reserve n for Nat;

theorem Th12:
  for f be sequence of REAL-NS n st f is Cauchy_sequence_by_Norm
  holds f is convergent
proof
  let vseq be sequence of REAL-NS n such that
A1: vseq is Cauchy_sequence_by_Norm;
  reconsider xvseq=vseq as sequence of REAL n by Def4;
  defpred P[set, set] means ex rseqi be Real_Sequence st
   for l be Nat holds rseqi.l = (xvseq.l).$1 & rseqi is convergent &
      $2 = lim rseqi;
A2: for i be Nat st i in Seg n ex y be Element of REAL st P[i,y]
  proof
    let i be Nat such that
A3: i in Seg n;
    deffunc F(Nat) = (xvseq.$1).i;
    consider rseqi be Real_Sequence such that
A4: for l be Nat holds rseqi.l = F(l) from SEQ_1:sch 1;
     reconsider lr = lim rseqi as Element of REAL by XREAL_0:def 1;
    take lr;
    now
      let e be Real such that
A5:   e > 0;
      thus ex k be Nat st for m be Nat st k<=m holds
      |.rseqi.m -rseqi.k.| < e
      proof
        consider k be Nat such that
A6:     for n, m be Nat st n >= k & m >= k holds ||.(vseq.n
        ) - (vseq.m).|| < e by A1,A5,RSSPACE3:8;
        take k;
        let m be Nat;
        assume k<=m;
        then
A7:     ||.(vseq.m) - (vseq.k).|| < e by A6;
        len ((xvseq.m) - (xvseq.k)) =n by CARD_1:def 7;
        then i in dom ((xvseq.m) - (xvseq.k)) by A3,FINSEQ_1:def 3;
        then (xvseq.m).i-(xvseq.k).i = ((xvseq.m) - (xvseq.k)).i by VALUED_1:13
;
        then
A8:     |.(xvseq.m).i-(xvseq.k).i.| <= ||.(vseq.m) - (vseq.k).|| by A3,Th5,Th9;
        rseqi.m=(xvseq.m).i & rseqi.k=(xvseq.k).i by A4;
        hence thesis by A7,A8,XXREAL_0:2;
      end;
    end;
    then rseqi is convergent by SEQ_4:41;
    hence thesis by A4;
  end;
  consider f be FinSequence of REAL such that
A9: dom f = Seg n & for k be Nat st k in Seg n holds P[k,f.k] from
  FINSEQ_1:sch 5(A2);
  reconsider tseq=f as Element of REAL n by A9,Th6;
  reconsider xseq=tseq as Point of REAL-NS n by Def4;
A10: xseq=tseq;
  for k be Nat st k in Seg n holds P[k,f.k] by A9;
  hence thesis by A10,Th11;
end;
