reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;

theorem
  for f1,f2 be PartFunc of M,COMPLEX holds
  (f1 + f2) (#) f3 = f1 (#) f3 + f2 (#) f3
proof
  let f1,f2 be PartFunc of M,COMPLEX;
A1: dom ((f1 + f2) (#) f3) = dom (f1 + f2) /\ dom f3 by Def1
    .= dom f1 /\ dom f2 /\ (dom f3 /\ dom f3) by VALUED_1:def 1
    .= dom f1 /\ dom f2 /\ dom f3 /\ dom f3 by XBOOLE_1:16
    .= dom f1 /\ dom f3 /\ dom f2 /\ dom f3 by XBOOLE_1:16
    .= dom f1 /\ dom f3 /\ (dom f2 /\ dom f3) by XBOOLE_1:16
    .= dom (f1 (#) f3) /\ (dom f2 /\ dom f3) by Def1
    .= dom (f1 (#) f3) /\ dom (f2 (#) f3) by Def1
    .= dom (f1 (#) f3 + f2 (#) f3) by VFUNCT_1:def 1;
A2: dom (f1 + f2) = dom f1 /\ dom f2 by VALUED_1:def 1;
  now
    let x be Element of M;
    assume
A3: x in dom ((f1 + f2)(#)f3);
    then
A4: x in dom (f1(#)f3) /\ dom (f2(#)f3) by A1,VFUNCT_1:def 1;
    then
A5: x in dom (f1(#)f3) by XBOOLE_0:def 4;
    x in dom (f1 + f2) /\ dom f3 by A3,Def1;
    then
A6: x in dom (f1 + f2) by XBOOLE_0:def 4;
    then x in dom f1 by A2,XBOOLE_0:def 4;
    then
A7: f1/.x = f1.x by PARTFUN1:def 6;
    x in dom f2 by A2,A6,XBOOLE_0:def 4;
    then
A8: f2/.x = f2.x by PARTFUN1:def 6;
A9: (f1 + f2)/.x = (f1 + f2).x by A6,PARTFUN1:def 6
      .= f1/.x + f2/.x by A6,A7,A8,VALUED_1:def 1;
A10: x in dom (f2 (#) f3) by A4,XBOOLE_0:def 4;
    thus ((f1 + f2) (#) f3)/.x = (f1 + f2)/.x * (f3/.x) by A3,Def1
      .= f1/.x * (f3/.x) + f2/.x * (f3/.x) by A9,CLVECT_1:def 3
      .= ((f1 (#) f3)/.x) + f2/.x* (f3/.x) by A5,Def1
      .= ((f1 (#) f3)/.x) + ((f2 (#) f3)/.x) by A10,Def1
      .= ((f1 (#) f3) + (f2 (#) f3))/.x by A1,A3,VFUNCT_1:def 1;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
