theorem
  |(p,q)| <= |(p,p)| + |(q,q)|
proof
  0 <= |(p,p)| & 0 <= |(q,q)| by Th119;
  then
A1: 0/2 <= (|(p,p)| + |(q,q)|)/2;
  |(p-q, p-q)| = |(p,p)| - 2*|(p,q)| + |(q,q)| by Th139
    .= |(p,p)| + |(q,q)| - 2*|(p,q)|;
  then 2*|(p,q)| <= |(p,p)| + |(q,q)| - 0 by Th119,XREAL_1:11;
  then (2*|(p,q)|)/2 <= (|(p,p)| + |(q,q)|)/2 by XREAL_1:72;
  then (0 qua Element of NAT) + |(p,q)|
   <= (|(p,p)| + |(q,q)|)/2 + (|(p,p)| + |(q,q)|)/2 by A1,XREAL_1:7;
  hence thesis;
end;
