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 Th20:
  for L be Lipschitzian LinearOperator of E,F
  holds
    for i be Nat
    holds
      diff(L,i,[#]E) is_differentiable_on [#]E
    & diff(L,i,[#]E) `| [#]E is_continuous_on [#]E
  proof
    let L be Lipschitzian LinearOperator of E,F;
    let i be Nat;

    per cases;
    suppose
      A1: i = 0;
      then
      A2: diff(L,i,[#]E)
       = L | [#]E by NDIFF_6:11
      .= L;

      A3: diff(L,i,[#]E) `| [#]E
       = diff(L,i+1,[#]E) by NDIFF_6:13
      .= [#]E --> L by A1,Th16;

      A4: diff_SP (0,E,F) = F by NDIFF_6:def 2;
      hence diff (L,i,[#]E) is_differentiable_on [#]E
        by A1,A2,FUNCT_2:def 1,NDIFF_7:26;

      A5: dom(diff(L,i,[#]E) `| [#]E) = [#]E by A3,FUNCOP_1:13;
      reconsider r = L as Point of
        R_NormSpace_of_BoundedLinearOperators(E,diff_SP (0,E,F))
        by A4,LOPBAN_1:def 9;
      rng (diff(L,i,[#]E) `| [#]E) = {r} by A3,FUNCOP_1:8;
      hence diff(L,i,[#]E) `| [#]E is_continuous_on [#]E
        by A1,A5,NFCONT_1:47;
    end;
    suppose
      i <> 0;
      then ex j be Nat st i = j+1 by NAT_1:6;
      hence thesis by Th19;
    end;
  end;
