 reserve X,Y,Z,E,F,G,S,T for RealLinearSpace;

theorem
  ex I be LinearOperator of R_VectorSpace_of_LinearOperators
    (X,R_VectorSpace_of_LinearOperators(Y,Z)),
    R_VectorSpace_of_MultilinearOperators(<*X,Y*>,Z)
  st I is bijective
   & for u be Point of R_VectorSpace_of_LinearOperators(X,
     R_VectorSpace_of_LinearOperators(Y,Z)) holds
     for x be Point of X,y be Point of Y holds (I.u).<*x,y*> = (u.x).y
  proof
    consider I be LinearOperator of
    R_VectorSpace_of_LinearOperators(X,R_VectorSpace_of_LinearOperators(Y,Z)),
    R_VectorSpace_of_BilinearOperators(X,Y,Z) such that
    A1: I is bijective
      & for u be Point of R_VectorSpace_of_LinearOperators(X,
        R_VectorSpace_of_LinearOperators(Y,Z)) holds
        for x be Point of X,y be Point of Y holds
        (I.u).(x,y) = (u.x).y by LOPBAN_9:26;
    consider J be LinearOperator of
      R_VectorSpace_of_BilinearOperators(X,Y,Z),
      R_VectorSpace_of_MultilinearOperators(<*X,Y*>,Z) such that
    A2: J is bijective
      & for u be Point of R_VectorSpace_of_BilinearOperators(X,Y,Z)
        holds J.u = u * (IsoCPRLSP(X,Y))" by IS03;
    reconsider K = J*I as LinearOperator of
      R_VectorSpace_of_LinearOperators(X,
        R_VectorSpace_of_LinearOperators(Y,Z)),
      R_VectorSpace_of_MultilinearOperators(<*X,Y*>,Z) by LOPBAN_2:1;
    take K;
    thus K is bijective by A1,A2,FUNCT_2:27;
    let u be Point of R_VectorSpace_of_LinearOperators(X,
        R_VectorSpace_of_LinearOperators(Y,Z));
    let x be Point of X, y be Point of Y;
A3: K.u = J.(I.u) by FUNCT_2:15;
    reconsider xy = <*x,y*> as Point of product <*X,Y*> by PRVECT_3:14;
    thus (K.u).<*x,y*> = ((I.u) * (IsoCPRLSP(X,Y))").xy by A2,A3
    .= (I.u).((IsoCPRLSP (X,Y))" .xy) by FUNCT_2:15
    .= (I.u).(x,y) by defISOA1
    .= (u.x).y by A1;
  end;
