let F be bool_DOMAIN of ExtREAL ; :: thesis: for S being non empty ext-real-membered set st S = union F holds
inf S is LowerBound of INF F
let S be non empty ext-real-membered set ; :: thesis: ( S = union F implies inf S is LowerBound of INF F )
assume A1: S = union F ; :: thesis: inf S is LowerBound of INF F
for x being ext-real number st x in INF F holds
inf S <= x

proof

let x be ext-real number ; :: thesis: ( x in INF F implies inf S <= x )
assume x in INF F ; :: thesis: inf S <= x
then consider A being non empty ext-real-membered set such that
A2: ( A in F & x = inf A ) by Def20;
A c= S

proof

let a be ext-real number ; :: according to MEMBERED:def 8 :: thesis: ( not a in A or a in S )
assume a in A ; :: thesis: a in S
hence a in S by A1, A2, TARSKI:def 4; :: thesis: verum

end;

hence inf S <= x by A2, XXREAL_2:60; :: thesis: verum

end;

hence inf S is LowerBound of INF F by XXREAL_2:def 2; :: thesis: verum