SET007 Axioms: SET007+897.ax
%------------------------------------------------------------------------------
% File : SET007+897 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Brouwer Fixed Point Theorem for Disks on the Plane
% 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 : brouwer [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 30 ( 0 unt; 0 def)
% Number of atoms : 292 ( 41 equ)
% Maximal formula atoms : 19 ( 9 avg)
% Number of connectives : 323 ( 61 ~; 6 |; 141 &)
% ( 5 <=>; 110 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 13 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of predicates : 22 ( 21 usr; 0 prp; 1-3 aty)
% Number of functors : 47 ( 47 usr; 5 con; 0-5 aty)
% Number of variables : 131 ( 124 !; 7 ?)
% 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_brouwer,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& ~ v3_realset2(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ~ v1_xboole_0(k1_brouwer(A,B)) ) ).
fof(fc2_brouwer,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& ~ v3_realset2(B)
& l1_pre_topc(B) )
=> ~ v1_xboole_0(k1_brouwer(A,B)) ) ).
fof(fc3_brouwer,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C)
& ~ v3_xreal_0(C) )
=> ( ~ v3_struct_0(k2_brouwer(A,B,C))
& v2_pre_topc(k2_brouwer(A,B,C)) ) ) ).
fof(fc4_brouwer,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C) )
=> ( v2_pre_topc(k2_brouwer(A,B,C))
& v1_topalg_2(k2_brouwer(A,B,C),A) ) ) ).
fof(t1_brouwer,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( r2_hidden(C,k1_brouwer(A,B))
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& C = k8_borsuk_1(A,B,D,E)
& D != E ) ) ) ) ) ).
fof(t2_brouwer,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> k2_topreal9(A,B,np__0) = k1_struct_0(k15_euclid(A),B) ) ) ).
fof(d2_brouwer,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( v1_xreal_0(C)
=> k2_brouwer(A,B,C) = k3_pre_topc(k15_euclid(A),k2_topreal9(A,B,C)) ) ) ) ).
fof(t3_brouwer,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> u1_struct_0(k2_brouwer(A,C,B)) = k2_topreal9(A,C,B) ) ) ) ).
fof(t4_brouwer,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ~ ( C != D
& m1_subset_1(C,u1_struct_0(k2_brouwer(A,E,B)))
& ~ m1_subset_1(C,u1_struct_0(k7_toprealb(A,E,B)))
& ! [F] :
( m1_subset_1(F,u1_struct_0(k7_toprealb(A,E,B)))
=> k1_tarski(F) != k5_subset_1(u1_struct_0(k15_euclid(A)),k4_topreal9(A,C,D),k3_topreal9(A,E,B)) ) ) ) ) ) ) ) ).
fof(t5_brouwer,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
=> ~ ( C != D
& r2_hidden(C,u1_struct_0(k7_toprealb(A,E,B)))
& m1_subset_1(D,u1_struct_0(k2_brouwer(A,E,B)))
& ! [F] :
( m1_subset_1(F,u1_struct_0(k7_toprealb(A,E,B)))
=> ~ ( F != C
& k2_tarski(C,F) = k5_subset_1(u1_struct_0(k15_euclid(A)),k4_topreal9(A,C,D),k3_topreal9(A,E,B)) ) ) ) ) ) ) ) ) ).
fof(d3_brouwer,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( ( v1_xreal_0(E)
& ~ v3_xreal_0(E) )
=> ( ( m1_subset_1(C,u1_struct_0(k2_brouwer(A,B,E)))
& m1_subset_1(D,u1_struct_0(k2_brouwer(A,B,E))) )
=> ( C = D
| ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
=> ( F = k3_brouwer(A,B,C,D,E)
<=> ( r2_hidden(F,k5_subset_1(u1_struct_0(k15_euclid(A)),k4_topreal9(A,C,D),k3_topreal9(A,B,E)))
& F != C ) ) ) ) ) ) ) ) ) ) ).
fof(t6_brouwer,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(B)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
=> ( ( m1_subset_1(C,u1_struct_0(k2_brouwer(B,D,A)))
& m1_subset_1(E,u1_struct_0(k2_brouwer(B,D,A))) )
=> ( C = E
| m1_subset_1(k3_brouwer(B,D,C,E,A),u1_struct_0(k7_toprealb(B,D,A))) ) ) ) ) ) ) ) ).
fof(t7_brouwer,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(C)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(C)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(C)))
=> ! [G] :
( m1_subset_1(G,k1_euclid(C))
=> ! [H] :
( m1_subset_1(H,k1_euclid(C))
=> ! [I] :
( m1_subset_1(I,k1_euclid(C))
=> ( ( G = D
& H = E
& I = F
& m1_subset_1(D,u1_struct_0(k2_brouwer(C,F,B)))
& m1_subset_1(E,u1_struct_0(k2_brouwer(C,F,B)))
& A = k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(k2_euclid_2(C,k20_euclid(C,E,D),k20_euclid(C,D,F))),k8_square_1(k6_xcmplx_0(k5_square_1(k2_euclid_2(C,k20_euclid(C,E,D),k20_euclid(C,D,F))),k3_xcmplx_0(k15_rvsum_1(k11_rvsum_1(k8_euclid(C,H,G))),k6_xcmplx_0(k15_rvsum_1(k11_rvsum_1(k8_euclid(C,G,I))),k5_square_1(B)))))),k15_rvsum_1(k11_rvsum_1(k8_euclid(C,H,G)))) )
=> ( D = E
| k3_brouwer(C,F,D,E,B) = k17_euclid(C,k18_euclid(k6_xcmplx_0(np__1,A),C,D),k18_euclid(A,C,E)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t8_brouwer,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ! [F] :
( v1_xreal_0(F)
=> ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(k15_euclid(np__2)))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(k15_euclid(np__2)))
=> ( ( m1_subset_1(G,u1_struct_0(k2_brouwer(np__2,I,B)))
& m1_subset_1(H,u1_struct_0(k2_brouwer(np__2,I,B)))
& C = k5_real_1(k21_euclid(H),k21_euclid(G))
& D = k5_real_1(k22_euclid(H),k22_euclid(G))
& E = k5_real_1(k21_euclid(G),k21_euclid(I))
& F = k5_real_1(k22_euclid(G),k22_euclid(I))
& A = k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(E,C),k3_xcmplx_0(F,D))),k8_square_1(k6_xcmplx_0(k5_square_1(k2_xcmplx_0(k3_xcmplx_0(E,C),k3_xcmplx_0(F,D))),k3_xcmplx_0(k2_xcmplx_0(k5_square_1(C),k5_square_1(D)),k6_xcmplx_0(k2_xcmplx_0(k5_square_1(E),k5_square_1(F)),k5_square_1(B)))))),k2_xcmplx_0(k5_square_1(C),k5_square_1(D))) )
=> ( G = H
| k3_brouwer(np__2,I,G,H,B) = k23_euclid(k2_xcmplx_0(k21_euclid(G),k3_xcmplx_0(A,C)),k2_xcmplx_0(k22_euclid(G),k3_xcmplx_0(A,D))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_brouwer,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( ( v1_xreal_0(C)
& ~ v3_xreal_0(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k2_brouwer(A,B,C)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k2_brouwer(A,B,C)),u1_struct_0(k2_brouwer(A,B,C)))
& m2_relset_1(E,u1_struct_0(k2_brouwer(A,B,C)),u1_struct_0(k2_brouwer(A,B,C))) )
=> ( ~ r2_abian(u1_struct_0(k2_brouwer(A,B,C)),D,E)
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k7_toprealb(A,B,C)))
=> ( F = k4_brouwer(A,B,C,D,E)
<=> ? [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(A)))
& ? [H] :
( m1_subset_1(H,u1_struct_0(k15_euclid(A)))
& G = D
& H = k8_funct_2(u1_struct_0(k2_brouwer(A,B,C)),u1_struct_0(k2_brouwer(A,B,C)),E,D)
& F = k3_brouwer(A,B,H,G,C) ) ) ) ) ) ) ) ) ) ) ).
fof(t9_brouwer,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k2_brouwer(B,C,A)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A)))
& m2_relset_1(E,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A))) )
=> ( m1_subset_1(D,u1_struct_0(k7_toprealb(B,C,A)))
=> ( r2_abian(u1_struct_0(k2_brouwer(B,C,A)),D,E)
| k4_brouwer(B,C,A,D,E) = D ) ) ) ) ) ) ) ).
fof(t10_brouwer,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_pre_topc(C,k2_brouwer(np__2,B,A)) )
=> ~ ( C = k7_toprealb(np__2,B,A)
& r1_borsuk_1(k2_brouwer(np__2,B,A),C) ) ) ) ) ).
fof(d5_brouwer,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( ( v1_xreal_0(B)
& ~ v3_xreal_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k2_brouwer(A,C,B)))
& m2_relset_1(D,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k2_brouwer(A,C,B))) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k7_toprealb(A,C,B)))
& m2_relset_1(E,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k7_toprealb(A,C,B))) )
=> ( E = k5_brouwer(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_struct_0(k2_brouwer(A,C,B)))
=> k8_funct_2(u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k7_toprealb(A,C,B)),E,F) = k4_brouwer(A,C,B,F,D) ) ) ) ) ) ) ) ).
fof(t11_brouwer,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k2_brouwer(B,C,A)))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A)))
& m2_relset_1(E,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A))) )
=> ( m1_subset_1(D,u1_struct_0(k7_toprealb(B,C,A)))
=> ( r2_abian(u1_struct_0(k2_brouwer(B,C,A)),D,E)
| k8_funct_2(u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k7_toprealb(B,C,A)),k5_brouwer(B,A,C,E),D) = D ) ) ) ) ) ) ) ).
fof(t12_brouwer,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_subset_1(B,k1_numbers,k5_numbers) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(B)))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A)))
& v5_pre_topc(D,k2_brouwer(B,C,A),k2_brouwer(B,C,A))
& m2_relset_1(D,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A))) )
=> ( ~ r3_abian(D)
=> k2_partfun1(u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k7_toprealb(B,C,A)),k5_brouwer(B,A,C,D),k3_topreal9(B,C,A)) = k7_grcat_1(k7_toprealb(B,C,A)) ) ) ) ) ) ).
fof(t13_brouwer,axiom,
! [A] :
( ( v1_xreal_0(A)
& v2_xreal_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A)))
& v5_pre_topc(C,k2_brouwer(np__2,B,A),k2_brouwer(np__2,B,A))
& m2_relset_1(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A))) )
=> ( ~ r3_abian(C)
=> v5_pre_topc(k5_brouwer(np__2,A,B,C),k2_brouwer(np__2,B,A),k7_toprealb(np__2,B,A)) ) ) ) ) ).
fof(t14_brouwer,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A)))
& v5_pre_topc(C,k2_brouwer(np__2,B,A),k2_brouwer(np__2,B,A))
& m2_relset_1(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A))) )
=> r3_abian(C) ) ) ) ).
fof(t15_brouwer,axiom,
! [A] :
( ( v1_xreal_0(A)
& ~ v3_xreal_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A)))
& v5_pre_topc(C,k2_brouwer(np__2,B,A),k2_brouwer(np__2,B,A))
& m2_relset_1(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A))) )
=> ? [D] :
( m1_subset_1(D,u1_struct_0(k2_brouwer(np__2,B,A)))
& k8_funct_2(u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A)),C,D) = D ) ) ) ) ).
fof(dt_k1_brouwer,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A)
& ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> m1_subset_1(k1_brouwer(A,B),k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) ) ).
fof(dt_k2_brouwer,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C) )
=> m1_pre_topc(k2_brouwer(A,B,C),k15_euclid(A)) ) ).
fof(dt_k3_brouwer,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& m1_subset_1(C,u1_struct_0(k15_euclid(A)))
& m1_subset_1(D,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(E)
& ~ v3_xreal_0(E) )
=> m1_subset_1(k3_brouwer(A,B,C,D,E),u1_struct_0(k15_euclid(A))) ) ).
fof(dt_k4_brouwer,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,u1_struct_0(k15_euclid(A)))
& v1_xreal_0(C)
& ~ v3_xreal_0(C)
& m1_subset_1(D,u1_struct_0(k2_brouwer(A,B,C)))
& v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k2_brouwer(A,B,C)),u1_struct_0(k2_brouwer(A,B,C)))
& m1_relset_1(E,u1_struct_0(k2_brouwer(A,B,C)),u1_struct_0(k2_brouwer(A,B,C))) )
=> m1_subset_1(k4_brouwer(A,B,C,D,E),u1_struct_0(k7_toprealb(A,B,C))) ) ).
fof(dt_k5_brouwer,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& v1_xreal_0(B)
& ~ v3_xreal_0(B)
& m1_subset_1(C,u1_struct_0(k15_euclid(A)))
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k2_brouwer(A,C,B)))
& m1_relset_1(D,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k2_brouwer(A,C,B))) )
=> ( v1_funct_1(k5_brouwer(A,B,C,D))
& v1_funct_2(k5_brouwer(A,B,C,D),u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k7_toprealb(A,C,B)))
& m2_relset_1(k5_brouwer(A,B,C,D),u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k7_toprealb(A,C,B))) ) ) ).
fof(d1_brouwer,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> k1_brouwer(A,B) = a_2_0_brouwer(A,B) ) ) ).
fof(fraenkel_a_2_0_brouwer,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B)
& ~ v3_struct_0(C)
& v2_pre_topc(C)
& l1_pre_topc(C) )
=> ( r2_hidden(A,a_2_0_brouwer(B,C))
<=> ? [D,E] :
( m1_subset_1(D,u1_struct_0(B))
& m1_subset_1(E,u1_struct_0(C))
& A = k8_borsuk_1(B,C,D,E)
& D != E ) ) ) ).
%------------------------------------------------------------------------------