Let $K$ be an algebraically closed field of characteristic zero and let $k \subsetneq K$ be a sub-field of $K$. Assume that the field extension $K(b_1,b_2)/K(a_1,a_2)$ has a finite degree $m_K$ and the field extension $k(b_1,b_2)/k(a_1,a_2)$ has a finite degree $m_k$. Also assume that $a_1,a_2$ and $b_1,b_2$ are algebraically independent over $k$.
Is it true that $m_k \geq m_K$? When $m_k = m_K$?