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 Th48:
  for E,F be RealNormSpace,
      Z be non empty Subset of E,
      L1 be PartFunc of E,F,
      L0 be Point of F
   st Z is open & L1 = Z --> L0
  holds
    L1 is_differentiable_on Z
  & L1`|Z is_continuous_on Z
  & L1`|Z = Z --> 0.R_NormSpace_of_BoundedLinearOperators(E,F)
proof
  let E,F be RealNormSpace,
      Z be non empty Subset of E,
      L1 be PartFunc of E,F,
      L0 be Point of F;
  assume
  A1: Z is open & L1 = Z --> L0;
  A2: dom L1 = Z by A1,FUNCOP_1:13;
  A3: rng(L1) = {L0} by A1,FUNCOP_1:8;
  then
  A4: L1 is_differentiable_on Z
    & for x be Point of E st x in Z
      holds (L1`|Z)/.x = 0.R_NormSpace_of_BoundedLinearOperators(E,F)
      by A1,A2,NDIFF_1:33;

  A5: dom(L1`|Z) = Z by A4,NDIFF_1:def 9;
  A6: for z be object st z in dom (L1`|Z)
      holds (L1`|Z).z = 0.R_NormSpace_of_BoundedLinearOperators(E,F)
  proof
    let z be object;
    assume A7: z in dom(L1`|Z);
    then reconsider x = z as Point of E;
    thus (L1`|Z).z
     = (L1`|Z)/.x by PARTFUN1:def 6,A7
    .= 0.R_NormSpace_of_BoundedLinearOperators(E,F)
      by A1,A2,A3,A5,A7,NDIFF_1:33;
  end;
  then
  A8: L1`|Z = Z --> 0.R_NormSpace_of_BoundedLinearOperators(E,F)
      by A5,FUNCOP_1:11;

  rng(L1`|Z) = {0.R_NormSpace_of_BoundedLinearOperators(E,F)}
    by A8,FUNCOP_1:8;
  hence thesis by A1,A2,A3,A5,A6,FUNCOP_1:11,NDIFF_1:33,45;
end;
