
theorem Th8:
  for F being bool_DOMAIN of ExtREAL, S being ext-real-membered
  set st S = union F holds inf INF(F) is LowerBound 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 inf INF(F) <= x
  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 = inf Z;
    inf Z is LowerBound of Z & a in INF(F) by A3,Def4,XXREAL_2:def 4;
    hence thesis by A2,XXREAL_2:62,def 2;
  end;
  hence thesis by XXREAL_2:def 2;
end;
