
theorem Th5:
  for F being bool_DOMAIN of ExtREAL, S being ext-real-membered
  set st S = union F holds sup SUP(F) is UpperBound of S
proof
  let F be bool_DOMAIN of ExtREAL, S be ext-real-membered set;
  assume
A1: S = union F;
  for x being ExtReal st x in S holds x <= sup SUP(F)
  proof
    let x be ExtReal;
    assume x in S;
    then consider Z being set such that
A2: x in Z and
A3: Z in F by A1,TARSKI:def 4;
    reconsider Z as non empty ext-real-membered set by A2,A3;
    set a = sup Z;
    sup Z is UpperBound of Z & a in SUP(F) by A3,Def3,XXREAL_2:def 3;
    hence thesis by A2,XXREAL_2:61,def 1;
  end;
  hence thesis by XXREAL_2:def 1;
end;
