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,
      Z be Subset of E,
      u be PartFunc of E,F,
      L be Lipschitzian LinearOperator of F,G
  holds
    for i be Nat
     st u is_differentiable_on i,Z
      & diff(u,i,Z) is_continuous_on Z
    holds L*u is_differentiable_on i,Z
        & diff(L*u,i,Z) is_continuous_on Z
proof
  let E,F,G be RealNormSpace,
      Z be Subset of E,
      u be PartFunc of E,F,
      L be Lipschitzian LinearOperator of F,G;
  let i be Nat;

  assume
  A1: u is_differentiable_on i,Z
    & diff(u,i,Z) is_continuous_on Z;

  hence L*u is_differentiable_on i,Z by Th25;

  A2: L*u is_differentiable_on i,Z
    & diff(L*u,i,Z) = LTRN(i,L,E) * diff(u,i,Z) by A1,Th25;
  A3: diff(u,i,Z) .: Z c= [#] diff_SP(i,E,F);
  LTRN(i,L,E) is_continuous_on [#] diff_SP(i,E,F) by LOPBAN_5:4;
  hence diff(L*u,i,Z) is_continuous_on Z by A1,A2,A3,Th16;
end;
