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(#)f.|| = |.z.|(#)||.f.||
proof
A1: dom (||.z(#)f.||) = dom (z(#)f) by NORMSP_0:def 3
    .= dom f by Def2
    .= dom (||.f.||) by NORMSP_0:def 3
    .= dom (|.z.|(#)||.f.||) by VALUED_1:def 5;
  now
    let c be Element of M;
    assume
A2: c in dom (||.z(#)f.||);
    then
A3: c in dom (||.f.||) by A1,VALUED_1:def 5;
A4: c in dom (z(#)f) by A2,NORMSP_0:def 3;
    thus (||.z(#)f.||).c = ||.(z(#)f)/.c.|| by A2,NORMSP_0:def 3
      .=||.z*(f/.c).|| by A4,Def2
      .=|.z.|*||.f/.c.|| by CLVECT_1:def 13
      .=|.z.|*(||.f.||.c) by A3,NORMSP_0:def 3
      .=(|.z.|(#)||.f.||).c by A1,A2,VALUED_1:def 5;
  end;
  hence thesis by A1,PARTFUN1:5;
end;
