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 Th18:
  for L be Lipschitzian LinearOperator of E,F
  holds
    for i be Nat
    holds diff(L,i+2,[#]E) = [#]E --> 0.diff_SP(i+2,E,F)
  proof
    let L be Lipschitzian LinearOperator of E,F;

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

    A1: P[0]
    proof
      A2: 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
      0.diff_SP(2,E,F)
       = [#]E --> 0.(diff_SP(1,E,F)) by LOPBAN_1:31
      .= [#]E --> ([#]E --> 0.F) by A2,LOPBAN_1:31;

      hence diff(L,0+2,[#]E)
       = [#]E --> 0.diff_SP(0+2,E,F) by Th16;
    end;

    A4: for i be Nat st P[i] holds P[i+1]
    proof
      let i be Nat;
      assume
      A5: P[i];
      set L1 = diff(L,i+2,[#]E);
      A6: dom L1 = [#]E by A5,FUNCOP_1:13;
      A7: rng(L1) = {0.diff_SP(i+2,E,F)} by A5,FUNCOP_1:8;
      then
      L1 is_differentiable_on [#]E
        & for x be Point of E st x in [#]E
          holds (L1 `| [#]E ) /. x
            = 0.R_NormSpace_of_BoundedLinearOperators(E, diff_SP(i+2,E,F))
              by A6,NDIFF_1:33; then
      A9: dom (L1`| [#]E ) = [#]E by NDIFF_1:def 9;

      A10: for z be object st z in dom (L1 `| [#]E)
           holds (L1 `| [#]E).z = 0.diff_SP((i+2)+1,E,F)
      proof
        let z be object;
        assume
        A11: z in dom(L1 `| [#]E);
        then reconsider x = z as Point of E;
        thus (L1 `| [#]E). z = (L1 `| [#]E) /. x by A11,PARTFUN1:def 6
        .= 0.R_NormSpace_of_BoundedLinearOperators(E,diff_SP(i+2,E,F))
            by A6,A7,NDIFF_1:33
        .= 0.diff_SP((i+2)+1,E,F) by NDIFF_6:10;
      end;
      thus diff(L,(i+1)+2,[#]E)
       = diff(L,(i+2)+1,[#]E)
      .= diff(L,(i+2),[#]E) `| [#]E by NDIFF_6:13
      .= [#]E --> 0.diff_SP((i+1)+2,E,F) by A9,A10,FUNCOP_1:11;
    end;
    for i be Nat holds P[i] from NAT_1:sch 2(A1,A4);
    hence thesis;
  end;
