reserve V for non empty RealLinearSpace;
reserve S for Real_Sequence;
reserve k,n,m,m1 for Nat;
reserve g,h,r,x for Real;

theorem Th121:
  S is convergent implies S - x is convergent & lim (S-x) = (lim S) - x
proof
  assume B1: S is convergent;
  set g = lim S;
  set h = g - x;
X1: now let r;
  assume 0 < r;
  then consider m1 such that
A2: for n st m1 <= n holds |.(S.n) - g.| < r by B1,SEQ_2:def 7;
  take k = m1;
  let n;
  assume B3: k <= n;
  |.(S.n) - g.| = |.((S.n) - x) - h.|
    .= |.(S - x).n - h.| by Def14;
  hence |.(S - x).n - h.| < r by A2,B3;
  end;
  hence S - x is convergent;
  hence lim (S-x) = (lim S) - x by X1,SEQ_2:def 7;
end;
