
theorem Th15:
  for X,Y be RealNormSpace,
      f be Function of X,Y
  holds
    f is Lipschitzian LinearOperator of X,Y
    iff f * (IsoCPNrSP X)" is Lipschitzian LinearOperator of product <*X*>,Y
  proof
    let X,Y be RealNormSpace,
        f be Function of X,Y;

    thus
    f is Lipschitzian LinearOperator of X,Y implies
    f * (IsoCPNrSP X)" is Lipschitzian LinearOperator of product <*X*>,Y
      by LOPBAN_2:2;

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

    rng(IsoCPNrSP X) = the carrier of product <*X*> by FUNCT_2:def 3; then
    A1: (IsoCPNrSP X)" * (IsoCPNrSP X) = id the carrier of 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 LOPBAN_2:2;
  end;
