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

proof

let x be ext-real number ; :: thesis: ( x in S implies inf (INF F) <= x )
assume x in S ; :: thesis: inf (INF F) <= x
then consider Z being set such that
A2: ( x in Z & Z in F ) by A1, TARSKI:def 4;
reconsider Z = Z as non empty ext-real-membered set by A2;
consider a being set such that
A3: a = inf Z ;
reconsider a = a as ext-real number by A3;
A4: inf Z is LowerBound of Z by XXREAL_2:def 4;
a in INF F by A2, A3, Def20;
hence inf (INF F) <= x by A3, A4, A2, XXREAL_2:62, XXREAL_2:def 2; :: thesis: verum

end;

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