reserve A for Tolerance_Space,
  X, Y for Subset of A;
reserve A for Approximation_Space,
  X for Subset of A;
reserve A for finite Tolerance_Space,
  X for Subset of A,
  x for Element of A;
reserve A for finite Approximation_Space,
  X, Y for Subset of A,
  x for Element of A;

theorem
  UAp X = { x where x is Element of A : MemberFunc (X, A).x > 0 }
proof
  thus UAp X c= { x where x is Element of A : MemberFunc (X, A).x > 0 }
  proof
    let y be object;
    assume
A1: y in UAp X;
    then reconsider y9 = y as Element of A;
    not y in (UAp X)` by A1,XBOOLE_0:def 5;
    then MemberFunc (X, A).y9 <> 0 by Th41;
    then MemberFunc (X, A).y9 > 0 by Th38;
    hence thesis;
  end;
  let y be object;
  assume y in { x where x is Element of A : MemberFunc (X, A).x > 0 };
  then
A2: ex y9 being Element of A st y9 = y & MemberFunc (X, A). y9 > 0;
  then not y in (UAp X)` by Th41;
  hence thesis by A2,XBOOLE_0:def 5;
end;
