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 Th15:
  for f be Lipschitzian BilinearOperator of E,F,G
  holds
    f `| [#][:E,F:] is Lipschitzian LinearOperator of [:E,F:],
      R_NormSpace_of_BoundedLinearOperators([:E,F:],G)
  & f `| [#][:E,F:] is_differentiable_on [#][:E,F:]
  & f `| [#][:E,F:] is_continuous_on the carrier of [:E,F:]
  & for z be Point of [:E,F:]
    holds diff(f `| [#][:E,F:], z) = f `| [#][:E,F:]
  proof
    let f be Lipschitzian BilinearOperator of E,F,G;
    set Z = [#][:E,F:];
    set B = R_NormSpace_of_BoundedLinearOperators([:E,F:],G);
    A1: f is_differentiable_on Z
      & f `| Z is_continuous_on Z by Th14;
    set L = f `| [#][:E,F:];
    A2: rng L c= the carrier of B;
    dom L = Z by A1,NDIFF_1:def 9;
    then reconsider L as Function of [:E,F:],B by A2,FUNCT_2:2;
    for x, y be Element of [:E,F:]
    holds L.(x + y) = L.x + L.y
    proof
      let x, y be Element of [:E,F:];

      thus L.(x + y)
       = L/.(x + y)
      .= diff(f, x + y) by A1,NDIFF_1:def 9
      .= diff(f, x) + diff(f, y) by Th13
      .= L/.x + diff(f, y) by A1,NDIFF_1:def 9
      .= L/.x + L/.y by A1,NDIFF_1:def 9
      .= L.x + L.y;
    end;
    then
    A3: L is additive;
    for x be VECTOR of [:E,F:], a be Real
    holds L.(a * x) = a * L.x
    proof
      let x be VECTOR of [:E,F:], a be Real;
      thus L.(a * x)
       = L/.(a * x)
      .= diff(f, a * x) by A1,NDIFF_1:def 9
      .= a * diff(f, x) by Th13
      .= a * L/.x by A1,NDIFF_1:def 9
      .= a * L.x;
    end;
    then reconsider L as LinearOperator of [:E,F:],B
      by A3,LOPBAN_1:def 5;

    consider K be Real such that
    A4: 0 <= K
      & for z be Point of [:E,F:]
        holds
          f is_differentiable_in z
        & (for dx be Point of E, dy be Point of F
           holds diff(f,z).(dx,dy) = f.(dx,z `2)+ f.(z `1,dy))
        & ||.diff(f,z).|| <= K * ||.z.|| by Th11;

    for y be VECTOR of [:E,F:] holds ||.L.y.|| <= K * ||.y.||
    proof
      let y be VECTOR of [:E,F:];
      L.y
       = L/.y
      .= diff(f,y) by A1,NDIFF_1:def 9;
      hence
      ||.L.y.|| <= K * ||.y.|| by A4;
    end;
    then reconsider L as Lipschitzian LinearOperator of [:E,F:],B
      by A4,LOPBAN_1:def 8;
    for z be Point of [:E,F:]
      holds diff(L,z) = L by NDIFF_7:26;
    hence thesis by A1,NDIFF_1:def 9,NDIFF_7:26;
  end;
