
theorem Th16:
  for X,Y be RealNormSpace,
      f be Function of product <*X*>,Y
  holds
    f is LinearOperator of product <*X*>,Y
    iff f * (IsoCPNrSP X) is LinearOperator of X,Y
  proof
    let X,Y be RealNormSpace,
        f be Function of product <*X*>,Y;
    set g = f * (IsoCPNrSP X);
    rng(IsoCPNrSP X) = the carrier of product <*X*> by FUNCT_2:def 3; then
    A1: (IsoCPNrSP X) * (IsoCPNrSP X)" = id the carrier of product <*X*>
        by FUNCT_2:29;

    g * (IsoCPNrSP X)"
     = f* ((IsoCPNrSP X) * (IsoCPNrSP X)") by RELAT_1:36
    .= f by A1,FUNCT_2:17;

    hence thesis by Th14;
  end;
