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 :: EUCLID_2:33
  for x,y being Element of REAL n holds |.x+y.|^2 = |.x.|^2 + 2*|(x, y)|
  + |.y.|^2
proof
  let x,y be Element of REAL n;
  thus |.x+y.|^2 = |(x+y,x+y)| by EUCLID_2:4
    .= |(x,x)|+2*|(x,y)|+|(y,y)| by Th32
    .= |.x.|^2 + 2*|(x, y)| + |(y,y)| by EUCLID_2:4
    .= |.x.|^2 + 2*|(x, y)| + |.y.|^2 by EUCLID_2:4;
end;
