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
   st u is_differentiable_on Z
    & u `| Z is_continuous_on Z
  holds
     L*u is_differentiable_on Z
   & L*u `| 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;
  assume
  A1: u is_differentiable_on Z
    & u `| Z is_continuous_on Z;
  A2: u.:Z c= [#]F;

    L is_differentiable_on [#]F
  & L `| [#]F is_continuous_on [#]F by Th20;
  hence thesis by A1,A2,Th23;
end;
