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 Th48:
  w in cl_Ball(x,r) iff dist(x,w) <= r
proof
  thus w in cl_Ball(x,r) implies dist(x,w) <= r
  proof
    assume w in cl_Ball(x,r);
    then ||.x - w.|| <= r by Th47;
    hence thesis by CSSPACE:def 16;
  end;
  assume dist(x,w) <= r;
  then ||.x - w.|| <= r by CSSPACE:def 16;
  hence thesis;
end;
