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 Th14:
  seq1 is convergent & seq2 is convergent implies lim (seq1 - seq2
  ) = (lim seq1) - (lim seq2)
proof
  assume that
A1: seq1 is convergent and
A2: seq2 is convergent;
  set g2 = lim seq2;
  set g1 = lim seq1;
  set g = g1 - g2;
A3: now
    let r;
    assume r > 0;
    then
A4: r/2 > 0 by XREAL_1:215;
    then consider m1 such that
A5: for n st n >= m1 holds dist((seq1.n) , g1) < r/2 by A1,Def2;
    consider m2 such that
A6: for n st n >= m2 holds dist((seq2.n) , g2) < r/2 by A2,A4,Def2;
    take k = m1 + m2;
    let n such that
A7: n >= k;
    k >= m2 by NAT_1:12;
    then n >= m2 by A7,XXREAL_0:2;
    then
A8: dist((seq2.n) , g2) < r/2 by A6;
    dist((seq1 - seq2).n, g) = dist((seq1.n) - (seq2.n) , g1 - g2) by
NORMSP_1:def 3;
    then
A9: dist((seq1 - seq2).n, g) <= dist((seq1.n) , g1) + dist((seq2.n) , g2)
    by BHSP_1:41;
    m1 + m2 >= m1 by NAT_1:12;
    then n >= m1 by A7,XXREAL_0:2;
    then dist((seq1.n) , g1) < r/2 by A5;
    then dist((seq1.n) , g1) + dist((seq2.n) , g2) < r/2 + r/2 by A8,XREAL_1:8;
    hence dist((seq1 - seq2).n, g) < r by A9,XXREAL_0:2;
  end;
  seq1 - seq2 is convergent by A1,A2,Th4;
  hence thesis by A3,Def2;
end;
