theorem LmAddEll2: :: EC_PF_3:28

for p being 5 _or_greater Prime
for z being Element of EC_WParam p
for P, Q being Element of EC_SetProjCo ((z `1),(z `2),p)
for g2, g3, g4, g8, gf1, gf2, gf3, gf4 being Element of (GF p) st P _EQ_ Q & P `3_3 = 1 & Q `3_3 = 1 & g2 = 2 mod p & g3 = 3 mod p & g4 = 4 mod p & g8 = 8 mod p & gf1 = (z `1) + (g3 * ((P `1_3) |^ 2)) & gf2 = P `2_3 & gf3 = ((P `1_3) * (P `2_3)) * gf2 & gf4 = (gf1 |^ 2) - (g8 * gf3) holds
(addell_ProjCo (z,p)) . (P,Q) = [((g2 * gf4) * gf2),((gf1 * ((g4 * gf3) - gf4)) - ((g8 * ((P `2_3) |^ 2)) * (gf2 |^ 2))),(g8 * (gf2 |^ 3))]