reserve X for RealUnitarySpace;
reserve x, y, z, g, g1, g2 for Point of X;
reserve a, q, r for Real;
reserve seq, seq1, seq2, seq9 for sequence of X;
reserve k, n, m, m1, m2 for Nat;

theorem Th3:
  seq1 is convergent & seq2 is convergent implies seq1 + seq2 is convergent
proof
  assume that
A1: seq1 is convergent and
A2: seq2 is convergent;
  consider g1 such that
A3: for r st r > 0 ex m st for n st n >= m holds dist((seq1.n) , g1) < r
  by A1;
  consider g2 such that
A4: for r st r > 0 ex m st for n st n >= m holds dist((seq2.n) , g2) < r
  by A2;
  take g = g1 + g2;
  let r;
  assume
A5: r > 0;
  then consider m1 such that
A6: for n st n >= m1 holds dist((seq1.n) , g1) < r/2 by A3,XREAL_1:215;
  consider m2 such that
A7: for n st n >= m2 holds dist((seq2.n) , g2) < r/2 by A4,A5,XREAL_1:215;
   reconsider k = m1 + m2 as Nat;
  take k;
  let n be Nat such that
A8: n >= k;
  k >= m2 by NAT_1:12;
  then n >= m2 by A8,XXREAL_0:2;
  then
A9: dist((seq2.n) , g2) < r/2 by A7;
  dist((seq1 + seq2).n, g) = dist((seq1.n) + (seq2.n) , (g1 + g2)) by
NORMSP_1:def 2;
  then
A10: dist((seq1 + seq2).n, g) <= dist((seq1.n) , g1) + dist((seq2.n) , g2)
  by BHSP_1:40;
  m1 + m2 >= m1 by NAT_1:12;
  then n >= m1 by A8,XXREAL_0:2;
  then dist((seq1.n) , g1) < r/2 by A6;
  then dist((seq1.n) , g1) + dist((seq2.n) , g2) < r/2 + r/2 by A9,XREAL_1:8;
  hence thesis by A10,XXREAL_0:2;
end;
