reserve m,n,i,i2,j for Nat,
  r,r1,r2,s,t for Real,
  x,y,z for object;
reserve p,p1,p2,p3,q,q1,q2,q3,q4 for Point of TOP-REAL n;
reserve u for Point of Euclid n;
reserve R for Subset of TOP-REAL n;
reserve P,Q for Subset of TOP-REAL n;
reserve D for non vertical non horizontal non empty compact Subset of TOP-REAL
  2;

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;
