 reserve a,b,r for Real;
 reserve A for non empty set;
 reserve X,x for set;
 reserve f,g,F,G for PartFunc of REAL,REAL;
 reserve n for Element of NAT;

theorem Th9:
  for f be PartFunc of REAL,REAL
  holds f + f = 2(#)f
proof
  let f be PartFunc of REAL,REAL;

  A1: dom(f+f) = dom f /\ dom f by VALUED_1:def 1
              .= dom f;
  A2: dom(2(#)f) = dom f by VALUED_1:def 5;
  now
    let t be object;
    assume t in dom(f+f);
    hence (f+f).t = f.t + f.t by VALUED_1:def 1
                 .= 2 * f.t
                 .= (2(#)f).t by VALUED_1:6;
  end;
  hence thesis by A1,A2,FUNCT_1:2;
end;
