  We (re)derive the propagators and Feynman rules for the massless scalar and
vector multiplets in N=2 Projective Superspace ('Projective Hyperspace'). With
these, we are able to calculate both the divergent and finite parts of 2, 3 &
4-point functions at 1-loop for N=2 Super-Yang-Mills theory (SYM) explicitly in
Projective Hyperspace itself. We find that effectively only the coupling
constant needs to be renormalized unlike in the N=1 case where an independent
wavefunction renormalization is also required. This feature is similar to that
of the background field gauge, even though we are using ordinary Fermi-Feynman
gauge. The computation of 1-hoop beta-function is then straightforward and
matches with the known result. We also show that it receives no 2-hoops
contributions. All these calculations provide an alternative proof of the
finiteness of N=4 SYM.
