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) & w in Ball(x,r) implies dist(y,w) < 2 * r
proof
  assume that
A1: y in Ball(x,r) and
A2: w in Ball(x,r);
  dist(x,w) < r by A2,Th41;
  then
A3: r + dist(x,w) < r + r by XREAL_1:6;
A4: dist(y,w) <= dist(y,x) + dist(x,w) by CSSPACE:51;
  dist(x,y) < r by A1,Th41;
  then dist(x,y) + dist(x,w) < r + dist(x,w) by XREAL_1:6;
  then dist(x,y) + dist(x,w) < 2 * r by A3,XXREAL_0:2;
  hence thesis by A4,XXREAL_0:2;
end;
