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 Th21:
  for E,F,G be RealNormSpace,
      u be PartFunc of E,F,
      Z be Subset of E,
      L be Lipschitzian LinearOperator of F,G
   st u is_differentiable_on Z
  holds
      L*u is_differentiable_on Z
    & for x be Point of E st x in Z
      holds ((L*u) `| Z) /. x = L * ((u`| Z)/.x)
proof
  let E,F,G be RealNormSpace,
      u be PartFunc of E,F,
      Z be Subset of E,
      L be Lipschitzian LinearOperator of F,G;
  assume
  A1: u is_differentiable_on Z;
  A2: u.:Z c= [#]F;
  A3: L is_differentiable_on [#]F
    & for x be Point of F
      holds (L`| [#]F) /. x = L by Th20;
  hence L*u is_differentiable_on Z by A1,A2,Th19;
  let x be Point of E;

  A4: modetrans((L`| [#]F)/.(u /. x),F,G)
   = (L`| [#]F)/.(u /. x) by LOPBAN_1:def 11
  .= L by Th20;

  assume x in Z;
  then ((L*u) `| Z) /. x
   = ((L`| [#]F)/.(u /. x)) * ((u`| Z)/.x) by A1,A2,A3,Th19
  .= L * ((u`| Z)/.x) by A4,LOPBAN_1:def 11;
  hence thesis;
end;
