let p be 5 _or_greater Prime; :: thesis: for z being Element of EC_WParam p
for g2, gf1, gf2, gf3 being Element of (GF p)
for P, Q 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 & gf1 = ((Q `2_3) * (P `3_3)) - ((P `2_3) * (Q `3_3)) & gf2 = ((Q `1_3) * (P `3_3)) - ((P `1_3) * (Q `3_3)) & gf3 = ((((gf1 |^ 2) * (P `3_3)) * (Q `3_3)) - (gf2 |^ 3)) - (((g2 * (gf2 |^ 2)) * (P `1_3)) * (Q `3_3)) & R = [(gf2 * gf3),((gf1 * ((((gf2 |^ 2) * (P `1_3)) * (Q `3_3)) - gf3)) - (((gf2 |^ 3) * (P `2_3)) * (Q `3_3))),(((gf2 |^ 3) * (P `3_3)) * (Q `3_3))] holds
(gf2 * (P `3_3)) * (R `2_3) = - ((gf1 * (((R `1_3) * (P `3_3)) - ((P `1_3) * (R `3_3)))) + ((gf2 * (P `2_3)) * (R `3_3)))
let z be Element of EC_WParam p; :: thesis: for g2, gf1, gf2, gf3 being Element of (GF p)
for P, Q 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 & gf1 = ((Q `2_3) * (P `3_3)) - ((P `2_3) * (Q `3_3)) & gf2 = ((Q `1_3) * (P `3_3)) - ((P `1_3) * (Q `3_3)) & gf3 = ((((gf1 |^ 2) * (P `3_3)) * (Q `3_3)) - (gf2 |^ 3)) - (((g2 * (gf2 |^ 2)) * (P `1_3)) * (Q `3_3)) & R = [(gf2 * gf3),((gf1 * ((((gf2 |^ 2) * (P `1_3)) * (Q `3_3)) - gf3)) - (((gf2 |^ 3) * (P `2_3)) * (Q `3_3))),(((gf2 |^ 3) * (P `3_3)) * (Q `3_3))] holds
(gf2 * (P `3_3)) * (R `2_3) = - ((gf1 * (((R `1_3) * (P `3_3)) - ((P `1_3) * (R `3_3)))) + ((gf2 * (P `2_3)) * (R `3_3)))
let g2, gf1, gf2, gf3 be Element of (GF p); :: thesis: for P, Q 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 & gf1 = ((Q `2_3) * (P `3_3)) - ((P `2_3) * (Q `3_3)) & gf2 = ((Q `1_3) * (P `3_3)) - ((P `1_3) * (Q `3_3)) & gf3 = ((((gf1 |^ 2) * (P `3_3)) * (Q `3_3)) - (gf2 |^ 3)) - (((g2 * (gf2 |^ 2)) * (P `1_3)) * (Q `3_3)) & R = [(gf2 * gf3),((gf1 * ((((gf2 |^ 2) * (P `1_3)) * (Q `3_3)) - gf3)) - (((gf2 |^ 3) * (P `2_3)) * (Q `3_3))),(((gf2 |^ 3) * (P `3_3)) * (Q `3_3))] holds
(gf2 * (P `3_3)) * (R `2_3) = - ((gf1 * (((R `1_3) * (P `3_3)) - ((P `1_3) * (R `3_3)))) + ((gf2 * (P `2_3)) * (R `3_3)))
let P, Q be Element of EC_SetProjCo ((z `1),(z `2),p); :: thesis: 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 & gf1 = ((Q `2_3) * (P `3_3)) - ((P `2_3) * (Q `3_3)) & gf2 = ((Q `1_3) * (P `3_3)) - ((P `1_3) * (Q `3_3)) & gf3 = ((((gf1 |^ 2) * (P `3_3)) * (Q `3_3)) - (gf2 |^ 3)) - (((g2 * (gf2 |^ 2)) * (P `1_3)) * (Q `3_3)) & R = [(gf2 * gf3),((gf1 * ((((gf2 |^ 2) * (P `1_3)) * (Q `3_3)) - gf3)) - (((gf2 |^ 3) * (P `2_3)) * (Q `3_3))),(((gf2 |^ 3) * (P `3_3)) * (Q `3_3))] holds
(gf2 * (P `3_3)) * (R `2_3) = - ((gf1 * (((R `1_3) * (P `3_3)) - ((P `1_3) * (R `3_3)))) + ((gf2 * (P `2_3)) * (R `3_3)))
let R be Element of [: the carrier of (GF p), the carrier of (GF p), the carrier of (GF p):]; :: thesis: ( g2 = 2 mod p & gf1 = ((Q `2_3) * (P `3_3)) - ((P `2_3) * (Q `3_3)) & gf2 = ((Q `1_3) * (P `3_3)) - ((P `1_3) * (Q `3_3)) & gf3 = ((((gf1 |^ 2) * (P `3_3)) * (Q `3_3)) - (gf2 |^ 3)) - (((g2 * (gf2 |^ 2)) * (P `1_3)) * (Q `3_3)) & R = [(gf2 * gf3),((gf1 * ((((gf2 |^ 2) * (P `1_3)) * (Q `3_3)) - gf3)) - (((gf2 |^ 3) * (P `2_3)) * (Q `3_3))),(((gf2 |^ 3) * (P `3_3)) * (Q `3_3))] implies (gf2 * (P `3_3)) * (R `2_3) = - ((gf1 * (((R `1_3) * (P `3_3)) - ((P `1_3) * (R `3_3)))) + ((gf2 * (P `2_3)) * (R `3_3))) )
assume that
g2 = 2 mod p and
( gf1 = ((Q `2_3) * (P `3_3)) - ((P `2_3) * (Q `3_3)) & gf2 = ((Q `1_3) * (P `3_3)) - ((P `1_3) * (Q `3_3)) & gf3 = ((((gf1 |^ 2) * (P `3_3)) * (Q `3_3)) - (gf2 |^ 3)) - (((g2 * (gf2 |^ 2)) * (P `1_3)) * (Q `3_3)) ) and
A1: R = [(gf2 * gf3),((gf1 * ((((gf2 |^ 2) * (P `1_3)) * (Q `3_3)) - gf3)) - (((gf2 |^ 3) * (P `2_3)) * (Q `3_3))),(((gf2 |^ 3) * (P `3_3)) * (Q `3_3))] ; :: thesis: (gf2 * (P `3_3)) * (R `2_3) = - ((gf1 * (((R `1_3) * (P `3_3)) - ((P `1_3) * (R `3_3)))) + ((gf2 * (P `2_3)) * (R `3_3)))
((R `1_3) * (P `3_3)) - ((P `1_3) * (R `3_3)) = ((gf2 * gf3) * (P `3_3)) - ((P `1_3) * (R `3_3)) by A1
.= ((gf2 * gf3) * (P `3_3)) - ((P `1_3) * (((gf2 |^ 3) * (P `3_3)) * (Q `3_3))) by A1
.= ((gf2 * (P `3_3)) * gf3) - ((P `1_3) * (((gf2 |^ (2 + 1)) * (P `3_3)) * (Q `3_3))) by GROUP_1:def 3
.= ((gf2 * (P `3_3)) * gf3) - ((P `1_3) * ((((gf2 |^ 2) * gf2) * (P `3_3)) * (Q `3_3))) by EC_PF_1:24
.= ((gf2 * (P `3_3)) * gf3) - (((((gf2 |^ 2) * gf2) * (P `3_3)) * (P `1_3)) * (Q `3_3)) by GROUP_1:def 3
.= ((gf2 * (P `3_3)) * gf3) - ((((gf2 * (P `3_3)) * (gf2 |^ 2)) * (P `1_3)) * (Q `3_3)) by GROUP_1:def 3
.= ((gf2 * (P `3_3)) * gf3) - ((gf2 * (P `3_3)) * (((gf2 |^ 2) * (P `1_3)) * (Q `3_3))) by Th11
.= (gf2 * (P `3_3)) * (gf3 - (((gf2 |^ 2) * (P `1_3)) * (Q `3_3))) by VECTSP_1:11
.= (gf2 * (P `3_3)) * (- ((((gf2 |^ 2) * (P `1_3)) * (Q `3_3)) - gf3)) by VECTSP_1:17
.= - ((gf2 * (P `3_3)) * ((((gf2 |^ 2) * (P `1_3)) * (Q `3_3)) - gf3)) by VECTSP_1:8
.= (- (gf2 * (P `3_3))) * ((((gf2 |^ 2) * (P `1_3)) * (Q `3_3)) - gf3) by VECTSP_1:9 ;
then A2: gf1 * (((R `1_3) * (P `3_3)) - ((P `1_3) * (R `3_3))) = (- (gf2 * (P `3_3))) * (gf1 * ((((gf2 |^ 2) * (P `1_3)) * (Q `3_3)) - gf3)) by GROUP_1:def 3;
A3: (gf2 * (P `2_3)) * (R `3_3) = (gf2 * (P `2_3)) * (((gf2 |^ 3) * (P `3_3)) * (Q `3_3)) by A1
.= gf2 * ((P `2_3) * (((gf2 |^ 3) * (P `3_3)) * (Q `3_3))) by GROUP_1:def 3
.= gf2 * ((((P `3_3) * (gf2 |^ 3)) * (P `2_3)) * (Q `3_3)) by GROUP_1:def 3
.= (((gf2 * (P `3_3)) * (gf2 |^ 3)) * (P `2_3)) * (Q `3_3) by Th11
.= (gf2 * (P `3_3)) * (((gf2 |^ 3) * (P `2_3)) * (Q `3_3)) by Th11
.= (- (gf2 * (P `3_3))) * (- (((gf2 |^ 3) * (P `2_3)) * (Q `3_3))) by VECTSP_1:10 ;
thus - ((gf1 * (((R `1_3) * (P `3_3)) - ((P `1_3) * (R `3_3)))) + ((gf2 * (P `2_3)) * (R `3_3))) = - ((- (gf2 * (P `3_3))) * ((gf1 * ((((gf2 |^ 2) * (P `1_3)) * (Q `3_3)) - gf3)) - (((gf2 |^ 3) * (P `2_3)) * (Q `3_3)))) by A2, A3, VECTSP_1:def 7
.= - (- ((gf2 * (P `3_3)) * (R `2_3))) by A1, VECTSP_1:9
.= (gf2 * (P `3_3)) * (R `2_3) by RLVECT_1:17 ; :: thesis: verum