reserve n for Nat;

theorem Th11:
  for n be Nat for x be Point of REAL-NS n, xs be
Element of REAL n, seq be sequence of REAL-NS n, xseq be sequence of  REAL
  n st xs = x & xseq = seq holds ( seq is convergent & lim seq = x iff
   for i be Nat st i in Seg n ex rseqi be Real_Sequence st
    for k be Nat
    holds rseqi.k = (xseq.k).i & rseqi is convergent & xs.i = lim rseqi )
proof
  defpred P[Nat] means
   for x be Point of REAL-NS $1, xs be Element of REAL $1,
     seq be sequence of REAL-NS $1, xseq be sequence of  REAL $1 st
                 xs= x & xseq=seq
   holds ((seq is convergent & lim seq = x) iff
     (for i be Nat st i in Seg $1 ex rseqi be Real_Sequence st for k be Nat
  holds rseqi.k = (xseq.k).i & rseqi is convergent & xs.i= lim rseqi));
A1: now
    let n be Nat;
    assume
A2: P[n];
    now
      let x be Point of REAL-NS (n+1), xs be Element of REAL (n+1), seq be
      sequence of REAL-NS (n+1), xseq be sequence of  REAL (n+1);
      assume
A3:   xs= x & xseq=seq;
A4:   now
        assume
A5:     for i be Nat st i in Seg (n+1) ex rseqi be
Real_Sequence st for k be Nat holds rseqi.k = (xseq.k).i & rseqi is
        convergent & xs.i= lim rseqi;
        thus seq is convergent & lim seq = x
        proof
          len xs= n+1 by CARD_1:def 7;
          then len (xs|n) = n by FINSEQ_1:59,NAT_1:11;
          then dom(xs|n) = Seg n by FINSEQ_1:def 3;
          then reconsider xsn=xs|n as Element of REAL n by Th6;
          reconsider xn=xsn as Point of REAL-NS n by Def4;
          defpred P1[Nat,Element of REAL n] means $2=(xseq.$1) |n;
          set seq2 = ||.seq - x.|| (#) ||.seq - x.||;
          consider rseqn1 be Real_Sequence such that
A6:       for k be Nat holds rseqn1.k = (xseq.k).(n+1) &
          rseqn1 is convergent & xs.(n+1)= lim rseqn1 by A5,FINSEQ_1:4;
A7:       for i be Element of NAT ex y be Element of REAL n st P1[i ,y]
          proof
           let i be Element of NAT;
           take y=(xseq.i) |n;
            len (xseq.i)= n+1 by CARD_1:def 7;
            then len (xseq.i|n) = n by FINSEQ_1:59,NAT_1:11;
            then dom(xseq.i|n) = Seg n by FINSEQ_1:def 3;
           hence thesis by Th6;
          end;
          consider xseqn be sequence of REAL n such that
A8:       for i be Element of NAT holds P1[i,xseqn.i] from FUNCT_2:sch 3(A7);
          reconsider seqn=xseqn as sequence of REAL-NS n by Def4;
          set seqn2 = ||.seqn - xn.|| (#) ||.seqn - xn.||;
          deffunc F(Nat)=|.rseqn1.$1 -xs.(n+1).|;
          consider absrseq be Real_Sequence such that
A9:       for i be Nat holds absrseq.i =F(i) from SEQ_1:
          sch 1;
A10:      for i be Nat st i in Seg n ex rseqi be Real_Sequence
st for k be Nat holds rseqi.k = (xseqn.k).i & rseqi is convergent &
          xsn.i= lim rseqi
          proof
            let i be Nat such that
A11:        i in Seg n;
            n <= n+1 by NAT_1:11;
            then Seg n c= Seg (n+1) by FINSEQ_1:5;
            then consider rseqi be Real_Sequence such that
A12:        for k be Nat holds rseqi.k = (xseq.k).i &
            rseqi is convergent & xs.i= lim rseqi by A5,A11;
A13:        now
              let k be Nat;
A14:         k in NAT by ORDINAL1:def 12;
              thus rseqi.k=(xseq.k).i by A12
                .=((xseq.k) |Seg n).i by A11,FUNCT_1:49
                .=((xseq.k) |n).i by FINSEQ_1:def 16
                .=(xseqn.k).i by A8,A14;
            end;
            xsn.i = (xs|Seg n).i by FINSEQ_1:def 16
              .=xs.i by A11,FUNCT_1:49;
            hence thesis by A12,A13;
          end;
          then
A15:      xn=lim seqn by A2;
          set rseqn2 =absrseq (#) absrseq;
          xsn= xn;
          then
A16:      seqn is convergent by A2,A10;
          then
A17:      ||.seqn - xn.|| is convergent by A15,NORMSP_1:24;
          then
A18:      seqn2 is convergent by SEQ_2:14;
          now
            reconsider rxs= xs as Element of (n+1)-tuples_on REAL by
EUCLID:def 1;
            let k be Nat;
A19:         k in NAT by ORDINAL1:def 12;
A20:        ||.seq - x.||.k = ||.(seq - x).k.|| by NORMSP_0:def 4
              .= ||.seq.k - x.|| by NORMSP_1:def 4;
            reconsider rxseqk= (xseq.k) as Element of (n+1)-tuples_on REAL by
EUCLID:def 1;
A21:        ||.seqn - xn.||.k = ||.(seqn - xn).k.|| by NORMSP_0:def 4
              .= ||.seqn.k - xn.|| by NORMSP_1:def 4;
            len (xseqn.k -xsn) = n by CARD_1:def 7;
            then
A22:        dom(xseqn.k -xsn) = Seg n by FINSEQ_1:def 3;
A23:        (xseq.k-xs).(n+1) = rxseqk.(n+1)-rxs.(n+1) by RVSUM_1:27
              .= rseqn1.k - xs.(n+1) by A6;
            len (xseq.k-xs) = n+1 by CARD_1:def 7;
            then
A24:        len ((xseq.k-xs) |n) = n by FINSEQ_1:59,NAT_1:11;
A25:        now
              reconsider xseq2 = xseqn.k, xs2 = xsn as Element of n-tuples_on
              REAL by EUCLID:def 1;
              reconsider xseq1 = xseq.k, xs1 = xs as Element of (n+1)
              -tuples_on REAL by EUCLID:def 1;
              let i be Nat;
              assume i in dom ((xseq.k-xs) |n);
              then
A26:          i in Seg n by A24,FINSEQ_1:def 3;
A27:          (xseqn.k-xsn).i = xseq2.i-xs2.i by RVSUM_1:27;
A28:          (xseq.k-xs).i = xseq1.i-xs1.i by RVSUM_1:27;
              thus ((xseq.k-xs) |n).i =((xseq.k-xs) |Seg n).i
              by FINSEQ_1:def 16
                .= (xseq.k-xs).i by A26,FUNCT_1:49
                .= ((xseq.k) |Seg n).i - xs.i by A26,A28,FUNCT_1:49
                .= ((xseq.k) |Seg n).i - (xs|Seg n).i by A26,FUNCT_1:49
                .= ((xseq.k) |n).i- (xs|Seg n).i by FINSEQ_1:def 16
                .= ((xseq.k) |n).i-(xs|n).i by FINSEQ_1:def 16
                .= (xseqn.k -xsn).i by A8,A27,A19;
            end;
            dom ((xseq.k-xs) |n) = Seg n by A24,FINSEQ_1:def 3;
            then
A29:        (xseq.k-xs) |n=xseqn.k -xsn by A22,A25,FINSEQ_1:13;
A30:        0<= (rseqn1.k - xs.(n+1))^2 by XREAL_1:63;
A31:        absrseq.k = |.rseqn1.k - xs.(n+1).| by A9;
            ||.seq.k - x.|| = |.xseq.k-xs .| by A3,Th1,Th5;
            hence seq2.k = |.xseq.k-xs .|^2 by A20,SEQ_1:8
              .= |.xseqn.k-xsn .|^2 +(rseqn1.k - xs.(n+1))^2 by A23,A29,Th10
              .= ||. seqn.k-xn .||^2 +(rseqn1.k - xs.(n+1))^2 by Th1,Th5
              .= ( ||.seqn - xn.|| (#) ||.seqn - xn.||).k + (rseqn1.k - xs.(
            n+1))^2 by A21,SEQ_1:8
              .= seqn2.k + |.(rseqn1.k - xs.(n+1))*(rseqn1.k - xs.(n+1)).|
            by A30,ABSVALUE:def 1
              .= seqn2.k + |.rseqn1.k - xs.(n+1).| *|.rseqn1.k - xs.(n+1)
            .| by COMPLEX1:65
              .= seqn2.k + rseqn2.k by A31,SEQ_1:8;
          end;
          then
A32:      seq2=seqn2+rseqn2 by SEQ_1:7;
A33:      now
            let e be Real;
            assume e > 0;
            then consider m be Nat such that
A34:        for k be Nat st m <= k holds |.rseqn1.k -xs.
            (n+1).| < e by A6,SEQ_2:def 7;
            now
              let k be Nat;
              assume m <= k;
              then |.|.rseqn1.k -xs.(n+1).|-0 .| < e by A34;
              hence |.absrseq.k-0 .| < e by A9;
            end;
            hence ex m be Nat st for k be Nat st m <= k
            holds |.absrseq.k-0 .| < e;
          end;
          then
A35:      absrseq is convergent by SEQ_2:def 6;
          then lim absrseq = 0 by A33,SEQ_2:def 7;
          then
A36:      lim rseqn2 = 0 * 0 by A35,SEQ_2:15
            .=0;
A37:      rseqn2 is convergent by A35,SEQ_2:14;
          then
A38:      seq2 is convergent by A18,A32,SEQ_2:5;
          lim ||.seqn - xn.|| = 0 by A16,A15,NORMSP_1:24;
          then lim seqn2 = 0 * 0 by A17,SEQ_2:15
            .= 0;
          then
A39:      lim seq2 = 0+ 0 by A18,A37,A36,A32,SEQ_2:6
            .= 0;
A40:      for e be Real st e > 0 ex m be Nat st for k be
          Nat st k >= m holds ||.seq.k - x.|| < e
          proof
            let e be Real such that
A41:        e > 0;
            e*0 < e*e by A41,XREAL_1:97;
            then consider m be Nat such that
A42:        for k be Nat st m<=k holds |.seq2.k-0 .| < e*e
            by A38,A39,SEQ_2:def 7;
            now
              let k be Nat such that
A43:          m<= k;
              ||.seq - x.||.k = ||.(seq - x).k.|| by NORMSP_0:def 4
                .= ||.seq.k - x.|| by NORMSP_1:def 4;
              then seq2.k = ( ||.seq.k - x.|| )*( ||.seq.k - x.|| ) by SEQ_1:8;
              then
|.seq2.k-0 .| = ||.seq.k - x.|| * ||.seq.k - x.|| by ABSVALUE:def 1;
              then
A44:          ||.seq.k - x.|| * ||.seq.k - x.|| < e*e by A42,A43;
A45:          sqrt( ||.seq.k - x.|| * ||.seq.k - x.|| ) = sqrt( ||.seq.k
              - x.||^2)
                .= ||.seq.k - x.|| by SQUARE_1:22;
              sqrt(e*e) = sqrt(e^2) .= e by A41,SQUARE_1:22;
              hence ||.seq.k - x.|| < e by A44,A45,SQUARE_1:27;
            end;
            hence thesis;
          end;
          then seq is convergent by NORMSP_1:def 6;
          hence thesis by A40,NORMSP_1:def 7;
        end;
      end;
      now
        assume
A46:    seq is convergent & lim seq = x;
        now
          let i be Nat such that
A47:      i in Seg (n+1);
          deffunc F(Nat) = (xseq.$1).i;
          consider rseqi be Real_Sequence such that
A48:      for l be Nat holds rseqi.l = F(l) from SEQ_1:sch
          1;
A49:      now
            let e be Real such that
A50:        e > 0;
            thus ex k be Nat st for m be Nat st k<=m
            holds |.rseqi.m-xs.i.| < e
            proof
              consider k be Nat such that
A51:          for m be Nat st m >= k holds ||.(seq.m) - x
              .|| < e by A46,A50,NORMSP_1:def 7;
              take k;
              let m be Nat;
              assume k<=m;
              then
A52:          ||.(seq.m) - x.|| < e by A51;
              len ((xseq.m) - xs) = (n+1) by CARD_1:def 7;
              then i in dom ((xseq.m) - xs) by A47,FINSEQ_1:def 3;
              then (xseq.m).i-xs.i = ((xseq.m) - xs).i by VALUED_1:13;
              then
A53:          |.(xseq.m).i-xs.i.| <= ||.(seq.m) - x.|| by A3,A47,Th5,Th9;
              rseqi.m-xs.i=(xseq.m).i-xs.i by A48;
              hence thesis by A52,A53,XXREAL_0:2;
            end;
          end;
          then
A54:      rseqi is convergent by SEQ_2:def 6;
          then xs.i= lim rseqi by A49,SEQ_2:def 7;
          hence ex rseqi be Real_Sequence st for k be Nat holds
rseqi.k = (xseq.k).i & rseqi is convergent & xs.i= lim rseqi by A48,A54;
        end;
        hence for i be Nat st i in Seg (n+1) ex rseqi be
Real_Sequence st for k be Nat holds rseqi.k = (xseq.k).i & rseqi is
        convergent & xs.i= lim rseqi;
      end;
      hence
      seq is convergent & lim seq = x iff for i be Nat st i in
Seg (n+1) ex rseqi be Real_Sequence st for k be Nat holds rseqi.k =
      (xseq.k).i & rseqi is convergent & xs.i= lim rseqi by A4;
    end;
    hence P[n+1];
  end;
A55: P[0]
  proof
    let x be Point of REAL-NS 0, xs be Element of REAL 0, seq be sequence of
    REAL-NS 0, xseq be sequence of REAL 0;
    assume that
A56: xs= x and
A57: xseq=seq;
    now
      assume for i be Nat st i in Seg 0 ex rseqi be Real_Sequence st
for k be Nat holds rseqi.k = (xseq.k).i & rseqi is convergent & xs.i
      = lim rseqi;
A58:  for i be Nat holds seq.i=0.(REAL-NS 0)
      proof
        let i be Nat;
        xseq.i = 0.REAL 0;
        hence thesis by A57,Def4;
      end;
      xs = 0*0;
      then
A59:  x=0.(REAL-NS 0) by A56,Def4;
A60:  for r be Real st 0 < r ex m be Nat st
          for k be Nat st m <= k holds ||. seq.k - x .|| < r
      proof
        let r be Real;
        assume
A61:    0 < r;
        take m = 1;
        let k be Nat;
        assume m <= k;
        ||. seq.k - x .|| = ||. 0.(REAL-NS 0) - 0.(REAL-NS 0) .|| by A59,A58
          .= ||. 0.(REAL-NS 0) .|| by RLVECT_1:15
          .= 0;
        hence thesis by A61;
      end;
      then seq is convergent by NORMSP_1:def 6;
      hence seq is convergent & lim seq = x by A60,NORMSP_1:def 7;
    end;
    hence thesis;
  end;
  thus for n be Nat holds P[n] from NAT_1:sch 2(A55,A1);
end;
