theorem
  succ X \ {X} = X
proof
  thus succ X \ {X} c= X
  proof
    let x be object;
    assume
A1: x in succ X \ {X};
    then
A2: not x in {X} by XBOOLE_0:def 5;
    x in X or x = X by A1,Th4;
    hence thesis by A2,TARSKI:def 1;
  end;
  let x be object;
  assume
A3: x in X; then
  reconsider xx = x as set;
  not xx in xx;
  then x <> X by A3;
  then
A4: not x in {X} by TARSKI:def 1;
  x in succ X by A3,Th4;
  hence thesis by A4,XBOOLE_0:def 5;
end;
