Lipschitz properties for families of convex mappings

The aim of the present note is to give a new proof of the fact that a pointwisely bounded family of continuous convex mappings defined on an open convex subset Ohm of a barrelled locally convex space and with values in a locally convex space, ordered by a normal cone, is locally equi-Lipschitz on Ohm and equi-Lipschitz on every compact subset of Ohm.