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 ||.f1(#)f2.|| = |.f1.|(#)||.f2 .||
proof
  let f1 be PartFunc of M,COMPLEX;
A1: dom (f1 (#) f2) = dom f1 /\ dom f2 by Def1;
A2: dom f1 /\ dom f2 = dom f1 /\ dom (||.f2.||) by NORMSP_0:def 3
    .= dom (|.f1.|) /\ dom (||.f2.||) by VALUED_1:def 11
    .= dom (|.f1.|(#)||.f2.||) by VALUED_1:def 4;
A3: dom (||.f1 (#) f2.||) = dom (f1 (#) f2) by NORMSP_0:def 3;
  now
    let c be Element of M;
    assume
A4: c in dom (||.f1 (#) f2.||);
    then c in dom (|.f1.|) /\ dom (||.f2.||) by A3,A1,A2,VALUED_1:def 4;
    then
A5: c in dom (||.f2.||) by XBOOLE_0:def 4;
A6: c in dom (f1 (#) f2) by A4,NORMSP_0:def 3;
    then c in dom f1 by A1,XBOOLE_0:def 4;
    then
A7: f1/.c = f1.c by PARTFUN1:def 6;
    thus (||.f1(#)f2.||).c = ||.(f1(#)f2)/.c.|| by A4,NORMSP_0:def 3
      .= ||.f1/.c * (f2/.c).|| by A6,Def1
      .= |.(f1/.c).| * ||.(f2/.c).|| by CLVECT_1:def 13
      .= ((|.f1.|).c) * ||.(f2/.c).|| by A7,VALUED_1:18
      .= ((|.f1.|).c) * (||.f2.||).c by A5,NORMSP_0:def 3
      .= (|.f1.|(#)||.f2.||).c by VALUED_1:5;
  end;
  hence thesis by A3,A1,A2,PARTFUN1:5;
end;
