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
  for E,F,G be RealNormSpace,
      u be PartFunc of E,F,
      L be Lipschitzian LinearOperator of F,G,
      x be Point of E
  st u is_differentiable_in x
  holds
      L*u is_differentiable_in x
    & diff(L*u,x) = L * diff(u,x)
proof
  let E,F,G be RealNormSpace,
      u be PartFunc of E,F,
      L be Lipschitzian LinearOperator of F,G,
      x be Point of E;
  assume
  A1: u is_differentiable_in x;
  A2: L is_differentiable_in u/.x
    & diff(L,u/.x) = L by NDIFF_7:26;
  then
  A3: L*u is_differentiable_in x
    & diff(L*u,x) = diff(L,u/.x) * diff(u,x) by A1,NDIFF_2:13;
  modetrans(diff(u,x),E,F) = diff(u,x) by LOPBAN_1:def 11;
  hence thesis by A2,A3,LOPBAN_1:def 11;
end;
