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 Th58:
  for S,E,F be RealNormSpace,
      u be PartFunc of S,E,
      v be PartFunc of S,F,
      w be PartFunc of S,[:E,F:],
      Z be Subset of S
   st w = <:u,v:>
    & u is_differentiable_on Z
    & v is_differentiable_on Z
  holds
    diff(w,0,Z) is_differentiable_on Z
  & ex T be Lipschitzian LinearOperator of
      [:diff_SP(1,S,E),diff_SP(1,S,F):], diff_SP(1,S,[:E,F:])
    st T = CTP(S,diff_SP(0,S,E), diff_SP(0,S,F))
     & diff(w,1,Z) = T * <:diff(u,1,Z), diff(v,1,Z):>
proof
  let S,E,F be RealNormSpace,
      u be PartFunc of S,E,
      v be PartFunc of S,F,
      w be PartFunc of S,[:E,F:],
      Z be Subset of S;

  assume
  A1: w = <:u,v:>
    & u is_differentiable_on Z
    & v is_differentiable_on Z;

  then
  A2: w is_differentiable_on Z
    & w`|Z = CTP(S,diff_SP(0,S,E),diff_SP(0,S,F)) * <:u`|Z,v`|Z:> by Th57;

  A3: diff(w,1,Z)
   = (w|Z)`|Z by NDIFF_6:11
  .= w`|Z by A1,Th4,Th57;

  A4: diff(u,1,Z)
   = (u|Z)`|Z by NDIFF_6:11
  .= u`|Z by A1,Th4;

  A5: diff(v,1,Z)
   = (v|Z)`|Z by NDIFF_6:11
  .= v`|Z by A1,Th4;

  A6: diff(w,0,Z) = w|Z by NDIFF_6:11;

  A7: CTP(S,diff_SP (0,S,E),diff_SP(0,S,F)) is Lipschitzian LinearOperator
    of [:diff_SP(0+1,S,E),diff_SP(0+1,S,F):],
          R_NormSpace_of_BoundedLinearOperators
          (S,[:diff_SP(0,S,E),diff_SP(0,S,F):]) by Th56;

  A8: diff_SP(0,S,E) = E by NDIFF_6:7;
  A9: diff_SP(0,S,F) = F by NDIFF_6:7;

  R_NormSpace_of_BoundedLinearOperators(S,[:diff_SP(0,S,E),diff_SP(0,S,F):])
  = diff_SP(1,S,[:E,F:]) by A8,A9,NDIFF_6:7;

  then reconsider T = CTP(S,diff_SP(0,S,E),diff_SP(0,S,F))
    as Lipschitzian LinearOperator of
        [:diff_SP(1,S,E),diff_SP(1,S,F):],diff_SP(1,S,[:E,F:]) by A7;

  diff_SP(0,S,[:E,F:]) = [:E,F:] by NDIFF_6:7;
  hence diff(w,0,Z) is_differentiable_on Z by A2,A6,Th3;

  take T;
  thus T = CTP(S,diff_SP(0,S,E),diff_SP(0,S,F));
  thus thesis by A1,A3,A4,A5,Th57;
end;
