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 Th14:
  for f be Lipschitzian BilinearOperator of E,F,G,
      Z be Subset of [:E,F:]
   st Z is open
  holds
    f is_differentiable_on Z
  & f `| Z is_continuous_on Z
  proof
    let f be Lipschitzian BilinearOperator of E,F,G,
        Z be Subset of [:E,F:];
    assume
    A1: Z is open;
    consider K be Real such that
    A2: 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;

    A3: dom f = the carrier of [:E,F:] by FUNCT_2:def 1;

    for x be Point of [:E,F:] st x in Z
    holds f is_differentiable_in x by A2;
    hence
    A4: f is_differentiable_on Z by A1,A3,NDIFF_1:31;
    set g1 = f `| Z;
    A5: dom g1 = Z by A4,NDIFF_1:def 9;
    A6: 0 + 0 < K + 1 by A2,XREAL_1:8;
    A7: K + 0 < K + 1 by XREAL_1:8;

    for t0 be Point of [:E,F:]
    for r be Real
     st t0 in Z & 0 < r
    holds
      ex s be Real
      st 0 < s
       & for t1 be Point of [:E,F:]
          st t1 in Z & ||.t1 - t0.|| < s
         holds ||.(g1/.t1) - (g1/.t0).|| < r
    proof
      let t0 be Point of [:E,F:];
      let r0 be Real;
      assume
      A8: t0 in Z & 0 < r0;
      set r = r0 / 2;
      set s = r / (K+1);
      take s;
      A9: 0 < r & r < r0 by A8,XREAL_1:215,216;
      hence 0 < s by A6,XREAL_1:139;
      let t1 be Point of [:E,F:];
      assume
      A10: t1 in Z & ||.t1 - t0.|| < s; then
      A11: g1/.t1 = diff(f,t1) by A4,NDIFF_1:def 9;

      (g1/.t1) - (g1/.t0)
       = diff(f,t1) - diff(f,t0) by A4,A8,A11,NDIFF_1:def 9
      .= diff(f,t1-t0) by Th13;
      then
      A12: ||.(g1/.t1) - (g1/.t0).|| <= K * ||.t1 - t0.|| by A2;
      0 <= ||.t1 - t0.|| by NORMSP_1:4;
      then
      A13: K * ||.t1-t0.|| <= (K+1) * ||.t1 - t0.|| by A7,XREAL_1:64;
      (K+1) * ||.t1 - t0.|| <= (K+1) * s by A6,A10,XREAL_1:64;
      then (K+1) * ||.t1 - t0.|| <= r by A6,XCMPLX_1:87;
      then (K+1) * ||.t1 - t0.|| < r0 by A9,XXREAL_0:2;
      then K * ||.t1 - t0.|| < r0 by A13,XXREAL_0:2;
      hence ||.(g1/.t1) - (g1/.t0).|| < r0 by A12,XXREAL_0:2;
    end;
    hence
    f `| Z is_continuous_on Z by A5,NFCONT_1:19;
  end;
