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
  S is convergent implies lim (S - x) = (lim S) - x
proof
  set g = lim S;
  set h = g - x;
  assume
A1: S is convergent;
A2: now
    let r;
    assume 0 < r;
    then consider m1 such that
A3: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,Def16;
    take k = m1;
    let n;
    assume k <= n;
    then
A4: ||.(S.n) - g.|| < r by A3;
    ||.(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 ||.(S - x).n - h.|| < r by A4,NORMSP_1:def 4;
  end;
  S - x is convergent by A1,Th115;
  hence thesis by A2,Def16;
end;
