theorem
  (r+f)`| = f`|
  proof
    reconsider s = r as Element of REAL by XREAL_0:def 1;
    set g = REAL --> s;
A1: g`| = REAL --> 0 by Th11;
A2: dom(f`|) = REAL by FUNCT_2:def 1;
    dom f = REAL by FUNCT_2:def 1;
    then r+f = g+f by Th3;
    hence (r+f)`| = g`|+f`| by Th14
    .= f`| by A1,A2,Th6;
  end;
