reserve n for Nat;

theorem Th16:
  for n be Nat, seq1 be sequence of REAL-NS n, seq2 be
sequence of REAL-US n st seq1 = seq2 holds (seq1 is convergent implies seq2 is
  convergent & lim seq1 = lim seq2) & (seq2 is convergent implies seq1 is
  convergent & lim seq1 = lim seq2)
proof
  let n be Nat;
  let seq1 be sequence of REAL-NS n;
  let seq2 be sequence of REAL-US n;
  assume
A1: seq1 = seq2;
A2: the carrier of REAL-NS n = REAL n by Def4
    .= the carrier of REAL-US n by Def6;
  now
    reconsider LIMIT = lim seq1 as Point of REAL-US n by A2;
    assume
A3: seq1 is convergent;
    then consider RNg be Point of REAL-NS n such that
A4: for r be Real st 0 < r ex m be Nat st
     for k be Nat st m <= k holds ||.(seq1.k) - RNg.|| < r by NORMSP_1:def 6;
    reconsider RUg = RNg as Point of REAL-US n by A2;
    for r be Real st 0 < r
    ex m be Nat st for k be Nat st m <= k holds dist(seq2.k, RUg) < r
    proof
      let r be Real;
      assume 0 < r;
      then consider m be Nat such that
A5:   for k be Nat st m <= k holds ||.(seq1.k) - RNg.|| < r by A4;
      take m;
        let k be Nat;
        reconsider p = seq1.k - RNg as Element of REAL n by Def4;
        assume
A6:     m <= k;
        -RNg = -RUg by Th13;
        then
A7:     p = seq2.k - RUg by A1,Th13;
        ||.(seq1.k) - RNg.|| = |.p.| by Th1
          .= sqrt |(p,p)| by EUCLID_2:5
          .= sqrt Sum mlt(p,p) by RVSUM_1:def 16
          .= sqrt ((Euclid_scalar n).(p,p)) by Def5
          .= ||. seq2.k - RUg .|| by A7,Def6;
        hence thesis by A5,A6;
    end;
    hence
A8: seq2 is convergent by BHSP_2:def 1;
    for r be Real st r > 0
ex m be Nat st for k be Nat st k >= m holds dist((seq2.k), LIMIT) < r
    proof
      let r be Real;
      assume r > 0;
      then consider m be Nat such that
A9:   for k be Nat st m <= k holds ||.(seq1.k) - lim seq1
      .|| < r by A3,NORMSP_1:def 7;
      take m;
        let k be Nat;
        reconsider p = seq1.k - lim seq1 as Element of REAL n by Def4;
        assume
A10:    m <= k;
        -lim seq1 = -LIMIT by Th13;
        then
A11:    p = seq2.k - LIMIT by A1,Th13;
        ||.(seq1.k) - lim seq1.|| = |.p.| by Th1
          .= sqrt |(p,p)| by EUCLID_2:5
          .= sqrt Sum mlt(p,p) by RVSUM_1:def 16
          .= sqrt ((Euclid_scalar n).(p,p)) by Def5
          .= ||. seq2.k-LIMIT .|| by A11,Def6;
        hence thesis by A9,A10;
    end;
    hence lim seq2 = lim seq1 by A8,BHSP_2:def 2;
  end;
  hence seq1 is convergent implies seq2 is convergent & lim seq1 = lim seq2;
  now
    reconsider LIMIT = lim seq2 as Point of REAL-NS n by A2;
    assume
A12: seq2 is convergent;
    then consider RUg be Point of REAL-US n such that
A13: for r be Real st 0 < r ex m be Nat st
    for k be Nat st m <= k holds dist(seq2.k, RUg) < r by BHSP_2:def 1;
    reconsider RNg = RUg as Point of REAL-NS n by A2;
    for r be Real st 0 < r
      ex m be Nat st
       for k be Nat st m <= k holds ||.(seq1.k)-RNg.|| < r
    proof
      let r be Real;
      assume 0 < r;
      then consider m be Nat such that
A14:  for k be Nat st m <= k holds dist(seq2.k, RUg) < r by A13;
      take m;
      for k be Nat st m <= k holds ||.(seq1.k)-RNg.|| < r
      proof
        let k be Nat;
        reconsider p = seq2.k - RUg as Element of REAL n by Def6;
        assume m <= k;
        then
A15:    dist(seq2.k,RUg) < r by A14;
        -RNg = -RUg by Th13;
        then
A16:    p = seq1.k - RNg by A1,Th13;
        dist(seq2.k, RUg) = sqrt ((Euclid_scalar n).(p,p)) by Def6
          .= sqrt Sum mlt(p,p) by Def5
          .= sqrt |(p,p)| by RVSUM_1:def 16
          .= |.p.| by EUCLID_2:5;
        hence thesis by A15,A16,Th1;
      end;
      hence thesis;
    end;
    hence
A17: seq1 is convergent by NORMSP_1:def 6;
    for r be Real st r > 0
    ex m be Nat st
     for k be Nat st k >= m holds ||.(seq1.k)-LIMIT.|| < r
    proof
      let r be Real;
      assume r > 0;
      then consider m be Nat such that
A18:  for k be Nat st k >= m holds dist((seq2.k), lim seq2
      ) < r by A12,BHSP_2:def 2;
      take m;
      let k be Nat;
      assume k >= m;
      then
A19:  dist(seq2.k,lim seq2) < r by A18;
      reconsider p = seq2.k - lim seq2 as Element of REAL n by Def6;
      -lim seq2 = -LIMIT by Th13;
      then
A20:  p = seq1.k - LIMIT by A1,Th13;
      dist(seq2.k, lim seq2) = sqrt ((Euclid_scalar n).(p,p)) by Def6
        .= sqrt Sum mlt(p,p) by Def5
        .= sqrt |(p,p)| by RVSUM_1:def 16
        .= |.p.| by EUCLID_2:5;
      hence thesis by A19,A20,Th1;
    end;
    hence lim seq1 = lim seq2 by A17,NORMSP_1:def 7;
  end;
  hence thesis;
end;
