theorem Th62: :: EC_PF_2:62

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