reserve X,G for RealNormSpace-Sequence,
          Y for RealNormSpace;
reserve f for MultilinearOperator of X,Y;

theorem Th42:
  for X be RealNormSpace-Sequence, Y be RealNormSpace st Y is complete
  for seq be sequence of R_NormSpace_of_BoundedMultilinearOperators(X,Y)
    st seq is Cauchy_sequence_by_Norm
  holds seq is convergent
  proof
    let X be RealNormSpace-Sequence,Y be RealNormSpace such that
    A1: Y is complete;
    let vseq be sequence of R_NormSpace_of_BoundedMultilinearOperators(X,Y)
    such that
    A2: vseq is Cauchy_sequence_by_Norm;
    defpred P[set, set] means ex xseq be sequence of Y st
    (for n be Nat holds xseq.n = modetrans((vseq.n),X,Y).$1) &
    xseq is convergent & $2 = lim xseq;
    A3: for x be Element of product X ex y be Element of Y st P[x,y]
    proof
      let x be Element of product X;
      deffunc F(Nat) = modetrans((vseq.$1),X,Y).x;
      consider xseq be sequence of Y 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 = F(n)
      proof let n be Nat;
        n in NAT by ORDINAL1:def 12;
        hence thesis by A4;
      end;
      take lim xseq;
      A6: for m,k be Nat holds
          ||.xseq.m-xseq.k.|| <= ||.vseq.m - vseq.k.|| * NrProduct x
      proof
        let m,k be Nat;
        reconsider h1 = vseq.m-vseq.k
          as Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
        k in NAT by ORDINAL1:def 12; then
        A7: xseq.k = modetrans((vseq.k),X,Y).x by A4;
a8:     vseq.m is Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
a9:     vseq.k is Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
        m in NAT by ORDINAL1:def 12; then
        xseq.m = modetrans((vseq.m),X,Y).x by A4;
        then xseq.m - xseq.k = h1.x by A7,a8,a9,LOPBAN10:52;
        hence thesis by LOPBAN10:45;
      end;
      now
        let e be Real such that
        A10: e > 0;
        now
          per cases;
          case
            A11: ex i be Element of dom X st x.i = 0.(X.i);
            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;
              m in NAT by ORDINAL1:def 12; then
              A12: xseq.m = modetrans((vseq.m),X,Y).x by A4
              .= 0.Y by A11,LOPBAN10:36;
              n in NAT by ORDINAL1:def 12; then
              xseq.n = modetrans((vseq.n),X,Y).x by A4
              .= 0.Y by A11,LOPBAN10:36;
              hence thesis by A10,A12;
            end;
          end;
          case
            not ex i be Element of dom X st x.i = 0.(X.i); then
            A13: NrProduct x > 0 by LOPBAN10:27; then
            consider k be Nat such that
            A15: for n, m be Nat st n >= k & m >= k
            holds ||.(vseq.n) - (vseq.m).|| < e / (NrProduct x)
              by A2,A10,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
              A16: n >=k and
              A17: m >= k;
              ||.(vseq.n) - (vseq.m).|| < e / (NrProduct x)
                by A15,A16,A17; then
              A18: ||.(vseq.n) - (vseq.m).|| * (NrProduct x)
                < e / (NrProduct x) * (NrProduct x) by A13,XREAL_1:68;
              A19: e / (NrProduct x) * (NrProduct x)
               = e * ((NrProduct x)"* (NrProduct x))
              .= e * 1 by A13,XCMPLX_0:def 7
              .= e;
              ||.xseq.n - xseq.m.||
                <= ||.(vseq.n) - (vseq.m).|| * ( NrProduct x ) by A6;
              hence thesis by A18,A19,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 Cauchy_sequence_by_Norm by RSSPACE3:8;
      hence thesis by A1,A5;
    end;
    consider f be Function of the carrier of product X,
    the carrier of Y such that
    A20: for x be Element of product X holds P[x,f.x] from FUNCT_2:sch 3(A3);
    reconsider tseq = f as Function of product X,Y;
    A21: for u be Point of product X,
             i be Element of dom X,
             x be Point of X.i holds
            ex xseqi be sequence of Y
            st (for n be Nat holds
                xseqi.n = ( modetrans((vseq.n),X,Y) * reproj(i,u)).x)
             & xseqi is convergent
             & (tseq*reproj (i,u)).x = lim xseqi
    proof
      let u be Point of product X,
      i be Element of dom X,
      x be Point of X.i;
      reconsider v = reproj (i,u).x as Point of product X;
      consider xseq be sequence of Y such that
      A22: (for n be Nat holds xseq.n = modetrans((vseq.n),X,Y).v) &
          xseq is convergent & tseq.v = lim xseq by A20;
      A23: dom reproj (i,u) = the carrier of (X.i) by FUNCT_2:def 1;
      take xseq;
      thus for n be Nat holds xseq.n
        = (modetrans((vseq.n),X,Y) * reproj(i,u)).x
      proof
        let n be Nat;
        thus xseq.n = modetrans((vseq.n),X,Y).v by A22
        .= (vseq.n).(reproj(i,u).x) by LOPBAN10:def 13
        .= ((vseq.n) * reproj(i,u)).x by A23,FUNCT_1:13
        .= (modetrans((vseq.n),X,Y) * reproj(i,u)).x by LOPBAN10:def 13;
      end;
      thus xseq is convergent by A22;
      thus (tseq * reproj(i,u)).x = lim xseq by A22,A23,FUNCT_1:13;
    end;
    now
      let i be Element of dom X,u be Point of product X;
      set tseqiu = tseq * reproj(i,u);
      deffunc H(Nat) = modetrans((vseq.$1),X,Y) * reproj (i,u);
      A24: now
        let x,y be Point of X.i;
        consider xseq be sequence of Y such that
        A25: for n be Nat holds xseq.n = H(n).x and
        A26: xseq is convergent and
        A27: tseqiu.x = lim xseq by A21;
        consider zseq be sequence of Y such that
        A28: for n be Nat holds zseq.n = H(n).(x+y) and
        zseq is convergent and
        A29: tseqiu.(x+y) = lim zseq by A21;
        consider yseq be sequence of Y such that
        A30: for n be Nat holds yseq.n = H(n).y and
        A31: yseq is convergent and
        A32: tseqiu.y = lim yseq by A21;
        now
          let n be Nat;
          A33: H(n) is LinearOperator of X.i,Y by LOPBAN10:def 6;
          thus zseq.n = H(n).(x+y) by A28
          .= H(n).x+H(n).y by A33,VECTSP_1:def 20
          .= xseq.n + H(n).y by A25
          .= xseq.n +yseq.n by A30;
        end; then
        zseq = xseq + yseq by NORMSP_1:def 2;
        hence tseqiu.(x+y) = tseqiu.x+tseqiu.y
          by A26,A27,A29,A31,A32,NORMSP_1:25;
      end;
      now
        let x be Point of X.i;
        let a be Real;
        consider xseq be sequence of Y such that
        A34: for n be Nat holds xseq.n = H(n).x and
        A35: xseq is convergent and
        A36: tseqiu.x = lim xseq by A21;
        consider zseq be sequence of Y such that
        A37: for n be Nat holds zseq.n = H(n).(a*x) and
        zseq is convergent and
        A38: tseqiu.(a*x) = lim zseq by A21;
        now
          let n be Nat;
          A39: H(n) is LinearOperator of X.i,Y by LOPBAN10:def 6;
          thus zseq.n = H(n).(a*x) by A37
          .= a * H(n).x by A39,LOPBAN_1:def 5
          .= a * xseq.n by A34;
        end; then
        zseq = a * xseq by NORMSP_1:def 5;
        hence tseqiu.(a*x) = a * tseqiu.x by A35,A36,A38,NORMSP_1:28;
      end;
      hence tseq * reproj(i,u) is LinearOperator of X.i,Y
        by A24,LOPBAN_1:def 5,VECTSP_1:def 20;
    end; then
    reconsider tseq as MultilinearOperator of X,Y by LOPBAN10:def 6;
    B39: now
      let e1 be Real such that
      A40: e1 > 0;
      reconsider e = e1 as Real;
      consider k be Nat such that
      A41: for n, m be Nat st n >= k & m >= k
           holds ||.(vseq.n) - (vseq.m).|| < e by A2,A40,RSSPACE3:8;
      reconsider k as Nat;
      take k;
      now
        let m be Nat;
        assume m >= k; then
        A42: ||.(vseq.m) - (vseq.k).|| < e by A41;
        A43: ||.vseq.m.|| = ||.vseq.||.m by NORMSP_0:def 4;
        A44: ||.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 A42,A43,A44,XXREAL_0:2;
      end;
      hence for m be Nat st m >= k holds |.||.vseq.||.m - ||.vseq.||.k .| < e1;
    end; then
    A45: ||.vseq.|| is convergent by SEQ_4:41;
    A46: tseq is Lipschitzian
    proof
      take lim (||.vseq.||);
      A47: now
        let x be Point of product X;
        consider xseq be sequence of Y such that
        A48: for n be Nat holds xseq.n = modetrans((vseq.n),X,Y).x and
        A49: xseq is convergent and
        A50: tseq.x = lim xseq by A20;
        A51: ||.tseq.x.|| = lim ||.xseq.|| by A49,A50,LOPBAN_1:20;
        A52: for m be Nat holds ||.xseq.m.|| <= ||.vseq.m.|| * ( NrProduct x )
        proof
          let m be Nat;
          A53: xseq.m = modetrans((vseq.m),X,Y).x by A48;
          vseq.m is Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
          hence thesis by A53,LOPBAN10:45;
        end;
        A54: for n be Nat holds ||.xseq.||.n <= ((NrProduct x)(#)||.vseq.||).n
        proof
          let n be Nat;
          A55: ||.xseq.||.n = ||.(xseq.n).|| by NORMSP_0:def 4;
          A56: ||.vseq.n.|| = ||.vseq.||.n by NORMSP_0:def 4;
          ||.(xseq.n).|| <= ||.vseq.n.|| * (NrProduct x) by A52;
          hence thesis by A55,A56,SEQ_1:9;
        end;
        A58: lim ( (NrProduct x)(#)||.vseq.|| )
           = lim (||.vseq.||) * (NrProduct x) by B39,SEQ_2:8,SEQ_4:41;
        ||.xseq.|| is convergent by A49,A50,LOPBAN_1:20;
        hence ||.tseq.x.|| <= lim (||.vseq.||) * (NrProduct x)
          by A45,A51,A54,A58,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 B39,A47,SEQ_2:17,SEQ_4:41;
    end;
    A59: 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 product X holds
         ||.modetrans((vseq.n),X,Y).x - tseq.x.|| <= e * (NrProduct x)
    proof
      let e be Real;
      assume e > 0; then
      consider k be Nat such that
      A60: 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
        A61: n >= k;
        now
          let x be Point of product X;
          consider xseq be sequence of Y such that
          A62: for n be Nat holds xseq.n = modetrans((vseq.n),X,Y). x and
          A63: xseq is convergent and
          A64: tseq.x = lim xseq by A20;
          A65: for m,k be Nat holds
               ||.xseq.m - xseq.k.|| <= ||.vseq.m - vseq.k.|| * (NrProduct x)
          proof
            let m,k be Nat;
            reconsider h1 = vseq.m - vseq.k
            as Lipschitzian MultilinearOperator of X,Y by LOPBAN10:def 11;
            A66: xseq.k = modetrans((vseq.k),X,Y).x by A62;
a67:        vseq.m is Lipschitzian MultilinearOperator of X,Y
              by LOPBAN10:def 11;
a68:        vseq.k is Lipschitzian MultilinearOperator of X,Y
              by LOPBAN10:def 11;
            xseq.m = modetrans((vseq.m),X,Y).x by A62; then
            xseq.m - xseq.k =h1.x by A66,a67,a68,LOPBAN10:52;
            hence thesis by LOPBAN10:45;
          end;
          A69: for m be Nat st m >= k holds
              ||.xseq.n-xseq.m.|| <= e * (NrProduct x)
          proof
            let m be Nat;
            assume m >= k; then
            A70: ||.vseq.n - vseq.m.|| < e by A60,A61;
            A71: ||.xseq.n-xseq.m.||
              <= ||.vseq.n - vseq.m.|| * ( NrProduct x ) by A65;
            ||.vseq.n - vseq.m.|| * (NrProduct x)
              <= e * (NrProduct x) by A70,XREAL_1:64;
            hence thesis by A71,XXREAL_0:2;
          end;
          ||.xseq.n - tseq.x.|| <= e * (NrProduct x)
          proof
            deffunc F(Nat) = ||.xseq.$1 - xseq.n.||;
            consider rseq be Real_Sequence such that
            A72: for m be Nat holds rseq.m = F(m) from SEQ_1:sch 1;
            now
              let x be object;
              assume x in NAT; then
              reconsider k = x as Nat;
              thus rseq.x = ||.xseq.k - xseq.n.|| by A72
              .= ||.(xseq - xseq.n).k.|| by NORMSP_1:def 4
              .= ||.(xseq - xseq.n).||.x by NORMSP_0:def 4;
            end; then
            A73: rseq = ||.xseq - xseq.n.|| by FUNCT_2:12;
            A74: xseq - xseq.n is convergent by A63,NORMSP_1:21;
            lim (xseq-xseq.n)= tseq.x - xseq.n by A63,A64,NORMSP_1:27; then
            A75: lim rseq = ||.tseq.x-xseq.n.|| by A73,A74,LOPBAN_1:41;
            for m be Nat st m >= k holds rseq.m <= e * (NrProduct x)
            proof
              let m be Nat such that
              A76: m >=k;
              rseq.m = ||.xseq.m-xseq.n.|| by A72
              .= ||.xseq.n-xseq.m.|| by NORMSP_1:7;
              hence thesis by A69,A76;
            end; then
            lim rseq <= e * (NrProduct x)
              by A73,A74,LOPBAN_1:41,RSSPACE2:5;
            hence thesis by A75,NORMSP_1:7;
          end;
          hence ||.modetrans((vseq.n),X,Y).x - tseq.x.||
            <= e * (NrProduct x) by A62;
        end;
        hence for x be Point of product X holds
          ||.modetrans((vseq.n),X,Y).x - tseq.x.|| <= e * (NrProduct x);
      end;
      hence thesis;
    end;
    reconsider tseq as Lipschitzian MultilinearOperator of X,Y by A46;
    reconsider tv = tseq as Point of
    R_NormSpace_of_BoundedMultilinearOperators(X,Y) by LOPBAN10:def 11;
    A77: 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
      A78: e > 0;
      consider k be Nat such that
      A79: for n be Nat st n >= k holds for x be Point of product X
           holds ||.modetrans((vseq.n),X,Y).x - tseq.x.||
            <= e * (NrProduct x) by A59,A78;
      now
        set g1 = tseq;
        let n be Nat such that
        A80: n >= k;
        reconsider h1 = vseq.n - tv as Lipschitzian
        MultilinearOperator of X,Y by LOPBAN10:def 11;
        set f1 = modetrans((vseq.n),X,Y);
        A81: now
          let t be Point of product X;
          assume for i be Element of dom X holds
          ||.t.i.|| <= 1; then
          0 <= NrProduct t & NrProduct t <= 1 by LOPBAN10:35; then
          A82: e * (NrProduct t) <= e * 1 by A78,XREAL_1:64;
          A83: ||.f1.t-g1.t.|| <= e * (NrProduct t) by A79,A80;
          vseq.n is Lipschitzian MultilinearOperator of X,Y
            by LOPBAN10:def 11; then
          ||.h1.t.|| = ||.f1.t-g1.t.|| by LOPBAN10:52;
          hence ||.h1.t.|| <= e by A82,A83,XXREAL_0:2;
        end;
        A84: now
          let r be Real;
          assume r in PreNorms(h1); then
          ex t be Point of product X
          st r = ||.h1.t.||
           & for i be Element of dom X holds ||.t.i.|| <= 1;
          hence r <= e by A81;
        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 A84,LOPBAN10:43;
      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
      A86: e > 0;
      consider m be Nat such that
      A87: for n be Nat st n >= m holds
           ||.(vseq.n) - tv.|| <= e/2 by A77,A86;
      A88: e/2 < e by A86,XREAL_1:216;
      now
        let n be Nat;
        assume n >= m; then
        ||.(vseq.n) - tv.|| <= e/2 by A87;
        hence ||.(vseq.n) - tv.|| < e by A88,XXREAL_0:2;
      end;
      hence thesis;
    end;
    hence thesis by NORMSP_1:def 6;
  end;
