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 f be Lipschitzian BilinearOperator of E,F,G,
      Z be Subset of [:E,F:]
    st Z is open
  holds
    f is_partial_differentiable_on`1 Z
  & f is_partial_differentiable_on`2 Z
  & f `partial`1| Z is_continuous_on Z
  & f `partial`2| 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;
    A2: 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_partial_differentiable_in`1 x by Th4;
    hence
    A3: f is_partial_differentiable_on`1 Z by A1,A2,NDIFF_7:43;
    for x be Point of [:E,F:] st x in Z
    holds f is_partial_differentiable_in`2 x by Th4;
    hence
    A4: f is_partial_differentiable_on`2 Z by A1,A2,NDIFF_7:44;

    set g1 = f `partial`1| Z;
    set g2 = f `partial`2| Z;
    A5: dom g1 = Z by A3,NDIFF_7:def 10;
    A6: dom g2 = Z by A4,NDIFF_7:def 11;
    consider K be Real such that
    A7: 0 <=K
      & for z be Point of [:E,F:]
        holds
          ||. partdiff`1 (f,z) .|| <= K * ||.z.||
        & ||. partdiff`2 (f,z) .|| <= K * ||.z.|| by Th6;
    A8: 0 + 0 < K + 1 by A7,XREAL_1:8;
    A9: 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
      A10: t0 in Z & 0 < r0;
      set r = r0/2;
      set s = r/(K+1);
      take s;
      A11: 0 < r & r < r0 by A10,XREAL_1:215,XREAL_1:216;
      hence 0 < s by A8,XREAL_1:139;
      let t1 be Point of [:E,F:];
      assume
      A12: t1 in Z & ||.t1 - t0.|| < s;
      A13: g1/.t1 = partdiff`1(f,t1) by A3,A12,NDIFF_7:def 10;
      (g1/. t1) - (g1/. t0)
       = partdiff`1(f,t1) - partdiff`1(f,t0) by A3,A10,A13,NDIFF_7:def 10
      .= partdiff`1(f,t1-t0) by Th7;
      then
      A14: ||. (g1/. t1) - (g1/. t0) .|| <= K * ||.t1 - t0.|| by A7;
      0 <= ||.t1 - t0.|| by NORMSP_1:4;
      then
      A15: K * ||.t1 - t0.|| <= (K+1) * ||.t1 - t0.|| by A9,XREAL_1:64;
      (K + 1) * ||.t1 - t0.|| <= (K+1) * s by A8,A12,XREAL_1:64;
      then (K + 1) * ||.t1 - t0.|| <= r by A8,XCMPLX_1:87;
      then (K + 1) * ||.t1 - t0.|| < r0 by A11,XXREAL_0:2;
      then K * ||.t1 - t0.|| < r0 by A15,XXREAL_0:2;
      hence ||.(g1/. t1) - (g1/. t0).|| < r0 by A14,XXREAL_0:2;
    end;
    hence f `partial`1| Z is_continuous_on Z by A5,NFCONT_1:19;

    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 ||.(g2/. t1) - (g2/. t0).|| < r
    proof
      let t0 be Point of [:E,F:];
      let r0 be Real;
      assume
      A16: t0 in Z & 0 < r0;
      set r = r0/2;
      set s = r/(K+1);
      take s;
      A17: 0 < r & r < r0 by A16,XREAL_1:215,216;
      hence 0 < s by A8,XREAL_1:139;
      let t1 be Point of [:E,F:];
      assume
      A18: t1 in Z & ||.t1 - t0.|| < s;
      A19: g2/.t1 = partdiff`2(f,t1) by A4,A18,NDIFF_7:def 11;
      (g2/. t1) - (g2/. t0)
       = partdiff`2(f, t1) - partdiff`2(f, t0) by A4,A16,A19,NDIFF_7:def 11
      .= partdiff`2(f, t1 - t0) by Th8;
      then
      A20: ||. (g2/. t1) - (g2/. t0) .|| <= K * ||.t1 - t0.|| by A7;
      0 <= ||.t1 - t0.|| by NORMSP_1:4;
      then
      A21: K * ||.t1 - t0.|| <= (K + 1) * ||.t1 - t0.|| by A9,XREAL_1:64;
      (K + 1) * ||.t1 - t0.|| <= (K + 1) * s by A8,A18,XREAL_1:64;
      then (K + 1) * ||.t1 - t0.|| <= r by A8,XCMPLX_1:87;
      then (K + 1) * ||.t1 - t0.|| < r0 by A17,XXREAL_0:2;
      then K * ||.t1 - t0.|| < r0 by A21,XXREAL_0:2;
      hence ||.(g2/. t1) - (g2/. t0).|| < r0 by A20,XXREAL_0:2;
    end;
    hence f `partial`2| Z is_continuous_on Z by A6,NFCONT_1:19;
  end;
