
theorem Th7:
  for F being bool_DOMAIN of ExtREAL, S being non empty
  ext-real-membered set st S = union F holds inf S is LowerBound of INF(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 INF(F) holds inf S <= x
  proof
    let x be ExtReal;
    assume x in INF(F);
    then consider A being non empty ext-real-membered set such that
A2: A in F and
A3: x = inf A by Def4;
    A c= S
    by A1,A2,TARSKI:def 4;
    hence thesis by A3,XXREAL_2:60;
  end;
  hence thesis by XXREAL_2:def 2;
end;
