
theorem Th39:
  for X,Y be RealNormSpace holds
  ex I be Lipschitzian LinearOperator of
    R_NormSpace_of_BoundedLinearOperators(X,Y),
    R_NormSpace_of_BoundedLinearOperators(product <*X*>,Y)
  st I is one-to-one onto isometric
   & (for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
          x be Point of X
      holds (I.u).<*x*> = u.x)
   & for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
     holds ||.u.|| = ||.I.u.||
  proof
    let X,Y be RealNormSpace;
    set J = IsoCPNrSP(X);

    consider I be Lipschitzian LinearOperator of
      R_NormSpace_of_BoundedLinearOperators(X,Y),
      R_NormSpace_of_BoundedLinearOperators(product <*X*>,Y)
    such that
    A1: I is one-to-one onto isometric
      & for x be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
        holds I.x = x * J" by Th38;

    A2: for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
            x be Point of X
        holds (I.u).<*x*> = u.x
    proof
      let u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
          x be Point of X;

      A3: (IsoCPNrSP(X)).x = <*x*> by Def2;
      reconsider px = <*x*> as Point of product <*X*> by Th12;

      thus (I.u).<*x*>
       = (u * J").px by A1
      .= u.(J".px) by FUNCT_2:15
      .= u.x by A3,FUNCT_2:26;
    end;

    take I;
    thus I is one-to-one onto isometric by A1;
    thus
    for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
        x be Point of X
    holds (I.u).<*x*> = u.x by A2;

    let u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    thus thesis by A1,NDIFF_7:7;
  end;
