reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;
reserve z,z1,z2 for Element of COMPLEX;
reserve n for Nat,
  x, y, a for Real,
  p, p1, p2, p3, q, q1, q2 for Element of n-tuples_on REAL;

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;
