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 Th33: :: EUCLID_2:32
  for x,y being Element of REAL n holds |(x-y, x-y)| = |(x, x)| -
  2*|(x, y)| + |(y, y)|
proof
  let x,y be Element of REAL n;
  thus |(x-y, x-y)| = |(x,x)| - |(x,y)| - |(y,x)| + |(y,y)| by Th31
    .= |(x, x)| - 2*|(x, y)| + |(y, y)|;
end;
