reserve A for Tolerance_Space,
  X, Y for Subset of A;

theorem Th25:
  X c= Y implies UAp X c= UAp Y
proof
  assume
A1: X c= Y;
  let x be object;
  assume
A2: x in UAp X;
  then Class (the InternalRel of A, x) meets X by Th10;
  then Class (the InternalRel of A, x) meets Y by A1,XBOOLE_1:63;
  hence thesis by A2;
end;
