reserve a,b,s,t,u,lambda for Real,
  n for Nat;
reserve x,x1,x2,x3,y1,y2 for Element of REAL n;

theorem Th31: :: EUCLID_2:30
  for x1,x2,y1,y2 being Element of REAL n holds |(x1-x2, y1-y2)| =
  |(x1, y1)| - |(x1, y2)| - |(x2, y1)| + |(x2, y2)|
proof
  let x1,x2,y1,y2 being Element of REAL n;
  thus |(x1-x2, y1-y2)| = |(x1,y1-y2)| - |(x2,y1-y2)| by Th26
    .= |(x1,y1)| - |(x1,y2)| - |(x2,y1-y2)| by Th26
    .= |(x1,y1)| - |(x1,y2)| - (|(x2,y1)| - |(x2,y2)|) by Th26
    .= |(x1, y1)| - |(x1, y2)| - |(x2, y1)| + |(x2, y2)|;
end;
