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 f3 be PartFunc of M,COMPLEX holds f3(#)(f1 + f2) = f3(#)f1 + f3(#) f2
proof
  let f3 be PartFunc of M,COMPLEX;
A1: dom (f3 (#) (f1 + f2)) = dom f3 /\ dom (f1 + f2) by Def1
    .= dom f3 /\ (dom f1 /\ dom f2) by VFUNCT_1:def 1
    .= dom f3 /\ dom f3 /\ dom f1 /\ dom f2 by XBOOLE_1:16
    .= dom f3 /\ dom f1 /\ dom f3 /\ dom f2 by XBOOLE_1:16
    .= dom f3 /\ dom f1 /\ (dom f3 /\ dom f2) by XBOOLE_1:16
    .= dom (f3 (#) f1) /\ (dom f3 /\ dom f2) by Def1
    .= dom (f3 (#) f1) /\ dom (f3 (#) f2) by Def1
    .= dom (f3 (#) f1 + f3 (#) f2) by VFUNCT_1:def 1;
  now
    let x be Element of M;
    assume
A2: x in dom (f3(#)(f1 + f2));
    then x in dom f3 /\ dom (f1 + f2) by Def1;
    then
A3: x in dom (f1 + f2) by XBOOLE_0:def 4;
A4: x in dom (f3(#)f1) /\ dom (f3(#)f2) by A1,A2,VFUNCT_1:def 1;
    then
A5: x in dom (f3(#)f1) by XBOOLE_0:def 4;
A6: x in dom (f3 (#) f2) by A4,XBOOLE_0:def 4;
    thus (f3 (#) (f1 + f2))/.x = f3/.x * ((f1 + f2)/.x) by A2,Def1
      .= f3/.x * ((f1/.x) + (f2/.x)) by A3,VFUNCT_1:def 1
      .= f3/.x * (f1/.x) + f3/.x * (f2/.x) by CLVECT_1:def 2
      .= ((f3 (#) f1)/.x) + f3/.x* (f2/.x) by A5,Def1
      .= ((f3 (#) f1)/.x) + ((f3 (#) f2)/.x) by A6,Def1
      .= ((f3 (#) f1) + (f3 (#) f2))/.x by A1,A2,VFUNCT_1:def 1;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
