
theorem Th8:
  for X,Y be RealNormSpace,
      seqx be sequence of X,
      seqy be sequence of Y,
      x be Point of X,
      y be Point of Y holds
   ( seqx is convergent & lim seqx = x
            & seqy is convergent & lim seqy = y )
   iff
   <:seqx,seqy:> is convergent & lim <:seqx,seqy:> = [x,y]
proof
 let X,Y be RealNormSpace,
     seqx be sequence of X,
     seqy be sequence of Y,
     x be Point of X,
     y be Point of Y;
  set seq = <:seqx,seqy:>;
  set v = [x,y];
hereby assume A1:
    seqx is convergent & lim seqx = x & seqy is convergent & lim seqy = y;
  A2: for r be Real st 0 < r ex m be Nat
        st for n be Nat st m <= n holds ||.(seq.n) - v.|| < r
  proof
   let r1 be Real;
    assume A3: 0 < r1;
    set r=r1/2;
    A4: 0 < r & r < r1 by A3,XREAL_1:215,216;
    set r2=r/2;
    A5:0 < r2 & r2 < r by A4,XREAL_1:215,216;
    then consider m1 be Nat such that
    A6: for n be Nat
           st m1 <= n holds ||.(seqx.n) - x .|| < r2 by A1,NORMSP_1:def 7;
     consider m2 be Nat such that
     A7: for n be Nat
           st m2 <= n holds ||.(seqy.n) - y .|| < r2 by A1,A5,NORMSP_1:def 7;
     reconsider m=max(m1,m2) as Nat by TARSKI:1;
     take m;
     let n be Nat;
       assume A8: m <= n;
       m1 <= m by XXREAL_0:25; then
A9:   m1<= n by A8,XXREAL_0:2;
       m2 <= m by XXREAL_0:25; then
A10:   m2<= n by A8,XXREAL_0:2;
       n in NAT by ORDINAL1:def 12; then
  A11:  [seqx.n,seqy.n] = seq.n by FUNCT_3:59;
  A12:  - v = [-x, -y] by PRVECT_3:18;
       (seq.n) - v = [(seqx.n)-x,(seqy.n)-y] by A11,A12,PRVECT_3:18;
      then consider w be Element of REAL 2 such that
    A13: w = <* ||. (seqx.n)-x .||,||. (seqy.n)-y .|| *>
          & ||. (seq.n) - v .|| = |.w.| by PRVECT_3:18;
      now let i be Element of NAT;
       assume 1 <= i & i <= 2; then
    A14: i in Seg 2 by FINSEQ_1:1;
       per cases by A14,FINSEQ_1:2,TARSKI:def 2;
          suppose A15: i=1;
           A16:(proj (i,2)).w = w.1 by A15,PDIFF_1:def 1
                             .= ||. (seqx.n)-x .|| by A13,FINSEQ_1:44;
           |. (proj (i,2)).w .| = ||. (seqx.n)-x .||
                                by ABSVALUE:def 1, A16;
           hence |. (proj (i,2)).w .| <= r2 by A9, A6;
          end;
         suppose i=2; then
           A17:(proj (i,2)).w = w.2 by PDIFF_1:def 1
                  .= ||. (seqy.n)-y .|| by A13,FINSEQ_1:44;
           |. (proj (i,2)).w .| = ||. (seqy.n)-y .||
                                  by ABSVALUE:def 1, A17;
           hence |. (proj (i,2)).w .| <= r2 by A10,A7;
          end;
      end;
   then |.w.| <= 2*(r2) by PDIFF_8:17;
    hence ||. (seq.n) - v .|| < r1 by A13,A4,XXREAL_0:2;
  end;
   hence seq is convergent;
   hence lim seq = [x, y] by A2,NORMSP_1:def 7;
end;

assume
A18:seq is convergent & lim seq = [x,y];
   A19: for r be Real st 0 < r ex m be Nat
        st for n be Nat st m <= n holds
            ||.(seqx.n) - x.|| < r & ||. (seqy.n) - y.|| < r
  proof
   let r be Real;
    assume 0 < r;
    then consider m be Nat such that
A20: for n be Nat st m <= n holds ||.(seq.n) - v.|| < r by A18,NORMSP_1:def 7;
    take m;
    let n be Nat;
    assume m <= n; then
A21: ||.(seq.n) - v .|| < r by A20;
    n in NAT by ORDINAL1:def 12; then
A22: [seqx.n,seqy.n ] = seq.n by FUNCT_3:59;
A23: - v = [-x,-y] by PRVECT_3:18;
         (seq.n) - v = [(seqx.n)-x,(seqy.n)-y] by A22,A23,PRVECT_3:18;
    then consider w be Element of REAL 2 such that
A24: w = <* ||. (seqx.n)-x .||,||. (seqy.n)-y .|| *>
          & ||. (seq.n) - v .|| = |.w.| by PRVECT_3:18;
    (proj (1,2)).w = w.1 by PDIFF_1:def 1
                 .= ||. (seqx.n)-x .|| by A24,FINSEQ_1:44;
    then |. ||. (seqx.n)-x .|| .| <= |.w.| by PDIFF_8:5;
    then ||. (seqx.n)-x .|| <= |.w.| by ABSVALUE:def 1;
    hence ||.(seqx.n)-x.|| < r by A24,A21,XXREAL_0:2;
    (proj (2,2)).w = w.2 by PDIFF_1:def 1
                 .= ||. (seqy.n)-y .|| by A24,FINSEQ_1:44;
    then |. ||. (seqy.n)-y .|| .| <= |.w.| by PDIFF_8:5;
    then ||. (seqy.n)-y .|| <= |.w.| by ABSVALUE:def 1;
    hence ||.(seqy.n)-y.|| < r by A24,A21,XXREAL_0:2;
  end;
 A25 :now let r be Real;
      assume 0 < r;
      then consider m be Nat such that
      A26: for n be Nat st m <= n holds
       ||.(seqx.n)-x.|| < r & ||.(seqy.n)-y.|| < r by A19;
      take m;
      thus for n be Nat st m <= n holds
        ||.(seqx.n) - x.|| < r by A26;
  end;
   hence seqx is convergent;
   hence lim seqx = x by A25,NORMSP_1:def 7;
   A27: now let r be Real;
      assume 0 < r;
      then consider m be Nat such that
A28: for n be Nat st m <= n holds
        ||.(seqx.n)-x.|| < r & ||.(seqy.n)-y.|| < r by A19;
      take m;
      thus for n be Nat st m <= n holds
        ||.(seqy.n)-y.|| < r by A28;
   end;
   hence seqy is convergent;
   hence lim seqy = y by A27,NORMSP_1:def 7;
end;
