In generative syntax, a Cleft construction can be derived from a non-cleft construction by introducing one of its elements.
(i) We saw John (ii) a It was John [who we saw e] b It was John [that we saw e]
Smits, R.J.C. 1989. The relative and cleft constructions of the Germanic and Romance languages. Diss. KUB, Tilburg. Dordrecht: Foris.