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 Th32: :: EUCLID_2:31
  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 Th30
    .= |(x,x)|+2*|(x,y)|+|(y,y)|;
end;
