reserve E, F, G,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 Th32:
  for E,F be RealNormSpace,
      n be Nat,
      Z be Subset of E,
      g be PartFunc of E,F
   st g `| Z is_differentiable_on n,Z
    & g is_differentiable_on Z
  holds g is_differentiable_on n+1,Z
proof
  let E,F be RealNormSpace,
      n be Nat,
      Z be Subset of E,
      g be PartFunc of E,F;
  assume that
  A1: g `| Z is_differentiable_on n,Z and
  A2: g is_differentiable_on Z;

  set f = g `| Z;
  A3: dom(g `| Z) = Z by A2,NDIFF_1:def 9;
  A4: f|Z = (g|Z) `| Z by A2,A3,Th4;

  for i be Nat st i <= n + 1 - 1
  holds diff(g,i,Z) is_differentiable_on Z
  proof
    let i be Nat;
    assume A5: i <= n + 1 - 1;

    per cases;
    suppose
      A6: i = 0;
      A7: diff_SP(i,E,F) = F by A6,NDIFF_6:7;
      diff(g,i,Z) = g|Z by A6,NDIFF_6:11;
      hence diff(g,i,Z) is_differentiable_on Z by A2,A7,RELAT_1:62;
    end;
    suppose
      i <> 0;
      then 1-1 <= i -1 by XREAL_1:9,NAT_1:14;
      then i-1 in NAT by INT_1:3;
      then reconsider j = i-1 as Nat;

      A8: diff(g,j+1,Z) = diff(f,j,Z) by A4,Th31;
      diff(f,j,Z)is_differentiable_on Z by A1,A5,XREAL_1:9,NDIFF_6:14;
      hence diff(g,i,Z) is_differentiable_on Z by A8,Th30;
    end;
  end;
  hence thesis by A2,NDIFF_6:14;
end;
