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 Th139:
  |(p-q, p-q)| = |(p, p)| - 2*|(p, q)| + |(q, q)|
proof
  |(p-q, p-q)| = |(p,p)| - |(p,q)| - |(p,q)| + |(q, q)| by Th137
    .= |(p,p)| - 2*|(p,q)| + |(q, q)|;
  hence thesis;
end;
