reserve MS for non empty MidStr;
reserve a, b for Element of MS;

theorem Th4:
  for a,b,c,d being Element of Example holds a@a = a & a@b = b@a &
  (a@b)@(c@d) = (a@c)@(b@d) & ex x being Element of Example st x@a = b
proof
  let a,b,c,d be Element of Example;
  thus a@a = {} by TARSKI:def 1
    .= a by TARSKI:def 1;
  thus a@b = {} by TARSKI:def 1
    .= b@a by TARSKI:def 1;
  thus (a@b)@(c@d) = {} by TARSKI:def 1
    .= (a@c)@(b@d) by TARSKI:def 1;
  take x = a;
  thus x@a = {} by TARSKI:def 1
    .= b by TARSKI:def 1;
end;
