
theorem
  for F being bool_DOMAIN of ExtREAL, S being non empty
  ext-real-membered set st S = union F holds sup S = sup SUP(F)
proof
  let F be bool_DOMAIN of ExtREAL, S be non empty ext-real-membered set;
  set a = sup S;
  set b = sup SUP(F);
  assume
A1: S = union F;
  then sup S is UpperBound of SUP(F) by Th4;
  then
A2: b <= a by XXREAL_2:def 3;
  sup SUP(F) is UpperBound of S by A1,Th5;
  then a <= b by XXREAL_2:def 3;
  hence thesis by A2,XXREAL_0:1;
end;
