SET007 Axioms: SET007+799.ax
%------------------------------------------------------------------------------
% File : SET007+799 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Basic Properties of Rough Sets and Rough Membership Function
% 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 : roughs_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 125 ( 1 unt; 0 def)
% Number of atoms : 769 ( 55 equ)
% Maximal formula atoms : 12 ( 6 avg)
% Number of connectives : 776 ( 132 ~; 0 |; 358 &)
% ( 27 <=>; 259 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 45 ( 43 usr; 1 prp; 0-3 aty)
% Number of functors : 48 ( 48 usr; 5 con; 0-4 aty)
% Number of variables : 305 ( 290 !; 15 ?)
% 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_roughs_1,axiom,
! [A] :
( v1_orders_2(g1_orders_2(A,k6_partfun1(A)))
& v2_orders_2(g1_orders_2(A,k6_partfun1(A)))
& v3_orders_2(g1_orders_2(A,k6_partfun1(A)))
& v4_orders_2(g1_orders_2(A,k6_partfun1(A)))
& v1_orders_3(g1_orders_2(A,k6_partfun1(A))) ) ).
fof(fc2_roughs_1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ( ~ v3_struct_0(g1_orders_2(A,k1_eqrel_1(A)))
& v1_orders_2(g1_orders_2(A,k1_eqrel_1(A)))
& ~ v1_roughs_1(g1_orders_2(A,k1_eqrel_1(A))) ) ) ).
fof(cc1_roughs_1,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( v2_orders_2(A)
& ~ v1_orders_3(A) )
=> ( ~ v3_realset2(A)
& v2_orders_2(A) ) ) ) ).
fof(cc2_roughs_1,axiom,
! [A] :
( l1_orders_2(A)
=> ( ( v3_realset2(A)
& v2_orders_2(A) )
=> v1_orders_3(A) ) ) ).
fof(cc3_roughs_1,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_orders_3(A)
=> v1_roughs_1(A) ) ) ).
fof(cc4_roughs_1,axiom,
! [A] :
( l1_orders_2(A)
=> ( ~ v1_roughs_1(A)
=> ~ v1_orders_3(A) ) ) ).
fof(rc1_roughs_1,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& ~ v1_orders_3(A)
& ~ v1_roughs_1(A) ) ).
fof(cc5_roughs_1,axiom,
! [A] :
( ( v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v1_prob_2(A) ) ) ).
fof(rc2_roughs_1,axiom,
! [A] :
? [B] :
( m1_finseq_1(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B)
& v1_prob_2(B) ) ).
fof(rc3_roughs_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_finseq_1(B,A)
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B)
& v1_prob_2(B) ) ) ).
fof(fc3_roughs_1,axiom,
! [A,B] :
( ( v1_finset_1(A)
& m1_relset_1(B,A,A) )
=> ( v6_group_1(g1_orders_2(A,B))
& v1_orders_2(g1_orders_2(A,B)) ) ) ).
fof(cc6_roughs_1,axiom,
! [A] :
( l1_orders_2(A)
=> ( v2_roughs_1(A)
=> v3_roughs_1(A) ) ) ).
fof(fc4_roughs_1,axiom,
! [A] :
( v1_orders_2(g1_orders_2(A,k6_partfun1(A)))
& v2_orders_2(g1_orders_2(A,k6_partfun1(A)))
& v3_orders_2(g1_orders_2(A,k6_partfun1(A)))
& v4_orders_2(g1_orders_2(A,k6_partfun1(A)))
& v1_orders_3(g1_orders_2(A,k6_partfun1(A)))
& v1_roughs_1(g1_orders_2(A,k6_partfun1(A)))
& v2_roughs_1(g1_orders_2(A,k6_partfun1(A)))
& v3_roughs_1(g1_orders_2(A,k6_partfun1(A))) ) ).
fof(rc4_roughs_1,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v6_group_1(A)
& v1_orders_3(A)
& v1_roughs_1(A)
& v2_roughs_1(A)
& v3_roughs_1(A) ) ).
fof(rc5_roughs_1,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v6_group_1(A)
& ~ v1_orders_3(A)
& ~ v1_roughs_1(A)
& v2_roughs_1(A)
& v3_roughs_1(A) ) ).
fof(fc5_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ( v1_relat_1(u1_orders_2(A))
& v1_relat_2(u1_orders_2(A))
& v3_relat_2(u1_orders_2(A))
& v1_partfun1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(fc6_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ( v1_relat_1(u1_orders_2(A))
& v1_relat_2(u1_orders_2(A))
& v3_relat_2(u1_orders_2(A))
& v8_relat_2(u1_orders_2(A))
& v1_partfun1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(rc6_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v4_roughs_1(B,A) ) ) ).
fof(fc7_roughs_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v4_roughs_1(k3_roughs_1(A,B),A) ) ).
fof(fc8_roughs_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ~ v4_roughs_1(k4_roughs_1(A,B),A) ) ).
fof(rc7_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v1_roughs_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& v4_roughs_1(B,A) ) ) ).
fof(fc9_roughs_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v1_xboole_0(k1_card_1(k6_eqrel_1(u1_struct_0(A),u1_orders_2(A),B)))
& v1_card_1(k1_card_1(k6_eqrel_1(u1_struct_0(A),u1_orders_2(A),B))) ) ) ).
fof(cc7_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_orders_3(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ v4_roughs_1(B,A) ) ) ).
fof(t1_roughs_1,axiom,
! [A] :
( r1_tarski(k1_eqrel_1(A),k6_partfun1(A))
=> v1_realset1(A) ) ).
fof(d1_roughs_1,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_roughs_1(A)
<=> r1_tarski(u1_orders_2(A),k6_partfun1(u1_struct_0(A))) ) ) ).
fof(t2_roughs_1,axiom,
! [A] :
( ( v2_orders_2(A)
& l1_orders_2(A) )
=> r1_tarski(k6_partfun1(u1_struct_0(A)),u1_orders_2(A)) ) ).
fof(t3_roughs_1,axiom,
! [A,B] :
( ( v1_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> r1_tarski(k6_partfun1(A),B) ) ).
fof(t4_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v1_roughs_1(A)
& l1_orders_2(A) )
=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& B != C
& r2_hidden(k4_tarski(B,C),u1_orders_2(A)) ) ) ) ).
fof(t5_roughs_1,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> k3_card_3(k8_finseq_1(A,B,C)) = k2_xboole_0(k3_card_3(B),k3_card_3(C)) ) ) ).
fof(t6_roughs_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v1_prob_2(B)
& r1_tarski(A,B) )
=> v1_prob_2(A) ) ) ) ).
fof(t7_roughs_1,axiom,
! [A,B,C,D] :
( ( v1_relat_2(D)
& v3_relat_2(D)
& v1_partfun1(D,A,A)
& m2_relset_1(D,A,A) )
=> ( r2_hidden(B,k6_eqrel_1(A,D,C))
=> r2_hidden(C,k6_eqrel_1(A,D,B)) ) ) ).
fof(d2_roughs_1,axiom,
! [A] :
( l1_orders_2(A)
=> ( v2_roughs_1(A)
<=> ( v3_relat_2(u1_orders_2(A))
& v8_relat_2(u1_orders_2(A))
& v1_partfun1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A)) ) ) ) ).
fof(d3_roughs_1,axiom,
! [A] :
( l1_orders_2(A)
=> ( v3_roughs_1(A)
<=> ( v1_relat_2(u1_orders_2(A))
& v3_relat_2(u1_orders_2(A))
& v1_partfun1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A)) ) ) ) ).
fof(d6_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k5_roughs_1(A,B) = k6_subset_1(u1_struct_0(A),k4_roughs_1(A,B),k3_roughs_1(A,B)) ) ) ).
fof(d7_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v4_roughs_1(B,A)
<=> k5_roughs_1(A,B) != k1_xboole_0 ) ) ) ).
fof(t8_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( r2_hidden(C,k3_roughs_1(A,B))
=> r1_tarski(k6_eqrel_1(u1_struct_0(A),u1_orders_2(A),C),B) ) ) ) ).
fof(t9_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_tarski(k6_eqrel_1(u1_struct_0(A),u1_orders_2(A),C),B)
=> r2_hidden(C,k3_roughs_1(A,B)) ) ) ) ) ).
fof(t10_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
~ ( r2_hidden(C,k4_roughs_1(A,B))
& r1_xboole_0(k6_eqrel_1(u1_struct_0(A),u1_orders_2(A),C),B) ) ) ) ).
fof(t11_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ~ r1_xboole_0(k6_eqrel_1(u1_struct_0(A),u1_orders_2(A),C),B)
=> r2_hidden(C,k4_roughs_1(A,B)) ) ) ) ) ).
fof(t12_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k3_roughs_1(A,B),B) ) ) ).
fof(t13_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(B,k4_roughs_1(A,B)) ) ) ).
fof(t14_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k3_roughs_1(A,B),k4_roughs_1(A,B)) ) ) ).
fof(t15_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ~ v4_roughs_1(B,A)
<=> k3_roughs_1(A,B) = B ) ) ) ).
fof(t16_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( ~ v4_roughs_1(B,A)
<=> k4_roughs_1(A,B) = B ) ) ) ).
fof(t17_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( B = k3_roughs_1(A,B)
<=> B = k4_roughs_1(A,B) ) ) ) ).
fof(t18_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> k3_roughs_1(A,k1_pre_topc(A)) = k1_xboole_0 ) ).
fof(t19_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> k4_roughs_1(A,k1_pre_topc(A)) = k1_xboole_0 ) ).
fof(t20_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> k3_roughs_1(A,k2_pre_topc(A)) = k2_pre_topc(A) ) ).
fof(t21_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> k4_roughs_1(A,k2_pre_topc(A)) = k2_pre_topc(A) ) ).
fof(t22_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k3_roughs_1(A,k5_subset_1(u1_struct_0(A),B,C)) = k5_subset_1(u1_struct_0(A),k3_roughs_1(A,B),k3_roughs_1(A,C)) ) ) ) ).
fof(t23_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> k4_roughs_1(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k4_roughs_1(A,B),k4_roughs_1(A,C)) ) ) ) ).
fof(t24_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tarski(B,C)
=> r1_tarski(k3_roughs_1(A,B),k3_roughs_1(A,C)) ) ) ) ) ).
fof(t25_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_tarski(B,C)
=> r1_tarski(k4_roughs_1(A,B),k4_roughs_1(A,C)) ) ) ) ) ).
fof(t26_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k4_subset_1(u1_struct_0(A),k3_roughs_1(A,B),k3_roughs_1(A,C)),k3_roughs_1(A,k4_subset_1(u1_struct_0(A),B,C))) ) ) ) ).
fof(t27_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k4_roughs_1(A,k5_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),k4_roughs_1(A,B),k4_roughs_1(A,C))) ) ) ) ).
fof(t28_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_roughs_1(A,k3_subset_1(u1_struct_0(A),B)) = k3_subset_1(u1_struct_0(A),k4_roughs_1(A,B)) ) ) ).
fof(t29_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_roughs_1(A,k3_subset_1(u1_struct_0(A),B)) = k3_subset_1(u1_struct_0(A),k3_roughs_1(A,B)) ) ) ).
fof(t30_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_roughs_1(A,k3_roughs_1(A,k4_roughs_1(A,B))) = k4_roughs_1(A,B) ) ) ).
fof(t31_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_roughs_1(A,k4_roughs_1(A,k3_roughs_1(A,B))) = k3_roughs_1(A,B) ) ) ).
fof(t32_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k5_roughs_1(A,B) = k5_roughs_1(A,k3_subset_1(u1_struct_0(A),B)) ) ) ).
fof(t33_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_roughs_1(A,k3_roughs_1(A,B)) = k3_roughs_1(A,B) ) ) ).
fof(t34_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_roughs_1(A,k3_roughs_1(A,B)) = k4_roughs_1(A,k3_roughs_1(A,B)) ) ) ).
fof(t35_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_roughs_1(A,k4_roughs_1(A,B)) = k4_roughs_1(A,B) ) ) ).
fof(t36_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_roughs_1(A,k4_roughs_1(A,B)) = k3_roughs_1(A,k4_roughs_1(A,B)) ) ) ).
fof(t37_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C,D] :
( ( r2_hidden(C,k4_roughs_1(A,B))
& r2_hidden(k4_tarski(C,D),u1_orders_2(A)) )
=> r2_hidden(D,k4_roughs_1(A,B)) ) ) ) ).
fof(d8_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_roughs_1(C,A,B)
<=> C = k4_tarski(k3_roughs_1(A,B),k4_roughs_1(A,B)) ) ) ) ).
fof(d9_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),k1_numbers)
& m2_relset_1(C,u1_struct_0(A),k1_numbers) )
=> ( C = k6_roughs_1(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),k1_numbers,C,D) = k6_real_1(k4_card_1(k5_subset_1(u1_struct_0(A),B,k6_eqrel_1(u1_struct_0(A),u1_orders_2(A),D))),k4_card_1(k6_eqrel_1(u1_struct_0(A),u1_orders_2(A),D))) ) ) ) ) ) ).
fof(t38_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_xreal_0(np__0,k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),C))
& r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),C),np__1) ) ) ) ) ).
fof(t39_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r2_hidden(k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),C),k1_rcomp_1(np__0,np__1)) ) ) ) ).
fof(t40_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),C) = np__1
<=> r2_hidden(C,k3_roughs_1(A,B)) ) ) ) ) ).
fof(t41_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),C) = np__0
<=> r2_hidden(C,k3_subset_1(u1_struct_0(A),k4_roughs_1(A,B))) ) ) ) ) ).
fof(t42_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( ~ r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),C),np__0)
& ~ r1_xreal_0(np__1,k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),C)) )
<=> r2_hidden(C,k5_roughs_1(A,B)) ) ) ) ) ).
fof(t43_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_orders_3(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ~ v4_roughs_1(B,A) ) ) ).
fof(t44_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v1_orders_3(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k6_roughs_1(A,B) = k5_funct_3(B,u1_struct_0(A)) ) ) ).
fof(t45_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C,D] :
( r2_hidden(k4_tarski(C,D),u1_orders_2(A))
=> k1_funct_1(k6_roughs_1(A,B),C) = k1_funct_1(k6_roughs_1(A,B),D) ) ) ) ).
fof(t46_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,k3_subset_1(u1_struct_0(A),B)),C) = k5_real_1(np__1,k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),C)) ) ) ) ).
fof(t47_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_tarski(B,C)
=> r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),D),k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,C),D)) ) ) ) ) ) ).
fof(t48_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),D),k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,k4_subset_1(u1_struct_0(A),B,C)),D)) ) ) ) ) ).
fof(t49_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,k5_subset_1(u1_struct_0(A),B,C)),D),k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),D)) ) ) ) ) ).
fof(t50_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r1_xreal_0(k4_square_1(k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),D),k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,C),D)),k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,k4_subset_1(u1_struct_0(A),B,C)),D)) ) ) ) ) ).
fof(t51_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r1_xboole_0(B,C)
=> k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,k4_subset_1(u1_struct_0(A),B,C)),D) = k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),D),k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,C),D)) ) ) ) ) ) ).
fof(t52_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,k5_subset_1(u1_struct_0(A),B,C)),D),k3_square_1(k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,B),D),k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,C),D))) ) ) ) ) ).
fof(d10_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( D = k7_roughs_1(A,B,C)
<=> ( k4_relset_1(k5_numbers,k1_numbers,D) = k4_relset_1(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_relset_1(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B))
=> k1_funct_1(D,E) = k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,k1_roughs_1(u1_struct_0(A),B,E)),C) ) ) ) ) ) ) ) ) ).
fof(t53_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m2_finseq_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> k7_roughs_1(A,k8_finseq_1(k1_zfmisc_1(u1_struct_0(A)),B,k12_finseq_1(k1_zfmisc_1(u1_struct_0(A)),D)),C) = k8_finseq_1(k1_numbers,k7_roughs_1(A,B,C),k12_finseq_1(k1_numbers,k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,D),C))) ) ) ) ) ).
fof(t54_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,k1_pre_topc(A)),B) = np__0 ) ) ).
fof(t55_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_prob_2(C)
& m2_finseq_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> k8_funct_2(u1_struct_0(A),k1_numbers,k6_roughs_1(A,k2_roughs_1(u1_struct_0(A),C)),B) = k15_rvsum_1(k7_roughs_1(A,C,B)) ) ) ) ).
fof(d11_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r1_roughs_1(A,B,C)
<=> r1_tarski(k3_roughs_1(A,B),k3_roughs_1(A,C)) ) ) ) ) ).
fof(d12_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_roughs_1(A,B,C)
<=> r1_tarski(k4_roughs_1(A,B),k4_roughs_1(A,C)) ) ) ) ) ).
fof(d13_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r3_roughs_1(A,B,C)
<=> ( r1_roughs_1(A,B,C)
& r2_roughs_1(A,B,C) ) ) ) ) ) ).
fof(t59_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r1_roughs_1(A,B,C)
& r1_roughs_1(A,C,D) )
=> r1_roughs_1(A,B,D) ) ) ) ) ) ).
fof(t60_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r2_roughs_1(A,B,C)
& r2_roughs_1(A,C,D) )
=> r2_roughs_1(A,B,D) ) ) ) ) ) ).
fof(t61_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( ( r3_roughs_1(A,B,C)
& r3_roughs_1(A,C,D) )
=> r3_roughs_1(A,B,D) ) ) ) ) ) ).
fof(d14_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r4_roughs_1(A,B,C)
<=> k3_roughs_1(A,B) = k3_roughs_1(A,C) ) ) ) ) ).
fof(d15_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r5_roughs_1(A,B,C)
<=> k4_roughs_1(A,B) = k4_roughs_1(A,C) ) ) ) ) ).
fof(d16_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r6_roughs_1(A,B,C)
<=> ( k3_roughs_1(A,B) = k3_roughs_1(A,C)
& k4_roughs_1(A,B) = k4_roughs_1(A,C) ) ) ) ) ) ).
fof(d17_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r4_roughs_1(A,B,C)
<=> ( r1_roughs_1(A,B,C)
& r1_roughs_1(A,C,B) ) ) ) ) ) ).
fof(d18_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r5_roughs_1(A,B,C)
<=> ( r2_roughs_1(A,B,C)
& r2_roughs_1(A,C,B) ) ) ) ) ) ).
fof(d19_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( r6_roughs_1(A,B,C)
<=> ( r4_roughs_1(A,B,C)
& r5_roughs_1(A,B,C) ) ) ) ) ) ).
fof(dt_m1_roughs_1,axiom,
$true ).
fof(existence_m1_roughs_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ? [C] : m1_roughs_1(C,A,B) ) ).
fof(symmetry_r4_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r4_roughs_1(A,B,C)
=> r4_roughs_1(A,C,B) ) ) ).
fof(reflexivity_r4_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> r4_roughs_1(A,B,B) ) ).
fof(symmetry_r5_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r5_roughs_1(A,B,C)
=> r5_roughs_1(A,C,B) ) ) ).
fof(reflexivity_r5_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> r5_roughs_1(A,B,B) ) ).
fof(symmetry_r6_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ( r6_roughs_1(A,B,C)
=> r6_roughs_1(A,C,B) ) ) ).
fof(reflexivity_r6_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> r6_roughs_1(A,B,B) ) ).
fof(dt_k1_roughs_1,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers) )
=> m1_subset_1(k1_roughs_1(A,B,C),k1_zfmisc_1(A)) ) ).
fof(redefinition_k1_roughs_1,axiom,
! [A,B,C] :
( ( m1_finseq_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k5_numbers) )
=> k1_roughs_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k2_roughs_1,axiom,
! [A,B] :
( m1_finseq_1(B,k1_zfmisc_1(A))
=> m1_subset_1(k2_roughs_1(A,B),k1_zfmisc_1(A)) ) ).
fof(redefinition_k2_roughs_1,axiom,
! [A,B] :
( m1_finseq_1(B,k1_zfmisc_1(A))
=> k2_roughs_1(A,B) = k3_card_3(B) ) ).
fof(dt_k3_roughs_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k3_roughs_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k4_roughs_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k4_roughs_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k5_roughs_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> m1_subset_1(k5_roughs_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k6_roughs_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ( v1_funct_1(k6_roughs_1(A,B))
& v1_funct_2(k6_roughs_1(A,B),u1_struct_0(A),k1_numbers)
& m2_relset_1(k6_roughs_1(A,B),u1_struct_0(A),k1_numbers) ) ) ).
fof(dt_k7_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v3_roughs_1(A)
& l1_orders_2(A)
& m1_finseq_1(B,k1_zfmisc_1(u1_struct_0(A)))
& m1_subset_1(C,u1_struct_0(A)) )
=> m2_finseq_1(k7_roughs_1(A,B,C),k1_numbers) ) ).
fof(d4_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_roughs_1(A,B) = a_2_0_roughs_1(A,B) ) ) ).
fof(d5_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_roughs_1(A,B) = a_2_1_roughs_1(A,B) ) ) ).
fof(t56_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k3_roughs_1(A,B) = a_2_2_roughs_1(A,B) ) ) ).
fof(t57_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k4_roughs_1(A,B) = a_2_3_roughs_1(A,B) ) ) ).
fof(t58_roughs_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v6_group_1(A)
& v2_roughs_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> k5_roughs_1(A,B) = a_2_4_roughs_1(A,B) ) ) ).
fof(fraenkel_a_2_0_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v3_roughs_1(B)
& l1_orders_2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_0_roughs_1(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& r1_tarski(k6_eqrel_1(u1_struct_0(B),u1_orders_2(B),D),C) ) ) ) ).
fof(fraenkel_a_2_1_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v3_roughs_1(B)
& l1_orders_2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_1_roughs_1(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& ~ r1_xboole_0(k6_eqrel_1(u1_struct_0(B),u1_orders_2(B),D),C) ) ) ) ).
fof(fraenkel_a_2_2_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v6_group_1(B)
& v2_roughs_1(B)
& l1_orders_2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_2_roughs_1(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& k8_funct_2(u1_struct_0(B),k1_numbers,k6_roughs_1(B,C),D) = np__1 ) ) ) ).
fof(fraenkel_a_2_3_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v6_group_1(B)
& v2_roughs_1(B)
& l1_orders_2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_3_roughs_1(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& ~ r1_xreal_0(k8_funct_2(u1_struct_0(B),k1_numbers,k6_roughs_1(B,C),D),np__0) ) ) ) ).
fof(fraenkel_a_2_4_roughs_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v6_group_1(B)
& v2_roughs_1(B)
& l1_orders_2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r2_hidden(A,a_2_4_roughs_1(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& A = D
& ~ r1_xreal_0(k8_funct_2(u1_struct_0(B),k1_numbers,k6_roughs_1(B,C),D),np__0)
& ~ r1_xreal_0(np__1,k8_funct_2(u1_struct_0(B),k1_numbers,k6_roughs_1(B,C),D)) ) ) ) ).
%------------------------------------------------------------------------------