
theorem Th4:
  for F being bool_DOMAIN of ExtREAL, S being non empty
  ext-real-membered number st S = union F holds sup S is UpperBound of SUP(F)
proof
  let F be bool_DOMAIN of ExtREAL, S be non empty ext-real-membered set;
  assume
A1: S = union F;
  for x being ExtReal st x in SUP(F) holds x <= sup S
  proof
    let x be ExtReal;
    assume x in SUP(F);
    then consider A being non empty ext-real-membered set such that
A2: A in F and
A3: x = sup A by Def3;
    A c= S
    by A1,A2,TARSKI:def 4;
    hence thesis by A3,XXREAL_2:59;
  end;
  hence thesis by XXREAL_2:def 1;
end;
