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

theorem Th28:
  LAp X` = (UAp X)`
proof
  LAp X` misses UAp X
  proof
    assume LAp X` meets UAp X;
    then consider x being object such that
A1: x in LAp X` & x in UAp X by XBOOLE_0:3;
    Class (the InternalRel of A, x) meets X & Class (the InternalRel of A,
    x) c= X` by A1,Th8,Th10;
    hence thesis by XBOOLE_1:63,79;
  end;
  hence LAp X` c= (UAp X)` by SUBSET_1:23;
  let x be object;
  assume
A2: x in (UAp X)`;
  then not x in UAp X by XBOOLE_0:def 5;
  then Class (the InternalRel of A, x) misses X by A2;
  then Class (the InternalRel of A, x) c= X` by SUBSET_1:23;
  hence thesis by A2;
end;
