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 LM401:
  for X, Y, W be RealNormSpace,
  Z be Subset of [:X,Y:],
  f be PartFunc of [:X,Y:], W
  st f is_partial_differentiable_on`2 Z holds
  f `partial`2|Z = (f*(IsoCPNrSP(X,Y)"))`partial|(((IsoCPNrSP(X,Y)"))"Z,2)
  *IsoCPNrSP(X,Y)
  proof
    let X, Y, W be RealNormSpace,
    Z be Subset of [:X,Y:],
    f be PartFunc of [:X,Y:], W;
    assume AS0: f is_partial_differentiable_on`2 Z;
    set I = (IsoCPNrSP(X,Y)");
    set J = IsoCPNrSP(X,Y);
    set g = f*I;
    set E = I"Z;
    A2: dom J = the carrier of [:X,Y:] by FUNCT_2:def 1;
    set fpz = f `partial`2|Z;
    P1: dom fpz = Z &
    for z be Point of [:X,Y:] st z in Z holds fpz/.z =partdiff`2(f,z)
    by Def92,AS0;
    set gpe = g `partial|(E,2);
    P3X: g is_partial_differentiable_on E,2 by LM300,AS0; then
    P3: dom gpe = E &
    for x be Point of product <*X,Y*> st x in E holds
    gpe/.x = partdiff(g,x,2) by NDIFF_5:def 9;
    P4: dom (gpe*J) = J"E by P3,RELAT_1:147
    .= J"(J.:Z) by FUNCT_1:84
    .= Z by A2,FUNCT_1:94;
    now
      let x be object;
      assume P40: x in dom fpz;
      then reconsider z = x as Point of [:X,Y:];
      P44: J.z in J.: Z by P1,P40,FUNCT_2:35;
      I" = J by FUNCT_1:43; then
      P42: J.z in E by P44,FUNCT_1:85;
      thus fpz.x = fpz/.z by PARTFUN1:def 6,P40
      .= partdiff`2(f,z) by P1,P40
      .= partdiff(g,J.z,2) by LM210
      .= gpe/.(J.z) by P3X,P42,NDIFF_5:def 9
      .= gpe.(J.z) by P3,P42,PARTFUN1:def 6
      .= (gpe*J).x by A2,FUNCT_1:13;
    end;
    hence thesis by P1,P4,FUNCT_1:2;
  end;
