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 Th36:
  for S,E be RealNormSpace,
      Z be Subset of S,
      u be PartFunc of S,E,
      i be Nat
   st u is_differentiable_on i+1,Z
  holds u`|Z is_differentiable_on i,Z
proof
  let S,E be RealNormSpace,
      Z be Subset of S,
      u be PartFunc of S,E;

  defpred P[Nat] means
  u is_differentiable_on $1+1,Z
    implies
  u`|Z is_differentiable_on $1,Z;

  A1: P[0]
  proof
    assume u is_differentiable_on 0+1,Z;
    then A2: Z c= dom u & u|Z is_differentiable_on Z by NDIFF_6:15;
    u is_differentiable_on Z by A2;
    hence u`|Z is_differentiable_on 0,Z by NDIFF_1:def 9;
  end;

  A4: for i be Nat st P[i] holds P[i+1]
  proof
    let i be Nat;
    assume A5: P[i];
    assume A6: u is_differentiable_on (i + 1) + 1,Z;
    A7: 0 + (i + 1) <= (i + 1) + 1 by XREAL_1:7;

    0 <= i by NAT_1:2;
    then 0 + 2 <= i + 2 by XREAL_1:7;
    then A8: u is_differentiable_on 2,Z by A6,NDIFF_6:17;
    A9: Z c= dom u & u|Z is_differentiable_on Z by A8,NDIFF_6:16;
    A10: u is_differentiable_on Z by A9;
    then A11: dom (u`|Z) = Z by NDIFF_1:def 9;

    for k be Nat st k <= (i+1)-1
    holds diff(u`|Z,k,Z) is_differentiable_on Z
    proof
      let k be Nat;
      assume A12: k <= (i+1)-1;

      per cases;
      suppose
        A13: k = i;
        A14: diff_SP(k+1,S,E)
           = diff_SP(k,S,R_NormSpace_of_BoundedLinearOperators(S,E)) by Th30;

        (u|Z)`|Z = (u`|Z)|Z by A10,A11,Th4;
        then diff(u`|Z,k,Z) = diff(u,i+1,Z) by A13,Th31;
        hence diff(u`|Z,k,Z) is_differentiable_on Z by A6,A13,A14,NDIFF_6:14;
      end;
      suppose
        k <> i;
        then k < i by A12,XXREAL_0:1;
        then k + 1 <= i by NAT_1:13;
        then k + 1 - 1 <= i - 1 by XREAL_1:13;
        hence diff(u`|Z,k,Z) is_differentiable_on Z by A5,A6,A7,NDIFF_6:14,17;
      end;
    end;
    hence u`|Z is_differentiable_on i+1,Z by A11,NDIFF_6:14;
  end;
  for i be Nat holds P[i] from NAT_1:sch 2(A1,A4);
  hence thesis;
end;
