reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th9:
  for X,Y,Z be RealNormSpace,
          f be PartFunc of Y,Z,
          I be LinearOperator of X,Y,
          V be Subset of Y
   st f is_differentiable_on V
    & I is one-to-one onto isometric
  holds
    for y be Point of Y st y in V
    holds (f`| V).y = (((f*I)`| I"V )/.(I".y)) * I"
  proof
    let X,Y,Z be RealNormSpace,
            f be PartFunc of Y,Z,
            I be LinearOperator of X,Y,
            V be Subset of Y;
    assume that
    A1: f is_differentiable_on V and
    A2: I is one-to-one onto isometric;

    A3: V is open by A1,NDIFF_1:32;
    consider J be LinearOperator of Y,X such that
    A4: J = I" & J is one-to-one onto isometric by A2,NDIFF_7:9;
    A5: dom (f`| V) = V &
        for x be Point of Y st x in V holds (f`| V)/.x = diff(f,x)
        by A1,NDIFF_1:def 9;
    set g = f*I;
    set U = I"V;
    A6: g is_differentiable_on U by A1,A2,NDIFF_7:29;

    for y be Point of Y st y in dom (f`| V)
    holds (f`| V).y = ((g`| U )/.(J.y)) *I"
    proof
      let y be Point of Y;
      assume
      A7: y in dom (f`| V); then
      A8: (f`| V).y = (f`| V)/.y by PARTFUN1:def 6
                   .= diff(f,y) by A5,A7;
      A9: f is_differentiable_in y by A1,A3,A5,A7,NDIFF_1:31;
      consider x be Point of X such that
      A10: y = I.x by A2,FUNCT_2:113;
      reconsider I0 = I as Point of
        R_NormSpace_of_BoundedLinearOperators(X,Y) by A2,LOPBAN_1:def 9;
      A11: diff(f,y) * I0 * I0"
       = modetrans(diff(f,y),Y,Z) * (modetrans(I0,X,Y) * I0") by RELAT_1:36
      .= modetrans(diff(f,y),Y,Z) * (I * I") by LOPBAN_1:def 11
      .= modetrans(diff(f,y),Y,Z) * (id rng I0) by A2,FUNCT_1:39
      .= modetrans(diff(f,y),Y,Z) by A2,FUNCT_2:17
      .= (f`| V).y by A8,LOPBAN_1:def 11;
      x in U by A5,A10,A7,FUNCT_2:38; then
      A12: ((g`|U)/.x) * I0" = diff(g,x) * I0" by A6,NDIFF_1:def 9;
      x = J.y by A2,A4,A10,FUNCT_2:26;
      hence (f`| V).y = ((g`| U )/.(J.y)) * I"
        by A2,A9,A10,A11,A12,NDIFF_7:27;
    end;
    hence thesis by A4,A5;
  end;
