I'm with BOAC on this although for me it is more of the form
(½*d*v²)*S*Cl
Although it can be debated about downwash/circulation versus Bernoulli/pressure difference etc. this form of the lift equation points towards the use of dynamic pressure as defined by Bernoulli and that is what you see in the equation in brackets. You then have a pressure term, and an area term which when multiplied gives you a force. The non-dimensional coefficient gives a scaling factor to account for the aerofoil efficiency.