
theorem Th27:
  for X being non empty set,
      seq being sequence of C_Normed_Algebra_of_BoundedFunctions(X)
                    st seq is CCauchy holds seq is convergent
proof
  let X be non empty set,
      vseq be sequence of C_Normed_Algebra_of_BoundedFunctions(X);
  defpred S1[set, set] means ex xseq being Complex_Sequence st
  (for n being Nat holds xseq.n=modetrans(vseq.n,X).$1
          & xseq is convergent & $2 = lim xseq );
  assume
A1: vseq is CCauchy;
A2:for x being Element of X ex y being Element of COMPLEX st S1[x,y]
  proof
    let x be Element of X;
    deffunc H1(Nat)= modetrans((vseq.$1),X).x;
    consider xseq being Complex_Sequence such that
A3: for n being Element of NAT holds xseq.n=H1(n) from FUNCT_2:sch 4;
A4: for n being Nat holds xseq.n=H1(n)
     proof let n be Nat;
       n in NAT by ORDINAL1:def 12;
      hence thesis by A3;
     end;
     reconsider y = lim xseq as Element of COMPLEX by XCMPLX_0:def 2;
    take y;
A5: for m, k being Nat holds
                    |.(xseq.m)-(xseq.k).|<=||.((vseq.m)-(vseq.k)).||
    proof
      let m, k be Nat;
A6:    m in NAT & k in NAT by ORDINAL1:def 12;
      (vseq.m) - (vseq.k) in ComplexBoundedFunctions X;
      then consider h1 being Function of X,COMPLEX such that
A7:   h1 = (vseq.m) - (vseq.k) and
A8:   h1 | X is bounded;
      vseq.m in ComplexBoundedFunctions X;
      then ex vseqm being Function of X,COMPLEX st
      vseq.m = vseqm & vseqm | X is bounded; then
A9:   modetrans((vseq.m),X) = vseq.m by Th12;
      vseq.k in ComplexBoundedFunctions X;
      then ex vseqk being Function of X,COMPLEX st
      vseq.k = vseqk & vseqk | X is bounded; then
A10:   modetrans ((vseq.k),X) = vseq.k by Th12;
      ( xseq.m = (modetrans((vseq.m),X)).x
                & xseq.k = modetrans( (vseq.k),X).x ) by A3,A6;
      then (xseq.m) - (xseq.k) = h1.x by A9,A10,A7,Th26;
      hence |.(xseq.m) - (xseq.k).|
                    <= ||.((vseq.m) - (vseq.k)).|| by A7,A8,Th19;
    end;
    now
     let e be Real;
     assume e > 0;
     then consider k being Nat such that
A11:     for n, m being Nat st n >= k & m >= k holds
             ||.((vseq.n) - (vseq.m)).|| < e by A1,CSSPACE3:8;
     reconsider k as Nat;
     take k;
     hereby
      let n be Nat;
      assume n >= k; then
A12:  ||.((vseq.n) - (vseq.k)).|| < e by A11;
      |.(xseq.n)-(xseq.k).|<=||.((vseq.n)-(vseq.k)).|| by A5;
      hence |.(xseq.n) - (xseq.k).| < e by A12,XXREAL_0:2;
     end;
    end;
    then xseq is convergent by COMSEQ_3:46;
    hence S1[x,y] by A4;
  end;
  consider tseq being Function of X,COMPLEX such that
A13: for x being Element of X holds S1[x,(tseq.x)] from FUNCT_2:sch 3(A2);
  now
   let e1 be Real;
   assume
A14: e1 > 0;
   reconsider e = e1 as Real;
   consider k being Nat such that
A15:        for n, m being Nat st n >= k & m >= k holds
            ||.((vseq.n) - (vseq.m)).|| < e by A1,A14,CSSPACE3:8;
   take k;
   now
    let m be Nat;
A16:||.(vseq.m).|| = ||.vseq.||.m by NORMSP_0:def 4;
    assume m >= k;
    then
A17:||.((vseq.m) - (vseq.k)).|| < e by A15;
    ( |.(||.(vseq.m).|| - ||.(vseq.k).||) .|
                             <= ||.((vseq.m) - (vseq.k)).||
         & ||.(vseq.k).|| = ||.vseq.||.k )
                                        by CLVECT_1:110,NORMSP_0:def 4;
    hence |.((||.vseq.||.m) - (||.vseq.||.k)).| < e1
                                        by A17,A16,XXREAL_0:2;
   end;
   hence for m being Nat st m >= k holds
                  |. ((||.vseq.||.m) - (||.vseq.||.k)) .| < e1;
  end; then
A18:  ||.vseq.|| is convergent by SEQ_4:41;
  now
   let x be set;
   assume
A19:  x in X /\ (dom tseq);
   then consider xseq being Complex_Sequence such that
A20: for n being Nat holds xseq.n=modetrans((vseq.n),X).x and
A21: ( xseq is convergent & tseq.x = lim xseq ) by A13;
A22: for n being Nat holds |.xseq.|.n <= ||.vseq.||.n
   proof
     let n be Nat;
A23: xseq.n = modetrans((vseq.n),X).x by A20;
     vseq.n in ComplexBoundedFunctions X; then
A24: ex h1 being Function of X,COMPLEX st
        vseq.n = h1 & h1 | X is bounded;
     then modetrans((vseq.n),X) = vseq.n by Th12;
     then |.(xseq.n).| <= ||.(vseq.n).|| by A19,A24,A23,Th19;
     then |.xseq.|.n <= ||.(vseq.n).|| by VALUED_1:18;
     hence |.xseq.|.n <= ||.vseq.||.n by NORMSP_0:def 4;
   end;
   ( |.xseq.| is convergent & |.(tseq.x).| = lim (|.xseq.| ))
                                         by A21,SEQ_2:27;
   hence |.(tseq.x).| <= lim ||.vseq.|| by A18,A22,SEQ_2:18;
  end; then
   for x be Element of X st x in  X /\ (dom tseq) holds
     |.tseq /.x.| <= lim ||.vseq.||;
  then tseq | X is bounded by CFUNCT_1:69;
  then tseq in ComplexBoundedFunctions X;
  then reconsider tv = tseq
                   as Point of C_Normed_Algebra_of_BoundedFunctions(X);
A25:for e being Real st e > 0
    ex k being Nat st
       for n being Nat st n >= k holds
       for x being Element of X holds
                |.(modetrans((vseq.n),X).x - (tseq.x)).| <= e
  proof
    let e be Real;
    assume e > 0;
    then consider k being Nat such that
A26: for n, m being Nat st n >= k & m >= k holds
      ||.((vseq.n) - (vseq.m)).|| < e by A1,CSSPACE3:8;
    take k;
    now
     let n be Nat;
     assume
A27: n >= k;
     now
      let x be Element of X;
      consider xseq being Complex_Sequence such that
A28:  for n being Nat holds xseq.n = modetrans((vseq.n),X).x and
A29:  xseq is convergent and
A30:  tseq.x = lim xseq by A13;
      reconsider xn = xseq.n as Element of COMPLEX by XCMPLX_0:def 2;
      reconsider fseq = NAT --> xn as Complex_Sequence;
      set wseq = xseq - fseq;
      deffunc H1(Nat) = |.((xseq.$1) - (xseq.n)).|;
      consider rseq being Real_Sequence such that
A31:  for m being Nat holds rseq.m = H1(m) from SEQ_1:sch 1;
A32:  for m, k being Nat
           holds |.((xseq.m) - (xseq.k)).|
                                <= ||.((vseq.m) - (vseq.k)).||
      proof
        let m, k be Nat;
        (vseq.m) - (vseq.k) in ComplexBoundedFunctions(X);
        then consider h1 being Function of X,COMPLEX such that
A33:    h1 = (vseq.m) - (vseq.k) and
A34:    h1 | X is bounded;
        vseq.m in ComplexBoundedFunctions X;
        then ex vseqm being Function of X,COMPLEX st
                     vseq.m = vseqm & vseqm | X is bounded; then
A35:    modetrans((vseq.m),X) = vseq.m by Th12;
        vseq.k in ComplexBoundedFunctions(X);
        then ex vseqk being Function of X,COMPLEX st
                     vseq.k = vseqk & vseqk | X is bounded; then
A36:    modetrans((vseq.k),X) = vseq.k by Th12;
        ( xseq.m = modetrans((vseq.m),X).x
                    & xseq.k = modetrans((vseq.k),X).x ) by A28;
        then (xseq.m) - (xseq.k) = h1.x by A35,A36,A33,Th26;
        hence |.((xseq.m) - (xseq.k)).|
                  <= ||.((vseq.m) - (vseq.k)).|| by A33,A34,Th19;
      end;
A37:  for m being Nat st m >= k holds rseq.m <= e
      proof
        let m be Nat;
        assume m >= k; then
A38:    ||.((vseq.n) - (vseq.m)).|| < e by A26,A27;
        rseq.m = |.((xseq.m) - (xseq.n)).| by A31;
        then rseq.m = |.((xseq.n)-(xseq.m)).| by COMPLEX1:60;
        then rseq.m <= ||.((vseq.n) - (vseq.m)).|| by A32;
        hence rseq.m <= e by A38,XXREAL_0:2;
      end;
A39:  now
       let m be Element of NAT;
       (xseq - fseq).m = xseq.m + (-fseq).m   by VALUED_1:1
                      .= xseq.m - fseq.m by VALUED_1:8;
       hence (xseq - fseq).m = (xseq.m) - (xseq.n);
      end;
      now
       let x be object;
       assume x in NAT;
       then reconsider k = x as Element of NAT;
       rseq.x = |.((xseq.k) - (xseq.n)).| by A31;
       then rseq.x = |.((xseq - fseq).k).| by A39;
       hence rseq.x = (|.(xseq - fseq).|).x by VALUED_1:18;
      end; then
A40:  rseq = |.(xseq - fseq).| by FUNCT_2:12;
A41:  fseq is convergent by CFCONT_1:26;
A42:  lim rseq <= e by A41,A37,A40,A29,RSSPACE2:5;
      lim fseq = fseq.0 by CFCONT_1:28;
      then lim fseq = xseq.n;
      then lim (xseq-fseq)=(tseq.x)-(xseq.n) by A29,A30,A41,COMSEQ_2:26;
      then lim rseq = |.((tseq.x)-(xseq.n)).| by A41,A29,A40,SEQ_2:27;
      then |.((xseq.n)-(tseq.x)).|<= e by A42,COMPLEX1:60;
      hence |.(modetrans((vseq.n),X).x - tseq.x).| <= e by A28;
     end;
     hence for x being Element of X
                 holds |.(modetrans((vseq.n),X).x-tseq.x).|<= e;
    end;
    hence for n being Nat st n >= k holds
      for x being Element of X
        holds |.(modetrans((vseq.n),X).x-tseq.x).|<=e;
  end;
A43:for e being Real st e > 0 holds
       ex k being Nat st
            for n being Nat st n >= k holds ||.((vseq.n) - tv).|| <= e
  proof
    let e be Real;
    assume e > 0;
    then consider k being Nat such that
A44:for n being Nat st n >= k holds
           for x being Element of X holds
             |.(modetrans((vseq.n),X).x-tseq.x).|<= e by A25;
    take k;
    hereby
     let n be Nat;
     assume
A45:    n >= k;
     vseq.n in ComplexBoundedFunctions X;
     then consider f1 being Function of X,COMPLEX such that
A46:    f1 = vseq.n and
        f1 | X is bounded;
     (vseq.n) - tv in ComplexBoundedFunctions X;
     then consider h1 being Function of X,COMPLEX such that
A47:    h1 = (vseq.n) - tv and
A48:    h1 | X is bounded;
A49: now
      let t be Element of X;
      modetrans(vseq.n,X) = f1 & h1.t=(f1.t)-(tseq.t)
        by A46,A47,Def7,Th26;
      hence |.(h1.t).| <= e by A44,A45;
     end;
A50: now
      let r be Real;
      assume r in PreNorms h1;
      then ex t being Element of X st r = |.(h1.t).|;
      hence r <= e by A49;
     end;
     (for s being 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 A47,A48,A50,Th13;
    end;
  end;
  for e being Real st e > 0 holds ex m being Nat st
    for n being Nat st n >= m holds
         ||.((vseq.n) - tv).|| < e
  proof
    let e be Real;
    assume
A51:  e > 0;
    consider m being Nat such that
A52:for n being Nat st n >= m holds
                         ||.((vseq.n)-tv).||<= e / 2 by A43,A51;
    take m;
A53:e / 2 < e by A51,XREAL_1:216;
    hereby let n be Nat;
     assume n >= m;
     then ||.((vseq.n) - tv).|| <= e / 2 by A52;
     hence ||.((vseq.n) - tv).|| < e by A53,XXREAL_0:2;
    end;
  end;
  hence thesis;
end;
