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
  z(#)(f1 + f2) = z(#)f1 + z(#)f2
proof
A1: dom (z(#)(f1 + f2)) = dom (f1 + f2) by Def2
    .= dom f1 /\ dom f2 by VFUNCT_1:def 1
    .= dom f1 /\ dom (z(#)f2) by Def2
    .= dom (z(#)f1) /\ dom (z(#)f2) by Def2
    .= dom (z(#)f1 + z(#)f2) by VFUNCT_1:def 1;
  now
    let x be Element of M;
    assume
A2: x in dom (z(#)(f1 + f2));
    then
A3: x in dom (f1 + f2) by Def2;
A4: x in dom (z(#)f1) /\ dom (z(#)f2) by A1,A2,VFUNCT_1:def 1;
    then
A5: x in dom (z(#)f1) by XBOOLE_0:def 4;
A6: x in dom (z(#)f2) by A4,XBOOLE_0:def 4;
    thus (z(#)(f1 + f2))/.x = z * ((f1 + f2)/.x) by A2,Def2
      .= z * ((f1/.x) + (f2/.x)) by A3,VFUNCT_1:def 1
      .= z * (f1/.x) + z * (f2/.x) by CLVECT_1:def 2
      .= ((z(#)f1)/.x) + z * (f2/.x) by A5,Def2
      .= ((z(#)f1)/.x) + ((z(#)f2)/.x) by A6,Def2
      .= (z(#)f1 + z(#)f2)/.x by A1,A2,VFUNCT_1:def 1;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
