let F be bool_DOMAIN of ExtREAL; :: thesis: for S being ext-real-membered set st S = union F holds
sup (SUP F) is UpperBound of S
let S be ext-real-membered set ; :: thesis: ( S = union F implies sup (SUP F) is UpperBound of S )
assume A1: S = union F ; :: thesis: sup (SUP F) is UpperBound of S
for x being ExtReal st x in S holds
x <= sup (SUP F)

proof

let x be ExtReal; :: thesis: ( x in S implies x <= sup (SUP F) )
assume x in S ; :: thesis: x <= sup (SUP F)
then consider Z being set such that
A2: x in Z and
A3: Z in F by A1, TARSKI:def 4;
reconsider Z = Z as non empty ext-real-membered set by A2, A3;
set a = sup Z;
( sup Z is UpperBound of Z & sup Z in SUP F ) by A3, Def3, XXREAL_2:def 3;
hence x <= sup (SUP F) by A2, XXREAL_2:61, XXREAL_2:def 1; :: thesis: verum

end;

hence sup (SUP F) is UpperBound of S by XXREAL_2:def 1; :: thesis: verum