let b be Real; :: thesis: Intersection (ext_right_closed_sets b) is Element of Ext_Borel_Sets

for n being Nat holds (Complement (ext_right_closed_sets b)) . n is Element of Ext_Borel_Sets

then Union (Complement (ext_right_closed_sets b)) is Element of Ext_Borel_Sets by PROB_1:26;

hence Intersection (ext_right_closed_sets b) is Element of Ext_Borel_Sets by PROB_1:def 1; :: thesis: verum

for n being Nat holds (Complement (ext_right_closed_sets b)) . n is Element of Ext_Borel_Sets

proof

then
Complement (ext_right_closed_sets b) is SetSequence of Ext_Borel_Sets
by PROB_1:25;
let n be Nat; :: thesis: (Complement (ext_right_closed_sets b)) . n is Element of Ext_Borel_Sets

reconsider nn = n as Element of NAT by ORDINAL1:def 12;

((ext_right_closed_sets b) . nn) ` is Element of Ext_Borel_Sets

end;reconsider nn = n as Element of NAT by ORDINAL1:def 12;

((ext_right_closed_sets b) . nn) ` is Element of Ext_Borel_Sets

proof

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

end;proof

hence
((ext_right_closed_sets b) . nn) ` is Element of Ext_Borel_Sets
by PROB_1:def 1; :: thesis: verum
(ext_right_closed_sets b) . n = [.-infty,(b - n).]
by Def3000;

hence (ext_right_closed_sets b) . n is Element of Ext_Borel_Sets by Th3; :: thesis: verum

end;hence (ext_right_closed_sets b) . n is Element of Ext_Borel_Sets by Th3; :: thesis: verum

then Union (Complement (ext_right_closed_sets b)) is Element of Ext_Borel_Sets by PROB_1:26;

hence Intersection (ext_right_closed_sets b) is Element of Ext_Borel_Sets by PROB_1:def 1; :: thesis: verum