(a^2+b^2)(c^2+d^2)=(ac+be)^2+(ad-bc)^2 доказать тождество

Докажем тождество:

(a^2 + b^2) * (c^2 + d^2) = (a * c + b * d)^2 + (a * d - b * c)^2; 

(a^2 + b^2) * (c^2 + d^2) = (a * c)^2 + 2 * a * c * b * d + (b * d)^2 + (a * d)^2 - 2 * a * d * b * c + (b * c)^2; 

(a^2 + b^2) * (c^2 + d^2) = (a * c)^2 + (b * d)^2 + (a * d)^2 + (b * c)^2;  

(a^2 + b^2) * (c^2 + d^2) = (a^2 * c^2 + a^2 * d^2) + (b^2 * d^2 + b^2 * c^2); 

(a^2 + b^2) * (c^2 + d^2) = a^2 * (c^2 + d^2) + b^2 * (d^2 + c^2); 

(a^2 + b^2) * (c^2 + d^2) = (c^2 + d^2) * (a^2 + b^2); 

(a^2 + b^2) * (c^2 + d^2) = (a^2 + b^2) * (c^2 + d^2);  

Верно.