
theorem SFG:
  for f,g be complex-valued Function, X be set holds
  (f+g)|X = f|X + g|X
  proof
    let f,g be complex-valued Function, X be set;
    A1: dom (f|X) = dom f /\ X & dom (g|X) = dom g /\ X &
    dom ((f+g)|X) = dom (f+g) /\ X by RELAT_1:61;
    A2: dom (f + g) = dom f /\ dom g &
    dom (f|X + g|X) = dom (f|X) /\ dom (g|X) by VALUED_1:def 1; then
    A3: dom ((f+g)|X) = dom f /\ dom g /\ X by RELAT_1:61
    .= (dom f /\ X) /\ (dom g /\ X) by XBOOLE_1:116;
    for x be object st x in dom ((f+g)|X) holds
    ((f+g)|X).x = (f|X + g|X).x
    proof
      let x be object such that
      B1: x in dom ((f+g)|X);
      x in dom f & x in dom g & x in X by B1,A3,XBOOLE_0:def 4; then
      x in dom (f|X) & x in dom (g|X) by RELAT_1:57; then
      B3: (f|X).x = f.x & (g|X).x = g.x by FUNCT_1:47;
      ((f+g)|X).x = (f+g).x by B1,FUNCT_1:47
      .= f.x + g.x by B1,A1,VALUED_1:def 1
      .= ((f|X)+(g|X)).x by B1,B3,A1,A2,A3,VALUED_1:def 1;
      hence thesis;
    end;
    hence thesis by A1,A3,VALUED_1:def 1;
  end;
