reserve 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 LM090:
  for S, T be RealNormSpace,
  L be Lipschitzian LinearOperator of S, T,
  x0 be Point of S holds
  L is_differentiable_in x0 & diff(L,x0) = L
  proof
    let S, T be RealNormSpace,
    L0 be Lipschitzian LinearOperator of S, T,
    x0 be Point of S;
    the carrier of S c= the carrier of S; then
    reconsider Z = the carrier of S as Subset of S;
    reconsider E = {} as Subset of S by XBOOLE_1:2;
    reconsider L = L0
    as Point of R_NormSpace_of_BoundedLinearOperators(S,T) by LOPBAN_1:def 9;
    reconsider R = (the carrier of S) --> 0.T as PartFunc of S, T;
    A6: dom R = the carrier of S;
    now
      let h be (0.S)-convergent sequence of S;
      assume h is non-zero;
  A7: now
        let n be Nat;
        A8: R/.(h.n) = R.(h.n) by A6,PARTFUN1:def 6
        .= 0.T;
        A9: rng h c= dom R;
        A10: n in NAT by ORDINAL1:def 12;
        thus ((||.h.||")(#)(R/*h)).n = (||.h.||".n)*((R/*h).n)
        by NDIFF_1:def 2
        .= 0.T by A8,A9,A10,FUNCT_2:109,RLVECT_1:10;
      end; then
      A11: (||.h.||")(#)(R/*h) is constant by VALUED_0:def 18;
      hence (||.h.||")(#)(R/*h) is convergent by NDIFF_1:18;
      ((||.h.||")(#)(R/*h)).0 = 0.T by A7;
      hence lim ((||.h.||")(#)(R/*h)) = 0.T by A11,NDIFF_1:18;
    end;
    then reconsider R as RestFunc of S, T by NDIFF_1:def 5;
    set N = the Neighbourhood of x0;
    A15: for x be Point of S st x in N holds L0/.x-L0/.x0=L.(x-x0)+R/.(x-x0)
    proof
      let x be Point of S;
      A16: R/.(x-x0) =R.(x-x0) by A6,PARTFUN1:def 6
      .= 0.T;
      assume x in N;
      thus L0/.x-L0/.x0 = L.(x-x0) by LM001
      .= L.(x-x0)+R/.(x-x0) by A16,RLVECT_1:4;
    end;
    A17: dom L0 = the carrier of S by FUNCT_2:def 1;
    hence L0 is_differentiable_in x0 by A15;
    hence thesis by A17,A15,NDIFF_1:def 7;
  end;
