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 Th37:
  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
    & diff(u,i+1,Z) is_continuous_on Z
  holds
    u`|Z is_differentiable_on i,Z
  & diff(u`|Z,i,Z) is_continuous_on Z
proof
  let S,E be RealNormSpace,
      Z be Subset of S,
      u be PartFunc of S,E,
      i be Nat;

  assume
  A1: u is_differentiable_on i+1,Z
    & diff(u,i+1,Z) is_continuous_on Z;
  hence u`|Z is_differentiable_on i, Z by Th36;

  0 <= i by NAT_1:2;
  then 0 + 1 <= i + 1 by XREAL_1:7;
  then u is_differentiable_on 1 ,Z by A1,NDIFF_6:17;
  then Z c= dom u & u | Z is_differentiable_on Z by NDIFF_6:15;
  then A2: u is_differentiable_on Z;
  then A3: dom(u`|Z) = Z by NDIFF_1:def 9;

  A4: diff_SP(i + 1,S,E)
    = diff_SP(i,S,R_NormSpace_of_BoundedLinearOperators(S,E)) by Th30;

  (u|Z) `| Z = (u`|Z) | Z by A2,A3,Th4;
  hence diff(u`|Z,i,Z) is_continuous_on Z by A1,A4,Th31;
end;
