reserve i,n,m for Nat,
  x,y,X,Y for set,
  r,s for Real;

theorem Th5:
  for M be Reflexive symmetric triangle non empty MetrStruct for p
  be Point of M holds 0 <= r implies diameter cl_Ball(p,r) <= 2*r
proof
  let M be Reflexive symmetric triangle non empty MetrStruct;
  let p be Point of M;
A1: dist(p,p)=0 by METRIC_1:1;
  assume 0 <= r;
  then
A2: p in cl_Ball(p,r) by A1,METRIC_1:12;
A3: now
    let x,y be Point of M such that
A4: x in cl_Ball(p,r) and
A5: y in cl_Ball(p,r);
A6: dist(x,p)<=r by A4,METRIC_1:12;
A7: dist(x,y)<=dist(x,p)+dist(p,y) by METRIC_1:4;
    dist(p,y)<=r by A5,METRIC_1:12;
    then dist(x,p)+dist(p,y)<=r+r by A6,XREAL_1:7;
    hence dist(x,y)<=2*r by A7,XXREAL_0:2;
  end;
  cl_Ball(p,r) is bounded by TOPREAL6:59;
  hence thesis by A2,A3,TBSP_1:def 8;
end;
