reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;

theorem Th32:
  {1,2,3} \ {1} = {2,3}
  proof
    thus {1,2,3} \ {1} c= {2,3}
    proof
      let x be object;
      assume x in {1,2,3} \ {1};
      then x in {1,2,3} & not x in {1} by XBOOLE_0:def 5;
      then (x = 1 or x = 2 or x = 3) & not x = 1
        by ENUMSET1:def 1,TARSKI:def 1;
      hence thesis by TARSKI:def 2;
    end;
    let x be object;
    assume x in {2,3};
    then x = 2 or x = 3 by TARSKI:def 2;
    then x in {1,2,3} & not x in {1} by ENUMSET1:def 1,TARSKI:def 1;
    hence thesis by XBOOLE_0:def 5;
  end;
