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 Th115:
  S is convergent implies S - x is convergent
proof
  assume S is convergent;
  then consider g such that
A1: for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - g.|| < r;
  take h = g - x;
  let r;
  assume 0 < r;
  then consider m1 such that
A2: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1;
  take k = m1;
  let n;
  assume k <= n;
  then
A3: ||.(S.n) - g.|| < r by A2;
  ||.(S.n) - g.|| = ||.((S.n) - 09(CNS)) - g.|| by RLVECT_1:13
    .= ||.((S.n) - (x - x)) - g.|| by RLVECT_1:15
    .= ||.(((S.n) - x) + x) - g.|| by RLVECT_1:29
    .= ||.((S.n) - x) + ((-g) + x).|| by RLVECT_1:def 3
    .= ||.((S.n) - x) - h.|| by RLVECT_1:33;
  hence thesis by A3,NORMSP_1:def 4;
end;
