
theorem Th8:
  for R being non empty RelStr,
      X being Subset of R holds
    Uap X = UAp X
  proof
    let R be non empty RelStr,
        X be Subset of R;
A1: LAp X` misses UAp X
    proof
      assume LAp X` meets UAp X;
      then consider x being object such that
A2:   x in LAp X` & x in UAp X by XBOOLE_0:3;
      Class (the InternalRel of R, x) meets X &
        Class (the InternalRel of R, x) c= X` by A2,Th6,Th7;
      hence thesis by XBOOLE_1:63,79;
    end;
    (UAp X)` c= LAp X`
    proof let x be object;
      assume
A3:   x in (UAp X)`;
      then not x in UAp X by XBOOLE_0:def 5;
      then Class (the InternalRel of R, x) misses X by A3;
      then Class (the InternalRel of R, x) c= X` by SUBSET_1:23;
      hence x in LAp X` by A3;
    end;
    hence thesis by A1,SUBSET_1:17,23;
  end;
