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