
theorem Th7:
  for X,Y be RealLinearSpace,
      f be Function of product <*X*>,Y
  holds
    f is LinearOperator of product <*X*>,Y
    iff f * (IsoCPRLSP X) is LinearOperator of X,Y
  proof
    let X,Y be RealLinearSpace,
        f be Function of product <*X*>,Y;

    set g = f * (IsoCPRLSP X);
    rng(IsoCPRLSP X) = the carrier of product <*X*> by FUNCT_2:def 3; then
    A1: (IsoCPRLSP X) * (IsoCPRLSP X)"
     = id the carrier of product <*X*> by FUNCT_2:29;

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

    hence thesis by Th6;
  end;
