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 Th14:
  (z1*z2)(#)f = z1(#)(z2(#)f)
proof
A1: dom ((z1*z2) (#) f) = dom f by Def2
    .= dom (z2(#)f) by Def2
    .= dom (z1(#)(z2(#)f)) by Def2;
  now
    let x be Element of M;
    assume
A2: x in dom ((z1*z2)(#)f);
    then
A3: x in dom (z2(#)f) by A1,Def2;
    thus ((z1*z2)(#)f)/.x = z1*z2 * f/.x by A2,Def2
      .= z1*(z2 * f/.x) by CLVECT_1:def 4
      .= z1 * ((z2(#)f)/.x) by A3,Def2
      .= (z1(#)(z2(#)f))/.x by A1,A2,Def2;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
