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 Th137:
  |(p1-p2, q1-q2)| = |(p1, q1)| - |(p1, q2)| - |(p2, q1)| + |(p2, q2)|
proof
A1: |(p1,q1-q2)| = |(p1,q1)| - |(p1,q2)| & |(p2,q1-q2)| = |(p2,q1)| - |(p2,
  q2)| by Th134;
  |(p1-p2, q1-q2)| = |(p1, q1-q2)| - |(p2, q1-q2)| by Th134
    .= |(p1,q1)|-|(p1,q2)|-|(p2,q1)|+|(p2,q2)| by A1;
  hence thesis;
end;
