reserve X for non empty set;
reserve x for Element of X;
reserve d1,d2 for Element of X;
reserve A for BinOp of X;
reserve M for Function of [:X,X:],X;
reserve V for Ring;
reserve V1 for Subset of V;
reserve V for Algebra;
reserve V1 for Subset of V;
reserve MR for Function of [:REAL,X:],X;
reserve a for Real;
reserve F,G,H for VECTOR of R_Algebra_of_BoundedFunctions X;
reserve f,g,h for Function of X,REAL;
reserve F,G,H for Point of R_Normed_Algebra_of_BoundedFunctions X;

theorem Th35:
  for X be non empty set for seq be sequence of
R_Normed_Algebra_of_BoundedFunctions X st seq is Cauchy_sequence_by_Norm holds
  seq is convergent
proof
  let X be non empty set;
  let vseq be sequence of R_Normed_Algebra_of_BoundedFunctions X;
  defpred P[set,set] means ex xseq be Real_Sequence st
   (for n be Nat holds xseq.n=modetrans(vseq.n,X).$1) &
   xseq is convergent & $2= lim xseq;
  assume
A1: vseq is Cauchy_sequence_by_Norm;
A2: for x be Element of X ex y be Element of REAL st P[x,y]
  proof
    let x be Element of X;
    deffunc F(Nat) = modetrans(vseq.$1,X).x;
    consider xseq be Real_Sequence such that
A3: for n be Element of NAT holds xseq.n = F(n) from FUNCT_2:sch 4;
A4: for n be Nat holds xseq.n = F(n)
     proof let n be Nat;
       n in NAT by ORDINAL1:def 12;
      hence thesis by A3;
     end;
     reconsider lx = lim xseq as Element of REAL by XREAL_0:def 1;
    take lx;
A5: for m,k be Nat holds |.xseq.m-xseq.k.| <= ||.vseq.m - vseq.k.||
    proof
      let m,k be Nat;
      vseq.m-vseq.k in BoundedFunctions X;
      then consider h1 be Function of X,REAL such that
A6:   h1=vseq.m-vseq.k and
A7:   h1|X is bounded;
      vseq.m in BoundedFunctions X;
      then
      ex vseqm be Function of X,REAL st vseq.m=vseqm & vseqm|X is bounded;
      then
A8:   modetrans(vseq.m,X)=vseq.m by Th19;
      vseq.k in BoundedFunctions X;
      then
      ex vseqk be Function of X,REAL st vseq.k=vseqk & vseqk|X is bounded;
      then
A9:   modetrans(vseq.k,X)=vseq.k by Th19;
      xseq.m =modetrans(vseq.m,X).x & xseq.k =modetrans(vseq.k,X).x by A4;
      then xseq.m - xseq.k = h1.x by A8,A9,A6,Th34;
      hence thesis by A6,A7,Th26;
    end;
    now
      let e be Real such that
A10:   e > 0;
      consider k be Nat such that
A11:  for n, m be Nat st n >= k & m >= k holds ||.(vseq.n)
      - (vseq.m).|| < e by A1,A10,RSSPACE3:8;
      take k;
        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;
    then xseq is convergent by SEQ_4:41;
    hence thesis by A4;
  end;
  consider tseq be Function of X,REAL such that
A13: for x be Element of X holds P[x,tseq.x] from FUNCT_2:sch 3(A2);
  now
    let e1 be Real such that
A14: e1 >0;
    reconsider e =e1 as Real;
    consider k be Nat such that
A15: for n,m be Nat st n >= k & m >= k holds ||. vseq.n -
    vseq.m .|| < e by A1,A14,RSSPACE3:8;
    take k;
      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 NORMSP_0:def 4,NORMSP_1:9;
      hence |. ||.vseq.||.m - ||.vseq.||.k .| < e1 by A17,A16,XXREAL_0:2;
 end;
  then
A18: ||.vseq.|| is convergent by SEQ_4:41;
  now
    let x be object;
    assume
A19: x in X /\ dom tseq;
    then consider xseq be Real_Sequence such that
A20: for n be Nat holds xseq.n=modetrans(vseq.n,X).x and
A21: xseq is convergent & tseq.x = lim xseq by A13;
A22: for n be Nat holds abs(xseq).n <= ||.vseq.|| .n
    proof
      let n be Nat;
A23:  xseq.n =modetrans(vseq.n,X).x by A20;
      vseq.n in BoundedFunctions X;
      then
A24:  ex h1 be Function of X,REAL st vseq.n=h1 & h1|X is bounded;
      then modetrans(vseq.n,X)=vseq.n by Th19;
      then |.xseq.n.| <= ||.vseq.n.|| by A19,A24,A23,Th26;
      then abs(xseq).n <= ||.vseq.n.|| by VALUED_1:18;
      hence thesis by NORMSP_0:def 4;
    end;
    abs xseq is convergent & |.tseq.x.| = lim abs(xseq) by A21,SEQ_4:14;
    hence |.tseq.x.| <= lim ||.vseq.|| by A18,A22,SEQ_2:18;
  end;
  then tseq|X is bounded by RFUNCT_1:73;
  then tseq in BoundedFunctions X;
  then reconsider tv=tseq as Point of R_Normed_Algebra_of_BoundedFunctions X;
A25: for e be Real st e > 0
  ex k be Nat st
   for n be Nat st n >= k
     for x be Element of X holds |.modetrans(vseq.n,X).x -
  tseq.x.| <= e
  proof
    let e be Real;
    assume e > 0;
    then consider k be Nat such that
A26: for n,m be Nat st n >= k & m >= k holds ||. vseq.n -
    vseq.m .|| < e by A1,RSSPACE3:8;
    take k;
      let n be Nat such that
A27:  n >= k;
      now
        let x be Element of X;
        consider xseq be Real_Sequence such that
A28:    for n be Nat holds xseq.n=modetrans(vseq.n,X).x and
A29:    xseq is convergent and
A30:    tseq.x = lim xseq by A13;
        reconsider nn=n as Element of NAT by ORDINAL1:def 12;
        set fseq = seq_const xseq.n;
        set wseq = xseq-fseq;
        deffunc F(Nat) = |.xseq.$1 - xseq.n.|;
        consider rseq be Real_Sequence such that
A31:    for m be Nat holds rseq.m = F(m) from SEQ_1:sch 1;
A32:    for m,k be Nat holds |.xseq.m-xseq.k.| <= ||.vseq.m -
        vseq.k.||
        proof
          let m,k be Nat;
          vseq.m-vseq.k in BoundedFunctions X;
          then consider h1 be Function of X,REAL such that
A33:      h1 =vseq.m-vseq.k and
A34:      h1|X is bounded;
          vseq.m in BoundedFunctions X;
          then ex vseqm be Function of X,REAL st vseq.m=vseqm & vseqm|X is
          bounded;
          then
A35:      modetrans(vseq.m,X) = vseq.m by Th19;
          vseq.k in BoundedFunctions X;
          then ex vseqk be Function of X,REAL st vseq.k=vseqk & vseqk|X is
          bounded;
          then
A36:      modetrans(vseq.k,X)=vseq.k by Th19;
          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,Th34;
          hence thesis by A33,A34,Th26;
        end;
A37:    for m be 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 thesis by A38,XXREAL_0:2;
        end;
A39:    now
          let m be Element of NAT;
          wseq.m = xseq.m +(-fseq).m by SEQ_1:7;
          then wseq.m = xseq.m +-fseq.m by SEQ_1:10;
          hence wseq.m = xseq.m - xseq.n by SEQ_1:57;
        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 = |.wseq.k.| by A39;
          hence rseq.x = abs(wseq).x by VALUED_1:18;
        end;
        then
A40:    rseq = abs(wseq) by FUNCT_2:12;
A41:    wseq is convergent by A29;
        then rseq is convergent by A40;
        then
A42:    lim rseq <= e by A37,RSSPACE2:5;
        lim fseq = fseq.0 by SEQ_4:26;
        then lim fseq =xseq.n by SEQ_1:57;
        then lim wseq= tseq.x - xseq.n by A29,A30,SEQ_2:12;
        then lim rseq = |.tseq.x-xseq.n.| by A41,A40,SEQ_4:14;
        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 be Element of X holds |.modetrans(vseq.n,X).x - tseq.x.| <=
      e;
  end;
A43: for e be Real st e > 0
  ex k be Element of NAT st for n be Element of NAT
  st n >= k holds ||.vseq.n - tv.|| <= e
  proof
    let e be Real;
    assume e > 0;
    then consider k be Nat such that
A44: for n be Nat st n >= k holds for x be Element of X
    holds |.modetrans((vseq.n),X).x - tseq.x.| <= e by A25;
     reconsider k as Element of NAT by ORDINAL1:def 12;
    take k;
    hereby
      let n be Element of NAT such that
A45:  n >= k;
      vseq.n in BoundedFunctions X;
      then consider f1 be Function of X,REAL such that
A46:  f1=vseq.n and
      f1|X is bounded;
      vseq.n-tv in BoundedFunctions X;
      then consider h1 be Function of X,REAL 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,Def15,Th34;
        hence |.h1.t.| <=e by A44,A45;
      end;
A50:  now
        let r be Real;
        assume r in PreNorms h1;
        then ex t be Element of X st r=|.h1.t.|;
        hence r <=e by A49;
      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 A47,A48,A50,Th20;
    end;
  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
A51: e > 0;
    reconsider ee=e as Real;
    consider m be Element of NAT such that
A52: for n be Element of NAT st n >= m holds ||.(vseq.n) - tv.|| <= ee
    /2 by A43,A51,XREAL_1:215;
    take m;
A53: e/2<e by A51,XREAL_1:216;
      let n be Nat;
A54:    n in NAT by ORDINAL1:def 12;
      assume n >= m;
      then ||.(vseq.n) - tv.|| <= e/2 by A52,A54;
      hence ||.(vseq.n) - tv.|| < e by A53,XXREAL_0:2;
  end;
  hence thesis by NORMSP_1:def 6;
end;
