let b be Real; :: thesis:
for n being Nat holds (Complement (ext_half_open_sets b)) . n is Element of Ext_Borel_Sets

proof

let n be Nat; :: thesis:
reconsider nn = n as Element of NAT by ORDINAL1:def 12;
((ext_half_open_sets b) . nn) ` is Element of Ext_Borel_Sets

proof

(ext_half_open_sets b) . n is Element of Ext_Borel_Sets

proof

per cases by NAT_1:6;

suppose A1: n = 0 ; :: thesis:

(ext_half_open_sets b) . 0 = ].(b - 1),+infty.] by Def300;
hence (ext_half_open_sets b) . n is Element of Ext_Borel_Sets by A1, Th3; :: thesis:

end;

suppose ex k being Nat st n = k + 1 ; :: thesis:

then consider k being Nat such that
A2: n = k + 1 ;
reconsider k = k as Element of NAT by ORDINAL1:def 12;
(ext_half_open_sets b) . (k + 1) = ].(b - (1 / (k + 1))),+infty.] by Def300;
hence (ext_half_open_sets b) . n is Element of Ext_Borel_Sets by A2, Th3; :: thesis:

end;

end;

end;

hence ((ext_half_open_sets b) . nn) ` is Element of Ext_Borel_Sets by PROB_1:def 1; :: thesis:

end;

hence (Complement (ext_half_open_sets b)) . n is Element of Ext_Borel_Sets by PROB_1:def 2; :: thesis:

end;

then Complement (ext_half_open_sets b) is SetSequence of Ext_Borel_Sets by PROB_1:25;
then Union (Complement (ext_half_open_sets b)) is Element of Ext_Borel_Sets by PROB_1:26;
hence Intersection (ext_half_open_sets b) is Element of Ext_Borel_Sets by PROB_1:def 1; :: thesis: