
theorem Th6:
  for X,Y be RealLinearSpace,
      f be Function of X,Y
  holds
    f is LinearOperator of X,Y
    iff f * (IsoCPRLSP X)" is LinearOperator of product <*X*>,Y
  proof
    let X,Y be RealLinearSpace,
        f be Function of X,Y;

    thus
    f is LinearOperator of X,Y implies
    f * (IsoCPRLSP X)" is LinearOperator of product <*X*>,Y
      by LOPBAN_2:1;

    assume f * (IsoCPRLSP X)" is LinearOperator of product <*X*>,Y; then
    reconsider g = f * (IsoCPRLSP X)" as LinearOperator of product <*X*>,Y;

    rng (IsoCPRLSP X) = the carrier of product <*X*> by FUNCT_2:def 3; then
    A1: (IsoCPRLSP X)" * (IsoCPRLSP X) = id the carrier of 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 LOPBAN_2:1;
  end;
