rcomp_1(B,C) ) ) )
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,B)
& A = k1_rcomp_2(B,C) ) ) )
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,B)
& A = k2_rcomp_2(B,C) ) ) )
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,B)
& A = k2_rcomp_1(B,C) ) ) ) ) ) ).
fof(t42_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_jct_misc(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ~ ( ~ r2_hidden(A,B)
& ~ r1_xreal_0(A,k4_pscomp_1(B))
& ~ r1_xreal_0(k3_pscomp_1(B),A) ) ) ) ).
fof(t43_rcomp_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_seq_4(A)
& v1_jct_misc(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v3_seq_4(B)
& v1_jct_misc(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( ( r1_xreal_0(k4_pscomp_1(A),k4_pscomp_1(B))
& r1_xreal_0(k3_pscomp_1(B),k3_pscomp_1(A)) )
=> ( ( k4_pscomp_1(A) = k4_pscomp_1(B)
& r2_hidden(k4_pscomp_1(B),B)
& ~ r2_hidden(k4_pscomp_1(A),A) )
| ( k3_pscomp_1(A) = k3_pscomp_1(B)
& r2_hidden(k3_pscomp_1(B),B)
& ~ r2_hidden(k3_pscomp_1(A),A) )
| r1_tarski(B,A) ) ) ) ) ).
fof(t44_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
=> ( ( r1_xreal_0(A,B)
& C = k1_rcomp_1(A,B) )
=> k2_tops_1(k3_topalg_2,C) = k2_tarski(A,B) ) ) ) ) ).
fof(t45_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
=> ( C = k2_rcomp_1(A,B)
=> ( r1_xreal_0(B,A)
| k2_tops_1(k3_topalg_2,C) = k2_tarski(A,B) ) ) ) ) ) ).
fof(t46_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
=> ( C = k1_rcomp_2(A,B)
=> ( r1_xreal_0(B,A)
| k2_tops_1(k3_topalg_2,C) = k2_tarski(A,B) ) ) ) ) ) ).
fof(t47_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
=> ( C = k2_rcomp_2(A,B)
=> ( r1_xreal_0(B,A)
| k2_tops_1(k3_topalg_2,C) = k2_tarski(A,B) ) ) ) ) ) ).
fof(t48_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
=> ( C = k1_rcomp_1(A,B)
=> k1_tops_1(k3_topalg_2,C) = k2_rcomp_1(A,B) ) ) ) ) ).
fof(t49_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
=> ( C = k2_rcomp_1(A,B)
=> k1_tops_1(k3_topalg_2,C) = k2_rcomp_1(A,B) ) ) ) ) ).
fof(t50_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
=> ( C = k1_rcomp_2(A,B)
=> k1_tops_1(k3_topalg_2,C) = k2_rcomp_1(A,B) ) ) ) ) ).
fof(t51_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
=> ( C = k2_rcomp_2(A,B)
=> k1_tops_1(k3_topalg_2,C) = k2_rcomp_1(A,B) ) ) ) ) ).
fof(t52_rcomp_3,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topalg_2)))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ( A = B
=> ( v2_connsp_1(A,k3_topalg_2)
<=> v1_jct_misc(B) ) ) ) ) ).
fof(t53_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))
=> ( v3_seq_4(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) ) ) ) ) ) ).
fof(t54_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r1_xreal_0(A,B)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))
=> ( ( E = k1_rcomp_2(C,D)
& r1_xreal_0(D,B) )
=> ( r1_xreal_0(C,A)
| k1_tops_1(k4_topmetr(A,B),E) = k2_rcomp_1(C,D) ) ) ) ) ) ) ) ) ).
fof(t55_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( r1_xreal_0(A,B)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))
=> ( ( E = k2_rcomp_2(C,D)
& r1_xreal_0(A,C) )
=> ( r1_xreal_0(B,D)
| k1_tops_1(k4_topmetr(A,B),E) = k2_rcomp_1(C,D) ) ) ) ) ) ) ) ) ).
fof(t56_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_numbers))
=> ( C = D
=> ( v2_connsp_1(C,k4_topmetr(A,B))
<=> v1_jct_misc(D) ) ) ) ) ) ) ).
fof(t57_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( ( v3_pre_topc(C,k4_topmetr(A,B))
& v2_connsp_1(C,k4_topmetr(A,B))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))) )
=> ~ ( ~ v1_xboole_0(C)
& C != k1_rcomp_1(A,B)
& ! [D] :
( v1_xreal_0(D)
=> ~ ( ~ r1_xreal_0(D,A)
& r1_xreal_0(D,B)
& C = k1_rcomp_2(A,D) ) )
& ! [D] :
( v1_xreal_0(D)
=> ~ ( r1_xreal_0(A,D)
& ~ r1_xreal_0(B,D)
& C = k2_rcomp_2(D,B) ) )
& ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ~ ( r1_xreal_0(A,D)
& ~ r1_xreal_0(E,D)
& r1_xreal_0(E,B)
& C = k2_rcomp_1(D,E) ) ) ) ) ) ) ) ) ).
fof(t58_rcomp_3,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_pre_topc(A,B)
<=> r1_compts_1(A,B,k2_pre_topc(A)) ) ) ) ).
fof(t60_rcomp_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_realset2(A)
& l1_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( r1_pre_topc(A,C)
=> r2_hidden(k1_struct_0(A,B),C) ) ) ) ) ).
fof(d1_rcomp_3,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( v1_rcomp_3(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(C,B)
=> v2_connsp_1(C,A) ) ) ) ) ) ).
fof(d2_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B))
& r1_xreal_0(A,B) )
=> ! [D] :
( m2_finseq_1(D,k1_pcomps_1(k1_numbers))
=> ( m1_rcomp_3(D,A,B,C)
<=> ( r1_tarski(k5_relset_1(k5_numbers,k1_pcomps_1(k1_numbers),D),C)
& k5_setfam_1(k1_numbers,k5_relset_1(k5_numbers,k1_pcomps_1(k1_numbers),D)) = k1_rcomp_1(A,B)
& ! [E] :
( v4_ordinal2(E)
=> ( r1_xreal_0(np__1,E)
=> ( ~ ( r1_xreal_0(E,k3_finseq_1(D))
& v1_xboole_0(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)) )
& ( r1_xreal_0(k2_xcmplx_0(E,np__1),k3_finseq_1(D))
=> ( r1_xreal_0(k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,np__1))))
& r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,np__1))))
& ~ r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,np__1)))) ) )
& ( r1_xreal_0(k2_xcmplx_0(E,np__2),k3_finseq_1(D))
=> r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,np__2)))) ) ) ) )
& ( r2_hidden(k1_rcomp_1(A,B),C)
=> D = k13_binarith(k1_zfmisc_1(k1_numbers),k1_rcomp_1(A,B)) )
& ( ~ r2_hidden(k1_rcomp_1(A,B),C)
=> ( ? [E] :
( v1_xreal_0(E)
& ~ r1_xreal_0(E,A)
& r1_xreal_0(E,B)
& k1_funct_1(D,np__1) = k1_rcomp_2(A,E) )
& ? [E] :
( v1_xreal_0(E)
& r1_xreal_0(A,E)
& ~ r1_xreal_0(B,E)
& k1_funct_1(D,k3_finseq_1(D)) = k2_rcomp_2(E,B) )
& ! [E] :
( v4_ordinal2(E)
=> ~ ( ~ r1_xreal_0(E,np__1)
& ~ r1_xreal_0(k3_finseq_1(D),E)
& ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( v1_xreal_0(G)
=> ~ ( r1_xreal_0(A,F)
& ~ r1_xreal_0(G,F)
& r1_xreal_0(G,B)
& k1_funct_1(D,E) = k2_rcomp_1(F,G) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t63_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B))
& r1_xreal_0(A,B)
& r2_hidden(k1_rcomp_1(A,B),C) )
=> m1_rcomp_3(k13_binarith(k1_zfmisc_1(k1_numbers),k1_rcomp_1(A,B)),A,B,C) ) ) ) ) ).
fof(t64_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,A)))))
=> ! [C] :
( m1_rcomp_3(C,A,A,B)
=> ( ( r1_pre_topc(k4_topmetr(A,A),B)
& v1_tops_2(B,k4_topmetr(A,A))
& v1_rcomp_3(B,k4_topmetr(A,A)) )
=> C = k13_binarith(k1_zfmisc_1(k1_numbers),k1_rcomp_1(A,A)) ) ) ) ) ).
fof(t65_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [D] :
( m1_rcomp_3(D,A,B,C)
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B))
& r1_xreal_0(A,B) )
=> r1_xreal_0(np__1,k3_finseq_1(D)) ) ) ) ) ) ).
fof(t66_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [D] :
( m1_rcomp_3(D,A,B,C)
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B))
& r1_xreal_0(A,B)
& k3_finseq_1(D) = np__1 )
=> D = k13_binarith(k1_zfmisc_1(k1_numbers),k1_rcomp_1(A,B)) ) ) ) ) ) ).
fof(t67_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [F] :
( m1_rcomp_3(F,A,B,E)
=> ( ( r1_pre_topc(k4_topmetr(A,B),E)
& v1_tops_2(E,k4_topmetr(A,B))
& v1_rcomp_3(E,k4_topmetr(A,B))
& r1_xreal_0(A,B)
& r2_hidden(C,k4_finseq_1(F))
& r2_hidden(D,k4_finseq_1(F)) )
=> ( r1_xreal_0(D,C)
| r1_xreal_0(k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),F,C)),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),F,D))) ) ) ) ) ) ) ) ) ).
fof(t68_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [F] :
( m1_rcomp_3(F,A,B,E)
=> ( ( r1_pre_topc(k4_topmetr(A,B),E)
& v1_tops_2(E,k4_topmetr(A,B))
& v1_rcomp_3(E,k4_topmetr(A,B))
& r1_xreal_0(A,B)
& r2_hidden(C,k4_finseq_1(F))
& r2_hidden(D,k4_finseq_1(F)) )
=> ( r1_xreal_0(D,C)
| r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),F,C)),k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),F,D))) ) ) ) ) ) ) ) ) ).
fof(t69_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [E] :
( m1_rcomp_3(E,A,B,D)
=> ~ ( r1_pre_topc(k4_topmetr(A,B),D)
& v1_tops_2(D,k4_topmetr(A,B))
& v1_rcomp_3(D,k4_topmetr(A,B))
& r1_xreal_0(A,B)
& r1_xreal_0(np__1,C)
& r1_xreal_0(k2_xcmplx_0(C,np__1),k3_finseq_1(E))
& v1_xboole_0(k2_rcomp_1(k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),E,k2_xcmplx_0(C,np__1))),k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),E,C)))) ) ) ) ) ) ) ).
fof(t70_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [D] :
( m1_rcomp_3(D,A,B,C)
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B))
& r1_xreal_0(A,B) )
=> k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,np__1)) = A ) ) ) ) ) ).
fof(t71_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [D] :
( m1_rcomp_3(D,A,B,C)
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B))
& r1_xreal_0(A,B) )
=> r2_hidden(A,k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,np__1)) ) ) ) ) ) ).
fof(t72_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [D] :
( m1_rcomp_3(D,A,B,C)
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B))
& r1_xreal_0(A,B) )
=> k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k3_finseq_1(D))) = B ) ) ) ) ) ).
fof(t73_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [D] :
( m1_rcomp_3(D,A,B,C)
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B))
& r1_xreal_0(A,B) )
=> r2_hidden(B,k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k3_finseq_1(D))) ) ) ) ) ) ).
fof(d3_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [D] :
( m1_rcomp_3(D,A,B,C)
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B))
& r1_xreal_0(A,B) )
=> ! [E] :
( m2_finseq_1(E,k1_numbers)
=> ( m2_rcomp_3(E,A,B,C,D)
<=> ( k3_finseq_1(E) = k1_nat_1(k3_finseq_1(D),np__1)
& k2_seq_1(k5_numbers,k1_numbers,E,np__1) = A
& k2_seq_1(k5_numbers,k1_numbers,E,k3_finseq_1(E)) = B
& ! [F] :
( v4_ordinal2(F)
=> ( r1_xreal_0(np__1,F)
=> ( r1_xreal_0(k3_finseq_1(E),k2_xcmplx_0(F,np__1))
| r2_hidden(k2_seq_1(k5_numbers,k1_numbers,E,k2_xcmplx_0(F,np__1)),k2_rcomp_1(k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(F,np__1))),k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,F)))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t74_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [D] :
( m1_rcomp_3(D,A,B,C)
=> ! [E] :
( m2_rcomp_3(E,A,B,C,D)
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B))
& r1_xreal_0(A,B) )
=> r1_xreal_0(np__2,k3_finseq_1(E)) ) ) ) ) ) ) ).
fof(t75_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [D] :
( m1_rcomp_3(D,A,B,C)
=> ! [E] :
( m2_rcomp_3(E,A,B,C,D)
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B))
& r1_xreal_0(A,B)
& k3_finseq_1(D) = np__1 )
=> E = k10_finseq_1(A,B) ) ) ) ) ) ) ).
fof(t76_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [E] :
( m1_rcomp_3(E,A,B,D)
=> ! [F] :
( m2_rcomp_3(F,A,B,D,E)
=> ~ ( r1_pre_topc(k4_topmetr(A,B),D)
& v1_tops_2(D,k4_topmetr(A,B))
& v1_rcomp_3(D,k4_topmetr(A,B))
& r1_xreal_0(A,B)
& r1_xreal_0(np__1,C)
& ~ r1_xreal_0(k3_finseq_1(F),k2_xcmplx_0(C,np__1))
& r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),E,C)),k2_seq_1(k5_numbers,k1_numbers,F,k2_xcmplx_0(C,np__1))) ) ) ) ) ) ) ) ).
fof(t77_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [E] :
( m1_rcomp_3(E,A,B,D)
=> ! [F] :
( m2_rcomp_3(F,A,B,D,E)
=> ~ ( r1_pre_topc(k4_topmetr(A,B),D)
& v1_tops_2(D,k4_topmetr(A,B))
& v1_rcomp_3(D,k4_topmetr(A,B))
& r1_xreal_0(A,B)
& ~ r1_xreal_0(C,np__1)
& r1_xreal_0(C,k3_finseq_1(E))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,F,C),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),E,C))) ) ) ) ) ) ) ) ).
fof(t78_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [E] :
( m1_rcomp_3(E,A,B,D)
=> ! [F] :
( m2_rcomp_3(F,A,B,D,E)
=> ( ( r1_pre_topc(k4_topmetr(A,B),D)
& v1_tops_2(D,k4_topmetr(A,B))
& v1_rcomp_3(D,k4_topmetr(A,B))
& r1_xreal_0(A,B)
& r1_xreal_0(np__1,C) )
=> ( r1_xreal_0(k3_finseq_1(E),C)
| r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,F,C),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),E,k2_xcmplx_0(C,np__1)))) ) ) ) ) ) ) ) ) ).
fof(t79_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [D] :
( m1_rcomp_3(D,A,B,C)
=> ! [E] :
( m2_rcomp_3(E,A,B,C,D)
=> ( ( r1_pre_topc(k4_topmetr(A,B),C)
& v1_tops_2(C,k4_topmetr(A,B))
& v1_rcomp_3(C,k4_topmetr(A,B)) )
=> ( r1_xreal_0(B,A)
| v1_seqm_3(E) ) ) ) ) ) ) ) ).
fof(t80_rcomp_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
=> ! [E] :
( m1_rcomp_3(E,A,B,D)
=> ! [F] :
( m2_rcomp_3(F,A,B,D,E)
=> ( ( r1_pre_topc(k4_topmetr(A,B),D)
& v1_tops_2(D,k4_topmetr(A,B))
& v1_rcomp_3(D,k4_topmetr(A,B))
& r1_xreal_0(A,B)
& r1_xreal_0(np__1,C) )
=> ( r1_xreal_0(k3_finseq_1(F),C)
| r1_tarski(k1_rcomp_1(k2_seq_1(k5_numbers,k1_numbers,F,C),k2_seq_1(k5_numbers,k1_numbers,F,k2_xcmplx_0(C,np__1))),k1_funct_1(E,C)) ) ) ) ) ) ) ) ) ).
fof(dt_m1_rcomp_3,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) )
=> ! [D] :
( m1_rcomp_3(D,A,B,C)
=> m2_finseq_1(D,k1_pcomps_1(k1_numbers)) ) ) ).
fof(existence_m1_rcomp_3,axiom,
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) )
=> ? [D] : m1_rcomp_3(D,A,B,C) ) ).
fof(dt_m2_rcomp_3,axiom,
! [A,B,C,D] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
& m1_rcomp_3(D,A,B,C) )
=> ! [E] :
( m2_rcomp_3(E,A,B,C,D)
=> m2_finseq_1(E,k1_numbers) ) ) ).
fof(existence_m2_rcomp_3,axiom,
! [A,B,C,D] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))))
& m1_rcomp_3(D,A,B,C) )
=> ? [E] : m2_rcomp_3(E,A,B,C,D) ) ).
fof(t59_rcomp_3,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( r1_pre_topc(A,B)
& C = k3_tex_1(u1_struct_0(A),B,a_2_0_rcomp_3(A,B)) )
=> r1_pre_topc(A,C) ) ) ) ) ).
fof(t61_rcomp_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ( ( r1_pre_topc(A,B)
& v1_finset_1(B) )
=> ! [D] :
( ( C = k3_tex_1(u1_struct_0(A),B,a_2_1_rcomp_3(A,B))
& D = a_2_2_rcomp_3(A,C) )
=> r5_orders_1(k1_wellord2(D),D) ) ) ) ) ) ).
fof(t62_rcomp_3,axiom,
! [A] :
( ( v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B,C] :
( C = a_2_3_rcomp_3(A,B)
=> ! [D] :
( ( r7_orders_1(k1_wellord2(C),D)
& r2_hidden(D,k3_relat_1(k1_wellord2(C))) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(E,D)
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(F,D)
& r2_hidden(G,D)
& r1_tarski(E,k4_subset_1(u1_struct_0(A),F,G))
& E != F
& E != G ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_2_0_rcomp_3,axiom,
! [A,B,C] :
( ( l1_struct_0(B)
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) )
=> ( r2_hidden(A,a_2_0_rcomp_3(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& r2_hidden(D,C)
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& r2_hidden(E,C)
& r1_tarski(D,E)
& D != E ) ) ) ) ).
fof(fraenkel_a_2_1_rcomp_3,axiom,
! [A,B,C] :
( ( v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) )
=> ( r2_hidden(A,a_2_1_rcomp_3(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& A = D
& r2_hidden(D,C)
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& r2_hidden(E,C)
& r1_tarski(D,E)
& D != E ) ) ) ) ).
fof(fraenkel_a_2_2_rcomp_3,axiom,
! [A,B,C] :
( ( v2_pre_topc(B)
& l1_pre_topc(B)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) )
=> ( r2_hidden(A,a_2_2_rcomp_3(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
& A = D
& r1_pre_topc(B,D)
& r1_tarski(D,C) ) ) ) ).
fof(fraenkel_a_2_3_rcomp_3,axiom,
! [A,B,C] :
( ( v2_pre_topc(B)
& l1_pre_topc(B) )
=> ( r2_hidden(A,a_2_3_rcomp_3(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
& A = D
& r1_pre_topc(B,D)
& r1_tarski(D,C) ) ) ) ).
%------------------------------------------------------------------------------