reserve X for ComplexUnitarySpace;
reserve x, y, w, g, g1, g2 for Point of X;
reserve z for Complex;
reserve p, q, r, M, M1, M2 for Real;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve k,n,m for Nat;
reserve Nseq for increasing sequence of NAT;

theorem
  y in Ball(x,r) implies y - w in Ball(x - w,r)
proof
  assume y in Ball(x,r); then
A1: dist(x,y) < r by Th41;
  dist(x - w,y - w) = dist(x,y) by CSSPACE:58;
  hence thesis by A1,Th41;
end;
