
theorem
  for x being Real, A being Subset of REAL st A = REAL holds x ++ A = A
proof
  let x be Real, A be Subset of REAL;
  assume
A1: A = REAL;
  A c= x ++ A
  proof
    let y be object;
    assume y in A;
    then reconsider y as Real;
    reconsider z = y - x as Element of REAL by XREAL_0:def 1;
    y = x + z;
    hence thesis by A1,Lm1;
  end;
  hence thesis by A1;
end;
