field extensions degree theorem

Question

I have a theorem for field extensions wich says : let $K,L,M$ be field extensions and $M\supset L\supset K$ then $[M:K] = [M:L][L:K]$. In a prove about field extensions of prime degree I have come across this expression: $[L:K]=p=[L:K(a)][K(a):K]$. Is $L\supset K(a)$, if so why? Or is there another reason that this is true?