reserve PM for MetrStruct;
reserve x,y for Element of PM;
reserve r,p,q,s,t for Real;

theorem Th1:
  for r, p being Real st r <= p holds Ball(x,r) c= Ball(x,p)
proof
  let r, p be Real;
  assume
A1: r <= p;
  for y holds y in Ball(x,r) implies y in Ball(x,p)
  proof
    let y;
    assume
A2: y in Ball(x,r);
    then dist(x,y) < r by METRIC_1:11;
    then
A3: dist(x,y) < p by A1,XXREAL_0:2;
    PM is non empty by A2;
    hence thesis by A3,METRIC_1:11;
  end;
  hence thesis;
end;
