theorem Th87:
  for ep being Point of Euclid n,p,q being Point of TOP-REAL n st
  p=ep & q in Ball(ep,r) holds |.p-q.|<r & |.q-p.|<r
proof
  let ep be Point of Euclid n,p,q be Point of TOP-REAL n;
  assume that
A1: p=ep and
A2: q in Ball(ep,r);
  reconsider eq=q as Point of Euclid n by TOPREAL3:8;
  dist(ep,eq)<r by A2,METRIC_1:11;
  hence thesis by A1,JGRAPH_1:28;
end;
