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 Th51:
  w in Sphere(x,r) iff ||.x - w.|| = r
proof
  thus w in Sphere(x,r) implies ||.x - w.|| = r
  proof
    assume w in Sphere(x,r);
    then ex y be Point of X st w = y & ||.x - y.|| = r;
    hence thesis;
  end;
  assume ||.x - w.|| = r;
  hence thesis;
end;
