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 f being PartFunc of C,V holds
  ||.r(#)f.|| = |.r.|(#)||.f.||
proof
  let V be RealNormSpace-like non empty NORMSTR;
  let f be PartFunc of C,V;
A1: dom (||.r(#)f.||) = dom (r(#)f) by NORMSP_0:def 3
    .= dom f by Def4
    .= dom (||.f.||) by NORMSP_0:def 3
    .= dom (|.r.|(#)||.f.||) by VALUED_1:def 5;
  now
    let c;
    assume
A2: c in dom (||.r(#)f.||);
    then
A3: c in dom (||.f.||) by A1,VALUED_1:def 5;
A4: c in dom (r(#)f) by A2,NORMSP_0:def 3;
    thus (||.r(#)f.||).c = ||.(r(#)f)/.c.|| by A2,NORMSP_0:def 3
      .=||.r*(f/.c).|| by A4,Def4
      .=|.r.|*||.f/.c.|| by NORMSP_1:def 1
      .=|.r.|*(||.f.||.c) by A3,NORMSP_0:def 3
      .=(|.r.|(#)||.f.||).c by A1,A2,VALUED_1:def 5;
  end;
  hence thesis by A1,PARTFUN1:5;
end;
