reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem
  for V being RealNormSpace-like non empty NORMSTR
  for f1 being PartFunc of C,REAL 
  for f2 being PartFunc of C,V holds
  ||.f1(#)f2.|| = abs(f1)(#)||.f2.||
proof
  let V be RealNormSpace-like non empty NORMSTR;
  let f1 be PartFunc of C,REAL;
  let f2 be PartFunc of C,V;
A1: dom (||.f1 (#) f2.||) = dom (f1 (#) f2) by NORMSP_0:def 3
    .= dom f1 /\ dom f2 by Def3
    .= dom f1 /\ dom (||.f2.||) by NORMSP_0:def 3
    .= dom (abs(f1)) /\ dom (||.f2.||) by VALUED_1:def 11
    .= dom (abs(f1)(#)||.f2.||) by VALUED_1:def 4;
  now
    let c;
    assume
A2: c in dom (||.f1 (#) f2.||);
    then
A3: c in dom (f1 (#) f2) by NORMSP_0:def 3;
    c in dom (abs(f1)) /\ dom (||.f2.||) by A1,A2,VALUED_1:def 4;
    then
A4: c in dom (||.f2.||) by XBOOLE_0:def 4;
    thus (||.f1(#)f2.||).c = ||.(f1(#)f2)/.c.|| by A2,NORMSP_0:def 3
      .= ||.f1.c * (f2/.c).|| by A3,Def3
      .= |.f1.c.| * ||.(f2/.c).|| by NORMSP_1:def 1
      .= ((abs(f1)).c) * ||.(f2/.c).|| by VALUED_1:18
      .= ((abs(f1)).c) * (||.f2.||).c by A4,NORMSP_0:def 3
      .= (abs(f1)(#)||.f2.||).c by VALUED_1:5;
  end;
  hence thesis by A1,PARTFUN1:5;
end;
