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_2:34
  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 Th33
    .= |.x.|^2 - 2*|(x, y)| + |(y,y)| by EUCLID_2:4
    .= |.x.|^2 - 2*|(x, y)| + |.y.|^2 by EUCLID_2:4;
end;
