
theorem
  for F being bool_DOMAIN of ExtREAL, S being non empty
  ext-real-membered set st S = union F holds inf S = inf INF(F)
proof
  let F be bool_DOMAIN of ExtREAL, S be non empty ext-real-membered set;
  set a = inf S;
  set b = inf INF(F);
  assume
A1: S = union F;
  then inf S is LowerBound of INF(F) by Th7;
  then
A2: a <= b by XXREAL_2:def 4;
  inf INF(F) is LowerBound of S by A1,Th8;
  then b <= a by XXREAL_2:def 4;
  hence thesis by A2,XXREAL_0:1;
end;
