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)

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

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