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 Th19:
  for L be Lipschitzian LinearOperator of E,F
  holds
    for i be Nat
    holds
      diff(L,i+1,[#]E) is_differentiable_on [#]E
    & diff(L,i+1,[#]E) `| [#]E = [#]E --> 0.diff_SP(i+2,E,F)
    & diff(L,i+1,[#]E) `| [#]E is_continuous_on [#]E
  proof
    let L be Lipschitzian LinearOperator of E,F;

    defpred P[Nat] means
      diff(L,$1+1,[#]E) is_differentiable_on [#]E
    & diff(L,$1+1,[#]E) `| [#]E = [#]E --> 0.diff_SP($1+2,E,F)
    & diff(L,$1+1,[#]E) `| [#]E is_continuous_on [#]E;

    A1: L is Point of R_NormSpace_of_BoundedLinearOperators(E,F)
        by LOPBAN_1:def 9;

    A2: P[0]
    proof
      A3: diff_SP(1,E,F)
        = R_NormSpace_of_BoundedLinearOperators(E,F) by NDIFF_6:7;

      diff_SP (2,E,F)
       = diff_SP (1+1,E,F)
      .= R_NormSpace_of_BoundedLinearOperators(E,(diff_SP(1,E,F)))
          by NDIFF_6:10;
      then
      A4:
      0.diff_SP(2,E,F)
       = [#]E --> 0.(diff_SP(1,E,F)) by LOPBAN_1:31
      .= [#]E --> ([#]E --> 0.F) by A3,LOPBAN_1:31;
      set L1 = diff(L,0+1,[#]E);
      A5: diff(L,0+1,[#]E) = [#]E --> L by Th16;
      then
      A6: dom(L1) = [#]E by FUNCOP_1:13;
      rng(L1) = {L} by A5,FUNCOP_1:8;
      hence L1 is_differentiable_on [#]E by A1,A3,A6,NDIFF_1:33;

      thus
      A7: diff(L,0+1,[#]E) `| [#]E
       = diff(L,1+1,[#]E) by NDIFF_6:13
      .= [#]E --> 0.diff_SP(0+2,E,F) by A4,Th16;
      then
      A8: dom(diff(L,0+1,[#]E) `| [#]E) = [#]E by FUNCOP_1:13;
      reconsider r = 0.diff_SP((0+1)+1,E,F)
        as Point of R_NormSpace_of_BoundedLinearOperators(E,diff_SP(0+1,E,F))
        by NDIFF_6:10;
      rng(diff(L,0 +1,[#]E) `| [#]E) = {r} by A7,FUNCOP_1:8;
      hence diff(L,0+1,[#]E) `| [#]E is_continuous_on [#]E by A8,NFCONT_1:47;
    end;

    A9: for i be Nat st P[i] holds P[i+1]
    proof
      let i be Nat;
      assume
      A10: P[i];
      set L1 = diff(L,(i+1)+1,[#]E);
      A11: L1 = diff(L,i+1,[#]E) `| [#]E by NDIFF_6:13;
      then
      A12: dom L1 = [#]E by A10,FUNCOP_1:13;
      rng(L1) = {0.diff_SP(i+2,E,F)} by A10,A11,FUNCOP_1:8;
      hence L1 is_differentiable_on [#]E by A12,NDIFF_1:33;
      thus
      A13: diff(L,(i+1)+1,[#]E) `| [#]E
       = diff(L,i+2+1,[#]E) by NDIFF_6:13
      .= [#]E --> 0.diff_SP((i+1)+2,E,F) by Th18;
      then
      A14: dom(diff(L,(i+1)+1,[#]E) `| [#]E) = [#]E by FUNCOP_1:13;
      R_NormSpace_of_BoundedLinearOperators(E,diff_SP(i+2,E,F))
      = diff_SP((i+2)+1,E,F) by NDIFF_6:10;
      then
      reconsider r = 0.diff_SP((i+1)+2,E,F)
        as Point of R_NormSpace_of_BoundedLinearOperators(E,diff_SP(i+2,E,F));
      rng(diff(L,(i+1)+1,[#]E) `| [#]E) = {r} by A13,FUNCOP_1:8;
      hence diff(L,(i+1)+1,[#]E) `| [#]E is_continuous_on [#]E
        by A14,NFCONT_1:47;
    end;

    for i be Nat holds P[i] from NAT_1:sch 2(A2,A9);
    hence thesis;
  end;
