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 Th10:
  for f be Lipschitzian BilinearOperator of E,F,G
  holds
    ex K be Real
    st 0 <= K
     & for z be Point of [:E,F:]
       holds
        ex L be Lipschitzian LinearOperator of [:E,F:],G
        st ( for dx be Point of E, dy be Point of F
             holds L.(dx,dy) = f.(dx, z `2) + f.(z `1, dy) )
         & for s be Point of [:E,F:]
           holds ||.L.s.|| <= K * ||.z.|| * ||.s.||
  proof
    let f be Lipschitzian BilinearOperator of E,F,G;
    consider K be Real such that
    A1: 0 <= K
      & for x be VECTOR of E
        for y be VECTOR of F
        holds ||.f.(x,y).|| <= K * ||.x.|| * ||.y.|| by LOPBAN_9:def 3;
    set H = 2 * K;
    take H;
    thus 0 <= H by A1,XREAL_1:127;

    let z be Point of [:E,F:];

    deffunc F4(Element of E,Element of F) = f.($1,z `2) + f.(z `1,$2);

    consider L0 be Function of [:the carrier of E,the carrier of F:],
                               the carrier of G such that
    A2: for x be Element of the carrier of E
        for y be Element of the carrier of F
        holds L0.(x, y) = F4(x, y) from BINOP_1:sch 4;

    reconsider L = L0 as Function of [:E,F:],G;
    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:];
      consider Ex be Point of E, Fx be Point of F such that
      A3: x = [Ex,Fx] by PRVECT_3:18;

      consider Ey be Point of E, Fy be Point of F such that
      A4: y = [Ey,Fy] by PRVECT_3:18;
      A5: L.x
       = L.(Ex,Fx) by A3
      .= f.(Ex,z `2) + f.(z `1,Fx) by A2;
      A6: L.y
       = L.(Ey,Fy) by A4
      .= f.(Ey,z `2) + f.(z `1,Fy) by A2;

      thus
      L.(x + y)
       = L.(Ex + Ey,Fx + Fy) by A3,A4,PRVECT_3:18
      .= f.(Ex + Ey, z `2) + f.(z `1, Fx + Fy) by A2
      .= f.(Ex,z `2) + f.(Ey,z `2) + f.(z `1, Fx + Fy) by LOPBAN_8:12
      .= f.(Ex,z `2) + f.(Ey,z `2) + (f.(z `1, Fx) + f.(z `1, Fy))
          by LOPBAN_8:12
      .= (f.(Ex,z `2) + f.(Ey,z `2) + f.(z `1,Fx)) + f.(z `1,Fy)
          by RLVECT_1:def 3
      .= (f.(Ex,z `2) + f.(z `1,Fx) + f.(Ey,z `2)) + f.(z `1,Fy)
          by RLVECT_1:def 3
      .= L.x + L.y by A5,A6,RLVECT_1:def 3;
    end;
    then
    A7: 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;
      consider Ex be Point of E,Fx be Point of F such that
      A8: x = [Ex,Fx] by PRVECT_3:18;
      A9: L.x
       = L.(Ex,Fx) by A8
      .= f.(Ex,z `2) + f.(z `1,Fx) by A2;
      thus L.(a * x)
       = L.(a * Ex, a * Fx) by A8,PRVECT_3:18
      .= f.(a * Ex, z `2) + f.(z `1, a * Fx) by A2
      .= a * f.(Ex, z `2) + f.(z `1, a * Fx) by LOPBAN_8:12
      .= a * f.(Ex, z `2) + a * f.(z `1, Fx) by LOPBAN_8:12
      .= a * L.x by A9,RLVECT_1:def 5;
    end;
    then reconsider L as LinearOperator of [:E,F:],G
      by LOPBAN_1:def 5,A7;
    set K1 = 2 * K * ||.z.||;
    0 <= ||.z `2.|| by NORMSP_1:4;
    then
    A10: 0 <= K * ||.z `2.|| by A1,XREAL_1:127;
    0 <= ||.z `1.|| by NORMSP_1:4;
    then
    A11: 0 <= K * ||.z `1.|| by A1,XREAL_1:127;
    0 <= ||.z.|| by NORMSP_1:4;
    then
    0 <= K * ||.z.|| by A1,XREAL_1:127; then
    A13: 0 <= 2 * (K * ||.z.||) by XREAL_1:127;
    A14: for w be VECTOR of [:E,F:]
         holds ||.L . w.|| <= K1 * ||.w.||
    proof
      let w be Element of [:E,F:];
      consider x be Point of E,y be Point of F such that
      A15: w = [x,y] by PRVECT_3:18;
      L.w
       = L0.(x, y) by A15
      .= f.(x, z `2) + f.(z `1, y) by A2;
      then
      A16: ||.L.w.|| <= ||.f.(x, z `2).|| + ||.f.(z `1, y).||
          by NORMSP_1:def 1;
      A17: ||.f.(x,z `2).|| <= K * ||.x.|| * ||.z `2.|| by A1;
      ||.f.(z `1, y).|| <= K * ||.z `1.|| * ||.y.|| by A1;
      then ||.f.(x, z `2).|| + ||.f.(z `1,y).||
        <= K * ||.x.|| * ||.z `2.|| + K * ||.z `1.|| * ||.y.||
        by A17,XREAL_1:7;
      then
      A18: ||.L.w.|| <= K * ||.z `2.|| * ||.x.|| + K * ||.z `1.|| * ||.y.||
        by A16,XXREAL_0:2;
      A19: K * ||.z `2.|| * ||.x.|| <= K * ||.z `2.|| * ||.w.||
        by A10,A15,LOPBAN_7:15,XREAL_1:64;
      K * ||.z `1.|| * ||.y.|| <= K * ||.z `1.|| * ||.w.||
        by A11,A15,LOPBAN_7:15,XREAL_1:64;
      then K * ||.z `2.|| * ||.x.|| + K * ||.z `1.|| * ||.y.||
        <= K * ||.z `2.|| * ||.w.|| + K * ||.z `1.|| * ||.w.||
        by A19,XREAL_1:7;
      then
      A20: ||.L.w.|| <= (K * ||.z `2.|| + K * ||.z `1.||) * ||.w.||
        by A18,XXREAL_0:2;

      consider x1 be Point of E,y1 be Point of F such that
      A21: z = [x1, y1] by PRVECT_3:18;
      A22: K * ||.z `1.|| <= K * ||.z.|| by A1,A21,LOPBAN_7:15,XREAL_1:64;
      K * ||.z `2.|| <= K * ||.z.|| by A1,A21,LOPBAN_7:15,XREAL_1:64;
      then
      A23: K * ||.z `2.|| + K * ||.z `1.|| <= K * ||.z.|| + K * ||.z.||
        by A22,XREAL_1:7;
      0 <= ||.w.|| by NORMSP_1:4;
      then (K * ||.z `2.|| + K * ||.z `1.||) * ||.w.||
        <= (2 * K * ||.z.||) * ||.w.|| by A23,XREAL_1:64;
      hence ||.L . w.|| <= K1 * ||.w.|| by A20,XXREAL_0:2;
    end;
    then reconsider L as Lipschitzian LinearOperator of [:E,F:],G
      by A13,LOPBAN_1:def 8;
    take L;
    thus thesis by A2,A14;
  end;
