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 LM300:
  for X, Y, W be RealNormSpace,
  Z be Subset of [:X,Y:],
  f be PartFunc of [:X,Y:],W holds
  (f is_partial_differentiable_on`1 Z iff
  f*(IsoCPNrSP(X,Y)") is_partial_differentiable_on ((IsoCPNrSP(X,Y)"))"Z,1) &
  (f is_partial_differentiable_on`2 Z iff
  f*(IsoCPNrSP(X,Y)") is_partial_differentiable_on ((IsoCPNrSP(X,Y)"))"Z,2)
  proof
    let X, Y, W be RealNormSpace,
    Z be Subset of [:X,Y:],
    f be PartFunc of [:X,Y:], W;
    set I = (IsoCPNrSP(X,Y)");
    set J = IsoCPNrSP(X,Y);
    set g = f*I;
    set E = I"Z;
    X1: I = (IsoCPNrSP(X,Y)") & I is one-to-one onto &
    ( for x be Point of X, y be Point of Y holds I.<*x,y*> =[x,y] ) &
    0. [:X,Y:] = I.(0.product <*X,Y*>) & I is isometric by defISOA1,defISOA2;
    X2: J is one-to-one onto &
    ( for x be Point of X, y be Point of Y holds J.(x,y) =<*x,y*> ) &
    0.product <*X,Y*> = J.(0. [:X,Y:]) & J is isometric by defISO,ZeZe;
    A1: dom (f*I) = I"(dom f) by RELAT_1:147;
    A2: dom J = the carrier of [:X,Y:] by FUNCT_2:def 1;
    thus f is_partial_differentiable_on`1 Z iff
    g is_partial_differentiable_on E,1
    proof
      hereby
        assume P2: f is_partial_differentiable_on`1 Z;
        for w be Point of product <*X,Y*> st w in E
        holds g|E is_partial_differentiable_in w,1
        proof
          let w be Point of product <*X,Y*>;
          assume A4: w in E;
          consider z be Point of [:X,Y:] such that
          F4: w = IsoCPNrSP(X,Y).z by X2,FUNCT_2:113;
          I.w = z by A2,F4,FUNCT_1:34;
          then z in Z by A4,FUNCT_2:38;
          then (f|Z)*(IsoCPNrSP(X,Y)") is_partial_differentiable_in
          IsoCPNrSP(X,Y).z,1 by P2,LM200;
          hence thesis by F4,FX1;
        end;
        hence g is_partial_differentiable_on E,1 by A1,P2,RELAT_1:143;
      end;
      assume P2: g is_partial_differentiable_on E,1;
      then P3: I"Z c= I" (dom f) by RELAT_1:147;
      for z be Point of [:X,Y:] st z in Z holds
      f|Z is_partial_differentiable_in`1 z
      proof
        let z be Point of [:X,Y:];
        assume A4: z in Z;
        set w = IsoCPNrSP(X,Y).z;
        I.w = z by A2,FUNCT_1:34;
        then F4: w in E by FUNCT_2:38,A4;
        (f|Z)*I = g | E by FX1;
        hence thesis by F4,P2,LM200;
      end;
      hence f is_partial_differentiable_on`1 Z by P3,X1,FUNCT_1:88;
    end;
    thus f is_partial_differentiable_on`2 Z iff
    g is_partial_differentiable_on E,2
    proof
      hereby
        assume P2: f is_partial_differentiable_on`2 Z;
        for w be Point of product <*X,Y*> st w in E holds
        g|E is_partial_differentiable_in w,2
        proof
          let w be Point of product <*X,Y*>;
          assume A4: w in E;
          consider z be Point of [:X,Y:] such that
          F4: w = IsoCPNrSP(X,Y).z by X2,FUNCT_2:113;
          F8: I.w = z by A2,F4,FUNCT_1:34;
          I.w in Z by FUNCT_2:38,A4;
          then (f|Z)*(IsoCPNrSP(X,Y)") is_partial_differentiable_in
          IsoCPNrSP(X,Y).z,2 by F8,P2,LM200;
          hence thesis by F4,FX1;
        end;
        hence g is_partial_differentiable_on E,2 by A1,P2,RELAT_1:143;
      end;
      assume P2: g is_partial_differentiable_on E,2;
      P3: I"Z c= I" (dom f) by P2,RELAT_1:147;
      for z be Point of [:X,Y:] st z in Z holds
      f|Z is_partial_differentiable_in`2 z
      proof
        let z be Point of [:X,Y:];
        assume A4: z in Z;
        set w = IsoCPNrSP(X,Y).z;
        I.w = z by A2,FUNCT_1:34;
        then F4: w in E by FUNCT_2:38,A4;
        (f|Z)*I = g | E by FX1;
        hence thesis by F4,P2,LM200;
      end;
      hence f is_partial_differentiable_on`2 Z by P3,X1,FUNCT_1:88;
    end;
  end;
