theorem
  PM is triangle & y in Ball(x,r) /\ Ball(z,p) implies ex q st Ball(y,q)
  c= Ball(x,r) & Ball(y,q) c= Ball(z,p)
proof
  assume
A1: PM is triangle;
  assume
A2: y in Ball(x,r) /\ Ball(z,p);
  then y in Ball(x,r) by XBOOLE_0:def 4;
  then consider s such that
  s > 0 and
A3: Ball(y,s) c= Ball(x,r) by A1,Th27;
  y in Ball(z,p) by A2,XBOOLE_0:def 4;
  then consider t such that
  t > 0 and
A4: Ball(y,t) c= Ball(z,p) by A1,Th27;
  take q = min(s,t);
  Ball(y,q) c= Ball(y,s) by Th1,XXREAL_0:17;
  hence Ball(y,q) c= Ball(x,r) by A3;
  Ball(y,q) c= Ball(y,t) by Th1,XXREAL_0:17;
  hence thesis by A4;
end;
