theorem Th57:
  for r being Real for M be Reflexive symmetric triangle
  non empty MetrStruct for x be Element of M holds cl_Ball(x,r) is bounded
proof
  let r be Real;
  let M be Reflexive symmetric triangle non empty MetrStruct;
  let x be Element of M;
  cl_Ball(x,r) c= Ball(x,r+1)
  proof
    let y be object such that
A1: y in cl_Ball(x,r);
    reconsider Y=y as Point of M by A1;
A2: r+0<r+1 by XREAL_1:8;
    dist(x,Y)<=r by A1,METRIC_1:12;
    then dist(x,Y)<r+1 by A2,XXREAL_0:2;
    hence thesis by METRIC_1:11;
  end;
  hence thesis by TBSP_1:14;
end;
