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 be PartFunc of M,COMPLEX holds z (#) (f1 (#) f2) = z (#) f1 (#) f2
proof
  let f1 be PartFunc of M,COMPLEX;
A1: dom (f1 (#) f2) = dom f1 /\ dom f2 by Def1;
A2: dom (z(#)(f1 (#) f2)) = dom (f1 (#) f2) by Def2
    .= dom (z(#)f1) /\ dom f2 by A1,VALUED_1:def 5
    .= dom (z(#)f1(#)f2) by Def1;
  now
    let x be Element of M;
    assume
A3: x in dom (z(#)(f1(#)f2));
    then x in dom (z(#)f1) /\ dom f2 by A2,Def1;
    then x in dom (z(#)f1) by XBOOLE_0:def 4;
    then
A4: (z(#)f1)/.x = (z(#)f1).x by PARTFUN1:def 6;
A5: x in dom (f1(#)f2) by A3,Def2;
    then x in dom f1 by A1,XBOOLE_0:def 4;
    then
A6: f1/.x = f1.x by PARTFUN1:def 6;
    thus (z(#)(f1(#)f2))/.x = z * ((f1(#)f2)/.x) by A3,Def2
      .= z*(f1/.x * (f2/.x)) by A5,Def1
      .= (z*f1/.x) * (f2/.x) by CLVECT_1:def 4
      .= (z(#)f1)/.x * (f2/.x) by A4,A6,VALUED_1:6
      .= (z(#)f1 (#) f2)/.x by A2,A3,Def1;
  end;
  hence thesis by A2,PARTFUN2:1;
end;
