reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Subspace of V;
reserve x for set;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:COMPLEX,D:],D;
reserve B,C for Coset of W;
reserve CNS for ComplexNormSpace;
reserve x, y, w, g, g1, g2 for Point of CNS;
reserve S, S1, S2 for sequence of CNS;
reserve n, m, m1, m2 for Nat;
reserve r for Real;

theorem Th114:
  S1 is convergent & S2 is convergent implies S1 - S2 is convergent
proof
  assume that
A1: S1 is convergent and
A2: S2 is convergent;
  consider g1 such that
A3: for r st 0 < r ex m st for n st m <= n holds ||.(S1.n) - g1.|| < r
  by A1;
  consider g2 such that
A4: for r st 0 < r ex m st for n st m <= n holds ||.(S2.n) - g2.|| < r
  by A2;
  take g = g1 - g2;
  let r;
  assume
A5: 0 < r;
  then consider m1 such that
A6: for n st m1 <= n holds ||.(S1.n) - g1.|| < r/2 by A3;
  consider m2 such that
A7: for n st m2 <= n holds ||.(S2.n) - g2.|| < r/2 by A4,A5;
  take k = m1 + m2;
  let n such that
A8: k <= n;
  m2 <= k by NAT_1:12;
  then m2 <= n by A8,XXREAL_0:2;
  then
A9: ||.(S2.n) - g2.|| < r/2 by A7;
A10: ||.(S1 - S2).n - g.|| = ||.((S1.n) - (S2.n)) - (g1 - g2).|| by
NORMSP_1:def 3
    .= ||.(((S1.n) - (S2.n)) - g1) + g2.|| by RLVECT_1:29
    .= ||. ((S1.n) - (g1 + (S2.n))) + g2.|| by RLVECT_1:27
    .= ||. (((S1.n) - g1 ) - (S2.n)) + g2.|| by RLVECT_1:27
    .= ||. ((S1.n) - g1 ) - ((S2.n) - g2).|| by RLVECT_1:29;
A11: ||.((S1.n) - g1) - ((S2.n) - g2).|| <= ||.(S1.n) - g1.|| + ||.(S2.n) -
  g2.|| by Th104;
  m1 <= m1 + m2 by NAT_1:12;
  then m1 <= n by A8,XXREAL_0:2;
  then ||.(S1.n) - g1.|| < r/2 by A6;
  then ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| < r/2 + r/2 by A9,XREAL_1:8;
  hence thesis by A10,A11,XXREAL_0:2;
end;
