
theorem Th42:
  for X, Y, Z be RealNormSpace st Z is complete
  for seq be sequence of R_NormSpace_of_BoundedBilinearOperators(X,Y,Z)
  st seq is Cauchy_sequence_by_Norm
  holds seq is convergent
  proof
    let X, Y, Z be RealNormSpace such that
    A1: Z is complete;
    let vseq be sequence of R_NormSpace_of_BoundedBilinearOperators(X,Y,Z)
    such that
    A2: vseq is Cauchy_sequence_by_Norm;
    defpred P[set, set] means ex xseq be sequence of Z st
    (for n be Nat holds xseq.n = (vseq.n).$1) &
    xseq is convergent & $2= lim xseq;
    A3: for xy be Element of [:X,Y:] ex z be Element of Z st P[xy,z]
    proof
      let xy be Element of [:X,Y:];
      deffunc F(Nat) = modetrans((vseq.$1),X,Y,Z).xy;
      consider xseq be sequence of Z such that
      A4: for n be Element of NAT holds xseq.n = F(n)
          from FUNCT_2:sch 4;
      A5: for n be Nat holds xseq.n = (vseq.n).xy
      proof
        let n be Nat;
        n in NAT by ORDINAL1:def 12; then
        A6: xseq.n = modetrans((vseq.n),X,Y,Z).xy by A4;
        vseq.n is Lipschitzian BilinearOperator of X,Y,Z by Def9;
        hence thesis by A6;
      end;
      take lim xseq;
      consider x be Point of X, y be Point of Y such that
      A7: xy = [x,y] by PRVECT_3:18;
      A8: for m,k be Nat holds ||.xseq.m-xseq.k.||
        <= ||.vseq.m - vseq.k.|| * ( ||.x.||*||.y.|| )
      proof
        let m,k be Nat;
        reconsider h1 = vseq.m-vseq.k
          as Lipschitzian BilinearOperator of X,Y,Z by Def9;
        A9: xseq.m = (vseq.m).(x,y) by A5,A7;
        A10: xseq.k = (vseq.k).(x,y) by A5,A7;
        xseq.m - xseq.k = h1.(x,y) by A9,A10,Th40; then
        ||.xseq.m - xseq.k.||
          <= ||.vseq.m - vseq.k.|| * ||.x.|| * ||.y.|| by Th32;
        hence thesis;
      end;
      now
        let e be Real such that
        A11: e > 0;
        now
          per cases;
          case
            A12: x=0.X or y =0.Y;
            reconsider k=0 as Nat;
            take k;
            thus for n, m be Nat st n >= k & m >= k
                 holds ||.xseq.n - xseq.m.|| < e
            proof
              let n, m be Nat such that
              n >= k and
              m >= k;
              A13: xseq.m = (vseq.m).(x,y) by A5,A7;
              A14: vseq.m is Lipschitzian BilinearOperator of X,Y,Z
                & vseq.n is Lipschitzian BilinearOperator of X,Y,Z
                  by Def9;
              A15: x = 0*x or y = 0*y by A12; then
              A16: (vseq.m).(x,y) = 0*(vseq.m).(x,y) by A14,LOPBAN_8:12
              .= 0.Z by RLVECT_1:10;
              A17: xseq.n = (vseq.n).(x,y) by A5,A7;
              (vseq.n).(x,y) = 0*(vseq.n).(x,y) by A14,A15,LOPBAN_8:12
               .= 0.Z by RLVECT_1:10;
              hence thesis by A11,A13,A16,A17;
            end;
          end;
          case
            x <> 0.X & y <> 0.Y; then
            ||.x.|| <> 0 & ||.y.|| <> 0 by NORMSP_0:def 5; then
            ||.x.|| > 0 & ||.y.|| > 0; then
            A18: 0 < ||.x.|| * ||.y.|| by XREAL_1:129; then
            consider k be Nat such that
            A19: for n, m be Nat st n >= k & m >= k
                 holds ||.(vseq.n) - (vseq.m).|| < e / (||.x.||*||.y.||)
                 by A2,A11,XREAL_1:139,RSSPACE3:8;
            take k;
            thus for n, m be Nat st n >= k & m >= k
                 holds ||.xseq.n - xseq.m.|| < e
            proof
              let n,m be Nat such that
              A20: n >= k and
              A21: m >= k;
              ||.(vseq.n) - (vseq.m).|| < e / (||.x.|| * ||.y.||)
                by A19,A20,A21; then
              A22: ||.(vseq.n) - (vseq.m).|| * (||.x.|| * ||.y.|| )
                 < e / (||.x.|| * ||.y.|| ) * (||.x.|| * ||.y.|| )
                  by A18,XREAL_1:68;
              A23: e/(||.x.||*||.y.|| ) * (||.x.||*||.y.|| )
                = e*(||.x.||*||.y.||)"* (||.x.||*||.y.|| ) by XCMPLX_0:def 9
                .= e*((||.x.||*||.y.||)"* (||.x.||*||.y.|| ))
                .= e*1 by A18,XCMPLX_0:def 7
                .= e;
              ||.xseq.n-xseq.m.||
                <= ||.(vseq.n) - (vseq.m).|| * (||.x.||*||.y.|| ) by A8;
              hence thesis by A22,A23,XXREAL_0:2;
            end;
          end;
        end;
        hence ex k be Nat
              st for n, m be Nat st n >= k & m >= k
                 holds ||.xseq.n -xseq.m.|| < e;
      end; then
      xseq is convergent by A1,RSSPACE3:8;
      hence thesis by A5;
    end;
    consider f be Function of the carrier of [:X,Y:],
      the carrier of Z such that
    A24: for z be Element of [:X,Y:] holds P[z,f.z] from FUNCT_2:sch 3(A3);
    reconsider tseq = f as Function of [:X,Y:],Z;
    A25: for x1,x2 be Point of X, y be Point of Y
        holds tseq.(x1+x2,y) = tseq.(x1,y) + tseq.(x2,y)
    proof
      let x1,x2 be Point of X, y be Point of Y;
      reconsider x1y=[x1,y] as Point of [:X,Y:];
      reconsider x2y=[x2,y] as Point of [:X,Y:];
      reconsider x1x2y=[x1+x2,y] as Point of [:X,Y:];
      consider sqx1y be sequence of Z such that
      A26: for n be Nat holds sqx1y.n=(vseq.n).x1y
         & sqx1y is convergent
         & tseq.x1y = lim sqx1y by A24;
      consider sqx2y be sequence of Z such that
      A27: for n be Nat holds sqx2y.n=(vseq.n).x2y
         & sqx2y is convergent
         & tseq.x2y = lim sqx2y by A24;
      consider sqx1x2y be sequence of Z such that
      A28: for n be Nat holds sqx1x2y.n=(vseq.n).x1x2y
         & sqx1x2y is convergent
         & tseq.x1x2y = lim sqx1x2y by A24;
      for n be Nat holds sqx1x2y.n = sqx1y.n + sqx2y.n
      proof
        let n be Nat;
        A29: vseq.n is Lipschitzian BilinearOperator of X,Y,Z by Def9;
        A30: sqx1y.n = (vseq.n).(x1,y) by A26;
        A31: sqx2y.n = (vseq.n).(x2,y) by A27;
        thus sqx1x2y.n = (vseq.n).(x1+x2,y) by A28
         .= sqx1y.n + sqx2y.n by A29,A30,A31,LOPBAN_8:12;
      end; then
      A32: sqx1x2y = sqx1y + sqx2y by NORMSP_1:def 2;
      thus tseq. (x1+x2,y) = tseq. (x1,y) + tseq. (x2,y)
        by A26,A27,A28,A32,NORMSP_1:25;
    end;
    A33: for x be Point of X, y be Point of Y, a be Real
    holds tseq.(a*x,y) = a * tseq.(x,y)
    proof
      let x be Point of X, y be Point of Y, a be Real;
      reconsider xy = [x,y] as Point of [:X,Y:];
      reconsider axy = [a*x,y] as Point of [:X,Y:];
      consider sqxy be sequence of Z such that
      A34: for n be Nat holds sqxy.n = (vseq.n).xy
         & sqxy is convergent
         & tseq.xy = lim sqxy by A24;
      consider sqaxy be sequence of Z such that
      A35: for n be Nat holds sqaxy.n=(vseq.n).axy
         & sqaxy is convergent
         & tseq.axy = lim sqaxy by A24;
      for n be Nat holds sqaxy.n = a*sqxy.n
      proof
        let n be Nat;
        A36: vseq.n is Lipschitzian BilinearOperator of X,Y,Z by Def9;
        sqaxy.n = (vseq.n).(a*x,y) by A35
         .= a*(vseq.n).(x,y) by A36,LOPBAN_8:12;
        hence sqaxy.n = a*sqxy.n by A34;
      end;
      then sqaxy = a*sqxy by NORMSP_1:def 5;
      hence tseq.(a*x,y) = a * tseq.(x,y) by A34,A35,NORMSP_1:28;
    end;
    A40: for x be Point of X, y1,y2 be Point of Y
        holds tseq.(x,y1+y2) = tseq.(x,y1) + tseq.(x,y2)
    proof
      let x be Point of X, y1,y2 be Point of Y;
      reconsider x1y=[x,y1] as Point of [:X,Y:];
      reconsider x2y=[x,y2] as Point of [:X,Y:];
      reconsider x1x2y=[x,y1+y2] as Point of [:X,Y:];
      consider sqx1y be sequence of Z such that
      A41: for n be Nat holds sqx1y.n=(vseq.n).x1y
         & sqx1y is convergent
         & tseq.x1y = lim sqx1y by A24;
      consider sqx2y be sequence of Z such that
      A42: for n be Nat holds sqx2y.n=(vseq.n).x2y
         & sqx2y is convergent
         & tseq.x2y = lim sqx2y by A24;
      consider sqx1x2y be sequence of Z such that
      A43: for n be Nat holds sqx1x2y.n=(vseq.n).x1x2y
         & sqx1x2y is convergent
         & tseq.x1x2y = lim sqx1x2y by A24;
      for n be Nat holds sqx1x2y.n = sqx1y.n + sqx2y.n
      proof
        let n be Nat;
        A44: vseq.n is Lipschitzian BilinearOperator of X,Y,Z by Def9;
        A45: sqx1y.n = (vseq.n).(x,y1) by A41;
        A46: sqx2y.n = (vseq.n).(x,y2) by A42;
        thus sqx1x2y.n = (vseq.n).(x,y1+y2) by A43
         .= sqx1y.n + sqx2y.n by A44,A45,A46,LOPBAN_8:12;
      end;
      then sqx1x2y = sqx1y + sqx2y by NORMSP_1:def 2;
      hence tseq.(x,y1+y2) = tseq.(x,y1) + tseq.(x,y2)
        by A41,A42,A43,NORMSP_1:25;
    end;
    for x be Point of X, y be Point of Y, a be Real
    holds tseq.(x,a*y) = a * tseq.(x,y)
    proof
      let x be Point of X, y be Point of Y, a be Real;
      reconsider xy = [x,y] as Point of [:X,Y:];
      reconsider axy = [x,a*y] as Point of [:X,Y:];
      consider sqxy be sequence of Z such that
      A48: for n be Nat holds sqxy.n = (vseq.n).xy
         & sqxy is convergent
         & tseq.xy = lim sqxy by A24;
      consider sqaxy be sequence of Z such that
      A49: for n be Nat holds sqaxy.n = (vseq.n).axy
         & sqaxy is convergent
         & tseq.axy = lim sqaxy by A24;
      for n be Nat holds sqaxy.n = a*sqxy.n
      proof
        let n be Nat;
        A50: vseq.n is Lipschitzian BilinearOperator of X,Y,Z by Def9;
        sqaxy.n = (vseq.n).(x,a*y) by A49
         .= a*(vseq.n).(x,y) by A50,LOPBAN_8:12;
        hence sqaxy.n = a*sqxy.n by A48;
      end; then
      sqaxy = a*sqxy by NORMSP_1:def 5;
      hence tseq.(x,a*y) = a * tseq.(x,y) by A48,A49,NORMSP_1:28;
    end;
    then
    reconsider tseq as BilinearOperator of X,Y,Z by A25,A33,A40,LOPBAN_8:12;
    B53:
    now
      let e1 be Real such that
      A54: e1 >0;
      reconsider e =e1 as Real;
      consider k be Nat such that
      A55: for n, m be Nat st n >= k & m >= k holds
           ||.(vseq.n) - (vseq.m).|| < e by A2,A54,RSSPACE3:8;
      reconsider k as Nat;
      take k;
      now
        let m be Nat;
        assume m >= k; then
        A56: ||.(vseq.m) - (vseq.k).|| <e by A55;
        A57: ||.vseq.m.||= ||.vseq.||.m by NORMSP_0:def 4;
        A58: ||.vseq.k.||= ||.vseq.||.k by NORMSP_0:def 4;
        |. ||.vseq.m.||- ||.vseq.k.|| .|
          <= ||.(vseq.m) - (vseq.k).|| by NORMSP_1:9;
        hence |. ||.vseq.||.m - ||.vseq.||.k .| <e1 by A56,A57,A58,XXREAL_0:2;
      end;
      hence for m be Nat st m >=k
      holds |.||.vseq.||.m - ||.vseq.||.k .| < e1;
    end; then
    A59: ||.vseq.|| is convergent by SEQ_4:41;
    A60: tseq is Lipschitzian
    proof
      take lim (||.vseq.|| );
      A61: now
        let x be Point of X, y be Point of Y;
        reconsider xy = [x,y] as Point of [:X,Y:];
        consider xyseq be sequence of Z such that
        A62: for n be Nat holds xyseq.n = (vseq.n).xy and
        A63: xyseq is convergent and
        A64: tseq.xy = lim xyseq by A24;
        A65: ||.tseq.xy.|| = lim ||.xyseq.|| by A63,A64,LOPBAN_1:20;
        A66: for m be Nat
             holds ||.xyseq.m.|| <= ||.vseq.m.|| * ( ||.x.|| * ||.y.|| )
        proof
          let m be Nat;
          vseq.m is Lipschitzian BilinearOperator of X,Y,Z by Def9; then
          ||.(vseq.m).(x,y).|| <= ||.vseq.m.|| * ||.x.|| * ||.y.|| by Th32;
          hence ||.xyseq.m.|| <= ||.vseq.m.|| * ( ||.x.|| * ||.y.|| ) by A62;
        end;
        A68: for n be Nat holds
             ||.xyseq.||.n <= ( (||.x.|| * ||.y.||)(#)||.vseq .||).n
        proof
          let n be Nat;
          A69: ||.xyseq.||.n = ||.(xyseq.n).|| by NORMSP_0:def 4;
          A70: ||.vseq.n.|| = ||.vseq.||.n by NORMSP_0:def 4;
          ||.(xyseq.n).|| <= ||.vseq.n.|| * (||.x.|| * ||.y.||) by A66;
          hence thesis by A69,A70,SEQ_1:9;
        end;
        A72: lim ( ( ||.x.||*||.y.|| )(#)||.vseq.|| )
           = lim (||.vseq.|| ) * ( ||.x.||*||.y.|| ) by B53,SEQ_2:8,SEQ_4:41;
        ||.xyseq.|| is convergent by A63,A64,LOPBAN_1:20;
        hence ||.tseq.(x,y).|| <= lim (||.vseq.|| ) * ||.x.|| * ||.y.||
          by A59,A65,A68,A72,SEQ_2:18;
      end;
      now
        let n be Nat;
        ||.vseq.n.|| >= 0;
        hence ||.vseq.||.n >= 0 by NORMSP_0:def 4;
      end;
      hence thesis by B53,A61,SEQ_2:17,SEQ_4:41;
    end;
    A73: for e be Real
         st e > 0 ex k be Nat st for n be Nat st n >= k
         holds for x be Point of X, y be Point of Y
         holds ||.(vseq.n).(x,y) - tseq.(x,y) .|| <= e * ||.x.|| * ||.y.||
    proof
      let e be Real;
      assume e > 0; then
      consider k be Nat such that
      A74: for n, m be Nat st n >= k & m >= k
           holds ||.(vseq.n) - (vseq.m).|| < e by A2,RSSPACE3:8;
      take k;
      now
        let n be Nat such that
        A75: n >= k;
        now
          let x be Point of X, y be Point of Y;
          reconsider xy = [x,y] as Point of [:X,Y:];
          consider xyseq be sequence of Z such that
          A76: for n be Nat holds xyseq.n=(vseq.n). xy and
          A77: xyseq is convergent and
          A78: tseq.xy = lim xyseq by A24;
          A79: for m,k be Nat
               holds ||.xyseq.m-xyseq.k.||
                  <= ||.vseq.m - vseq.k.|| * ( ||.x.|| * ||.y.|| )
          proof
            let m,k be Nat;
            reconsider h1 = vseq.m-vseq.k
              as Lipschitzian BilinearOperator of X,Y,Z by Def9;
            xyseq.k = (vseq.k).xy by A76; then
            xyseq.m - xyseq.k = (vseq.m).(x,y)-(vseq.k).(x,y) by A76
             .= h1.(x,y) by Th40; then
            ||.xyseq.m-xyseq.k.||
              <= ||.vseq.m - vseq.k.|| * ||.x.|| * ||.y.|| by Th32;
            hence thesis;
          end;
          A81: for m be Nat st m >= k
               holds ||.xyseq.n - xyseq.m.|| <= e * ||.x.|| * ||.y.||
          proof
            let m be Nat;
            assume m >= k; then
            A82: ||.vseq.n - vseq.m.|| < e by A74,A75;
            A83: ||.xyseq.n - xyseq.m.||
              <= ||.vseq.n - vseq.m.|| * ( ||.x.||*||.y.|| ) by A79;
            ||.vseq.n - vseq.m.|| * ( ||.x.||*||.y.|| )
              <= e * ( ||.x.|| * ||.y.|| ) by A82,XREAL_1:64;
            hence thesis by A83,XXREAL_0:2;
          end;
          ||.xyseq.n - tseq.(x,y).|| <= e * ||.x.|| * ||.y.||
          proof
            deffunc F(Nat) = ||.xyseq.$1 - xyseq.n.||;
            consider rseq be Real_Sequence such that
            A84: for m be Nat holds rseq.m = F(m) from SEQ_1:sch 1;
            now
              let i be object;
              assume i in NAT; then
              reconsider k=i as Nat;
              thus rseq.i = ||.xyseq.k - xyseq.n.|| by A84
               .= ||.(xyseq - xyseq.n).k.|| by NORMSP_1:def 4
               .= ||.(xyseq - xyseq.n).||.i by NORMSP_0:def 4;
            end;
            then
            A85: rseq = ||.xyseq - xyseq.n.|| by FUNCT_2:12;
            A86: xyseq - xyseq.n is convergent by A77,NORMSP_1:21;
            lim (xyseq-xyseq.n) = tseq.(x,y) - xyseq.n
              by A77,A78,NORMSP_1:27; then
            A87: lim rseq = ||.tseq.(x,y)-xyseq.n.|| by A85,A86,LOPBAN_1:41;
            for m be Nat st m >= k holds rseq.m <= e * ||.x.||*||.y.||
            proof
              let m be Nat such that
              A88: m >= k;
              rseq.m = ||.xyseq.m - xyseq.n.|| by A84
               .= ||.xyseq.n - xyseq.m.|| by NORMSP_1:7;
              hence thesis by A81,A88;
            end; then
            lim rseq <= e * ||.x.||*||.y.|| by A85,A86,LOPBAN_1:41,Lm3;
            hence thesis by A87,NORMSP_1:7;
          end;
          hence ||.(vseq.n).(x,y) - tseq.(x,y).||
            <= e * ||.x.|| * ||.y.|| by A76;
        end;
        hence
        for x be Point of X, y be Point of Y
        holds ||.(vseq.n).(x,y) - tseq.(x,y).|| <= e * ||.x.|| * ||.y.||;
      end;
      hence thesis;
    end;
    reconsider tseq as Lipschitzian BilinearOperator of X,Y,Z by A60;
    reconsider tv = tseq as Point of
      R_NormSpace_of_BoundedBilinearOperators(X,Y,Z) by Def9;
    A89: for e be Real st e > 0
         ex k be Nat
         st for n be Nat st n >= k holds ||.vseq.n - tv.|| <= e
    proof
      let e be Real such that
      A90: e > 0;
      consider k be Nat such that
      A91: for n be Nat st n >= k holds
      for x be Point of X,y be Point of Y
      holds ||.(vseq.n).(x,y) - tseq.(x,y) .||
          <= e * ||.x.|| * ||.y.|| by A73,A90;
      now
        set g1 = tseq;
        let n be Nat such that
        A92: n >= k;
        reconsider h1 = vseq.n-tv
          as Lipschitzian BilinearOperator of X,Y,Z by Def9;
        set f1 = vseq.n;
        A93: now
          let t be Point of X, s be Point of Y;
          assume ||.t.|| <= 1 & ||.s.|| <= 1; then
          ||.t.|| * ||.s.|| <= 1 * 1 by XREAL_1:66; then
          A94: e * ( ||.t.|| * ||.s.|| ) <= e * 1 by A90,XREAL_1:64;
          A95: ||.f1.(t,s)-g1.(t,s).|| <= e * ||.t.|| * ||.s.|| by A91,A92;
          ||.h1.(t,s).|| = ||.f1.(t,s)-g1.(t,s).|| by Th40;
          hence ||.h1.(t,s) .|| <= e by A94,A95,XXREAL_0:2;
        end;
        A96: now
          let r be Real;
          assume r in PreNorms(h1); then
          ex t be VECTOR of X, s be VECTOR of Y
          st r = ||.h1.(t,s).|| & ||.t.|| <= 1 & ||.s.|| <= 1;
          hence r <= e by A93;
        end;
        (for s be Real st s in PreNorms(h1) holds s <= e)
        implies upper_bound PreNorms(h1) <= e by SEQ_4:45;
        hence ||.vseq.n-tv.|| <=e by A96,Th30;
      end;
      hence thesis;
    end;
    for e be Real st e > 0
    ex m be Nat
    st for n be Nat st n >= m holds ||.(vseq.n) - tv.|| < e
    proof
      let e be Real such that
      A98: e > 0;
      consider m be Nat such that
      A99: for n be Nat st n >= m holds ||.(vseq.n) - tv.|| <= e/2
        by A89,A98,XREAL_1:215;
      A100: e/2 < e by A98,XREAL_1:216;
      now
        let n be Nat;
        assume n >= m;
        then ||.(vseq.n) - tv.|| <= e/2 by A99;
        hence ||.(vseq.n) - tv.|| < e by A100,XXREAL_0:2;
      end;
      hence thesis;
    end;
    hence thesis by NORMSP_1:def 6;
  end;
