  We develop some tools for manipulating and constructing projections in
C*-algebras. These are then applied to give short proofs of some standard
projection homotopy results, as well as strengthen some fundamental classical
results for C*-algebras of real rank zero, specifically on liftings, excising
pure states and Kadison's transitivity theorem. Lastly, we investigate some
order properties of the set of projections in C*-algebras of real rank zero,
building on our previous work.
