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 F,G be RealNormSpace,
      L be Lipschitzian LinearOperator of F,G
  holds
    L is_differentiable_on [#]F
  & L`| [#]F is_continuous_on [#]F
  & for x be Point of F holds (L`| [#]F) /. x = L
proof
  let F,G be RealNormSpace,
      L be Lipschitzian LinearOperator of F,G;
  A1: diff(L,0,[#]F)
     = L | [#]F by NDIFF_6:11
    .= L;
  (diff_SP(F,G)).0 = G by NDIFF_6:7;
  hence
  A3: L is_differentiable_on [#]F
    & L`| [#]F is_continuous_on [#]F by A1,NDIFF12:20;
  let x be Point of F;
  thus (L`| [#]F) /. x
     = diff(L,x) by A3,NDIFF_1:def 9
    .= L by NDIFF_7:26;
end;
