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 S,E,F,G be RealNormSpace,
      Z be Subset of S,
      B be Lipschitzian BilinearOperator of E,F,G,
      W be PartFunc of S,G,
      w be PartFunc of S,[:E,F:],
      u be PartFunc of S,E,
      v be PartFunc of S,F
   st u is_differentiable_on Z
    & u `| Z is_continuous_on Z
    & v is_differentiable_on Z
    & v `| Z is_continuous_on Z
    & W = B * w & w = <:u,v:>
  holds
     W is_differentiable_on Z
   & W `| Z is_continuous_on Z
proof
  let S,E,F,G be RealNormSpace,
      Z be Subset of S,
      B be Lipschitzian BilinearOperator of E,F,G,
      W be PartFunc of S,G,
      w be PartFunc of S,[:E,F:],
      u be PartFunc of S,E,
      v be PartFunc of S,F;

  assume that
  A1: u is_differentiable_on Z & u `| Z is_continuous_on Z and
  A2: v is_differentiable_on Z & v `| Z is_continuous_on Z and
  A3: W = B * w & w = <:u,v:>;
  A4: w is_differentiable_on Z
    & w `| Z is_continuous_on Z by A1,A2,A3,Th29;
  A5: B is_differentiable_on [#][:E,F:]
    & B `| [#][:E,F:] is_continuous_on [#][:E,F:] by NDIFF12:14;
  w.:Z c= [#][:E,F:];
  hence W is_differentiable_on Z & W `| Z is_continuous_on Z
    by A3,A4,A5,Th23;
end;
