ec_pf_2.miz
begin
reserve x for
set;
reserve i,j for
Integer;
reserve n,n1,n2,n3 for
Nat;
reserve p for
Prime;
reserve a,b,c,d for
Element of (
GF p);
reserve K for
Ring;
reserve a1,a2,a3,a4,a5,a6 for
Element of K;
theorem ::
EC_PF_2:1
Th1: for F be
add-associative
right_zeroed
right_complementable
distributive
unital non
empty
doubleLoopStr holds for a be
Element of F holds (a
|^ 2)
= ((
- a)
|^ 2)
proof
let F be
add-associative
right_zeroed
right_complementable
distributive
unital non
empty
doubleLoopStr;
let a be
Element of F;
set a2 = (
- a);
thus (a
|^ 2)
= (a
* a) by
GROUP_1: 51
.= (a2
* a2) by
VECTSP_1: 10
.= (a2
|^ 2) by
GROUP_1: 51;
end;
theorem ::
EC_PF_2:2
for K be
associative
commutative
well-unital
almost_left_invertible non
degenerated
doubleLoopStr holds ((
1. K)
" )
= (
1. K)
proof
let K be
associative
commutative
well-unital
almost_left_invertible non
degenerated
doubleLoopStr;
(
1. K)
<> (
0. K);
then (((
1. K)
" )
* (
1. K))
= (
1. K) by
VECTSP_1:def 10;
hence thesis;
end;
theorem ::
EC_PF_2:3
Th3: for K be
Field, a1,a2,a3,a4 be
Element of K holds a2
<> (
0. K) & a4
<> (
0. K) & (a1
* (a2
" ))
= (a3
* (a4
" )) implies (a1
* a4)
= (a2
* a3)
proof
let K be
Field, a1,a2,a3,a4 be
Element of K;
assume
A1: a2
<> (
0. K) & a4
<> (
0. K);
assume
A2: (a1
* (a2
" ))
= (a3
* (a4
" ));
(a1
* ((a2
" )
* a2))
= ((a3
* (a4
" ))
* a2) by
A2,
GROUP_1:def 3;
then (a1
* (
1. K))
= ((a3
* (a4
" ))
* a2) by
A1,
VECTSP_1:def 10;
then a1
= ((a3
* a2)
* (a4
" )) by
GROUP_1:def 3;
then (a1
* a4)
= ((a3
* a2)
* ((a4
" )
* a4)) by
GROUP_1:def 3
.= ((a3
* a2)
* (
1. K)) by
A1,
VECTSP_1:def 10
.= (a3
* a2);
hence thesis;
end;
theorem ::
EC_PF_2:4
Th4: for K be
Field, a1,a2,a3,a4 be
Element of K holds a2
<> (
0. K) & a4
<> (
0. K) & (a1
* a4)
= (a2
* a3) implies (a1
* (a2
" ))
= (a3
* (a4
" ))
proof
let K be
Field, a1,a2,a3,a4 be
Element of K;
assume
A1: a2
<> (
0. K) & a4
<> (
0. K);
assume
A2: (a1
* a4)
= (a2
* a3);
((a2
* a3)
* (a4
" ))
= (a1
* ((a4
" )
* a4)) by
A2,
GROUP_1:def 3
.= (a1
* (
1. K)) by
A1,
VECTSP_1:def 10
.= a1;
then a1
= ((a3
* (a4
" ))
* a2) by
GROUP_1:def 3;
then (a1
* (a2
" ))
= ((a3
* (a4
" ))
* ((a2
" )
* a2)) by
GROUP_1:def 3
.= ((a3
* (a4
" ))
* (
1. K)) by
A1,
VECTSP_1:def 10
.= (a3
* (a4
" ));
hence thesis;
end;
theorem ::
EC_PF_2:5
Th5: n
>= 1 implies ((
0. K)
|^ n)
= (
0. K)
proof
set a1 = (
0. K);
assume
A2: n
>= 1;
(n
- 1)
in
NAT by
A2,
INT_1: 5;
then
consider n1 be
Nat such that
A3: n1
= (n
- 1);
(a1
|^ n)
= (a1
|^ (n1
+ 1)) by
A3
.= ((a1
|^ n1)
* a1) by
EC_PF_1: 24;
hence (a1
|^ n)
= (
0. K);
end;
theorem ::
EC_PF_2:6
for K be
add-associative
right_zeroed
right_complementable
Abelian non
empty
addLoopStr, a1,a2 be
Element of K holds a1
= (
- a2) implies (
- a1)
= a2
proof
let K be
add-associative
right_zeroed
right_complementable
Abelian non
empty
addLoopStr, a1,a2 be
Element of K;
assume a1
= (
- a2);
then (a1
+ a2)
= (
0. K) by
VECTSP_1: 16;
hence thesis by
VECTSP_1: 16;
end;
theorem ::
EC_PF_2:7
Th7: for K be
Abelian
AddGroup, a1,a2,a3,a4 be
Element of K holds (((a1
+ a2)
+ a3)
+ a4)
= (((a4
+ a2)
+ a3)
+ a1) & (((a1
+ a2)
+ a3)
+ a4)
= (((a1
+ a4)
+ a3)
+ a2)
proof
let K be
Abelian
AddGroup, a1,a2,a3,a4 be
Element of K;
thus (((a1
+ a2)
+ a3)
+ a4)
= (((a2
+ a3)
+ a1)
+ a4) by
ALGSTR_1: 7
.= ((a4
+ (a2
+ a3))
+ a1) by
ALGSTR_1: 8
.= (((a4
+ a2)
+ a3)
+ a1) by
ALGSTR_1: 7;
thus (((a1
+ a2)
+ a3)
+ a4)
= (((a1
+ a3)
+ a2)
+ a4) by
ALGSTR_1: 8
.= (((a1
+ a3)
+ a4)
+ a2) by
ALGSTR_1: 8
.= (((a1
+ a4)
+ a3)
+ a2) by
ALGSTR_1: 8;
end;
theorem ::
EC_PF_2:8
Th8: for K be
Abelian
AddGroup, a1,a2,a3,a4,a5 be
Element of K holds (((a1
+ a2)
+ a3)
+ a4)
= (a1
+ ((a2
+ a3)
+ a4)) & ((((a1
+ a2)
+ a3)
+ a4)
+ a5)
= (a1
+ (((a2
+ a3)
+ a4)
+ a5))
proof
let K be
Abelian
AddGroup, a1,a2,a3,a4,a5 be
Element of K;
thus (((a1
+ a2)
+ a3)
+ a4)
= ((a1
+ a2)
+ (a3
+ a4)) by
ALGSTR_1: 7
.= (a1
+ (a2
+ (a3
+ a4))) by
ALGSTR_1: 7
.= (a1
+ ((a2
+ a3)
+ a4)) by
ALGSTR_1: 7;
thus ((((a1
+ a2)
+ a3)
+ a4)
+ a5)
= (((a1
+ a2)
+ a3)
+ (a4
+ a5)) by
ALGSTR_1: 7
.= ((a1
+ a2)
+ (a3
+ (a4
+ a5))) by
ALGSTR_1: 7
.= (a1
+ (a2
+ (a3
+ (a4
+ a5)))) by
ALGSTR_1: 7
.= (a1
+ ((a2
+ a3)
+ (a4
+ a5))) by
ALGSTR_1: 7
.= (a1
+ (((a2
+ a3)
+ a4)
+ a5)) by
ALGSTR_1: 7;
end;
theorem ::
EC_PF_2:9
for K be
Abelian
AddGroup, a1,a2,a3,a4,a5,a6 be
Element of K holds (((((a1
+ a2)
+ a3)
+ a4)
+ a5)
+ a6)
= (a1
+ ((((a2
+ a3)
+ a4)
+ a5)
+ a6))
proof
let K be
Abelian
AddGroup, a1,a2,a3,a4,a5,a6 be
Element of K;
thus (((((a1
+ a2)
+ a3)
+ a4)
+ a5)
+ a6)
= ((((a1
+ a2)
+ a3)
+ a4)
+ (a5
+ a6)) by
ALGSTR_1: 7
.= (a1
+ (((a2
+ a3)
+ a4)
+ (a5
+ a6))) by
Th8
.= (a1
+ ((((a2
+ a3)
+ a4)
+ a5)
+ a6)) by
ALGSTR_1: 7;
end;
theorem ::
EC_PF_2:10
Th10: for K be
comRing, a1,a2,a3,a4 be
Element of K holds (((a1
* a2)
* a3)
* a4)
= (((a4
* a2)
* a3)
* a1) & (((a1
* a2)
* a3)
* a4)
= (((a1
* a4)
* a3)
* a2)
proof
let K be
comRing, a1,a2,a3,a4 be
Element of K;
thus (((a1
* a2)
* a3)
* a4)
= (((a2
* a3)
* a1)
* a4) by
GROUP_1:def 3
.= ((a4
* (a2
* a3))
* a1) by
GROUP_1:def 3
.= (((a4
* a2)
* a3)
* a1) by
GROUP_1:def 3;
thus (((a1
* a2)
* a3)
* a4)
= (((a1
* a3)
* a2)
* a4) by
GROUP_1:def 3
.= (((a1
* a3)
* a4)
* a2) by
GROUP_1:def 3
.= (((a1
* a4)
* a3)
* a2) by
GROUP_1:def 3;
end;
theorem ::
EC_PF_2:11
Th11: (((a1
* a2)
* a3)
* a4)
= (a1
* ((a2
* a3)
* a4)) & ((((a1
* a2)
* a3)
* a4)
* a5)
= (a1
* (((a2
* a3)
* a4)
* a5))
proof
thus (((a1
* a2)
* a3)
* a4)
= ((a1
* a2)
* (a3
* a4)) by
GROUP_1:def 3
.= (a1
* (a2
* (a3
* a4))) by
GROUP_1:def 3
.= (a1
* ((a2
* a3)
* a4)) by
GROUP_1:def 3;
thus ((((a1
* a2)
* a3)
* a4)
* a5)
= (((a1
* a2)
* a3)
* (a4
* a5)) by
GROUP_1:def 3
.= ((a1
* a2)
* (a3
* (a4
* a5))) by
GROUP_1:def 3
.= (a1
* (a2
* (a3
* (a4
* a5)))) by
GROUP_1:def 3
.= (a1
* ((a2
* a3)
* (a4
* a5))) by
GROUP_1:def 3
.= (a1
* (((a2
* a3)
* a4)
* a5)) by
GROUP_1:def 3;
end;
theorem ::
EC_PF_2:12
Th12: (((((a1
* a2)
* a3)
* a4)
* a5)
* a6)
= (a1
* ((((a2
* a3)
* a4)
* a5)
* a6)) & (((((a1
* a2)
* a3)
* a4)
* a5)
* a6)
= (((a1
* ((a2
* a3)
* a4))
* a5)
* a6)
proof
thus
A1: (((((a1
* a2)
* a3)
* a4)
* a5)
* a6)
= ((((a1
* a2)
* a3)
* a4)
* (a5
* a6)) by
GROUP_1:def 3
.= (a1
* (((a2
* a3)
* a4)
* (a5
* a6))) by
Th11
.= (a1
* ((((a2
* a3)
* a4)
* a5)
* a6)) by
GROUP_1:def 3;
thus (((((a1
* a2)
* a3)
* a4)
* a5)
* a6)
= (a1
* (((a2
* a3)
* a4)
* (a5
* a6))) by
A1,
GROUP_1:def 3
.= ((a1
* ((a2
* a3)
* a4))
* (a5
* a6)) by
GROUP_1:def 3
.= (((a1
* ((a2
* a3)
* a4))
* a5)
* a6) by
GROUP_1:def 3;
end;
theorem ::
EC_PF_2:13
Th13: K is
commutative implies (((a1
* a2)
* a3)
|^ n)
= (((a1
|^ n)
* (a2
|^ n))
* (a3
|^ n))
proof
assume
A1: K is
commutative;
hence (((a1
* a2)
* a3)
|^ n)
= (((a1
* a2)
|^ n)
* (a3
|^ n)) by
BINOM: 9
.= (((a1
|^ n)
* (a2
|^ n))
* (a3
|^ n)) by
A1,
BINOM: 9;
end;
theorem ::
EC_PF_2:14
Th14: (a1
* ((a2
+ a3)
+ a4))
= (((a1
* a2)
+ (a1
* a3))
+ (a1
* a4)) & (a1
* ((a2
+ a3)
- a4))
= (((a1
* a2)
+ (a1
* a3))
- (a1
* a4)) & (a1
* ((a2
- a3)
+ a4))
= (((a1
* a2)
- (a1
* a3))
+ (a1
* a4)) & (a1
* ((a2
- a3)
- a4))
= (((a1
* a2)
- (a1
* a3))
- (a1
* a4)) & (a1
* (((
- a2)
+ a3)
+ a4))
= (((
- (a1
* a2))
+ (a1
* a3))
+ (a1
* a4)) & (a1
* (((
- a2)
+ a3)
- a4))
= (((
- (a1
* a2))
+ (a1
* a3))
- (a1
* a4)) & (a1
* (((
- a2)
- a3)
+ a4))
= (((
- (a1
* a2))
- (a1
* a3))
+ (a1
* a4)) & (a1
* (((
- a2)
- a3)
- a4))
= (((
- (a1
* a2))
- (a1
* a3))
- (a1
* a4))
proof
thus (a1
* ((a2
+ a3)
+ a4))
= ((a1
* (a2
+ a3))
+ (a1
* a4)) by
VECTSP_1:def 7
.= (((a1
* a2)
+ (a1
* a3))
+ (a1
* a4)) by
VECTSP_1:def 7;
thus (a1
* ((a2
+ a3)
- a4))
= ((a1
* (a2
+ a3))
+ (a1
* (
- a4))) by
VECTSP_1:def 7
.= (((a1
* a2)
+ (a1
* a3))
+ (a1
* (
- a4))) by
VECTSP_1:def 7
.= (((a1
* a2)
+ (a1
* a3))
- (a1
* a4)) by
VECTSP_1: 8;
thus (a1
* ((a2
- a3)
+ a4))
= ((a1
* (a2
+ (
- a3)))
+ (a1
* a4)) by
VECTSP_1:def 7
.= (((a1
* a2)
+ (a1
* (
- a3)))
+ (a1
* a4)) by
VECTSP_1:def 7
.= (((a1
* a2)
- (a1
* a3))
+ (a1
* a4)) by
VECTSP_1: 8;
thus (a1
* ((a2
- a3)
- a4))
= ((a1
* (a2
+ (
- a3)))
+ (a1
* (
- a4))) by
VECTSP_1:def 7
.= (((a1
* a2)
+ (a1
* (
- a3)))
+ (a1
* (
- a4))) by
VECTSP_1:def 7
.= (((a1
* a2)
- (a1
* a3))
+ (a1
* (
- a4))) by
VECTSP_1: 8
.= (((a1
* a2)
- (a1
* a3))
- (a1
* a4)) by
VECTSP_1: 8;
thus (a1
* (((
- a2)
+ a3)
+ a4))
= ((a1
* ((
- a2)
+ a3))
+ (a1
* a4)) by
VECTSP_1:def 7
.= (((a1
* (
- a2))
+ (a1
* a3))
+ (a1
* a4)) by
VECTSP_1:def 7
.= (((
- (a1
* a2))
+ (a1
* a3))
+ (a1
* a4)) by
VECTSP_1: 8;
thus (a1
* (((
- a2)
+ a3)
- a4))
= ((a1
* ((
- a2)
+ a3))
+ (a1
* (
- a4))) by
VECTSP_1:def 7
.= (((a1
* (
- a2))
+ (a1
* a3))
+ (a1
* (
- a4))) by
VECTSP_1:def 7
.= (((
- (a1
* a2))
+ (a1
* a3))
+ (a1
* (
- a4))) by
VECTSP_1: 8
.= (((
- (a1
* a2))
+ (a1
* a3))
- (a1
* a4)) by
VECTSP_1: 8;
thus (a1
* (((
- a2)
- a3)
+ a4))
= ((a1
* ((
- a2)
+ (
- a3)))
+ (a1
* a4)) by
VECTSP_1:def 7
.= (((a1
* (
- a2))
+ (a1
* (
- a3)))
+ (a1
* a4)) by
VECTSP_1:def 7
.= (((
- (a1
* a2))
+ (a1
* (
- a3)))
+ (a1
* a4)) by
VECTSP_1: 8
.= (((
- (a1
* a2))
- (a1
* a3))
+ (a1
* a4)) by
VECTSP_1: 8;
thus (a1
* (((
- a2)
- a3)
- a4))
= ((a1
* ((
- a2)
+ (
- a3)))
+ (a1
* (
- a4))) by
VECTSP_1:def 7
.= (((a1
* (
- a2))
+ (a1
* (
- a3)))
+ (a1
* (
- a4))) by
VECTSP_1:def 7
.= (((
- (a1
* a2))
+ (a1
* (
- a3)))
+ (a1
* (
- a4))) by
VECTSP_1: 8
.= (((
- (a1
* a2))
- (a1
* a3))
+ (a1
* (
- a4))) by
VECTSP_1: 8
.= (((
- (a1
* a2))
- (a1
* a3))
- (a1
* a4)) by
VECTSP_1: 8;
end;
theorem ::
EC_PF_2:15
Th15: for K be
comRing, a1,a2 be
Element of K holds ((a1
+ a2)
* (a1
- a2))
= ((a1
|^ 2)
- (a2
|^ 2))
proof
let K be
comRing, a1,a2 be
Element of K;
thus ((a1
+ a2)
* (a1
- a2))
= ((a1
* (a1
- a2))
+ (a2
* (a1
- a2))) by
VECTSP_1:def 7
.= (((a1
* a1)
- (a1
* a2))
+ (a2
* (a1
- a2))) by
VECTSP_1: 11
.= (((a1
|^ 2)
- (a1
* a2))
+ (a2
* (a1
- a2))) by
GROUP_1: 51
.= (((a1
|^ 2)
- (a1
* a2))
+ ((a2
* a1)
- (a2
* a2))) by
VECTSP_1: 11
.= (((a1
|^ 2)
- (a1
* a2))
+ ((a1
* a2)
- (a2
|^ 2))) by
GROUP_1: 51
.= ((a1
|^ 2)
+ ((
- (a1
* a2))
+ ((a1
* a2)
- (a2
|^ 2)))) by
ALGSTR_1: 7
.= ((a1
|^ 2)
+ (((
- (a1
* a2))
+ (a1
* a2))
- (a2
|^ 2))) by
ALGSTR_1: 7
.= ((a1
|^ 2)
+ ((
0. K)
- (a2
|^ 2))) by
VECTSP_1: 19
.= ((a1
|^ 2)
- (a2
|^ 2)) by
VECTSP_1: 18;
end;
theorem ::
EC_PF_2:16
for K be
comRing, a1,a2 be
Element of K holds ((a1
+ a2)
* (((a1
|^ 2)
- (a1
* a2))
+ (a2
|^ 2)))
= ((a1
|^ 3)
+ (a2
|^ 3))
proof
let K be
comRing, a1,a2 be
Element of K;
thus ((a1
+ a2)
* (((a1
|^ 2)
- (a1
* a2))
+ (a2
|^ 2)))
= ((a1
* (((a1
|^ 2)
- (a1
* a2))
+ (a2
|^ 2)))
+ (a2
* (((a1
|^ 2)
- (a1
* a2))
+ (a2
|^ 2)))) by
VECTSP_1:def 7
.= (((a1
* ((a1
|^ 2)
- (a1
* a2)))
+ (a1
* (a2
|^ 2)))
+ (a2
* (((a1
|^ 2)
- (a1
* a2))
+ (a2
|^ 2)))) by
VECTSP_1:def 7
.= ((((a1
* (a1
|^ 2))
- (a1
* (a1
* a2)))
+ (a1
* (a2
|^ 2)))
+ (a2
* (((a1
|^ 2)
- (a1
* a2))
+ (a2
|^ 2)))) by
VECTSP_1: 11
.= (((((a1
|^ 2)
* a1)
- ((a1
* a1)
* a2))
+ (a1
* (a2
|^ 2)))
+ (a2
* (((a1
|^ 2)
- (a1
* a2))
+ (a2
|^ 2)))) by
GROUP_1:def 3
.= (((((a1
|^ 2)
* a1)
- ((a1
* a1)
* a2))
+ (a1
* (a2
|^ 2)))
+ ((a2
* ((a1
|^ 2)
- (a1
* a2)))
+ (a2
* (a2
|^ 2)))) by
VECTSP_1:def 7
.= ((((a1
|^ (2
+ 1))
- ((a1
* a1)
* a2))
+ (a1
* (a2
|^ 2)))
+ ((a2
* ((a1
|^ 2)
- (a1
* a2)))
+ ((a2
|^ 2)
* a2))) by
EC_PF_1: 24
.= ((((a1
|^ 3)
- ((a1
* a1)
* a2))
+ (a1
* (a2
|^ 2)))
+ (((a2
* (a1
|^ 2))
- (a2
* (a1
* a2)))
+ ((a2
|^ 2)
* a2))) by
VECTSP_1: 11
.= ((((a1
|^ 3)
- ((a1
|^ 2)
* a2))
+ (a1
* (a2
|^ 2)))
+ (((a2
* (a1
|^ 2))
- (a2
* (a1
* a2)))
+ ((a2
|^ 2)
* a2))) by
GROUP_1: 51
.= ((((a1
|^ 3)
- ((a1
|^ 2)
* a2))
+ (a1
* (a2
|^ 2)))
+ ((((a1
|^ 2)
* a2)
- ((a1
* a2)
* a2))
+ (a2
|^ (2
+ 1)))) by
EC_PF_1: 24
.= ((((a1
|^ 3)
- ((a1
|^ 2)
* a2))
+ (a1
* (a2
|^ 2)))
+ ((((a1
|^ 2)
* a2)
- (a1
* (a2
* a2)))
+ (a2
|^ 3))) by
GROUP_1:def 3
.= (((a1
|^ 3)
+ ((
- ((a1
|^ 2)
* a2))
+ (a1
* (a2
|^ 2))))
+ ((((a1
|^ 2)
* a2)
- (a1
* (a2
* a2)))
+ (a2
|^ 3))) by
ALGSTR_1: 7
.= (((a1
|^ 3)
+ ((
- ((a1
|^ 2)
* a2))
+ (a1
* (a2
|^ 2))))
+ ((((a1
|^ 2)
* a2)
- (a1
* (a2
|^ 2)))
+ (a2
|^ 3))) by
GROUP_1: 51
.= ((((a1
|^ 3)
+ ((a1
* (a2
|^ 2))
- ((a1
|^ 2)
* a2)))
+ (((a1
|^ 2)
* a2)
- (a1
* (a2
|^ 2))))
+ (a2
|^ 3)) by
ALGSTR_1: 8
.= (((a1
|^ 3)
+ (((a1
* (a2
|^ 2))
- ((a1
|^ 2)
* a2))
+ (((a1
|^ 2)
* a2)
- (a1
* (a2
|^ 2)))))
+ (a2
|^ 3)) by
ALGSTR_1: 7
.= (((a1
|^ 3)
+ ((((a1
* (a2
|^ 2))
- ((a1
|^ 2)
* a2))
+ ((a1
|^ 2)
* a2))
- (a1
* (a2
|^ 2))))
+ (a2
|^ 3)) by
ALGSTR_1: 7
.= (((a1
|^ 3)
+ (((a1
* (a2
|^ 2))
+ ((
- ((a1
|^ 2)
* a2))
+ ((a1
|^ 2)
* a2)))
- (a1
* (a2
|^ 2))))
+ (a2
|^ 3)) by
ALGSTR_1: 7
.= (((a1
|^ 3)
+ (((a1
* (a2
|^ 2))
+ (
0. K))
- (a1
* (a2
|^ 2))))
+ (a2
|^ 3)) by
VECTSP_1: 19
.= (((a1
|^ 3)
+ (
0. K))
+ (a2
|^ 3)) by
VECTSP_1: 19,
ALGSTR_1: 7
.= ((a1
|^ 3)
+ (a2
|^ 3)) by
ALGSTR_1: 7;
end;
theorem ::
EC_PF_2:17
Th17: for K be
comRing, a1,a2 be
Element of K holds ((a1
- a2)
* (((a1
|^ 2)
+ (a1
* a2))
+ (a2
|^ 2)))
= ((a1
|^ 3)
- (a2
|^ 3))
proof
let K be
comRing, a1,a2 be
Element of K;
thus ((a1
- a2)
* (((a1
|^ 2)
+ (a1
* a2))
+ (a2
|^ 2)))
= ((a1
* (((a1
|^ 2)
+ (a1
* a2))
+ (a2
|^ 2)))
- (a2
* (((a1
|^ 2)
+ (a1
* a2))
+ (a2
|^ 2)))) by
VECTSP_1: 13
.= (((a1
* ((a1
|^ 2)
+ (a1
* a2)))
+ (a1
* (a2
|^ 2)))
- (a2
* (((a1
|^ 2)
+ (a1
* a2))
+ (a2
|^ 2)))) by
VECTSP_1:def 7
.= ((((a1
* (a1
|^ 2))
+ (a1
* (a1
* a2)))
+ (a1
* (a2
|^ 2)))
- (a2
* (((a1
|^ 2)
+ (a1
* a2))
+ (a2
|^ 2)))) by
VECTSP_1:def 7
.= (((((a1
|^ 2)
* a1)
+ ((a1
* a1)
* a2))
+ (a1
* (a2
|^ 2)))
- (a2
* (((a1
|^ 2)
+ (a1
* a2))
+ (a2
|^ 2)))) by
GROUP_1:def 3
.= (((((a1
|^ 2)
* a1)
+ ((a1
* a1)
* a2))
+ (a1
* (a2
|^ 2)))
- ((a2
* ((a1
|^ 2)
+ (a1
* a2)))
+ (a2
* (a2
|^ 2)))) by
VECTSP_1:def 7
.= ((((a1
|^ (2
+ 1))
+ ((a1
* a1)
* a2))
+ (a1
* (a2
|^ 2)))
- ((a2
* ((a1
|^ 2)
+ (a1
* a2)))
+ ((a2
|^ 2)
* a2))) by
EC_PF_1: 24
.= ((((a1
|^ 3)
+ ((a1
* a1)
* a2))
+ (a1
* (a2
|^ 2)))
- (((a2
* (a1
|^ 2))
+ (a2
* (a1
* a2)))
+ ((a2
|^ 2)
* a2))) by
VECTSP_1:def 7
.= ((((a1
|^ 3)
+ ((a1
|^ 2)
* a2))
+ (a1
* (a2
|^ 2)))
- (((a2
* (a1
|^ 2))
+ (a2
* (a1
* a2)))
+ ((a2
|^ 2)
* a2))) by
GROUP_1: 51
.= ((((a1
|^ 3)
+ ((a1
|^ 2)
* a2))
+ (a1
* (a2
|^ 2)))
- (((a2
* (a1
|^ 2))
+ (a2
* (a1
* a2)))
+ (a2
|^ (2
+ 1)))) by
EC_PF_1: 24
.= (((((a1
|^ 3)
+ ((a1
|^ 2)
* a2))
+ (a1
* (a2
|^ 2)))
- ((a2
* (a1
|^ 2))
+ (a2
* (a1
* a2))))
- (a2
|^ 3)) by
VECTSP_1: 17
.= ((((a1
|^ 3)
+ (((a1
|^ 2)
* a2)
+ (a1
* (a2
|^ 2))))
- ((a2
* (a1
|^ 2))
+ (a2
* (a1
* a2))))
- (a2
|^ 3)) by
ALGSTR_1: 7
.= (((a1
|^ 3)
+ ((((a1
|^ 2)
* a2)
+ (a1
* (a2
|^ 2)))
- ((a2
* (a1
|^ 2))
+ (a2
* (a1
* a2)))))
- (a2
|^ 3)) by
ALGSTR_1: 7
.= (((a1
|^ 3)
+ ((((a1
|^ 2)
* a2)
+ (a1
* (a2
|^ 2)))
+ ((
- (a2
* (a1
* a2)))
- (a2
* (a1
|^ 2)))))
- (a2
|^ 3)) by
VECTSP_1: 17
.= (((a1
|^ 3)
+ ((((a1
|^ 2)
* a2)
+ (a1
* (a2
|^ 2)))
+ ((
- (a1
* (a2
* a2)))
- ((a1
|^ 2)
* a2))))
- (a2
|^ 3)) by
GROUP_1:def 3
.= (((a1
|^ 3)
+ ((((a1
|^ 2)
* a2)
+ (a1
* (a2
|^ 2)))
+ ((
- (a1
* (a2
|^ 2)))
- ((a1
|^ 2)
* a2))))
- (a2
|^ 3)) by
GROUP_1: 51
.= (((a1
|^ 3)
+ (((((a1
|^ 2)
* a2)
+ (a1
* (a2
|^ 2)))
- (a1
* (a2
|^ 2)))
- ((a1
|^ 2)
* a2)))
- (a2
|^ 3)) by
ALGSTR_1: 7
.= (((a1
|^ 3)
+ ((((a1
|^ 2)
* a2)
+ ((a1
* (a2
|^ 2))
- (a1
* (a2
|^ 2))))
- ((a1
|^ 2)
* a2)))
- (a2
|^ 3)) by
ALGSTR_1: 7
.= (((a1
|^ 3)
+ ((((a1
|^ 2)
* a2)
+ (
0. K))
- ((a1
|^ 2)
* a2)))
- (a2
|^ 3)) by
VECTSP_1: 19
.= (((a1
|^ 3)
+ (
0. K))
- (a2
|^ 3)) by
VECTSP_1: 19,
ALGSTR_1: 7
.= ((a1
|^ 3)
- (a2
|^ 3)) by
ALGSTR_1: 7;
end;
definition
let n,p be
Nat;
::
EC_PF_2:def1
attr p is n
_or_greater means
:
Def1: n
<= p;
end
registration
cluster 5
_or_greater
prime for
Nat;
existence by
Def1,
PEPIN: 59;
end
theorem ::
EC_PF_2:18
Th18: for gi,gj,gij,a be
Element of (
GF p) st gi
= (i
mod p) & gj
= (j
mod p) & gij
= ((i
+ j)
mod p) holds ((gi
* a)
+ (gj
* a))
= (gij
* a)
proof
let gi,gj,gij,a be
Element of (
GF p) such that
A1: gi
= (i
mod p) & gj
= (j
mod p) & gij
= ((i
+ j)
mod p);
((gi
+ gj)
* a)
= (gij
* a) by
A1,
EC_PF_1: 15;
hence thesis by
VECTSP_1:def 7;
end;
theorem ::
EC_PF_2:19
Th19: for gi,gj,a be
Element of (
GF p) st gi
= (i
mod p) & gj
= (j
mod p) & j
= (i
+ 1) holds ((gi
* a)
+ a)
= (gj
* a)
proof
let gi,gj,a be
Element of (
GF p) such that
A1: gi
= (i
mod p) & gj
= (j
mod p) & j
= (i
+ 1);
reconsider g1 = (1
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
p
> 1 by
INT_2:def 4;
then g1
= 1 by
NAT_D: 63
.= (
1. (
GF p)) by
EC_PF_1: 12;
then ((gi
* a)
+ ((
1. (
GF p))
* a))
= (gj
* a) by
A1,
Th18;
hence thesis;
end;
theorem ::
EC_PF_2:20
Th20: for g2,a be
Element of (
GF p) st g2
= (2
mod p) holds (a
+ a)
= (g2
* a)
proof
let g2,a be
Element of (
GF p) such that
A1: g2
= (2
mod p);
reconsider g1 = (1
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
A2: g2
= ((1
+ 1)
mod p) by
A1;
p
> 1 by
INT_2:def 4;
then g1
= 1 by
NAT_D: 63
.= (
1. (
GF p)) by
EC_PF_1: 12;
then (((
1. (
GF p))
* a)
+ ((
1. (
GF p))
* a))
= (g2
* a) by
A2,
Th18;
hence thesis;
end;
theorem ::
EC_PF_2:21
Th21: for gi,gj,gij,a be
Element of (
GF p) st gi
= (i
mod p) & gj
= (j
mod p) & gij
= ((i
- j)
mod p) holds ((gi
* a)
- (gj
* a))
= (gij
* a)
proof
let gi,gj,gij,a be
Element of (
GF p) such that
A1: gi
= (i
mod p) & gj
= (j
mod p) & gij
= ((i
- j)
mod p);
(gj
+ gij)
= ((j
+ (i
- j))
mod p) by
A1,
EC_PF_1: 15
.= gi by
A1;
then (((gj
* a)
+ (gij
* a))
- (gj
* a))
= ((gi
* a)
- (gj
* a)) by
VECTSP_1:def 7;
then ((gij
* a)
+ ((gj
* a)
+ (
- (gj
* a))))
= ((gi
* a)
- (gj
* a)) by
ALGSTR_1: 7;
then ((gij
* a)
+ (
0. (
GF p)))
= ((gi
* a)
- (gj
* a)) by
VECTSP_1: 19;
hence thesis by
ALGSTR_1: 7;
end;
theorem ::
EC_PF_2:22
Th22: for gi,gj,a be
Element of (
GF p) st gi
= (i
mod p) & gj
= (j
mod p) & i
= (j
+ 1) holds ((gi
* a)
- (gj
* a))
= a
proof
let gi,gj,a be
Element of (
GF p) such that
A1: gi
= (i
mod p) & gj
= (j
mod p) & i
= (j
+ 1);
reconsider g1 = (1
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
A2: g1
= ((i
- j)
mod p) by
A1;
p
> 1 by
INT_2:def 4;
then g1
= 1 by
NAT_D: 63
.= (
1. (
GF p)) by
EC_PF_1: 12;
then ((gi
* a)
- (gj
* a))
= ((
1. (
GF p))
* a) by
A1,
A2,
Th21;
hence thesis;
end;
theorem ::
EC_PF_2:23
Th23: for gi,gj,a be
Element of (
GF p) st gi
= (i
mod p) & gj
= (j
mod p) & i
= (j
+ 1) holds ((gi
* a)
- a)
= (gj
* a)
proof
let gi,gj,a be
Element of (
GF p) such that
A1: gi
= (i
mod p) & gj
= (j
mod p) & i
= (j
+ 1);
reconsider g1 = (1
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
A2: gj
= ((i
- 1)
mod p) by
A1;
p
> 1 by
INT_2:def 4;
then g1
= 1 by
NAT_D: 63
.= (
1. (
GF p)) by
EC_PF_1: 12;
then ((gi
* a)
- ((
1. (
GF p))
* a))
= (gj
* a) by
A1,
A2,
Th21;
hence thesis;
end;
theorem ::
EC_PF_2:24
Th24: for g2,a be
Element of (
GF p) st g2
= (2
mod p) holds ((g2
* a)
- a)
= a
proof
let g2,a be
Element of (
GF p) such that
A1: g2
= (2
mod p);
reconsider g1 = (1
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
A2: g1
= ((2
- 1)
mod p);
p
> 1 by
INT_2:def 4;
then g1
= 1 by
NAT_D: 63
.= (
1. (
GF p)) by
EC_PF_1: 12;
then ((g2
* a)
- ((
1. (
GF p))
* a))
= ((
1. (
GF p))
* a) by
A1,
A2,
Th21;
hence thesis;
end;
theorem ::
EC_PF_2:25
Th25: for g2,a,b be
Element of (
GF p) st g2
= (2
mod p) holds ((a
+ b)
|^ 2)
= (((a
|^ 2)
+ ((g2
* a)
* b))
+ (b
|^ 2))
proof
let g2,a,b be
Element of (
GF p) such that
A1: g2
= (2
mod p);
thus ((a
+ b)
|^ 2)
= ((a
+ b)
* (a
+ b)) by
EC_PF_1: 22
.= ((a
* (a
+ b))
+ (b
* (a
+ b))) by
VECTSP_1:def 7
.= (((a
* a)
+ (a
* b))
+ (b
* (a
+ b))) by
VECTSP_1:def 7
.= (((a
|^ 2)
+ (a
* b))
+ (b
* (a
+ b))) by
EC_PF_1: 22
.= (((a
|^ 2)
+ (a
* b))
+ ((b
* a)
+ (b
* b))) by
VECTSP_1:def 7
.= (((a
|^ 2)
+ (a
* b))
+ ((a
* b)
+ (b
|^ 2))) by
EC_PF_1: 22
.= ((a
|^ 2)
+ ((a
* b)
+ ((a
* b)
+ (b
|^ 2)))) by
ALGSTR_1: 7
.= ((a
|^ 2)
+ (((a
* b)
+ (a
* b))
+ (b
|^ 2))) by
ALGSTR_1: 7
.= ((a
|^ 2)
+ ((g2
* (a
* b))
+ (b
|^ 2))) by
A1,
Th20
.= (((a
|^ 2)
+ (g2
* (a
* b)))
+ (b
|^ 2)) by
ALGSTR_1: 7
.= (((a
|^ 2)
+ ((g2
* a)
* b))
+ (b
|^ 2)) by
GROUP_1:def 3;
end;
theorem ::
EC_PF_2:26
Th26: for g2,a,b be
Element of (
GF p) st g2
= (2
mod p) holds ((a
- b)
|^ 2)
= (((a
|^ 2)
- ((g2
* a)
* b))
+ (b
|^ 2))
proof
let g2,a,b be
Element of (
GF p) such that
A1: g2
= (2
mod p);
thus ((a
- b)
|^ 2)
= ((a
- b)
* (a
- b)) by
EC_PF_1: 22
.= ((a
* (a
- b))
- (b
* (a
- b))) by
VECTSP_1: 13
.= (((a
* a)
- (a
* b))
- (b
* (a
- b))) by
VECTSP_1: 11
.= (((a
|^ 2)
- (a
* b))
- (b
* (a
- b))) by
EC_PF_1: 22
.= (((a
|^ 2)
- (a
* b))
- ((b
* a)
- (b
* b))) by
VECTSP_1: 11
.= (((a
|^ 2)
- (a
* b))
- ((a
* b)
- (b
|^ 2))) by
EC_PF_1: 22
.= ((a
|^ 2)
+ ((
- (a
* b))
- ((a
* b)
- (b
|^ 2)))) by
ALGSTR_1: 7
.= ((a
|^ 2)
+ ((
- (a
* b))
+ ((
- (a
* b))
+ (b
|^ 2)))) by
VECTSP_1: 17
.= ((a
|^ 2)
+ (((
- (a
* b))
- (a
* b))
+ (b
|^ 2))) by
ALGSTR_1: 7
.= ((a
|^ 2)
+ ((g2
* (
- (a
* b)))
+ (b
|^ 2))) by
A1,
Th20
.= (((a
|^ 2)
+ (g2
* (
- (a
* b))))
+ (b
|^ 2)) by
ALGSTR_1: 7
.= (((a
|^ 2)
- (g2
* (a
* b)))
+ (b
|^ 2)) by
VECTSP_1: 8
.= (((a
|^ 2)
- ((g2
* a)
* b))
+ (b
|^ 2)) by
GROUP_1:def 3;
end;
theorem ::
EC_PF_2:27
Th27: for g2,a,b,c,d be
Element of (
GF p) st g2
= (2
mod p) holds (((a
* c)
+ (b
* d))
|^ 2)
= ((((a
|^ 2)
* (c
|^ 2))
+ ((((g2
* a)
* b)
* c)
* d))
+ ((b
|^ 2)
* (d
|^ 2)))
proof
let g2,a,b,c,d be
Element of (
GF p) such that
A1: g2
= (2
mod p);
thus (((a
* c)
+ (b
* d))
|^ 2)
= ((((a
* c)
|^ 2)
+ ((g2
* (a
* c))
* (b
* d)))
+ ((b
* d)
|^ 2)) by
A1,
Th25
.= ((((a
|^ 2)
* (c
|^ 2))
+ ((g2
* (a
* c))
* (b
* d)))
+ ((b
* d)
|^ 2)) by
BINOM: 9
.= ((((a
|^ 2)
* (c
|^ 2))
+ ((g2
* (a
* c))
* (b
* d)))
+ ((b
|^ 2)
* (d
|^ 2))) by
BINOM: 9
.= ((((a
|^ 2)
* (c
|^ 2))
+ (g2
* ((a
* c)
* (b
* d))))
+ ((b
|^ 2)
* (d
|^ 2))) by
GROUP_1:def 3
.= ((((a
|^ 2)
* (c
|^ 2))
+ (g2
* (((a
* c)
* b)
* d)))
+ ((b
|^ 2)
* (d
|^ 2))) by
GROUP_1:def 3
.= ((((a
|^ 2)
* (c
|^ 2))
+ (g2
* (((a
* b)
* c)
* d)))
+ ((b
|^ 2)
* (d
|^ 2))) by
GROUP_1:def 3
.= ((((a
|^ 2)
* (c
|^ 2))
+ ((((g2
* a)
* b)
* c)
* d))
+ ((b
|^ 2)
* (d
|^ 2))) by
Th11;
end;
theorem ::
EC_PF_2:28
Th28: for p be
Prime, n be
Nat, g2 be
Element of (
GF p) st p
> 2 & g2
= (2
mod p) holds g2
<> (
0. (
GF p)) & (g2
|^ n)
<> (
0. (
GF p))
proof
let p be
Prime, n be
Nat, g2 be
Element of (
GF p) such that
A1: p
> 2 and
A2: g2
= (2
mod p);
A3: g2
<>
0 by
A1,
A2,
NAT_D: 63;
hence g2
<> (
0. (
GF p)) by
EC_PF_1: 11;
(g2
|^ n)
<>
0 by
A3,
EC_PF_1: 25;
hence (g2
|^ n)
<> (
0. (
GF p)) by
EC_PF_1: 11;
end;
theorem ::
EC_PF_2:29
for p be
Prime, n be
Nat, g2,g3 be
Element of (
GF p) st p
> 3 & g3
= (3
mod p) holds g3
<> (
0. (
GF p)) & (g3
|^ n)
<> (
0. (
GF p))
proof
let p be
Prime, n be
Nat, g2,g3 be
Element of (
GF p) such that
A1: p
> 3;
assume
A2: g3
= (3
mod p);
A3: g3
<>
0 by
A1,
A2,
NAT_D: 63;
hence g3
<> (
0. (
GF p)) by
EC_PF_1: 11;
(g3
|^ n)
<>
0 by
A3,
EC_PF_1: 25;
hence (g3
|^ n)
<> (
0. (
GF p)) by
EC_PF_1: 11;
end;
begin
definition
let p be 5
_or_greater
Prime;
::
EC_PF_2:def2
func
EC_WParam p ->
Subset of
[:the
carrier of (
GF p), the
carrier of (
GF p):] equals {
[a, b] where a,b be
Element of (
GF p) : (
Disc (a,b,p))
<> (
0. (
GF p)) };
correctness
proof
now
let x be
object;
assume x
in {
[a, b] where a,b be
Element of (
GF p) : (
Disc (a,b,p))
<> (
0. (
GF p)) };
then ex a,b be
Element of (
GF p) st x
=
[a, b] & (
Disc (a,b,p))
<> (
0. (
GF p));
hence x
in
[:the
carrier of (
GF p), the
carrier of (
GF p):];
end;
hence thesis by
TARSKI:def 3;
end;
end
registration
let p be 5
_or_greater
Prime;
cluster (
EC_WParam p) -> non
empty;
coherence
proof
ex a,b be
Element of (
GF p) st
[a, b]
in (
EC_WParam p)
proof
set a = (
1. (
GF p));
set b = (
0. (
GF p));
reconsider g2 = (2
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
reconsider g4 = (4
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
reconsider g27 = (27
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
A1: g4
= ((2
* 2)
mod p)
.= (g2
* g2) by
EC_PF_1: 18
.= (g2
|^ 2) by
EC_PF_1: 22;
p
>= (4
+ 1) by
Def1;
then
A2: p
> 2 & p
> 3 by
XXREAL_0: 2;
(
Disc (a,b,p))
= ((g4
* (a
|^ (2
+ 1)))
+ (g27
* (b
|^ 2))) by
EC_PF_1:def 7
.= ((g4
* ((a
|^ 2)
* a))
+ (g27
* (b
|^ 2))) by
EC_PF_1: 24
.= ((g4
* ((a
|^ 2)
* a))
+ (g27
* (b
* b))) by
EC_PF_1: 22
.= ((g4
* ((a
* a)
* (
1. (
GF p))))
+ (g27
* ((
0. (
GF p))
* (
0. (
GF p))))) by
EC_PF_1: 22
.= (g4
* (
1. (
GF p))) by
ALGSTR_1: 7
.= g4;
then
A3: (
Disc (a,b,p))
<> (
0. (
GF p)) by
A1,
A2,
Th28;
take a, b;
thus thesis by
A3;
end;
hence thesis;
end;
end
definition
let p be 5
_or_greater
Prime;
let z be
Element of (
EC_WParam p);
:: original:
`1
redefine
func z
`1 ->
Element of (
GF p) ;
correctness
proof
thus (z
`1 ) is
Element of (
GF p);
end;
:: original:
`2
redefine
func z
`2 ->
Element of (
GF p) ;
correctness
proof
thus (z
`2 ) is
Element of (
GF p);
end;
end
theorem ::
EC_PF_2:30
Th30: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p) holds p
> 3 & (
Disc ((z
`1 ),(z
`2 ),p))
<> (
0. (
GF p))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p);
p
>= (4
+ 1) by
Def1;
hence p
> 3 by
XXREAL_0: 2;
z
in {
[a, b] where a,b be
Element of (
GF p) : (
Disc (a,b,p))
<> (
0. (
GF p)) };
then
consider a,b be
Element of (
GF p) such that
A1: z
=
[a, b] & (
Disc (a,b,p))
<> (
0. (
GF p));
thus (
Disc ((z
`1 ),(z
`2 ),p))
<> (
0. (
GF p)) by
A1;
end;
reserve px,py,pz for
object;
reserve Px,Py,Pz for
Element of (
GF p);
reserve P for
Element of (
ProjCo (
GF p));
reserve O for
Element of (
EC_SetProjCo (a,b,p));
definition
let p be
Prime;
let a,b be
Element of (
GF p);
let P be
Element of (
EC_SetProjCo (a,b,p));
::
EC_PF_2:def3
func P
`1_3 ->
Element of (
GF p) means
:
Def3: P
=
[px, py, pz] implies it
= px;
existence
proof
P is
Element of (
ProjCo (
GF p));
then
consider Px,Py,Pz be
object such that
A1: Px
in the
carrier of (
GF p) & Py
in the
carrier of (
GF p) & Pz
in the
carrier of (
GF p) and
A2: P
=
[Px, Py, Pz] by
MCART_1: 68;
take Px;
thus thesis by
A1,
A2,
XTUPLE_0: 3;
end;
uniqueness
proof
let y,z be
Element of (
GF p);
assume
A3: P
=
[px, py, pz] implies y
= px;
assume
A4: P
=
[px, py, pz] implies z
= px;
P is
Element of (
ProjCo (
GF p));
then
consider Px,Py,Pz be
object such that Px
in the
carrier of (
GF p) & Py
in the
carrier of (
GF p) & Pz
in the
carrier of (
GF p) and
A5: P
=
[Px, Py, Pz] by
MCART_1: 68;
y
= Px by
A3,
A5;
hence thesis by
A4,
A5;
end;
::
EC_PF_2:def4
func P
`2_3 ->
Element of (
GF p) means
:
Def4: P
=
[px, py, pz] implies it
= py;
existence
proof
P is
Element of (
ProjCo (
GF p));
then
consider Px,Py,Pz be
object such that
A6: Px
in the
carrier of (
GF p) & Py
in the
carrier of (
GF p) & Pz
in the
carrier of (
GF p) and
A7: P
=
[Px, Py, Pz] by
MCART_1: 68;
take Py;
thus thesis by
A6,
A7,
XTUPLE_0: 3;
end;
uniqueness
proof
let y,z be
Element of (
GF p);
assume
A8: P
=
[px, py, pz] implies y
= py;
assume
A9: P
=
[px, py, pz] implies z
= py;
P is
Element of (
ProjCo (
GF p));
then
consider Px,Py,Pz be
object such that Px
in the
carrier of (
GF p) & Py
in the
carrier of (
GF p) & Pz
in the
carrier of (
GF p) and
A10: P
=
[Px, Py, Pz] by
MCART_1: 68;
y
= Py by
A8,
A10;
hence thesis by
A9,
A10;
end;
::
EC_PF_2:def5
func P
`3_3 ->
Element of (
GF p) means
:
Def5: P
=
[px, py, pz] implies it
= pz;
existence
proof
P is
Element of (
ProjCo (
GF p));
then
consider Px,Py,Pz be
object such that
A11: Px
in the
carrier of (
GF p) & Py
in the
carrier of (
GF p) & Pz
in the
carrier of (
GF p) and
A12: P
=
[Px, Py, Pz] by
MCART_1: 68;
take Pz;
thus thesis by
A11,
A12,
XTUPLE_0: 3;
end;
uniqueness
proof
let y,z be
Element of (
GF p);
assume
A13: P
=
[px, py, pz] implies y
= pz;
assume
A14: P
=
[px, py, pz] implies z
= pz;
P is
Element of (
ProjCo (
GF p));
then
consider Px,Py,Pz be
object such that Px
in the
carrier of (
GF p) & Py
in the
carrier of (
GF p) & Pz
in the
carrier of (
GF p) and
A15: P
=
[Px, Py, Pz] by
MCART_1: 68;
y
= Pz by
A13,
A15;
hence thesis by
A14,
A15;
end;
end
theorem ::
EC_PF_2:31
Th31: for p be
Prime, a,b be
Element of (
GF p), P be
Element of (
EC_SetProjCo (a,b,p)) holds P
=
[(P
`1_3 ), (P
`2_3 ), (P
`3_3 )]
proof
let p be
Prime, a,b be
Element of (
GF p), P be
Element of (
EC_SetProjCo (a,b,p));
P is
Element of (
ProjCo (
GF p));
then
consider Px,Py,Pz be
object such that Px
in the
carrier of (
GF p) & Py
in the
carrier of (
GF p) & Pz
in the
carrier of (
GF p) and
A1: P
=
[Px, Py, Pz] by
MCART_1: 68;
thus P
=
[(P
`1_3 ), Py, Pz] by
A1,
Def3
.=
[(P
`1_3 ), (P
`2_3 ), Pz] by
A1,
Def4
.=
[(P
`1_3 ), (P
`2_3 ), (P
`3_3 )] by
A1,
Def5;
end;
AA: for X1,X2,X3 be non
empty
set holds for x be
Element of
[:X1, X2, X3:] holds x
=
[(x
`1_3 ), (x
`2_3 ), (x
`3_3 )];
theorem ::
EC_PF_2:32
Th32: for p be
Prime, a,b be
Element of (
GF p), P be
Element of (
EC_SetProjCo (a,b,p)), Q be
Element of (
ProjCo (
GF p)) holds P
= Q iff (P
`1_3 )
= (Q
`1_3 ) & (P
`2_3 )
= (Q
`2_3 ) & (P
`3_3 )
= (Q
`3_3 )
proof
let p be
Prime, a,b be
Element of (
GF p), P be
Element of (
EC_SetProjCo (a,b,p)), Q be
Element of (
ProjCo (
GF p));
A1: P
=
[(P
`1_3 ), (P
`2_3 ), (P
`3_3 )] by
Th31;
A2: Q
=
[(Q
`1_3 ), (Q
`2_3 ), (Q
`3_3 )] by
AA;
thus P
= Q implies (P
`1_3 )
= (Q
`1_3 ) & (P
`2_3 )
= (Q
`2_3 ) & (P
`3_3 )
= (Q
`3_3 ) by
A1;
assume
A3: (P
`1_3 )
= (Q
`1_3 ) & (P
`2_3 )
= (Q
`2_3 ) & (P
`3_3 )
= (Q
`3_3 );
thus P
= Q by
A2,
A3,
Th31;
end;
theorem ::
EC_PF_2:33
for p be
Prime, a,b,Px,Py,Pz be
Element of (
GF p), P be
Element of (
EC_SetProjCo (a,b,p)) st P
=
[Px, Py, Pz] holds (P
`1_3 )
= Px & (P
`2_3 )
= Py & (P
`3_3 )
= Pz by
Def3,
Def4,
Def5;
definition
let p be
Prime;
let P be
Element of (
ProjCo (
GF p));
let CEQ be
Function of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):], (
GF p);
::
EC_PF_2:def6
pred P
is_on_curve CEQ means (CEQ
. P)
= (
0. (
GF p));
correctness ;
end
theorem ::
EC_PF_2:34
Th34: P
is_on_curve (
EC_WEqProjCo (a,b,p)) iff P is
Element of (
EC_SetProjCo (a,b,p))
proof
hereby
assume P
is_on_curve (
EC_WEqProjCo (a,b,p));
then P
in { Q where Q be
Element of (
ProjCo (
GF p)) : ((
EC_WEqProjCo (a,b,p))
. Q)
= (
0. (
GF p)) };
hence P is
Element of (
EC_SetProjCo (a,b,p)) by
EC_PF_1:def 9;
end;
assume P is
Element of (
EC_SetProjCo (a,b,p));
then P
in (
EC_SetProjCo (a,b,p));
then P
in { Q where Q be
Element of (
ProjCo (
GF p)) : ((
EC_WEqProjCo (a,b,p))
. Q)
= (
0. (
GF p)) } by
EC_PF_1:def 9;
then ex Q be
Element of (
ProjCo (
GF p)) st P
= Q & ((
EC_WEqProjCo (a,b,p))
. Q)
= (
0. (
GF p));
hence P
is_on_curve (
EC_WEqProjCo (a,b,p));
end;
theorem ::
EC_PF_2:35
Th35: for p be
Prime, a,b be
Element of (
GF p), P be
Element of (
EC_SetProjCo (a,b,p)) holds ((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3))))
= (
0. (
GF p))
proof
let p be
Prime, a,b be
Element of (
GF p), P be
Element of (
EC_SetProjCo (a,b,p));
consider PP be
Element of (
ProjCo (
GF p)) such that
A1: PP
= P & PP
in (
EC_SetProjCo (a,b,p));
A2: (PP
`1_3 )
= (P
`1_3 ) & (PP
`2_3 )
= (P
`2_3 ) & (PP
`3_3 )
= (P
`3_3 ) by
A1,
Th32;
P
is_on_curve (
EC_WEqProjCo (a,b,p)) by
Th34;
hence thesis by
A2,
EC_PF_1:def 8,
A1;
end;
definition
let p be
Prime;
let P be
Element of (
ProjCo (
GF p));
::
EC_PF_2:def7
func
rep_pt (P) ->
Element of (
ProjCo (
GF p)) equals
:
Def7:
[((P
`1_3 )
* ((P
`3_3 )
" )), ((P
`2_3 )
* ((P
`3_3 )
" )), 1] if (P
`3_3 )
<>
0 ,
[
0 , 1,
0 ] if (P
`3_3 )
=
0 ;
coherence
proof
per cases ;
suppose
A1: (P
`3_3 )
<>
0 ;
[((P
`1_3 )
* ((P
`3_3 )
" )), ((P
`2_3 )
* ((P
`3_3 )
" )), (
1. (
GF p))]
<>
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))] by
XTUPLE_0: 3;
then not
[((P
`1_3 )
* ((P
`3_3 )
" )), ((P
`2_3 )
* ((P
`3_3 )
" )), (
1. (
GF p))]
in
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]} by
TARSKI:def 1;
then
[((P
`1_3 )
* ((P
`3_3 )
" )), ((P
`2_3 )
* ((P
`3_3 )
" )), (
1. (
GF p))]
in (
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):]
\
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]}) by
XBOOLE_0:def 5;
then
[((P
`1_3 )
* ((P
`3_3 )
" )), ((P
`2_3 )
* ((P
`3_3 )
" )), (
1. (
GF p))]
in (
ProjCo (
GF p)) by
EC_PF_1:def 6;
hence thesis by
A1,
EC_PF_1: 12;
end;
suppose
A2: (P
`3_3 )
=
0 ;
set Q =
[
0 , 1,
0 ];
thus thesis by
A2,
EC_PF_1: 42;
end;
end;
consistency ;
end
theorem ::
EC_PF_2:36
Th36: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds (
rep_pt P)
_EQ_ P & (
rep_pt P)
in (
EC_SetProjCo ((z
`1 ),(z
`2 ),p))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
set a = (z
`1 );
set b = (z
`2 );
A1: p
> 3 & (
Disc (a,b,p))
<> (
0. (
GF p)) by
Th30;
consider PP be
Element of (
ProjCo (
GF p)) such that
A2: PP
= P & PP
in (
EC_SetProjCo (a,b,p));
per cases ;
suppose (P
`3_3 )
=
0 ;
then
A3: (PP
`3_3 )
=
0 by
A2,
Th32;
consider Q be
Element of (
ProjCo (
GF p)) such that
A4: Q
in (
EC_SetProjCo (a,b,p)) & Q
_EQ_ PP and
A5: (Q
`1_3 )
=
0 & (Q
`2_3 )
= 1 & (Q
`3_3 )
=
0 by
A1,
A2,
A3,
EC_PF_1: 49;
(
rep_pt PP)
=
[
0 , 1,
0 ] by
A3,
Def7;
hence thesis by
A2,
A4,
A5,
AA;
end;
suppose (P
`3_3 )
<>
0 ;
then
A6: (PP
`3_3 )
<>
0 by
A2,
Th32;
consider Q be
Element of (
ProjCo (
GF p)) such that
A7: Q
in (
EC_SetProjCo (a,b,p)) & Q
_EQ_ PP and
A8: (Q
`3_3 )
= 1 by
A1,
A2,
A6,
EC_PF_1: 48;
consider d be
Element of (
GF p) such that d
<> (
0. (
GF p)) and
A9: (Q
`1_3 )
= (d
* (PP
`1_3 )) & (Q
`2_3 )
= (d
* (PP
`2_3 )) & (Q
`3_3 )
= (d
* (PP
`3_3 )) by
A7,
EC_PF_1:def 10;
A10: (d
* (PP
`3_3 ))
= (
1. (
GF p)) by
A8,
A9,
EC_PF_1: 12;
(PP
`3_3 )
<> (
0. (
GF p)) by
A6,
EC_PF_1: 11;
then d
= ((PP
`3_3 )
" ) by
A10,
VECTSP_1:def 10;
then Q
=
[((PP
`1_3 )
* ((PP
`3_3 )
" )), ((PP
`2_3 )
* ((PP
`3_3 )
" )), 1] by
A8,
A9,
AA;
hence thesis by
A2,
A6,
A7,
Def7;
end;
end;
theorem ::
EC_PF_2:37
Th37: for p be
Prime, a,b be
Element of (
GF p), P be
Element of (
ProjCo (
GF p)) holds ((
rep_pt P)
`3_3 )
=
0 implies (
rep_pt P)
=
[
0 , 1,
0 ] & (P
`3_3 )
=
0
proof
let p be
Prime, a,b be
Element of (
GF p), P be
Element of (
ProjCo (
GF p));
assume
A1: ((
rep_pt P)
`3_3 )
=
0 ;
hereby
assume
A2: (
rep_pt P)
<>
[
0 , 1,
0 ];
(
rep_pt P)
=
[((P
`1_3 )
* ((P
`3_3 )
" )), ((P
`2_3 )
* ((P
`3_3 )
" )), 1] by
A2,
Def7;
hence contradiction by
A1;
end;
assume
A3: (P
`3_3 )
<>
0 ;
(
rep_pt P)
=
[((P
`1_3 )
* ((P
`3_3 )
" )), ((P
`2_3 )
* ((P
`3_3 )
" )), 1] by
A3,
Def7;
hence contradiction by
A1;
end;
theorem ::
EC_PF_2:38
Th38: for p be
Prime, a,b be
Element of (
GF p), P be
Element of (
ProjCo (
GF p)) holds ((
rep_pt P)
`3_3 )
<>
0 implies (
rep_pt P)
=
[((P
`1_3 )
* ((P
`3_3 )
" )), ((P
`2_3 )
* ((P
`3_3 )
" )), 1] & (P
`3_3 )
<>
0
proof
let p be
Prime, a,b be
Element of (
GF p), P be
Element of (
ProjCo (
GF p));
assume
A1: ((
rep_pt P)
`3_3 )
<>
0 ;
hereby
assume
A2: (
rep_pt P)
<>
[((P
`1_3 )
* ((P
`3_3 )
" )), ((P
`2_3 )
* ((P
`3_3 )
" )), 1];
(
rep_pt P)
=
[
0 , 1,
0 ] by
A2,
Def7;
hence contradiction by
A1;
end;
assume
A3: (P
`3_3 )
=
0 ;
(
rep_pt P)
=
[
0 , 1,
0 ] by
A3,
Def7;
hence contradiction by
A1;
end;
theorem ::
EC_PF_2:39
Th39: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds P
_EQ_ Q iff (
rep_pt P)
= (
rep_pt Q)
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
set a = (z
`1 );
set b = (z
`2 );
consider PP be
Element of (
ProjCo (
GF p)) such that
A1: PP
= P & PP
in (
EC_SetProjCo (a,b,p));
consider QQ be
Element of (
ProjCo (
GF p)) such that
A2: QQ
= Q & QQ
in (
EC_SetProjCo (a,b,p));
set RP = (
rep_pt PP);
set RQ = (
rep_pt QQ);
hereby
assume
A3: P
_EQ_ Q;
RP
_EQ_ P by
A1,
Th36;
then
A4: RP
_EQ_ Q by
A3,
EC_PF_1: 44;
RQ
_EQ_ Q by
A2,
Th36;
then RP
_EQ_ RQ by
A4,
EC_PF_1: 44;
then
consider a be
Element of (
GF p) such that
A5: a
<> (
0. (
GF p)) and
A6: (RP
`1_3 )
= (a
* (RQ
`1_3 )) & (RP
`2_3 )
= (a
* (RQ
`2_3 )) & (RP
`3_3 )
= (a
* (RQ
`3_3 )) by
EC_PF_1:def 10;
per cases ;
suppose
A7: (PP
`3_3 )
=
0 ;
then RP
=
[
0 , 1,
0 ] by
Def7;
then (RP
`3_3 )
= (
0. (
GF p)) by
EC_PF_1: 11;
then (RQ
`3_3 )
= (
0. (
GF p)) by
A5,
A6,
VECTSP_1: 12
.=
0 by
EC_PF_1: 11;
then RQ
=
[
0 , 1,
0 ] by
Th37;
hence (
rep_pt P)
= (
rep_pt Q) by
A1,
A2,
A7,
Def7;
end;
suppose (PP
`3_3 )
<>
0 ;
then RP
=
[((PP
`1_3 )
* ((PP
`3_3 )
" )), ((PP
`2_3 )
* ((PP
`3_3 )
" )), 1] by
Def7;
then
A8: (RP
`3_3 )
= (
1. (
GF p)) by
EC_PF_1: 12;
then (RQ
`3_3 )
<> (
0. (
GF p)) by
A6;
then (RQ
`3_3 )
<>
0 by
EC_PF_1: 11;
then RQ
=
[((QQ
`1_3 )
* ((QQ
`3_3 )
" )), ((QQ
`2_3 )
* ((QQ
`3_3 )
" )), 1] by
Th38;
then (RQ
`3_3 )
= (
1. (
GF p)) by
EC_PF_1: 12;
then a
= (
1. (
GF p)) by
A6,
A8;
then (RP
`1_3 )
= (RQ
`1_3 ) & (RP
`2_3 )
= (RQ
`2_3 ) & (RP
`3_3 )
= (RQ
`3_3 ) by
A6;
then RP
=
[(RQ
`1_3 ), (RQ
`2_3 ), (RQ
`3_3 )] by
AA
.= RQ by
AA;
hence (
rep_pt P)
= (
rep_pt Q) by
A1,
A2;
end;
end;
assume
A9: (
rep_pt P)
= (
rep_pt Q);
A10: RQ
_EQ_ P by
A2,
A9,
Th36;
RQ
_EQ_ Q by
A2,
Th36;
hence thesis by
A10,
EC_PF_1: 44;
end;
begin
Lm1: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds
[(P
`1_3 ), (
- (P
`2_3 )), (P
`3_3 )] is
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
set a = (z
`1 );
set b = (z
`2 );
set R =
[(P
`1_3 ), (
- (P
`2_3 )), (P
`3_3 )];
P
in (
ProjCo (
GF p));
then P
in (
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):]
\
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]}) by
EC_PF_1:def 6;
then not P
in
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]} by
XBOOLE_0:def 5;
then P
<>
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))] by
TARSKI:def 1;
then not ((P
`1_3 )
= (
0. (
GF p)) & (P
`2_3 )
= (
0. (
GF p)) & (P
`3_3 )
= (
0. (
GF p))) by
Th31;
then not ((P
`1_3 )
= (
0. (
GF p)) & (
- (P
`2_3 ))
= (
0. (
GF p)) & (P
`3_3 )
= (
0. (
GF p))) by
VECTSP_2: 3;
then R
<>
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))] by
XTUPLE_0: 3;
then not R
in
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]} by
TARSKI:def 1;
then R
in (
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):]
\
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]}) by
XBOOLE_0:def 5;
then
reconsider R as
Element of (
ProjCo (
GF p)) by
EC_PF_1:def 6;
A1: ((R
`2_3 )
|^ 2)
= ((P
`2_3 )
|^ 2) by
Th1;
((((R
`2_3 )
|^ 2)
* (R
`3_3 ))
- ((((R
`1_3 )
|^ 3)
+ ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (b
* ((R
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
A1,
Th35;
then R
is_on_curve (
EC_WEqProjCo (a,b,p)) by
EC_PF_1:def 8;
hence thesis by
Th34;
end;
definition
let p be 5
_or_greater
Prime;
let z be
Element of (
EC_WParam p);
::
EC_PF_2:def8
func
compell_ProjCo (z,p) ->
Function of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) means
:
Def8: for P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds (it
. P)
=
[(P
`1_3 ), (
- (P
`2_3 )), (P
`3_3 )];
existence
proof
set a = (z
`1 );
set b = (z
`2 );
defpred
P[
Element of (
EC_SetProjCo (a,b,p)),
set] means $2
=
[($1
`1_3 ), (
- ($1
`2_3 )), ($1
`3_3 )];
A1: for P be
Element of (
EC_SetProjCo (a,b,p)) holds ex R be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st
P[P, R]
proof
let P be
Element of (
EC_SetProjCo (a,b,p));
set R =
[(P
`1_3 ), (
- (P
`2_3 )), (P
`3_3 )];
[(P
`1_3 ), (
- (P
`2_3 )), (P
`3_3 )] is
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) by
Lm1;
hence thesis;
end;
consider f be
Function of (
EC_SetProjCo (a,b,p)), (
EC_SetProjCo (a,b,p)) such that
A2: for P be
Element of (
EC_SetProjCo (a,b,p)) holds
P[P, (f
. P)] from
FUNCT_2:sch 3(
A1);
take f;
thus thesis by
A2;
end;
uniqueness
proof
set a = (z
`1 );
set b = (z
`2 );
deffunc
F(
Element of (
EC_SetProjCo (a,b,p))) =
[($1
`1_3 ), (
- ($1
`2_3 )), ($1
`3_3 )];
for f1,f2 be
Function of (
EC_SetProjCo (a,b,p)), (
EC_SetProjCo (a,b,p)) st (for x be
Element of (
EC_SetProjCo (a,b,p)) holds (f1
. x)
=
F(x)) & (for x be
Element of (
EC_SetProjCo (a,b,p)) holds (f2
. x)
=
F(x)) holds f1
= f2
proof
let f1,f2 be
Function of (
EC_SetProjCo (a,b,p)), (
EC_SetProjCo (a,b,p)) such that
A3: for x be
Element of (
EC_SetProjCo (a,b,p)) holds (f1
. x)
=
F(x) and
A4: for x be
Element of (
EC_SetProjCo (a,b,p)) holds (f2
. x)
=
F(x);
now
let x be
Element of (
EC_SetProjCo (a,b,p));
thus (f1
. x)
=
F(x) by
A3
.= (f2
. x) by
A4;
end;
hence thesis by
FUNCT_2: 63;
end;
hence thesis;
end;
end
definition
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p);
let F be
Function of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
let P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
:: original:
.
redefine
func F
. P ->
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) ;
correctness
proof
(F
. P)
in (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
hence thesis;
end;
end
theorem ::
EC_PF_2:40
Th40: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), O be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st O
=
[
0 , 1,
0 ] holds ((
compell_ProjCo (z,p))
. O)
_EQ_ O
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), O be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) such that
A1: O
=
[
0 , 1,
0 ];
set a = (z
`1 );
set b = (z
`2 );
A2: (O
`1_3 )
=
0 & (O
`2_3 )
= 1 & (O
`3_3 )
=
0 by
A1,
Def3,
Def4,
Def5;
consider OO be
Element of (
ProjCo (
GF p)) such that
A3: OO
= O & OO
in (
EC_SetProjCo (a,b,p));
A4: (OO
`1_3 )
=
0 & (OO
`2_3 )
= 1 & (OO
`3_3 )
=
0 by
A2,
A3,
Th32;
set CO = ((
compell_ProjCo (z,p))
. O);
consider COO be
Element of (
ProjCo (
GF p)) such that
A5: COO
= CO & COO
in (
EC_SetProjCo (a,b,p));
A6: (COO
`1_3 )
= (CO
`1_3 ) & (COO
`2_3 )
= (CO
`2_3 ) & (COO
`3_3 )
= (CO
`3_3 ) by
A5,
Th32;
CO
=
[(O
`1_3 ), (
- (O
`2_3 )), (O
`3_3 )] by
Def8;
then (COO
`3_3 )
=
0 by
A2,
A6,
Def5;
then
A7: (
rep_pt CO)
=
[
0 , 1,
0 ] by
A5,
Def7;
(
rep_pt O)
=
[
0 , 1,
0 ] by
A3,
A4,
Def7;
hence thesis by
A7,
Th39;
end;
theorem ::
EC_PF_2:41
Th41: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds ((
compell_ProjCo (z,p))
. ((
compell_ProjCo (z,p))
. P))
= P
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
set Q = ((
compell_ProjCo (z,p))
. P);
Q
=
[(P
`1_3 ), (
- (P
`2_3 )), (P
`3_3 )] by
Def8;
then
A1: (Q
`1_3 )
= (P
`1_3 ) & (Q
`2_3 )
= (
- (P
`2_3 )) & (Q
`3_3 )
= (P
`3_3 ) by
Def3,
Def4,
Def5;
set R = ((
compell_ProjCo (z,p))
. Q);
R
=
[(Q
`1_3 ), (
- (Q
`2_3 )), (Q
`3_3 )] by
Def8;
then (R
`1_3 )
= (P
`1_3 ) & (R
`2_3 )
= (
- (
- (P
`2_3 ))) & (R
`3_3 )
= (P
`3_3 ) by
A1,
Def3,
Def4,
Def5;
then (R
`1_3 )
= (P
`1_3 ) & (R
`2_3 )
= (P
`2_3 ) & (R
`3_3 )
= (P
`3_3 ) by
RLVECT_1: 17;
then R
=
[(P
`1_3 ), (P
`2_3 ), (P
`3_3 )] by
Th31;
hence thesis by
Th31;
end;
theorem ::
EC_PF_2:42
Th42: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st (P
`3_3 )
<>
0 holds (
rep_pt ((
compell_ProjCo (z,p))
. P))
= ((
compell_ProjCo (z,p))
. (
rep_pt P))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) such that
A1: (P
`3_3 )
<>
0 ;
set a = (z
`1 );
set b = (z
`2 );
consider PP be
Element of (
ProjCo (
GF p)) such that
A2: PP
= P & PP
in (
EC_SetProjCo (a,b,p));
set CP = ((
compell_ProjCo (z,p))
. P);
CP
=
[(P
`1_3 ), (
- (P
`2_3 )), (P
`3_3 )] by
Def8;
then
A3: (CP
`1_3 )
= (P
`1_3 ) & (CP
`2_3 )
= (
- (P
`2_3 )) & (CP
`3_3 )
= (P
`3_3 ) by
Def3,
Def4,
Def5;
set RP = (
rep_pt PP);
reconsider RP as
Element of (
EC_SetProjCo (a,b,p)) by
A2,
Th36;
(PP
`3_3 )
<>
0 by
A1,
A2,
Th32;
then RP
=
[((PP
`1_3 )
* ((PP
`3_3 )
" )), ((PP
`2_3 )
* ((PP
`3_3 )
" )), 1] by
Def7;
then (RP
`1_3 )
= ((PP
`1_3 )
* ((PP
`3_3 )
" )) & (RP
`2_3 )
= ((PP
`2_3 )
* ((PP
`3_3 )
" )) & (RP
`3_3 )
= 1 by
Def3,
Def4,
Def5;
then (RP
`1_3 )
= ((P
`1_3 )
* ((PP
`3_3 )
" )) & (RP
`2_3 )
= ((P
`2_3 )
* ((PP
`3_3 )
" )) & (RP
`3_3 )
= 1 by
A2,
Th32;
then
A4: (RP
`1_3 )
= ((P
`1_3 )
* ((P
`3_3 )
" )) & (RP
`2_3 )
= ((P
`2_3 )
* ((P
`3_3 )
" )) & (RP
`3_3 )
= 1 by
A2,
Th32;
consider CPP be
Element of (
ProjCo (
GF p)) such that
A5: CPP
= CP & CPP
in (
EC_SetProjCo (a,b,p));
set RCP = (
rep_pt CPP);
reconsider RCP as
Element of (
EC_SetProjCo (a,b,p)) by
A5,
Th36;
(CPP
`3_3 )
<>
0 by
A1,
A3,
A5,
Th32;
then RCP
=
[((CPP
`1_3 )
* ((CPP
`3_3 )
" )), ((CPP
`2_3 )
* ((CPP
`3_3 )
" )), 1] by
Def7;
then (RCP
`1_3 )
= ((CPP
`1_3 )
* ((CPP
`3_3 )
" )) & (RCP
`2_3 )
= ((CPP
`2_3 )
* ((CPP
`3_3 )
" )) & (RCP
`3_3 )
= 1 by
Def3,
Def4,
Def5;
then (RCP
`1_3 )
= ((CP
`1_3 )
* ((CPP
`3_3 )
" )) & (RCP
`2_3 )
= ((CP
`2_3 )
* ((CPP
`3_3 )
" )) & (RCP
`3_3 )
= 1 by
A5,
Th32;
then (RCP
`1_3 )
= ((CP
`1_3 )
* ((CP
`3_3 )
" )) & (RCP
`2_3 )
= ((CP
`2_3 )
* ((CP
`3_3 )
" )) & (RCP
`3_3 )
= 1 by
A5,
Th32;
then
A6: RCP
=
[((P
`1_3 )
* ((P
`3_3 )
" )), ((
- (P
`2_3 ))
* ((P
`3_3 )
" )), 1] by
A3,
Th31;
set CRP = ((
compell_ProjCo (z,p))
. RP);
CRP
=
[((P
`1_3 )
* ((P
`3_3 )
" )), (
- ((P
`2_3 )
* ((P
`3_3 )
" ))), 1] by
A4,
Def8;
hence thesis by
A2,
A5,
A6,
VECTSP_1: 9;
end;
theorem ::
EC_PF_2:43
Th43: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds P
= Q iff ((
compell_ProjCo (z,p))
. P)
= ((
compell_ProjCo (z,p))
. Q)
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
thus P
= Q implies ((
compell_ProjCo (z,p))
. P)
= ((
compell_ProjCo (z,p))
. Q);
assume
A1: ((
compell_ProjCo (z,p))
. P)
= ((
compell_ProjCo (z,p))
. Q);
thus P
= ((
compell_ProjCo (z,p))
. ((
compell_ProjCo (z,p))
. Q)) by
A1,
Th41
.= Q by
Th41;
end;
theorem ::
EC_PF_2:44
Th44: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st (P
`3_3 )
<>
0 holds P
_EQ_ ((
compell_ProjCo (z,p))
. P) iff (P
`2_3 )
=
0
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) such that
A1: (P
`3_3 )
<>
0 ;
set a = (z
`1 );
set b = (z
`2 );
consider PP be
Element of (
ProjCo (
GF p)) such that
A2: PP
= P & PP
in (
EC_SetProjCo (a,b,p));
A3: (PP
`1_3 )
= (P
`1_3 ) & (PP
`2_3 )
= (P
`2_3 ) & (PP
`3_3 )
= (P
`3_3 ) by
A2,
Th32;
set CP = ((
compell_ProjCo (z,p))
. P);
CP
=
[(P
`1_3 ), (
- (P
`2_3 )), (P
`3_3 )] by
Def8;
then
A4: (CP
`1_3 )
= (P
`1_3 ) & (CP
`2_3 )
= (
- (P
`2_3 )) & (CP
`3_3 )
= (P
`3_3 ) by
Def3,
Def4,
Def5;
set RP = (
rep_pt PP);
reconsider RP as
Element of (
EC_SetProjCo (a,b,p)) by
A2,
Th36;
RP
=
[((P
`1_3 )
* ((P
`3_3 )
" )), ((P
`2_3 )
* ((P
`3_3 )
" )), 1] by
A1,
A3,
Def7;
then
A5: (RP
`2_3 )
= ((P
`2_3 )
* ((P
`3_3 )
" )) by
Def4;
consider CPP be
Element of (
ProjCo (
GF p)) such that
A6: CPP
= CP & CPP
in (
EC_SetProjCo (a,b,p));
A7: (CPP
`1_3 )
= (P
`1_3 ) & (CPP
`2_3 )
= (
- (P
`2_3 )) & (CPP
`3_3 )
= (P
`3_3 ) by
A4,
A6,
Th32;
set RCP = (
rep_pt CPP);
reconsider RCP as
Element of (
EC_SetProjCo (a,b,p)) by
A6,
Th36;
A8: RCP
=
[((P
`1_3 )
* ((P
`3_3 )
" )), ((
- (P
`2_3 ))
* ((P
`3_3 )
" )), 1] by
A1,
A7,
Def7;
hereby
assume
A9: P
_EQ_ ((
compell_ProjCo (z,p))
. P);
(P
`3_3 )
<> (
0. (
GF p)) by
A1,
EC_PF_1: 11;
then
A10: ((P
`3_3 )
" )
<> (
0. (
GF p)) by
VECTSP_1: 25;
A11: p
> 2 by
Th30,
XXREAL_0: 2;
RP
=
[((P
`1_3 )
* ((P
`3_3 )
" )), ((
- (P
`2_3 ))
* ((P
`3_3 )
" )), 1] by
A2,
A6,
A8,
A9,
Th39;
then ((P
`2_3 )
* ((P
`3_3 )
" ))
= (((P
`3_3 )
" )
* (
- (P
`2_3 ))) by
A5,
Def4;
then (P
`2_3 )
= (
- (P
`2_3 )) by
A10,
VECTSP_1: 5;
then ((P
`2_3 )
+ (P
`2_3 ))
= (
0. (
GF p)) by
VECTSP_1: 19;
then (P
`2_3 )
= (
0. (
GF p)) by
A11,
EC_PF_1: 27;
hence (P
`2_3 )
=
0 by
EC_PF_1: 11;
end;
assume
A12: (P
`2_3 )
=
0 ;
then (P
`2_3 )
= (
0. (
GF p)) by
EC_PF_1: 11;
then (
- (P
`2_3 ))
= (
0. (
GF p)) by
VECTSP_2: 3;
then (CPP
`1_3 )
= (P
`1_3 ) & (CPP
`2_3 )
=
0 & (CPP
`3_3 )
= (P
`3_3 ) by
A7,
EC_PF_1: 11;
hence P
_EQ_ ((
compell_ProjCo (z,p))
. P) by
A6,
A12,
Th32;
end;
theorem ::
EC_PF_2:45
Th45: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st (P
`3_3 )
<>
0 holds ((P
`1_3 )
= (Q
`1_3 ) & (P
`3_3 )
= (Q
`3_3 )) iff P
= Q or P
= ((
compell_ProjCo (z,p))
. Q)
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) such that
A1: (P
`3_3 )
<>
0 ;
set a = (z
`1 );
set b = (z
`2 );
A2: (P
`3_3 )
<> (
0. (
GF p)) by
A1,
EC_PF_1: 11;
hereby
assume
A3: (P
`1_3 )
= (Q
`1_3 ) & (P
`3_3 )
= (Q
`3_3 );
A4: ((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
Th35;
((((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))
- ((((Q
`1_3 )
|^ 3)
+ ((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2)))
+ (b
* ((Q
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
Th35;
then
A5: (((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))
= ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((Q
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3))) by
A3,
VECTSP_1: 19
.= (((P
`2_3 )
|^ 2)
* (P
`3_3 )) by
A3,
A4,
VECTSP_1: 19;
((P
`2_3 )
* (P
`2_3 ))
= ((P
`2_3 )
|^ 2) by
EC_PF_1: 22
.= ((Q
`2_3 )
|^ 2) by
A2,
A3,
A5,
VECTSP_1: 5
.= ((Q
`2_3 )
* (Q
`2_3 )) by
EC_PF_1: 22;
then (P
`2_3 )
= (Q
`2_3 ) or (P
`2_3 )
= (
- (Q
`2_3 )) by
EC_PF_1: 26;
then P
=
[(Q
`1_3 ), (Q
`2_3 ), (Q
`3_3 )] or P
=
[(Q
`1_3 ), (
- (Q
`2_3 )), (Q
`3_3 )] by
A3,
Th31;
hence P
= Q or P
= ((
compell_ProjCo (z,p))
. Q) by
Th31,
Def8;
end;
assume
A6: P
= Q or P
= ((
compell_ProjCo (z,p))
. Q);
P
=
[(Q
`1_3 ), (Q
`2_3 ), (Q
`3_3 )] or P
=
[(Q
`1_3 ), (
- (Q
`2_3 )), (Q
`3_3 )] by
A6,
Th31,
Def8;
hence (P
`1_3 )
= (Q
`1_3 ) & (P
`3_3 )
= (Q
`3_3 ) by
Def3,
Def5;
end;
theorem ::
EC_PF_2:46
Th46: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds P
_EQ_ Q iff ((
compell_ProjCo (z,p))
. P)
_EQ_ ((
compell_ProjCo (z,p))
. Q)
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
set a = (z
`1 );
set b = (z
`2 );
consider PP be
Element of (
ProjCo (
GF p)) such that
A1: PP
= P & PP
in (
EC_SetProjCo (a,b,p));
A2: (PP
`1_3 )
= (P
`1_3 ) & (PP
`2_3 )
= (P
`2_3 ) & (PP
`3_3 )
= (P
`3_3 ) by
A1,
Th32;
consider QQ be
Element of (
ProjCo (
GF p)) such that
A3: QQ
= Q & QQ
in (
EC_SetProjCo (a,b,p));
A4: (QQ
`1_3 )
= (Q
`1_3 ) & (QQ
`2_3 )
= (Q
`2_3 ) & (QQ
`3_3 )
= (Q
`3_3 ) by
A3,
Th32;
set CP = ((
compell_ProjCo (z,p))
. P);
consider CPP be
Element of (
ProjCo (
GF p)) such that
A5: CPP
= CP & CPP
in (
EC_SetProjCo (a,b,p));
A6: (CPP
`1_3 )
= (CP
`1_3 ) & (CPP
`2_3 )
= (CP
`2_3 ) & (CPP
`3_3 )
= (CP
`3_3 ) by
A5,
Th32;
set CQ = ((
compell_ProjCo (z,p))
. Q);
consider CQQ be
Element of (
ProjCo (
GF p)) such that
A7: CQQ
= CQ & CQQ
in (
EC_SetProjCo (a,b,p));
A8: (CQQ
`1_3 )
= (CQ
`1_3 ) & (CQQ
`2_3 )
= (CQ
`2_3 ) & (CQQ
`3_3 )
= (CQ
`3_3 ) by
A7,
Th32;
hereby
assume
A9: P
_EQ_ Q;
A10: (
rep_pt P)
= (
rep_pt Q) by
A9,
Th39;
per cases ;
suppose
A11: (P
`3_3 )
=
0 ;
(PP
`3_3 )
=
0 by
A1,
A11,
Th32;
then (
rep_pt QQ)
=
[
0 , 1,
0 ] by
A1,
A3,
A10,
Def7;
then ((
rep_pt QQ)
`3_3 )
=
0 ;
then
A12: (Q
`3_3 )
=
0 by
A4,
Th37;
CP
=
[(P
`1_3 ), (
- (P
`2_3 )),
0 ] by
A11,
Def8;
then
A13: (CPP
`3_3 )
=
0 by
A6,
Def5;
CQ
=
[(Q
`1_3 ), (
- (Q
`2_3 )),
0 ] by
A12,
Def8;
then (CQQ
`3_3 )
=
0 by
A8,
Def5;
then (
rep_pt CQ)
=
[
0 , 1,
0 ] by
A7,
Def7
.= (
rep_pt CP) by
A5,
A13,
Def7;
hence ((
compell_ProjCo (z,p))
. P)
_EQ_ ((
compell_ProjCo (z,p))
. Q) by
Th39;
end;
suppose
A14: (P
`3_3 )
<>
0 ;
(
rep_pt QQ)
=
[((PP
`1_3 )
* ((PP
`3_3 )
" )), ((PP
`2_3 )
* ((PP
`3_3 )
" )), 1] by
A1,
A2,
A3,
A10,
A14,
Def7;
then
A15: ((
rep_pt QQ)
`3_3 )
<>
0 ;
((
compell_ProjCo (z,p))
. (
rep_pt P))
= (
rep_pt ((
compell_ProjCo (z,p))
. Q)) by
A4,
A10,
A15,
Th38,
Th42;
then (
rep_pt ((
compell_ProjCo (z,p))
. P))
= (
rep_pt ((
compell_ProjCo (z,p))
. Q)) by
A14,
Th42;
hence ((
compell_ProjCo (z,p))
. P)
_EQ_ ((
compell_ProjCo (z,p))
. Q) by
Th39;
end;
end;
assume
A16: ((
compell_ProjCo (z,p))
. P)
_EQ_ ((
compell_ProjCo (z,p))
. Q);
per cases ;
suppose
A17: (P
`3_3 )
=
0 ;
CP
=
[(P
`1_3 ), (
- (P
`2_3 )),
0 ] by
A17,
Def8;
then (CPP
`3_3 )
=
0 by
A6,
Def5;
then
A18: (
rep_pt CPP)
=
[
0 , 1,
0 ] by
Def7;
(
rep_pt CQQ)
=
[
0 , 1,
0 ] by
A5,
A7,
A16,
A18,
Th39;
then ((
rep_pt CQQ)
`3_3 )
=
0 ;
then
A19: (CQ
`3_3 )
=
0 by
A8,
Th37;
CQ
=
[(Q
`1_3 ), (
- (Q
`2_3 )), (Q
`3_3 )] by
Def8;
then (Q
`3_3 )
=
0 by
A19,
Def5;
then
A20: (
rep_pt QQ)
=
[
0 , 1,
0 ] by
A4,
Def7;
(
rep_pt PP)
=
[
0 , 1,
0 ] by
A2,
A17,
Def7;
hence P
_EQ_ Q by
A1,
A3,
A20,
Th39;
end;
suppose
A21: (P
`3_3 )
<>
0 ;
CP
=
[(P
`1_3 ), (
- (P
`2_3 )), (P
`3_3 )] by
Def8;
then
A22: (CPP
`3_3 )
<>
0 by
A6,
A21,
Def5;
set RP = (
rep_pt P);
reconsider RP as
Element of (
EC_SetProjCo (a,b,p)) by
Th36;
set RQ = (
rep_pt Q);
reconsider RQ as
Element of (
EC_SetProjCo (a,b,p)) by
Th36;
A23: (
rep_pt CP)
= (
rep_pt CQ) by
A16,
Th39;
(
rep_pt CPP)
=
[((CPP
`1_3 )
* ((CPP
`3_3 )
" )), ((CPP
`2_3 )
* ((CPP
`3_3 )
" )), 1] by
A22,
Def7;
then ((
rep_pt CQQ)
`3_3 )
<>
0 by
A5,
A7,
A23;
then
A24: (CQ
`3_3 )
<>
0 by
A8,
Th38;
CQ
=
[(Q
`1_3 ), (
- (Q
`2_3 )), (Q
`3_3 )] by
Def8;
then
A25: (Q
`3_3 )
<>
0 by
A24,
Def5;
A26: (
rep_pt CP)
= ((
compell_ProjCo (z,p))
. RP) by
A21,
Th42;
(
rep_pt CQ)
= ((
compell_ProjCo (z,p))
. RQ) by
A25,
Th42;
then RP
= RQ by
A23,
A26,
Th43;
hence P
_EQ_ Q by
Th39;
end;
end;
theorem ::
EC_PF_2:47
Th47: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds P
_EQ_ ((
compell_ProjCo (z,p))
. Q) iff ((
compell_ProjCo (z,p))
. P)
_EQ_ Q
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
set a = (z
`1 );
set b = (z
`2 );
set CP = ((
compell_ProjCo (z,p))
. P);
reconsider CP as
Element of (
EC_SetProjCo (a,b,p));
set CQ = ((
compell_ProjCo (z,p))
. Q);
reconsider CQ as
Element of (
EC_SetProjCo (a,b,p));
hereby
assume
A1: P
_EQ_ ((
compell_ProjCo (z,p))
. Q);
((
compell_ProjCo (z,p))
. P)
_EQ_ ((
compell_ProjCo (z,p))
. CQ) by
A1,
Th46;
hence ((
compell_ProjCo (z,p))
. P)
_EQ_ Q by
Th41;
end;
assume
A2: ((
compell_ProjCo (z,p))
. P)
_EQ_ Q;
((
compell_ProjCo (z,p))
. CP)
_EQ_ ((
compell_ProjCo (z,p))
. Q) by
A2,
Th46;
hence P
_EQ_ ((
compell_ProjCo (z,p))
. Q) by
Th41;
end;
theorem ::
EC_PF_2:48
Th48: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st (P
`3_3 )
<>
0 & (Q
`3_3 )
<>
0 holds (
rep_pt P)
= ((
compell_ProjCo (z,p))
. (
rep_pt Q)) iff P
_EQ_ ((
compell_ProjCo (z,p))
. Q)
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) such that
A1: (P
`3_3 )
<>
0 & (Q
`3_3 )
<>
0 ;
set a = (z
`1 );
set b = (z
`2 );
set CQ = ((
compell_ProjCo (z,p))
. Q);
reconsider CQ as
Element of (
EC_SetProjCo (a,b,p));
hereby
assume
A2: (
rep_pt P)
= ((
compell_ProjCo (z,p))
. (
rep_pt Q));
(
rep_pt P)
= (
rep_pt CQ) by
A1,
A2,
Th42;
hence P
_EQ_ ((
compell_ProjCo (z,p))
. Q) by
Th39;
end;
assume P
_EQ_ ((
compell_ProjCo (z,p))
. Q);
hence (
rep_pt P)
= (
rep_pt CQ) by
Th39
.= ((
compell_ProjCo (z,p))
. (
rep_pt Q)) by
A1,
Th42;
end;
theorem ::
EC_PF_2:49
for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds P
_EQ_ Q implies ((P
`2_3 )
* (Q
`3_3 ))
= ((Q
`2_3 )
* (P
`3_3 ))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
set a = (z
`1 );
set b = (z
`2 );
consider PP be
Element of (
ProjCo (
GF p)) such that
A1: PP
= P & PP
in (
EC_SetProjCo (a,b,p));
A2: (PP
`1_3 )
= (P
`1_3 ) & (PP
`2_3 )
= (P
`2_3 ) & (PP
`3_3 )
= (P
`3_3 ) by
A1,
Th32;
consider QQ be
Element of (
ProjCo (
GF p)) such that
A3: QQ
= Q & QQ
in (
EC_SetProjCo (a,b,p));
A4: (QQ
`1_3 )
= (Q
`1_3 ) & (QQ
`2_3 )
= (Q
`2_3 ) & (QQ
`3_3 )
= (Q
`3_3 ) by
A3,
Th32;
assume
A5: P
_EQ_ Q;
A6: (
rep_pt PP)
= (
rep_pt QQ) by
A1,
A3,
A5,
Th39;
per cases ;
suppose
A7: (PP
`3_3 )
=
0 ;
then (P
`3_3 )
= (
0. (
GF p)) by
A2,
EC_PF_1: 11;
then
A8: ((Q
`2_3 )
* (P
`3_3 ))
= (
0. (
GF p));
(
rep_pt QQ)
=
[
0 , 1,
0 ] by
A6,
A7,
Def7;
then ((
rep_pt QQ)
`3_3 )
=
0 ;
then (Q
`3_3 )
=
0 by
A4,
Th37
.= (
0. (
GF p)) by
EC_PF_1: 11;
hence ((P
`2_3 )
* (Q
`3_3 ))
= ((Q
`2_3 )
* (P
`3_3 )) by
A8;
end;
suppose
A9: (PP
`3_3 )
<>
0 ;
then
A10: (PP
`3_3 )
<> (
0. (
GF p)) by
EC_PF_1: 11;
A11: (
rep_pt QQ)
=
[((PP
`1_3 )
* ((PP
`3_3 )
" )), ((PP
`2_3 )
* ((PP
`3_3 )
" )), 1] by
A6,
A9,
Def7;
then
A12: ((
rep_pt QQ)
`2_3 )
= ((PP
`2_3 )
* ((PP
`3_3 )
" ));
A13: ((
rep_pt QQ)
`3_3 )
<>
0 by
A11;
then (QQ
`3_3 )
<>
0 by
Th38;
then
A14: (QQ
`3_3 )
<> (
0. (
GF p)) by
EC_PF_1: 11;
(
rep_pt QQ)
=
[((QQ
`1_3 )
* ((QQ
`3_3 )
" )), ((QQ
`2_3 )
* ((QQ
`3_3 )
" )), 1] by
A13,
Th38;
then ((PP
`2_3 )
* ((PP
`3_3 )
" ))
= ((QQ
`2_3 )
* ((QQ
`3_3 )
" )) by
A12;
hence ((P
`2_3 )
* (Q
`3_3 ))
= ((Q
`2_3 )
* (P
`3_3 )) by
A2,
A4,
A10,
A14,
Th3;
end;
end;
theorem ::
EC_PF_2:50
Th50: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st (P
`3_3 )
<>
0 & (Q
`3_3 )
<>
0 holds P
_EQ_ Q or P
_EQ_ ((
compell_ProjCo (z,p))
. Q) iff ((P
`1_3 )
* (Q
`3_3 ))
= ((Q
`1_3 )
* (P
`3_3 ))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) such that
A1: (P
`3_3 )
<>
0 & (Q
`3_3 )
<>
0 ;
A2: (P
`3_3 )
<> (
0. (
GF p)) & (Q
`3_3 )
<> (
0. (
GF p)) by
A1,
EC_PF_1: 11;
set a = (z
`1 );
set b = (z
`2 );
consider PP be
Element of (
ProjCo (
GF p)) such that
A3: PP
= P & PP
in (
EC_SetProjCo (a,b,p));
A4: (PP
`1_3 )
= (P
`1_3 ) & (PP
`2_3 )
= (P
`2_3 ) & (PP
`3_3 )
= (P
`3_3 ) by
A3,
Th32;
consider QQ be
Element of (
ProjCo (
GF p)) such that
A5: QQ
= Q & QQ
in (
EC_SetProjCo (a,b,p));
A6: (QQ
`1_3 )
= (Q
`1_3 ) & (QQ
`2_3 )
= (Q
`2_3 ) & (QQ
`3_3 )
= (Q
`3_3 ) by
A5,
Th32;
A7: (PP
`3_3 )
<>
0 & (QQ
`3_3 )
<>
0 by
A1,
A3,
A5,
Th32;
set RP = (
rep_pt PP);
reconsider RP as
Element of (
EC_SetProjCo (a,b,p)) by
A3,
Th36;
set RQ = (
rep_pt QQ);
reconsider RQ as
Element of (
EC_SetProjCo (a,b,p)) by
A5,
Th36;
A8: RP
=
[((PP
`1_3 )
* ((PP
`3_3 )
" )), ((PP
`2_3 )
* ((PP
`3_3 )
" )), 1] by
A7,
Def7;
RQ
=
[((QQ
`1_3 )
* ((QQ
`3_3 )
" )), ((QQ
`2_3 )
* ((QQ
`3_3 )
" )), 1] by
A7,
Def7;
then
A9: (RQ
`1_3 )
= ((QQ
`1_3 )
* ((QQ
`3_3 )
" )) & (RQ
`3_3 )
= 1 by
Def3,
Def5;
then
A10: (RP
`3_3 )
= (RQ
`3_3 ) by
A8,
Def5;
A11: (RP
`3_3 )
<>
0 by
A8,
Def5;
then (RP
`1_3 )
= (RQ
`1_3 ) implies (
rep_pt P)
= (
rep_pt Q) or RP
= ((
compell_ProjCo (z,p))
. RQ) by
A3,
A5,
A10,
Th45;
then
A12: ((P
`1_3 )
* ((PP
`3_3 )
" ))
= ((Q
`1_3 )
* ((QQ
`3_3 )
" )) implies P
_EQ_ Q or RP
= ((
compell_ProjCo (z,p))
. RQ) by
A4,
A6,
A8,
A9,
Def3,
Th39;
RP
= RQ or RP
= ((
compell_ProjCo (z,p))
. RQ) implies (RP
`1_3 )
= (RQ
`1_3 ) by
A11,
Th45;
then P
_EQ_ Q or P
_EQ_ ((
compell_ProjCo (z,p))
. Q) implies ((P
`1_3 )
* ((P
`3_3 )
" ))
= ((Q
`1_3 )
* ((Q
`3_3 )
" )) by
A1,
A3,
A4,
A5,
A6,
A8,
A9,
Def3,
Th39,
Th48;
hence thesis by
A2,
A1,
A3,
A4,
A5,
A6,
A12,
Th3,
Th4,
Th48;
end;
theorem ::
EC_PF_2:51
Th51: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st (P
`3_3 )
<>
0 & (Q
`3_3 )
<>
0 & (P
`2_3 )
<>
0 holds P
_EQ_ ((
compell_ProjCo (z,p))
. Q) implies ((P
`2_3 )
* (Q
`3_3 ))
<> ((Q
`2_3 )
* (P
`3_3 ))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) such that
A1: (P
`3_3 )
<>
0 & (Q
`3_3 )
<>
0 & (P
`2_3 )
<>
0 ;
A2: (P
`3_3 )
<> (
0. (
GF p)) & (Q
`3_3 )
<> (
0. (
GF p)) by
A1,
EC_PF_1: 11;
set a = (z
`1 );
set b = (z
`2 );
consider PP be
Element of (
ProjCo (
GF p)) such that
A3: PP
= P & PP
in (
EC_SetProjCo (a,b,p));
A4: (PP
`1_3 )
= (P
`1_3 ) & (PP
`2_3 )
= (P
`2_3 ) & (PP
`3_3 )
= (P
`3_3 ) by
A3,
Th32;
consider QQ be
Element of (
ProjCo (
GF p)) such that
A5: QQ
= Q & QQ
in (
EC_SetProjCo (a,b,p));
A6: (QQ
`1_3 )
= (Q
`1_3 ) & (QQ
`2_3 )
= (Q
`2_3 ) & (QQ
`3_3 )
= (Q
`3_3 ) by
A5,
Th32;
assume
A7: P
_EQ_ ((
compell_ProjCo (z,p))
. Q);
assume
A8: ((P
`2_3 )
* (Q
`3_3 ))
= ((Q
`2_3 )
* (P
`3_3 ));
((P
`1_3 )
* (Q
`3_3 ))
= ((Q
`1_3 )
* (P
`3_3 )) by
A1,
A7,
Th50;
then
A9: ((P
`1_3 )
* ((P
`3_3 )
" ))
= ((Q
`1_3 )
* ((Q
`3_3 )
" )) by
A2,
Th4;
A10: ((P
`2_3 )
* ((P
`3_3 )
" ))
= ((Q
`2_3 )
* ((Q
`3_3 )
" )) by
A2,
A8,
Th4;
(
rep_pt P)
=
[((PP
`1_3 )
* ((PP
`3_3 )
" )), ((PP
`2_3 )
* ((PP
`3_3 )
" )), 1] by
A1,
A3,
A4,
Def7
.= (
rep_pt Q) by
A1,
A4,
A5,
A6,
A9,
A10,
Def7;
then
A11: P
_EQ_ Q by
Th39;
((
compell_ProjCo (z,p))
. P)
_EQ_ Q by
A7,
Th47;
then P
_EQ_ ((
compell_ProjCo (z,p))
. P) by
A11,
EC_PF_1: 44;
hence contradiction by
A1,
Th44;
end;
theorem ::
EC_PF_2:52
Th52: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds ( not P
_EQ_ Q) & P
_EQ_ ((
compell_ProjCo (z,p))
. Q) implies ((P
`2_3 )
* (Q
`3_3 ))
<> ((Q
`2_3 )
* (P
`3_3 ))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
set a = (z
`1 );
set b = (z
`2 );
A1: p
> 3 & (
Disc (a,b,p))
<> (
0. (
GF p)) by
Th30;
assume
A2: not P
_EQ_ Q;
assume
A3: P
_EQ_ ((
compell_ProjCo (z,p))
. Q);
consider PP be
Element of (
ProjCo (
GF p)) such that
A4: PP
= P & PP
in (
EC_SetProjCo (a,b,p));
A5: (P
`3_3 )
<>
0
proof
assume
A6: (P
`3_3 )
=
0 ;
set CQ = ((
compell_ProjCo (z,p))
. Q);
reconsider CQ as
Element of (
EC_SetProjCo (a,b,p));
A7: (PP
`3_3 )
=
0 by
A6,
A4,
Th32;
consider OO be
Element of (
ProjCo (
GF p)) such that
A8: OO
in (
EC_SetProjCo (a,b,p)) and
A9: PP
_EQ_ OO and
A10: (OO
`1_3 )
=
0 & (OO
`2_3 )
= 1 & (OO
`3_3 )
=
0 by
A1,
A4,
A7,
EC_PF_1: 49;
consider O be
Element of (
EC_SetProjCo (a,b,p)) such that
A11: O
= OO by
A8;
A12: (O
`1_3 )
=
0 & (O
`2_3 )
= 1 & (O
`3_3 )
=
0 by
A10,
A11,
Th32;
CQ
_EQ_ O by
A3,
A4,
A9,
A11,
EC_PF_1: 44;
then ((
compell_ProjCo (z,p))
. CQ)
_EQ_ ((
compell_ProjCo (z,p))
. O) by
Th46;
then
A13: Q
_EQ_ ((
compell_ProjCo (z,p))
. O) by
Th41;
((
compell_ProjCo (z,p))
. O)
_EQ_ O by
A12,
Th31,
Th40;
then Q
_EQ_ O by
A13,
EC_PF_1: 44;
hence contradiction by
A2,
A4,
A9,
A11,
EC_PF_1: 44;
end;
A14: (Q
`3_3 )
<>
0
proof
assume
A15: (Q
`3_3 )
=
0 ;
set CQ = ((
compell_ProjCo (z,p))
. Q);
reconsider CQ as
Element of (
ProjCo (
GF p));
CQ
=
[(Q
`1_3 ), (
- (Q
`2_3 )), (Q
`3_3 )] by
Def8;
then
A16: (CQ
`3_3 )
=
0 by
A15;
consider OO be
Element of (
ProjCo (
GF p)) such that
A17: OO
in (
EC_SetProjCo (a,b,p)) and
A18: CQ
_EQ_ OO and
A19: (OO
`1_3 )
=
0 & (OO
`2_3 )
= 1 & (OO
`3_3 )
=
0 by
A1,
A16,
EC_PF_1: 49;
consider O be
Element of (
EC_SetProjCo (a,b,p)) such that
A20: O
= OO by
A17;
A21: (O
`1_3 )
=
0 & (O
`2_3 )
= 1 & (O
`3_3 )
=
0 by
A19,
A20,
Th32;
P
_EQ_ O by
A3,
A18,
A20,
EC_PF_1: 44;
then
A22: ((
compell_ProjCo (z,p))
. P)
_EQ_ ((
compell_ProjCo (z,p))
. O) by
Th46;
((
compell_ProjCo (z,p))
. O)
_EQ_ O by
A21,
Th31,
Th40;
then ((
compell_ProjCo (z,p))
. P)
_EQ_ O by
A22,
EC_PF_1: 44;
then CQ
_EQ_ ((
compell_ProjCo (z,p))
. P) by
A18,
A20,
EC_PF_1: 44;
hence contradiction by
A2,
Th46;
end;
(P
`2_3 )
<>
0
proof
assume
A23: (P
`2_3 )
=
0 ;
P
_EQ_ ((
compell_ProjCo (z,p))
. P) by
A5,
A23,
Th44;
then ((
compell_ProjCo (z,p))
. P)
_EQ_ ((
compell_ProjCo (z,p))
. Q) by
A3,
EC_PF_1: 44;
hence contradiction by
A2,
Th46;
end;
hence thesis by
A3,
A5,
A14,
Th51;
end;
theorem ::
EC_PF_2:53
Th53: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g3 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st g3
= (3
mod p) & (P
`2_3 )
=
0 & (P
`3_3 )
<>
0 holds (((z
`1 )
* ((P
`3_3 )
|^ 2))
+ (g3
* ((P
`1_3 )
|^ 2)))
<>
0
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g3 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) such that
A1: g3
= (3
mod p) and
A2: (P
`2_3 )
=
0 and
A3: (P
`3_3 )
<>
0 ;
set a = (z
`1 );
set b = (z
`2 );
A4: p
> 3 & (
Disc (a,b,p))
<> (
0. (
GF p)) by
Th30;
((P
`3_3 )
|^ 2)
<>
0 by
A3,
EC_PF_1: 25;
then
A5: ((P
`3_3 )
|^ 2)
<> (
0. (
GF p)) by
EC_PF_1: 11;
0
= (
0. (
GF p)) by
EC_PF_1: 11;
then
A6: ((P
`2_3 )
|^ 2)
= (
0. (
GF p)) by
A2,
Th5;
reconsider g2 = (2
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
reconsider g4 = (4
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
reconsider g9 = (9
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
reconsider g27 = (27
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
A7: 3
= (2
+ 1);
A8: (g2
|^ 2)
= (g2
* g2) by
EC_PF_1: 22
.= ((2
* 2)
mod p) by
EC_PF_1: 18
.= g4;
A9: (g3
|^ 2)
= (g3
* g3) by
EC_PF_1: 22
.= ((3
* 3)
mod p) by
A1,
EC_PF_1: 18
.= g9;
A10: (g3
|^ 3)
= (g3
|^ (2
+ 1))
.= (g9
* g3) by
A9,
EC_PF_1: 24
.= ((9
* 3)
mod p) by
A1,
EC_PF_1: 18
.= g27;
assume
A11: ((a
* ((P
`3_3 )
|^ 2))
+ (g3
* ((P
`1_3 )
|^ 2)))
=
0 ;
((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
Th35;
then (((P
`2_3 )
|^ 2)
* (P
`3_3 ))
= ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3))) by
VECTSP_1: 19;
then (g3
* ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
A6;
then ((g3
* (((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))))
+ (g3
* (b
* ((P
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
VECTSP_1:def 7;
then (((g3
* ((P
`1_3 )
|^ 3))
+ (g3
* ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))))
+ (g3
* (b
* ((P
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
VECTSP_1:def 7;
then (((g3
* ((P
`1_3 )
|^ (2
+ 1)))
+ (g3
* (a
* ((P
`1_3 )
* ((P
`3_3 )
|^ 2)))))
+ (g3
* (b
* ((P
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
GROUP_1:def 3;
then (((g3
* ((P
`1_3 )
|^ (2
+ 1)))
+ ((g3
* a)
* ((P
`1_3 )
* ((P
`3_3 )
|^ 2))))
+ (g3
* (b
* ((P
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
GROUP_1:def 3;
then (((g3
* ((P
`1_3 )
|^ (2
+ 1)))
+ (((g3
* a)
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (g3
* (b
* ((P
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
GROUP_1:def 3;
then (((g3
* ((P
`1_3 )
|^ (2
+ 1)))
+ (((g3
* a)
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ ((g3
* b)
* ((P
`3_3 )
|^ 3)))
= (
0. (
GF p)) by
GROUP_1:def 3;
then (((g3
* (((P
`1_3 )
|^ 2)
* (P
`1_3 )))
+ (((g3
* a)
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ ((g3
* b)
* ((P
`3_3 )
|^ (2
+ 1))))
= (
0. (
GF p)) by
EC_PF_1: 24;
then (((g3
* (((P
`1_3 )
|^ 2)
* (P
`1_3 )))
+ (((g3
* a)
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ ((g3
* b)
* (((P
`3_3 )
|^ 2)
* (P
`3_3 ))))
= (
0. (
GF p)) by
EC_PF_1: 24;
then ((((g3
* ((P
`1_3 )
|^ 2))
* (P
`1_3 ))
+ (((g3
* a)
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ ((g3
* b)
* ((P
`3_3 )
* ((P
`3_3 )
|^ 2))))
= (
0. (
GF p)) by
GROUP_1:def 3;
then ((((g3
* ((P
`1_3 )
|^ 2))
* (P
`1_3 ))
+ (((g3
* a)
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (((g3
* b)
* (P
`3_3 ))
* ((P
`3_3 )
|^ 2)))
= (
0. (
GF p)) by
GROUP_1:def 3;
then (((g3
* ((P
`1_3 )
|^ 2))
* (P
`1_3 ))
+ ((((P
`3_3 )
|^ 2)
* ((g3
* a)
* (P
`1_3 )))
+ (((P
`3_3 )
|^ 2)
* ((g3
* b)
* (P
`3_3 )))))
= (
0. (
GF p)) by
ALGSTR_1: 7;
then
A12: (((g3
* ((P
`1_3 )
|^ 2))
* (P
`1_3 ))
+ (((P
`3_3 )
|^ 2)
* (((g3
* a)
* (P
`1_3 ))
+ ((g3
* b)
* (P
`3_3 )))))
= (
0. (
GF p)) by
VECTSP_1:def 7;
A13: ((a
* ((P
`3_3 )
|^ 2))
+ (g3
* ((P
`1_3 )
|^ 2)))
= (
0. (
GF p)) by
A11,
EC_PF_1: 11;
then (((
- (a
* ((P
`3_3 )
|^ 2)))
* (P
`1_3 ))
+ (((P
`3_3 )
|^ 2)
* (((g3
* a)
* (P
`1_3 ))
+ ((g3
* b)
* (P
`3_3 )))))
= (
0. (
GF p)) by
A12,
VECTSP_1: 16;
then ((((
- a)
* ((P
`3_3 )
|^ 2))
* (P
`1_3 ))
+ (((P
`3_3 )
|^ 2)
* (((g3
* a)
* (P
`1_3 ))
+ ((g3
* b)
* (P
`3_3 )))))
= (
0. (
GF p)) by
VECTSP_1: 9;
then ((((P
`3_3 )
|^ 2)
* ((
- a)
* (P
`1_3 )))
+ (((P
`3_3 )
|^ 2)
* (((g3
* a)
* (P
`1_3 ))
+ ((g3
* b)
* (P
`3_3 )))))
= (
0. (
GF p)) by
GROUP_1:def 3;
then (((P
`3_3 )
|^ 2)
* (((
- a)
* (P
`1_3 ))
+ (((g3
* a)
* (P
`1_3 ))
+ ((g3
* b)
* (P
`3_3 )))))
= (
0. (
GF p)) by
VECTSP_1:def 7;
then (((
- a)
* (P
`1_3 ))
+ (((g3
* a)
* (P
`1_3 ))
+ ((g3
* b)
* (P
`3_3 ))))
= (
0. (
GF p)) by
A5,
VECTSP_1: 12;
then ((((
- a)
* (P
`1_3 ))
+ ((g3
* a)
* (P
`1_3 )))
+ ((g3
* b)
* (P
`3_3 )))
= (
0. (
GF p)) by
ALGSTR_1: 7;
then (((
- (a
* (P
`1_3 )))
+ ((g3
* a)
* (P
`1_3 )))
+ ((g3
* b)
* (P
`3_3 )))
= (
0. (
GF p)) by
VECTSP_1: 9;
then (((g3
* (a
* (P
`1_3 )))
- (a
* (P
`1_3 )))
+ ((g3
* b)
* (P
`3_3 )))
= (
0. (
GF p)) by
GROUP_1:def 3;
then ((g2
* (a
* (P
`1_3 )))
+ ((g3
* b)
* (P
`3_3 )))
= (
0. (
GF p)) by
A1,
A7,
Th23;
then (g2
* (a
* (P
`1_3 )))
= (
- ((g3
* b)
* (P
`3_3 ))) by
VECTSP_1: 16;
then ((g2
* (a
* (P
`1_3 )))
|^ 2)
= (((g3
* b)
* (P
`3_3 ))
|^ 2) by
Th1;
then ((g2
|^ 2)
* ((a
* (P
`1_3 ))
|^ 2))
= (((g3
* b)
* (P
`3_3 ))
|^ 2) by
BINOM: 9;
then ((g2
|^ 2)
* ((a
* (P
`1_3 ))
|^ 2))
= ((g3
* (b
* (P
`3_3 )))
|^ 2) by
GROUP_1:def 3;
then ((g2
|^ 2)
* ((a
|^ 2)
* ((P
`1_3 )
|^ 2)))
= ((g3
* (b
* (P
`3_3 )))
|^ 2) by
BINOM: 9;
then ((g2
|^ 2)
* ((a
|^ 2)
* ((P
`1_3 )
|^ 2)))
= ((g3
|^ 2)
* ((b
* (P
`3_3 ))
|^ 2)) by
BINOM: 9;
then ((g2
|^ 2)
* ((a
|^ 2)
* ((P
`1_3 )
|^ 2)))
= ((g3
|^ 2)
* ((b
|^ 2)
* ((P
`3_3 )
|^ 2))) by
BINOM: 9;
then (((g2
|^ 2)
* (a
|^ 2))
* ((P
`1_3 )
|^ 2))
= ((g3
|^ 2)
* ((b
|^ 2)
* ((P
`3_3 )
|^ 2))) by
GROUP_1:def 3;
then
A14: (((g2
|^ 2)
* (a
|^ 2))
* ((P
`1_3 )
|^ 2))
= (((g3
|^ 2)
* (b
|^ 2))
* ((P
`3_3 )
|^ 2)) by
GROUP_1:def 3;
(((g2
|^ 2)
* (a
|^ 2))
* ((a
* ((P
`3_3 )
|^ 2))
+ (g3
* ((P
`1_3 )
|^ 2))))
= (
0. (
GF p)) by
A13;
then ((((g2
|^ 2)
* (a
|^ 2))
* (a
* ((P
`3_3 )
|^ 2)))
+ (((g2
|^ 2)
* (a
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2))))
= (
0. (
GF p)) by
VECTSP_1:def 7;
then (((g2
|^ 2)
* ((a
|^ 2)
* (a
* ((P
`3_3 )
|^ 2))))
+ (((g2
|^ 2)
* (a
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2))))
= (
0. (
GF p)) by
GROUP_1:def 3;
then (((g2
|^ 2)
* (((a
|^ 2)
* a)
* ((P
`3_3 )
|^ 2)))
+ (((g2
|^ 2)
* (a
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2))))
= (
0. (
GF p)) by
GROUP_1:def 3;
then (((g2
|^ 2)
* ((a
|^ (2
+ 1))
* ((P
`3_3 )
|^ 2)))
+ ((g3
* ((P
`1_3 )
|^ 2))
* ((g2
|^ 2)
* (a
|^ 2))))
= (
0. (
GF p)) by
EC_PF_1: 24;
then (((g2
|^ 2)
* ((a
|^ 3)
* ((P
`3_3 )
|^ 2)))
+ (g3
* (((P
`1_3 )
|^ 2)
* ((g2
|^ 2)
* (a
|^ 2)))))
= (
0. (
GF p)) by
GROUP_1:def 3;
then ((((g2
|^ 2)
* (a
|^ 3))
* ((P
`3_3 )
|^ 2))
+ (g3
* (((g3
|^ 2)
* (b
|^ 2))
* ((P
`3_3 )
|^ 2))))
= (
0. (
GF p)) by
A14,
GROUP_1:def 3;
then ((((g2
|^ 2)
* (a
|^ 3))
* ((P
`3_3 )
|^ 2))
+ (g3
* ((g3
|^ 2)
* ((b
|^ 2)
* ((P
`3_3 )
|^ 2)))))
= (
0. (
GF p)) by
GROUP_1:def 3;
then ((((g2
|^ 2)
* (a
|^ 3))
* ((P
`3_3 )
|^ 2))
+ ((g3
* (g3
|^ 2))
* ((b
|^ 2)
* ((P
`3_3 )
|^ 2))))
= (
0. (
GF p)) by
GROUP_1:def 3;
then ((((P
`3_3 )
|^ 2)
* ((g2
|^ 2)
* (a
|^ 3)))
+ ((g3
|^ (2
+ 1))
* ((b
|^ 2)
* ((P
`3_3 )
|^ 2))))
= (
0. (
GF p)) by
EC_PF_1: 24;
then ((((P
`3_3 )
|^ 2)
* ((g2
|^ 2)
* (a
|^ 3)))
+ (((g3
|^ 3)
* (b
|^ 2))
* ((P
`3_3 )
|^ 2)))
= (
0. (
GF p)) by
GROUP_1:def 3;
then (((P
`3_3 )
|^ 2)
* (((g2
|^ 2)
* (a
|^ 3))
+ ((g3
|^ 3)
* (b
|^ 2))))
= (
0. (
GF p)) by
VECTSP_1:def 7;
then ((g4
* (a
|^ 3))
+ (g27
* (b
|^ 2)))
= (
0. (
GF p)) by
A5,
A8,
A10,
VECTSP_1: 12;
hence contradiction by
A4,
EC_PF_1:def 7;
end;
theorem ::
EC_PF_2:54
Th54: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,gf1,gf2,gf3 be
Element of (
GF p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
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 ))))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,gf1,gf2,gf3 be
Element of (
GF p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):] such 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 ))];
(((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;
end;
theorem ::
EC_PF_2:55
Th55: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,gf1,gf2,gf3 be
Element of (
GF p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
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
|^ 2)
* (((((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`3_3 )))))
+ ((((P
`3_3 )
* (Q
`3_3 ))
* (R
`3_3 ))
* (gf1
|^ 2)))
= (
0. (
GF p))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,gf1,gf2,gf3 be
Element of (
GF p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):] such that
A1: g2
= (2
mod p) and
A2: 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
A3: 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 ))];
A4: ((((P
`3_3 )
* (Q
`3_3 ))
* (R
`3_3 ))
* (gf1
|^ 2))
= ((R
`3_3 )
* (((P
`3_3 )
* (Q
`3_3 ))
* (gf1
|^ 2))) by
GROUP_1:def 3
.= ((R
`3_3 )
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))) by
GROUP_1:def 3;
A5: ((((P
`3_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`3_3 )))
= ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (R
`3_3 )) by
VECTSP_1:def 7
.= (((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
+ (
0. (
GF p)))
* (R
`3_3 )) by
ALGSTR_1: 7
.= (((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
+ (((Q
`3_3 )
* (P
`1_3 ))
- ((Q
`3_3 )
* (P
`1_3 ))))
* (R
`3_3 )) by
VECTSP_1: 19
.= ((((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
+ ((Q
`3_3 )
* (P
`1_3 )))
- ((Q
`3_3 )
* (P
`1_3 )))
* (R
`3_3 )) by
ALGSTR_1: 7
.= ((((((Q
`3_3 )
* (P
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
+ ((P
`3_3 )
* (Q
`1_3 )))
- ((Q
`3_3 )
* (P
`1_3 )))
* (R
`3_3 )) by
ALGSTR_1: 7
.= ((((g2
* ((Q
`3_3 )
* (P
`1_3 )))
+ ((Q
`1_3 )
* (P
`3_3 )))
- ((P
`1_3 )
* (Q
`3_3 )))
* (R
`3_3 )) by
A1,
Th20
.= ((R
`3_3 )
* ((g2
* ((Q
`3_3 )
* (P
`1_3 )))
+ gf2)) by
A2,
ALGSTR_1: 7;
A6: ((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 )))
= ((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`3_3 ))
* (gf2
* gf3))) by
A3
.= ((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (gf2
* gf3)) by
Th11
.= (((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* gf2)
* gf3) by
GROUP_1:def 3
.= ((((gf2
* (gf2
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
* gf3) by
Th11
.= ((((gf2
|^ (2
+ 1))
* (P
`3_3 ))
* (Q
`3_3 ))
* gf3) by
EC_PF_1: 24
.= ((R
`3_3 )
* gf3) by
A3;
A7: ((gf2
|^ 2)
* (((((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`3_3 ))))
= ((gf2
|^ 2)
* ((((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 ))
+ ((R
`3_3 )
* ((g2
* ((Q
`3_3 )
* (P
`1_3 )))
+ gf2)))) by
A5,
ALGSTR_1: 7
.= (((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 )))
+ ((gf2
|^ 2)
* ((R
`3_3 )
* ((g2
* ((Q
`3_3 )
* (P
`1_3 )))
+ gf2)))) by
VECTSP_1:def 7
.= (((R
`3_3 )
* gf3)
+ ((R
`3_3 )
* ((gf2
|^ 2)
* ((g2
* ((Q
`3_3 )
* (P
`1_3 )))
+ gf2)))) by
A6,
GROUP_1:def 3
.= ((R
`3_3 )
* (gf3
+ ((gf2
|^ 2)
* ((g2
* ((Q
`3_3 )
* (P
`1_3 )))
+ gf2)))) by
VECTSP_1:def 7;
gf3
= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- ((((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 ))
+ (gf2
|^ (2
+ 1)))) by
A2,
VECTSP_1: 17
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- (((((gf2
|^ 2)
* g2)
* (P
`1_3 ))
* (Q
`3_3 ))
+ ((gf2
|^ 2)
* gf2))) by
EC_PF_1: 24
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- ((((gf2
|^ 2)
* g2)
* ((P
`1_3 )
* (Q
`3_3 )))
+ ((gf2
|^ 2)
* gf2))) by
GROUP_1:def 3
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- (((gf2
|^ 2)
* (g2
* ((P
`1_3 )
* (Q
`3_3 ))))
+ ((gf2
|^ 2)
* gf2))) by
GROUP_1:def 3
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- ((gf2
|^ 2)
* ((g2
* ((Q
`3_3 )
* (P
`1_3 )))
+ gf2))) by
VECTSP_1:def 7;
then ((gf2
|^ 2)
* (((((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`3_3 ))))
= ((R
`3_3 )
* ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
+ ((
- ((gf2
|^ 2)
* ((g2
* ((Q
`3_3 )
* (P
`1_3 )))
+ gf2)))
+ ((gf2
|^ 2)
* ((g2
* ((Q
`3_3 )
* (P
`1_3 )))
+ gf2))))) by
A7,
ALGSTR_1: 7
.= ((R
`3_3 )
* ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
+ (
0. (
GF p)))) by
VECTSP_1: 19
.= ((R
`3_3 )
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))) by
ALGSTR_1: 7;
hence ((
- ((gf2
|^ 2)
* (((((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`3_3 )))))
+ ((((P
`3_3 )
* (Q
`3_3 ))
* (R
`3_3 ))
* (gf1
|^ 2)))
= (
0. (
GF p)) by
A4,
RLVECT_1: 5;
end;
theorem ::
EC_PF_2:56
Th56: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,gf1,gf2,gf3 be
Element of (
GF p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
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 (((((z
`2 )
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
= ((
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 )))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,gf1,gf2,gf3 be
Element of (
GF p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):] such that
A1: g2
= (2
mod p) and
A2: 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
A3: 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 ))];
set a = (z
`1 );
set b = (z
`2 );
A4: ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
= (((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (P
`2_3 )))
- ((((Q
`2_3 )
* (P
`3_3 ))
- ((P
`2_3 )
* (Q
`3_3 )))
* (P
`1_3 ))) by
A2,
VECTSP_1: 11
.= (((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (P
`2_3 )))
- ((((Q
`2_3 )
* (P
`3_3 ))
* (P
`1_3 ))
- (((P
`2_3 )
* (Q
`3_3 ))
* (P
`1_3 )))) by
VECTSP_1: 11
.= (((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (P
`2_3 )))
+ ((((P
`2_3 )
* (Q
`3_3 ))
* (P
`1_3 ))
- (((Q
`2_3 )
* (P
`3_3 ))
* (P
`1_3 )))) by
VECTSP_1: 17
.= (((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
- ((P
`1_3 )
* ((Q
`3_3 )
* (P
`2_3 ))))
+ ((((P
`2_3 )
* (Q
`3_3 ))
* (P
`1_3 ))
- (((Q
`2_3 )
* (P
`3_3 ))
* (P
`1_3 )))) by
GROUP_1:def 3
.= ((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
+ ((
- ((P
`1_3 )
* ((Q
`3_3 )
* (P
`2_3 ))))
+ ((((P
`2_3 )
* (Q
`3_3 ))
* (P
`1_3 ))
- (((Q
`2_3 )
* (P
`3_3 ))
* (P
`1_3 ))))) by
ALGSTR_1: 7
.= ((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
+ (((
- ((P
`1_3 )
* ((P
`2_3 )
* (Q
`3_3 ))))
+ (((P
`2_3 )
* (Q
`3_3 ))
* (P
`1_3 )))
- (((Q
`2_3 )
* (P
`3_3 ))
* (P
`1_3 )))) by
ALGSTR_1: 7
.= ((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
+ ((
0. (
GF p))
- (((P
`3_3 )
* (Q
`2_3 ))
* (P
`1_3 )))) by
VECTSP_1: 19
.= ((((P
`3_3 )
* (Q
`1_3 ))
* (P
`2_3 ))
- (((P
`3_3 )
* (Q
`2_3 ))
* (P
`1_3 ))) by
VECTSP_1: 18
.= (((P
`3_3 )
* ((Q
`1_3 )
* (P
`2_3 )))
- (((P
`3_3 )
* (Q
`2_3 ))
* (P
`1_3 ))) by
GROUP_1:def 3
.= (((P
`3_3 )
* ((P
`2_3 )
* (Q
`1_3 )))
- ((P
`3_3 )
* ((Q
`2_3 )
* (P
`1_3 )))) by
GROUP_1:def 3
.= ((P
`3_3 )
* (((P
`2_3 )
* (Q
`1_3 ))
- ((Q
`2_3 )
* (P
`1_3 )))) by
VECTSP_1: 11;
((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 )))
= (((((Q
`1_3 )
* (P
`3_3 ))
* (Q
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (Q
`2_3 )))
- ((((Q
`2_3 )
* (P
`3_3 ))
- ((P
`2_3 )
* (Q
`3_3 )))
* (Q
`1_3 ))) by
A2,
VECTSP_1: 11
.= (((((Q
`1_3 )
* (P
`3_3 ))
* (Q
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (Q
`2_3 )))
- ((((Q
`2_3 )
* (P
`3_3 ))
* (Q
`1_3 ))
- (((P
`2_3 )
* (Q
`3_3 ))
* (Q
`1_3 )))) by
VECTSP_1: 11
.= (((((Q
`1_3 )
* (P
`3_3 ))
* (Q
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (Q
`2_3 )))
+ ((((P
`2_3 )
* (Q
`3_3 ))
* (Q
`1_3 ))
- (((Q
`2_3 )
* (P
`3_3 ))
* (Q
`1_3 )))) by
VECTSP_1: 17
.= ((((((Q
`1_3 )
* (P
`3_3 ))
* (Q
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (Q
`2_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* (Q
`1_3 )))
- (((Q
`2_3 )
* (P
`3_3 ))
* (Q
`1_3 ))) by
ALGSTR_1: 7
.= ((((((Q
`1_3 )
* (P
`3_3 ))
* (Q
`2_3 ))
- (((Q
`2_3 )
* (P
`3_3 ))
* (Q
`1_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* (Q
`1_3 )))
- (((P
`1_3 )
* (Q
`3_3 ))
* (Q
`2_3 ))) by
Th7
.= (((((Q
`1_3 )
* ((P
`3_3 )
* (Q
`2_3 )))
- ((Q
`1_3 )
* ((P
`3_3 )
* (Q
`2_3 ))))
+ (((Q
`3_3 )
* (P
`2_3 ))
* (Q
`1_3 )))
- (((Q
`3_3 )
* (P
`1_3 ))
* (Q
`2_3 ))) by
GROUP_1:def 3
.= (((
0. (
GF p))
+ (((Q
`3_3 )
* (P
`2_3 ))
* (Q
`1_3 )))
- (((Q
`3_3 )
* (P
`1_3 ))
* (Q
`2_3 ))) by
VECTSP_1: 19
.= ((((Q
`3_3 )
* (P
`2_3 ))
* (Q
`1_3 ))
- (((Q
`3_3 )
* (P
`1_3 ))
* (Q
`2_3 ))) by
ALGSTR_1: 7
.= (((Q
`3_3 )
* ((P
`2_3 )
* (Q
`1_3 )))
- (((Q
`3_3 )
* (P
`1_3 ))
* (Q
`2_3 ))) by
GROUP_1:def 3
.= (((Q
`3_3 )
* ((P
`2_3 )
* (Q
`1_3 )))
- ((Q
`3_3 )
* ((P
`1_3 )
* (Q
`2_3 )))) by
GROUP_1:def 3
.= ((Q
`3_3 )
* (((P
`2_3 )
* (Q
`1_3 ))
- ((P
`1_3 )
* (Q
`2_3 )))) by
VECTSP_1: 11;
then
A5: (((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 )))
* (P
`3_3 ))
= (((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
* (Q
`3_3 )) by
A4,
GROUP_1:def 3;
A6: (
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 )))
= (
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* gf3)
* gf2)) by
GROUP_1:def 3,
A3
.= (
- (((((gf2
* (gf2
|^ 2))
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* gf3)) by
Th12
.= (
- (((((gf2
|^ (2
+ 1))
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* gf3)) by
EC_PF_1: 24
.= (
- (((gf2
|^ 3)
* (P
`3_3 ))
* (((P
`1_3 )
* (Q
`1_3 ))
* gf3))) by
Th11
.= (((gf2
|^ 3)
* (P
`3_3 ))
* (
- (((P
`1_3 )
* (Q
`1_3 ))
* gf3))) by
VECTSP_1: 8;
(
- (gf1
* (((P
`1_3 )
* (Q
`2_3 ))
+ ((Q
`1_3 )
* (P
`2_3 )))))
= (
- ((((Q
`2_3 )
* (P
`3_3 ))
* (((P
`1_3 )
* (Q
`2_3 ))
+ ((Q
`1_3 )
* (P
`2_3 ))))
- (((P
`2_3 )
* (Q
`3_3 ))
* (((P
`1_3 )
* (Q
`2_3 ))
+ ((Q
`1_3 )
* (P
`2_3 )))))) by
A2,
VECTSP_1: 13
.= (
- (((((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))
+ (((Q
`2_3 )
* (P
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((P
`2_3 )
* (Q
`3_3 ))
* (((P
`1_3 )
* (Q
`2_3 ))
+ ((Q
`1_3 )
* (P
`2_3 )))))) by
VECTSP_1:def 7
.= (
- (((((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))
+ (((Q
`2_3 )
* (P
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- ((((P
`2_3 )
* (Q
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 )))))) by
VECTSP_1:def 7
.= (((((P
`2_3 )
* (Q
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- ((((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))
+ (((Q
`2_3 )
* (P
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))) by
VECTSP_1: 17
.= ((((((P
`2_3 )
* (Q
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
VECTSP_1: 17
.= ((((((P
`2_3 )
* (Q
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
ALGSTR_1: 8
.= ((((((P
`2_3 )
* (Q
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))
- (((P
`3_3 )
* ((Q
`1_3 )
* (P
`2_3 )))
* (Q
`2_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
GROUP_1:def 3
.= (((((((P
`2_3 )
* (Q
`3_3 ))
* (P
`1_3 ))
* (Q
`2_3 ))
- (((P
`3_3 )
* ((Q
`1_3 )
* (P
`2_3 )))
* (Q
`2_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
GROUP_1:def 3
.= (((((((P
`1_3 )
* (Q
`3_3 ))
* (P
`2_3 ))
* (Q
`2_3 ))
- (((P
`3_3 )
* ((Q
`1_3 )
* (P
`2_3 )))
* (Q
`2_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
GROUP_1:def 3
.= (((((((P
`1_3 )
* (Q
`3_3 ))
* (P
`2_3 ))
* (Q
`2_3 ))
- ((((P
`3_3 )
* (Q
`1_3 ))
* (P
`2_3 ))
* (Q
`2_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
GROUP_1:def 3
.= ((((((P
`1_3 )
* (Q
`3_3 ))
* ((P
`2_3 )
* (Q
`2_3 )))
- ((((P
`3_3 )
* (Q
`1_3 ))
* (P
`2_3 ))
* (Q
`2_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
GROUP_1:def 3
.= ((((((P
`1_3 )
* (Q
`3_3 ))
* ((P
`2_3 )
* (Q
`2_3 )))
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`2_3 )
* (Q
`2_3 ))))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
GROUP_1:def 3
.= ((((((P
`1_3 )
* (Q
`3_3 ))
- ((P
`3_3 )
* (Q
`1_3 )))
* ((P
`2_3 )
* (Q
`2_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
VECTSP_1: 13
.= ((((
- (((P
`1_3 )
* (Q
`3_3 ))
- ((P
`3_3 )
* (Q
`1_3 ))))
* (
- ((P
`2_3 )
* (Q
`2_3 ))))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
VECTSP_1: 10
.= ((((((P
`3_3 )
* (Q
`1_3 ))
- ((P
`1_3 )
* (Q
`3_3 )))
* (
- ((P
`2_3 )
* (Q
`2_3 ))))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
VECTSP_1: 17
.= (((
- ((((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 )))
* ((P
`2_3 )
* (Q
`2_3 ))))
+ (((P
`2_3 )
* (Q
`3_3 ))
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
VECTSP_1: 8
.= (((
- (gf2
* ((P
`2_3 )
* (Q
`2_3 ))))
+ (((Q
`3_3 )
* ((Q
`1_3 )
* (P
`2_3 )))
* (P
`2_3 )))
- (((Q
`2_3 )
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`2_3 )))) by
A2,
GROUP_1:def 3
.= (((
- (gf2
* ((P
`2_3 )
* (Q
`2_3 ))))
+ (((Q
`3_3 )
* ((Q
`1_3 )
* (P
`2_3 )))
* (P
`2_3 )))
- (((P
`3_3 )
* ((P
`1_3 )
* (Q
`2_3 )))
* (Q
`2_3 ))) by
GROUP_1:def 3
.= (((
- ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ (((Q
`3_3 )
* ((Q
`1_3 )
* (P
`2_3 )))
* (P
`2_3 )))
- (((P
`3_3 )
* ((P
`1_3 )
* (Q
`2_3 )))
* (Q
`2_3 ))) by
GROUP_1:def 3
.= (((
- ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ ((((Q
`3_3 )
* (Q
`1_3 ))
* (P
`2_3 ))
* (P
`2_3 )))
- (((P
`3_3 )
* ((P
`1_3 )
* (Q
`2_3 )))
* (Q
`2_3 ))) by
GROUP_1:def 3
.= (((
- ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ (((Q
`3_3 )
* (Q
`1_3 ))
* ((P
`2_3 )
* (P
`2_3 ))))
- (((P
`3_3 )
* ((P
`1_3 )
* (Q
`2_3 )))
* (Q
`2_3 ))) by
GROUP_1:def 3
.= (((
- ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ (((Q
`3_3 )
* (Q
`1_3 ))
* ((P
`2_3 )
|^ 2)))
- (((P
`3_3 )
* ((P
`1_3 )
* (Q
`2_3 )))
* (Q
`2_3 ))) by
EC_PF_1: 22
.= (((
- ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ (((Q
`3_3 )
* (Q
`1_3 ))
* ((P
`2_3 )
|^ 2)))
- ((((P
`3_3 )
* (P
`1_3 ))
* (Q
`2_3 ))
* (Q
`2_3 ))) by
GROUP_1:def 3
.= (((
- ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ (((Q
`3_3 )
* (Q
`1_3 ))
* ((P
`2_3 )
|^ 2)))
- (((P
`3_3 )
* (P
`1_3 ))
* ((Q
`2_3 )
* (Q
`2_3 )))) by
GROUP_1:def 3
.= (((
- ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ (((Q
`3_3 )
* (Q
`1_3 ))
* ((P
`2_3 )
|^ 2)))
- (((P
`3_3 )
* (P
`1_3 ))
* ((Q
`2_3 )
|^ 2))) by
EC_PF_1: 22
.= (((
- ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ (((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 )))
- (((P
`3_3 )
* (P
`1_3 ))
* ((Q
`2_3 )
|^ 2))) by
GROUP_1:def 3
.= (((
- ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ (((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 )))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* (P
`3_3 ))) by
GROUP_1:def 3;
then
A7: (((gf2
* (P
`2_3 ))
* (Q
`2_3 ))
- (gf1
* (((P
`1_3 )
* (Q
`2_3 ))
+ ((Q
`1_3 )
* (P
`2_3 )))))
= (((((gf2
* (P
`2_3 ))
* (Q
`2_3 ))
- ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ (((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 )))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* (P
`3_3 ))) by
Th8
.= (((
0. (
GF p))
+ (((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 )))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* (P
`3_3 ))) by
RLVECT_1: 5
.= ((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* (P
`3_3 ))) by
RLVECT_1: 4;
A8: ((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
= ((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (Q
`3_3 )) by
EC_PF_1: 22
.= (((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
* (((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
* (Q
`3_3 ))) by
GROUP_1:def 3
.= ((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
* ((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 ))))
* (P
`3_3 )) by
A5,
GROUP_1:def 3
.= ((((gf2
* (P
`2_3 ))
* ((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 ))))
- ((gf1
* (P
`1_3 ))
* ((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 )))))
* (P
`3_3 )) by
VECTSP_1: 13
.= (((((gf2
* (P
`2_3 ))
* (gf2
* (Q
`2_3 )))
- ((gf2
* (P
`2_3 ))
* (gf1
* (Q
`1_3 ))))
- ((gf1
* (P
`1_3 ))
* ((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 )))))
* (P
`3_3 )) by
VECTSP_1: 11
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
- ((gf2
* (P
`2_3 ))
* (gf1
* (Q
`1_3 ))))
- ((gf1
* (P
`1_3 ))
* ((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 )))))
* (P
`3_3 )) by
GROUP_1:def 3
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
- ((gf2
* (gf1
* (Q
`1_3 )))
* (P
`2_3 )))
- ((gf1
* (P
`1_3 ))
* ((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 )))))
* (P
`3_3 )) by
GROUP_1:def 3
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
- (((gf2
* gf1)
* (Q
`1_3 ))
* (P
`2_3 )))
- ((gf1
* (P
`1_3 ))
* ((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 )))))
* (P
`3_3 )) by
GROUP_1:def 3
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
- ((gf2
* gf1)
* ((Q
`1_3 )
* (P
`2_3 ))))
- ((gf1
* (P
`1_3 ))
* ((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 )))))
* (P
`3_3 )) by
GROUP_1:def 3
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
- ((gf2
* gf1)
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((gf1
* (P
`1_3 ))
* (gf2
* (Q
`2_3 )))
- ((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 )))))
* (P
`3_3 )) by
VECTSP_1: 11
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
- ((gf2
* gf1)
* ((Q
`1_3 )
* (P
`2_3 ))))
+ (((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 )))
- ((gf1
* (P
`1_3 ))
* (gf2
* (Q
`2_3 )))))
* (P
`3_3 )) by
VECTSP_1: 17
.= (((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
- ((gf2
* gf1)
* ((Q
`1_3 )
* (P
`2_3 ))))
+ ((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 ))))
- ((gf1
* (P
`1_3 ))
* (gf2
* (Q
`2_3 ))))
* (P
`3_3 )) by
ALGSTR_1: 7
.= (((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
- ((gf2
* gf1)
* ((Q
`1_3 )
* (P
`2_3 ))))
- ((gf1
* (P
`1_3 ))
* (gf2
* (Q
`2_3 ))))
+ ((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 ))))
* (P
`3_3 )) by
ALGSTR_1: 8
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ ((
- ((gf2
* gf1)
* ((Q
`1_3 )
* (P
`2_3 ))))
- ((gf1
* (P
`1_3 ))
* (gf2
* (Q
`2_3 )))))
+ ((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 ))))
* (P
`3_3 )) by
ALGSTR_1: 7
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ ((
- ((gf2
* gf1)
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((gf1
* (P
`1_3 ))
* gf2)
* (Q
`2_3 ))))
+ ((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 ))))
* (P
`3_3 )) by
GROUP_1:def 3
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ ((
- ((gf2
* gf1)
* ((Q
`1_3 )
* (P
`2_3 ))))
- (((gf2
* gf1)
* (P
`1_3 ))
* (Q
`2_3 ))))
+ ((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 ))))
* (P
`3_3 )) by
GROUP_1:def 3
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
+ ((
- ((gf2
* gf1)
* ((Q
`1_3 )
* (P
`2_3 ))))
- ((gf2
* gf1)
* ((P
`1_3 )
* (Q
`2_3 )))))
+ ((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 ))))
* (P
`3_3 )) by
GROUP_1:def 3
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
- (((gf2
* gf1)
* ((P
`1_3 )
* (Q
`2_3 )))
+ ((gf2
* gf1)
* ((Q
`1_3 )
* (P
`2_3 )))))
+ ((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 ))))
* (P
`3_3 )) by
VECTSP_1: 17
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
- ((gf2
* gf1)
* (((P
`1_3 )
* (Q
`2_3 ))
+ ((Q
`1_3 )
* (P
`2_3 )))))
+ ((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 ))))
* (P
`3_3 )) by
VECTSP_1:def 7
.= ((((gf2
* ((gf2
* (P
`2_3 ))
* (Q
`2_3 )))
- (gf2
* (gf1
* (((P
`1_3 )
* (Q
`2_3 ))
+ ((Q
`1_3 )
* (P
`2_3 ))))))
+ ((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 ))))
* (P
`3_3 )) by
GROUP_1:def 3
.= (((gf2
* (((gf2
* (P
`2_3 ))
* (Q
`2_3 ))
- (gf1
* (((P
`1_3 )
* (Q
`2_3 ))
+ ((Q
`1_3 )
* (P
`2_3 ))))))
+ ((gf1
* (P
`1_3 ))
* (gf1
* (Q
`1_3 ))))
* (P
`3_3 )) by
VECTSP_1: 11
.= (((gf2
* ((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* (P
`3_3 ))))
+ (((gf1
* (P
`1_3 ))
* gf1)
* (Q
`1_3 )))
* (P
`3_3 )) by
A7,
GROUP_1:def 3
.= (((gf2
* ((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* (P
`3_3 ))))
+ (((gf1
* gf1)
* (P
`1_3 ))
* (Q
`1_3 )))
* (P
`3_3 )) by
GROUP_1:def 3
.= (((gf2
* ((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* (P
`3_3 ))))
+ (((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 )))
* (P
`3_3 )) by
EC_PF_1: 22
.= (((gf2
* ((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* (P
`3_3 ))))
* (P
`3_3 ))
+ ((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))) by
VECTSP_1:def 7
.= ((gf2
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* (P
`3_3 )))
* (P
`3_3 )))
+ ((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))) by
GROUP_1:def 3
.= ((gf2
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 ))
* (P
`3_3 ))
- ((((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* (P
`3_3 ))
* (P
`3_3 ))))
+ ((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))) by
VECTSP_1: 13
.= ((gf2
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 ))
* (P
`3_3 ))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* ((P
`3_3 )
* (P
`3_3 )))))
+ ((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))) by
GROUP_1:def 3
.= ((gf2
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (Q
`3_3 ))
* (P
`3_3 ))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* ((P
`3_3 )
|^ 2))))
+ ((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))) by
EC_PF_1: 22
.= ((gf2
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((Q
`2_3 )
|^ 2))
* ((P
`3_3 )
|^ 2))))
+ ((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))) by
GROUP_1:def 3
.= ((gf2
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))))
+ ((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))) by
GROUP_1:def 3;
A9: ((((gf2
|^ 3)
* (P
`3_3 ))
* (
- (((P
`1_3 )
* (Q
`1_3 ))
* gf3)))
+ (((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (R
`3_3 )))
= ((((gf2
|^ 3)
* (P
`3_3 ))
* (((P
`1_3 )
* (Q
`1_3 ))
* (
- gf3)))
+ (((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (R
`3_3 ))) by
VECTSP_1: 8
.= (((((gf2
|^ 3)
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`1_3 )))
* (
- gf3))
+ (((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (R
`3_3 ))) by
GROUP_1:def 3
.= (((((gf2
|^ 3)
* ((P
`1_3 )
* (Q
`1_3 )))
* (P
`3_3 ))
* (
- gf3))
+ (((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (R
`3_3 ))) by
GROUP_1:def 3
.= ((((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (
- gf3))
+ (((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (R
`3_3 ))) by
GROUP_1:def 3
.= ((((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (
- gf3))
+ (((gf1
|^ 2)
* (((P
`1_3 )
* (Q
`1_3 ))
* (P
`3_3 )))
* (((gf2
|^ 3)
* (P
`3_3 ))
* (Q
`3_3 )))) by
Th11,
A3
.= ((((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (
- gf3))
+ (((((gf1
|^ 2)
* (((P
`1_3 )
* (Q
`1_3 ))
* (P
`3_3 )))
* (gf2
|^ 3))
* (P
`3_3 ))
* (Q
`3_3 ))) by
Th11
.= ((((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (
- gf3))
+ (((((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (gf1
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))) by
Th11
.= ((((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (
- gf3))
+ (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
Th11
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((
- gf3)
+ (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
VECTSP_1:def 7
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((
- ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
+ ((
- (gf2
|^ 3))
- (((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 )))))
+ (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
A2,
ALGSTR_1: 7
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (((
- ((
- (gf2
|^ 3))
- (((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 ))))
- (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))
+ (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
VECTSP_1: 17
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((
- ((
- (gf2
|^ 3))
- (((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 ))))
+ ((
- (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))
+ (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))) by
ALGSTR_1: 7
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((
- ((
- (gf2
|^ 3))
- (((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 ))))
+ (
0. (
GF p)))) by
RLVECT_1: 5
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (
- ((
- (gf2
|^ 3))
- (((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 ))))) by
RLVECT_1: 4
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 ))
+ (gf2
|^ (2
+ 1)))) by
VECTSP_1: 17
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 ))
+ ((gf2
|^ 2)
* gf2))) by
EC_PF_1: 24
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (((gf2
|^ 2)
* ((g2
* (P
`1_3 ))
* (Q
`3_3 )))
+ ((gf2
|^ 2)
* gf2))) by
Th11
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((gf2
|^ 2)
* (((g2
* (P
`1_3 ))
* (Q
`3_3 ))
+ gf2))) by
VECTSP_1:def 7
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((gf2
|^ 2)
* ((((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 )))
+ (g2
* ((P
`1_3 )
* (Q
`3_3 )))))) by
A2,
GROUP_1:def 3
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((gf2
|^ 2)
* ((((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))))) by
A1,
Th20
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((gf2
|^ 2)
* (((((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 )))
+ ((P
`1_3 )
* (Q
`3_3 )))
+ ((P
`1_3 )
* (Q
`3_3 ))))) by
ALGSTR_1: 7
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((gf2
|^ 2)
* ((((Q
`1_3 )
* (P
`3_3 ))
+ ((
- ((P
`1_3 )
* (Q
`3_3 )))
+ ((P
`1_3 )
* (Q
`3_3 ))))
+ ((P
`1_3 )
* (Q
`3_3 ))))) by
ALGSTR_1: 7
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((gf2
|^ 2)
* ((((Q
`1_3 )
* (P
`3_3 ))
+ (
0. (
GF p)))
+ ((P
`1_3 )
* (Q
`3_3 ))))) by
RLVECT_1: 5
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((gf2
* gf2)
* ((((Q
`1_3 )
* (P
`3_3 ))
+ (
0. (
GF p)))
+ ((P
`1_3 )
* (Q
`3_3 ))))) by
EC_PF_1: 22
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((gf2
* (((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 ))))
* (((Q
`1_3 )
* (P
`3_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))) by
A2,
RLVECT_1: 4
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (gf2
* ((((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 )))
* (((Q
`1_3 )
* (P
`3_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))))) by
GROUP_1:def 3
.= (((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (gf2
* ((((Q
`1_3 )
* (P
`3_3 ))
|^ 2)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))) by
Th15
.= ((((((gf2
|^ 3)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* gf2)
* ((((Q
`1_3 )
* (P
`3_3 ))
|^ 2)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 2))) by
GROUP_1:def 3
.= (((((gf2
* (gf2
|^ 3))
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((((Q
`1_3 )
* (P
`3_3 ))
|^ 2)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 2))) by
Th11
.= (((((gf2
|^ (3
+ 1))
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* ((((Q
`1_3 )
* (P
`3_3 ))
|^ 2)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 2))) by
EC_PF_1: 24
.= ((((gf2
|^ 4)
* ((P
`1_3 )
* (Q
`1_3 )))
* (P
`3_3 ))
* ((((Q
`1_3 )
* (P
`3_3 ))
|^ 2)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 2))) by
GROUP_1:def 3
.= ((((gf2
|^ 4)
* (P
`3_3 ))
* ((P
`1_3 )
* (Q
`1_3 )))
* ((((Q
`1_3 )
* (P
`3_3 ))
|^ 2)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 2))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((P
`1_3 )
* (Q
`1_3 ))
* ((((Q
`1_3 )
* (P
`3_3 ))
|^ 2)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((P
`1_3 )
* (Q
`1_3 ))
* (((Q
`1_3 )
* (P
`3_3 ))
|^ 2))
- (((P
`1_3 )
* (Q
`1_3 ))
* (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))) by
VECTSP_1: 11
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((P
`1_3 )
* (Q
`1_3 ))
* (((Q
`1_3 )
|^ 2)
* ((P
`3_3 )
|^ 2)))
- (((P
`1_3 )
* (Q
`1_3 ))
* (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))) by
BINOM: 9
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((P
`1_3 )
* (Q
`1_3 ))
* ((Q
`1_3 )
|^ 2))
* ((P
`3_3 )
|^ 2))
- (((P
`1_3 )
* (Q
`1_3 ))
* (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 2))
* (Q
`1_3 ))
- (((P
`1_3 )
* (Q
`1_3 ))
* (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))) by
Th10
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((Q
`1_3 )
|^ 2)
* (Q
`1_3 )))
- (((P
`1_3 )
* (Q
`1_3 ))
* (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ (2
+ 1)))
- (((P
`1_3 )
* (Q
`1_3 ))
* (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))) by
EC_PF_1: 24
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3))
- (((P
`1_3 )
* (Q
`1_3 ))
* (((P
`1_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))))) by
BINOM: 9
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3))
- ((((P
`1_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
|^ 2))
* ((Q
`3_3 )
|^ 2)))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3))
- (((((Q
`3_3 )
|^ 2)
* (Q
`1_3 ))
* ((P
`1_3 )
|^ 2))
* (P
`1_3 )))) by
Th10
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3))
- ((((Q
`3_3 )
|^ 2)
* (Q
`1_3 ))
* (((P
`1_3 )
|^ 2)
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3))
- (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((P
`1_3 )
|^ (2
+ 1))))) by
EC_PF_1: 24
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3))
- (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((P
`1_3 )
|^ 3))));
A10: ((gf2
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))))
* (R
`3_3 ))
= (((gf2
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))))
* ((gf2
|^ 3)
* (P
`3_3 )))
* (Q
`3_3 )) by
A3,
GROUP_1:def 3
.= (((gf2
* ((gf2
|^ 3)
* (P
`3_3 )))
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))))
* (Q
`3_3 )) by
GROUP_1:def 3
.= ((((gf2
* (gf2
|^ 3))
* (P
`3_3 ))
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))))
* (Q
`3_3 )) by
GROUP_1:def 3
.= ((((gf2
|^ (3
+ 1))
* (P
`3_3 ))
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))))
* (Q
`3_3 )) by
EC_PF_1: 24
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2)))
* (Q
`3_3 ))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
* (Q
`3_3 ))
- ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))
* (Q
`3_3 )))) by
VECTSP_1: 13
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* ((Q
`3_3 )
* (Q
`3_3 )))
- ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))
* (Q
`3_3 )))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* ((Q
`3_3 )
|^ 2))
- ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))
* (Q
`3_3 )))) by
EC_PF_1: 22
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((Q
`1_3 )
* (((P
`2_3 )
|^ 2)
* (P
`3_3 )))
* ((Q
`3_3 )
|^ 2))
- ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))
* (Q
`3_3 )))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (((P
`2_3 )
|^ 2)
* (P
`3_3 )))
- ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))
* (Q
`3_3 )))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (((P
`2_3 )
|^ 2)
* (P
`3_3 )))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))))) by
GROUP_1:def 3;
(
0. (
GF p))
= ((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3)))) by
Th35
.= ((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- (((P
`1_3 )
|^ 3)
+ (((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
+ (b
* ((P
`3_3 )
|^ 3))))) by
ALGSTR_1: 7
.= (((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((P
`1_3 )
|^ 3))
- (((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
+ (b
* ((P
`3_3 )
|^ 3)))) by
VECTSP_1: 17;
then
A11: ((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((P
`1_3 )
|^ 3))
= (((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
+ (b
* ((P
`3_3 )
|^ 3))) by
VECTSP_1: 19;
A12: (
0. (
GF p))
= ((((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))
- ((((Q
`1_3 )
|^ 3)
+ ((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2)))
+ (b
* ((Q
`3_3 )
|^ 3)))) by
Th35
.= ((((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))
- (((Q
`1_3 )
|^ 3)
+ (((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
+ (b
* ((Q
`3_3 )
|^ 3))))) by
ALGSTR_1: 7
.= (((((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))
- ((Q
`1_3 )
|^ 3))
- (((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
+ (b
* ((Q
`3_3 )
|^ 3)))) by
VECTSP_1: 17;
thus ((
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 )))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )))
= ((((gf2
|^ 3)
* (P
`3_3 ))
* (
- (((P
`1_3 )
* (Q
`1_3 ))
* gf3)))
+ (((gf2
* (((((Q
`1_3 )
* ((P
`2_3 )
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`2_3 )
|^ 2))))
* (R
`3_3 ))
+ (((((gf1
|^ 2)
* (P
`1_3 ))
* (Q
`1_3 ))
* (P
`3_3 ))
* (R
`3_3 )))) by
A6,
A8,
VECTSP_1:def 7
.= ((((gf2
|^ 4)
* (P
`3_3 ))
* ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3))
- (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((P
`1_3 )
|^ 3))))
+ (((gf2
|^ 4)
* (P
`3_3 ))
* ((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (((P
`2_3 )
|^ 2)
* (P
`3_3 )))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 )))))) by
A9,
A10,
ALGSTR_1: 7
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3))
- (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((P
`1_3 )
|^ 3)))
+ ((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (((P
`2_3 )
|^ 2)
* (P
`3_3 )))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 )))))) by
VECTSP_1:def 7
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3))
+ (
- (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((P
`1_3 )
|^ 3))))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (((P
`2_3 )
|^ 2)
* (P
`3_3 ))))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))))) by
ALGSTR_1: 7
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (((P
`2_3 )
|^ 2)
* (P
`3_3 )))
- (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((P
`1_3 )
|^ 3)))
+ (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3)))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))))) by
ALGSTR_1: 8
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((P
`1_3 )
|^ 3)))
+ (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3)))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))))) by
VECTSP_1: 11
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((P
`1_3 )
|^ 3)))
+ ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
|^ 3))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 )))))) by
ALGSTR_1: 7
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((P
`1_3 )
|^ 3)))
+ (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((Q
`1_3 )
|^ 3)
- (((Q
`2_3 )
|^ 2)
* (Q
`3_3 )))))) by
VECTSP_1: 11
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((P
`1_3 )
|^ 3)))
- (
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((Q
`1_3 )
|^ 3)
- (((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))))))) by
RLVECT_1: 17
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((P
`1_3 )
|^ 3)))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (
- (((Q
`1_3 )
|^ 3)
- (((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))))))) by
VECTSP_1: 8
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((P
`1_3 )
|^ 3)))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))
- ((Q
`1_3 )
|^ 3))))) by
VECTSP_1: 17
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
+ (b
* ((P
`3_3 )
|^ 3))))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
+ (b
* ((Q
`3_3 )
|^ 3)))))) by
A11,
A12,
VECTSP_1: 19
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (b
* ((P
`3_3 )
|^ 3))))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
+ (b
* ((Q
`3_3 )
|^ 3)))))) by
VECTSP_1:def 7
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (b
* ((P
`3_3 )
|^ 3))))
- ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2)))
+ (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3)))))) by
VECTSP_1:def 7
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (b
* ((P
`3_3 )
|^ 3))))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))))) by
VECTSP_1: 17
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (b
* ((P
`3_3 )
|^ 3))))) by
Th7
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* ((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (b
* ((P
`3_3 )
|^ 3))))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (a
* ((Q
`1_3 )
* ((Q
`3_3 )
|^ 2)))))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (b
* ((P
`3_3 )
|^ 3))))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
- ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* a)
* ((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (b
* ((P
`3_3 )
|^ 3))))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
- (((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
* ((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (b
* ((P
`3_3 )
|^ 3))))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
- ((((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2)))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (b
* ((P
`3_3 )
|^ 3))))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((
0. (
GF p))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (b
* ((P
`3_3 )
|^ 3))))) by
RLVECT_1: 5
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))
+ (((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* (b
* ((P
`3_3 )
|^ 3))))) by
RLVECT_1: 4
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* b)
* ((P
`3_3 )
|^ (2
+ 1)))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* b)
* (((P
`3_3 )
|^ 2)
* (P
`3_3 )))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))) by
EC_PF_1: 24
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((((Q
`1_3 )
* ((Q
`3_3 )
|^ 2))
* b)
* ((P
`3_3 )
|^ 2))
* (P
`3_3 ))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))
* b)
* (Q
`1_3 ))
* (P
`3_3 ))
- (((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* (b
* ((Q
`3_3 )
|^ 3))))) by
Th10
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))
* b)
* (Q
`1_3 ))
* (P
`3_3 ))
- ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* b)
* ((Q
`3_3 )
|^ (2
+ 1))))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))
* b)
* (Q
`1_3 ))
* (P
`3_3 ))
- ((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* b)
* (((Q
`3_3 )
|^ 2)
* (Q
`3_3 ))))) by
EC_PF_1: 24
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))
* b)
* (Q
`1_3 ))
* (P
`3_3 ))
- (((((P
`1_3 )
* ((P
`3_3 )
|^ 2))
* b)
* ((Q
`3_3 )
|^ 2))
* (Q
`3_3 )))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))
* b)
* (Q
`1_3 ))
* (P
`3_3 ))
- ((((((Q
`3_3 )
|^ 2)
* ((P
`3_3 )
|^ 2))
* b)
* (P
`1_3 ))
* (Q
`3_3 )))) by
Th10
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))
* b)
* ((Q
`1_3 )
* (P
`3_3 )))
- ((((((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))
* b)
* (P
`1_3 ))
* (Q
`3_3 )))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* ((((((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))
* b)
* ((Q
`1_3 )
* (P
`3_3 )))
- (((((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))
* b)
* ((P
`1_3 )
* (Q
`3_3 ))))) by
GROUP_1:def 3
.= (((gf2
|^ 4)
* (P
`3_3 ))
* (((((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))
* b)
* (((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 ))))) by
VECTSP_1: 11
.= ((((((gf2
|^ 4)
* (P
`3_3 ))
* ((Q
`3_3 )
|^ 2))
* ((P
`3_3 )
|^ 2))
* b)
* gf2) by
A2,
Th11
.= ((((((gf2
|^ 4)
* (P
`3_3 ))
* ((Q
`3_3 )
* (Q
`3_3 )))
* ((P
`3_3 )
|^ 2))
* b)
* gf2) by
EC_PF_1: 22
.= (((((((gf2
|^ (3
+ 1))
* (P
`3_3 ))
* (Q
`3_3 ))
* (Q
`3_3 ))
* ((P
`3_3 )
|^ 2))
* b)
* gf2) by
GROUP_1:def 3
.= ((((((((gf2
|^ 3)
* gf2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (Q
`3_3 ))
* ((P
`3_3 )
|^ 2))
* b)
* gf2) by
EC_PF_1: 24
.= (((((gf2
* (((gf2
|^ 3)
* (P
`3_3 ))
* (Q
`3_3 )))
* (Q
`3_3 ))
* ((P
`3_3 )
|^ 2))
* b)
* gf2) by
Th11
.= (gf2
* (gf2
* ((((R
`3_3 )
* (Q
`3_3 ))
* ((P
`3_3 )
|^ 2))
* b))) by
Th12,
A3
.= ((gf2
* gf2)
* ((((R
`3_3 )
* (Q
`3_3 ))
* ((P
`3_3 )
|^ 2))
* b)) by
GROUP_1:def 3
.= ((gf2
|^ 2)
* ((((R
`3_3 )
* (Q
`3_3 ))
* ((P
`3_3 )
|^ 2))
* b)) by
EC_PF_1: 22
.= ((gf2
|^ 2)
* (((b
* (Q
`3_3 ))
* ((P
`3_3 )
|^ 2))
* (R
`3_3 ))) by
Th10
.= (((((gf2
|^ 2)
* b)
* (Q
`3_3 ))
* ((P
`3_3 )
|^ 2))
* (R
`3_3 )) by
Th11
.= ((((b
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 )) by
GROUP_1:def 3;
end;
theorem ::
EC_PF_2:57
Th57: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,gf1,gf2,gf3 be
Element of (
GF p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
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 (((((z
`1 )
* (gf2
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
= (((gf2
|^ 2)
* (((((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`1_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`1_3 ))))
+ ((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,gf1,gf2,gf3 be
Element of (
GF p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):] such that
A1: g2
= (2
mod p) and
A2: 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
A3: 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 ))];
set a = (z
`1 );
set b = (z
`2 );
A4: ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
= (((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (P
`2_3 )))
- ((((Q
`2_3 )
* (P
`3_3 ))
- ((P
`2_3 )
* (Q
`3_3 )))
* (P
`1_3 ))) by
A2,
VECTSP_1: 11
.= (((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (P
`2_3 )))
- ((((Q
`2_3 )
* (P
`3_3 ))
* (P
`1_3 ))
- (((P
`2_3 )
* (Q
`3_3 ))
* (P
`1_3 )))) by
VECTSP_1: 11
.= (((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (P
`2_3 )))
+ ((((P
`2_3 )
* (Q
`3_3 ))
* (P
`1_3 ))
- (((Q
`2_3 )
* (P
`3_3 ))
* (P
`1_3 )))) by
VECTSP_1: 17
.= (((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
- ((P
`1_3 )
* ((Q
`3_3 )
* (P
`2_3 ))))
+ ((((P
`2_3 )
* (Q
`3_3 ))
* (P
`1_3 ))
- (((Q
`2_3 )
* (P
`3_3 ))
* (P
`1_3 )))) by
GROUP_1:def 3
.= ((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
+ ((
- ((P
`1_3 )
* ((Q
`3_3 )
* (P
`2_3 ))))
+ ((((P
`2_3 )
* (Q
`3_3 ))
* (P
`1_3 ))
- (((Q
`2_3 )
* (P
`3_3 ))
* (P
`1_3 ))))) by
ALGSTR_1: 7
.= ((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
+ (((
- ((P
`1_3 )
* ((P
`2_3 )
* (Q
`3_3 ))))
+ (((P
`2_3 )
* (Q
`3_3 ))
* (P
`1_3 )))
- (((Q
`2_3 )
* (P
`3_3 ))
* (P
`1_3 )))) by
ALGSTR_1: 7
.= ((((Q
`1_3 )
* (P
`3_3 ))
* (P
`2_3 ))
+ ((
0. (
GF p))
- (((P
`3_3 )
* (Q
`2_3 ))
* (P
`1_3 )))) by
VECTSP_1: 19
.= ((((P
`3_3 )
* (Q
`1_3 ))
* (P
`2_3 ))
- (((P
`3_3 )
* (Q
`2_3 ))
* (P
`1_3 ))) by
VECTSP_1: 18
.= (((P
`3_3 )
* ((Q
`1_3 )
* (P
`2_3 )))
- (((P
`3_3 )
* (Q
`2_3 ))
* (P
`1_3 ))) by
GROUP_1:def 3
.= (((P
`3_3 )
* ((P
`2_3 )
* (Q
`1_3 )))
- ((P
`3_3 )
* ((Q
`2_3 )
* (P
`1_3 )))) by
GROUP_1:def 3
.= ((P
`3_3 )
* (((P
`2_3 )
* (Q
`1_3 ))
- ((Q
`2_3 )
* (P
`1_3 )))) by
VECTSP_1: 11;
((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 )))
= (((((Q
`1_3 )
* (P
`3_3 ))
* (Q
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (Q
`2_3 )))
- ((((Q
`2_3 )
* (P
`3_3 ))
- ((P
`2_3 )
* (Q
`3_3 )))
* (Q
`1_3 ))) by
A2,
VECTSP_1: 11
.= (((((Q
`1_3 )
* (P
`3_3 ))
* (Q
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (Q
`2_3 )))
- ((((Q
`2_3 )
* (P
`3_3 ))
* (Q
`1_3 ))
- (((P
`2_3 )
* (Q
`3_3 ))
* (Q
`1_3 )))) by
VECTSP_1: 11
.= (((((Q
`1_3 )
* (P
`3_3 ))
* (Q
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (Q
`2_3 )))
+ ((((P
`2_3 )
* (Q
`3_3 ))
* (Q
`1_3 ))
- (((Q
`2_3 )
* (P
`3_3 ))
* (Q
`1_3 )))) by
VECTSP_1: 17
.= ((((((Q
`1_3 )
* (P
`3_3 ))
* (Q
`2_3 ))
- (((P
`1_3 )
* (Q
`3_3 ))
* (Q
`2_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* (Q
`1_3 )))
- (((Q
`2_3 )
* (P
`3_3 ))
* (Q
`1_3 ))) by
ALGSTR_1: 7
.= ((((((Q
`1_3 )
* (P
`3_3 ))
* (Q
`2_3 ))
- (((Q
`2_3 )
* (P
`3_3 ))
* (Q
`1_3 )))
+ (((P
`2_3 )
* (Q
`3_3 ))
* (Q
`1_3 )))
- (((P
`1_3 )
* (Q
`3_3 ))
* (Q
`2_3 ))) by
Th7
.= (((((Q
`1_3 )
* ((P
`3_3 )
* (Q
`2_3 )))
- ((Q
`1_3 )
* ((P
`3_3 )
* (Q
`2_3 ))))
+ (((Q
`3_3 )
* (P
`2_3 ))
* (Q
`1_3 )))
- (((Q
`3_3 )
* (P
`1_3 ))
* (Q
`2_3 ))) by
GROUP_1:def 3
.= (((
0. (
GF p))
+ (((Q
`3_3 )
* (P
`2_3 ))
* (Q
`1_3 )))
- (((Q
`3_3 )
* (P
`1_3 ))
* (Q
`2_3 ))) by
VECTSP_1: 19
.= ((((Q
`3_3 )
* (P
`2_3 ))
* (Q
`1_3 ))
- (((Q
`3_3 )
* (P
`1_3 ))
* (Q
`2_3 ))) by
ALGSTR_1: 7
.= (((Q
`3_3 )
* ((P
`2_3 )
* (Q
`1_3 )))
- (((Q
`3_3 )
* (P
`1_3 ))
* (Q
`2_3 ))) by
GROUP_1:def 3
.= (((Q
`3_3 )
* ((P
`2_3 )
* (Q
`1_3 )))
- ((Q
`3_3 )
* ((P
`1_3 )
* (Q
`2_3 )))) by
GROUP_1:def 3
.= ((Q
`3_3 )
* (((P
`2_3 )
* (Q
`1_3 ))
- ((P
`1_3 )
* (Q
`2_3 )))) by
VECTSP_1: 11;
then
A5: (((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 )))
* (P
`3_3 ))
= (((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
* (Q
`3_3 )) by
A4,
GROUP_1:def 3;
A6: (((((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`1_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`1_3 )))
= ((((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ ((((P
`3_3 )
* (Q
`1_3 ))
* (R
`1_3 ))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`1_3 )))) by
ALGSTR_1: 7
.= ((((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (R
`1_3 ))) by
VECTSP_1:def 7
.= ((((P
`1_3 )
* (Q
`1_3 ))
* ((gf2
|^ 3)
* ((P
`3_3 )
* (Q
`3_3 ))))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (gf2
* gf3))) by
GROUP_1:def 3,
A3
.= (((((P
`1_3 )
* (Q
`1_3 ))
* (gf2
|^ (2
+ 1)))
* ((P
`3_3 )
* (Q
`3_3 )))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (gf2
* gf3))) by
GROUP_1:def 3
.= (((((P
`1_3 )
* (Q
`1_3 ))
* ((gf2
|^ 2)
* gf2))
* ((P
`3_3 )
* (Q
`3_3 )))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (gf2
* gf3))) by
EC_PF_1: 24
.= (((((P
`1_3 )
* (Q
`1_3 ))
* ((gf2
|^ 2)
* gf2))
* ((P
`3_3 )
* (Q
`3_3 )))
+ (((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* gf2)
* gf3)) by
GROUP_1:def 3
.= ((((((P
`1_3 )
* (Q
`1_3 ))
* (gf2
|^ 2))
* gf2)
* ((P
`3_3 )
* (Q
`3_3 )))
+ ((gf2
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* gf3)) by
GROUP_1:def 3
.= (((gf2
* (((P
`1_3 )
* (Q
`1_3 ))
* (gf2
|^ 2)))
* ((P
`3_3 )
* (Q
`3_3 )))
+ (gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* gf3))) by
GROUP_1:def 3
.= ((gf2
* ((((P
`1_3 )
* (Q
`1_3 ))
* (gf2
|^ 2))
* ((P
`3_3 )
* (Q
`3_3 ))))
+ (gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* gf3))) by
GROUP_1:def 3
.= (gf2
* (((((P
`1_3 )
* (Q
`1_3 ))
* (gf2
|^ 2))
* ((P
`3_3 )
* (Q
`3_3 )))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* gf3))) by
VECTSP_1:def 7;
A7: gf3
= (((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- (gf2
|^ (2
+ 1)))
- (((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 ))) by
A2
.= (((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- ((gf2
|^ 2)
* gf2))
- (((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 ))) by
EC_PF_1: 24
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
+ ((
- ((gf2
|^ 2)
* gf2))
- (((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 )))) by
ALGSTR_1: 7
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
+ ((
- ((gf2
|^ 2)
* gf2))
- ((g2
* (gf2
|^ 2))
* ((P
`1_3 )
* (Q
`3_3 ))))) by
GROUP_1:def 3
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
+ ((
- ((gf2
|^ 2)
* gf2))
- ((gf2
|^ 2)
* (g2
* ((P
`1_3 )
* (Q
`3_3 )))))) by
GROUP_1:def 3
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
+ (((
- (gf2
|^ 2))
* gf2)
- ((gf2
|^ 2)
* (g2
* ((P
`1_3 )
* (Q
`3_3 )))))) by
VECTSP_1: 9
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
+ (((
- (gf2
|^ 2))
* gf2)
+ ((
- (gf2
|^ 2))
* (g2
* ((P
`1_3 )
* (Q
`3_3 )))))) by
VECTSP_1: 9
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
+ ((
- (gf2
|^ 2))
* ((((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 )))
+ (g2
* ((P
`1_3 )
* (Q
`3_3 )))))) by
A2,
VECTSP_1:def 7
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- ((gf2
|^ 2)
* ((((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 )))
+ (g2
* ((P
`1_3 )
* (Q
`3_3 )))))) by
VECTSP_1: 9
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- ((gf2
|^ 2)
* (((Q
`1_3 )
* (P
`3_3 ))
+ ((
- ((P
`1_3 )
* (Q
`3_3 )))
+ (g2
* ((P
`1_3 )
* (Q
`3_3 ))))))) by
ALGSTR_1: 7
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- ((gf2
|^ 2)
* (((Q
`1_3 )
* (P
`3_3 ))
+ ((g2
* ((P
`1_3 )
* (Q
`3_3 )))
- ((P
`1_3 )
* (Q
`3_3 ))))))
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- ((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))) by
A1,
Th24;
(((((P
`1_3 )
* (Q
`1_3 ))
* (gf2
|^ 2))
* ((P
`3_3 )
* (Q
`3_3 )))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* gf3))
= (((gf2
|^ 2)
* (((P
`1_3 )
* (Q
`1_3 ))
* ((P
`3_3 )
* (Q
`3_3 ))))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* gf3)) by
GROUP_1:def 3
.= (((gf2
|^ 2)
* ((P
`3_3 )
* ((Q
`3_3 )
* ((P
`1_3 )
* (Q
`1_3 )))))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* gf3)) by
GROUP_1:def 3
.= (((gf2
|^ 2)
* ((P
`3_3 )
* (((Q
`3_3 )
* (P
`1_3 ))
* (Q
`1_3 ))))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* gf3)) by
GROUP_1:def 3
.= (((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 ))))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* gf3)) by
GROUP_1:def 3
.= (((
- (gf2
|^ 2))
* (
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 )))))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- ((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))))) by
A7,
VECTSP_1: 10
.= (((
- (gf2
|^ 2))
* (
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 )))))
+ (((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))
- ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* ((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))))) by
VECTSP_1: 11
.= (((
- (gf2
|^ 2))
* (
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 )))))
+ ((
- ((gf2
|^ 2)
* ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))) by
GROUP_1:def 3
.= (((
- (gf2
|^ 2))
* (
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 )))))
+ ((
- ((gf2
|^ 2)
* ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
|^ 2)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))) by
EC_PF_1: 22
.= (((
- (gf2
|^ 2))
* (
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 )))))
+ (((
- (gf2
|^ 2))
* ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
|^ 2))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))) by
VECTSP_1: 9
.= ((((
- (gf2
|^ 2))
* (
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 )))))
+ ((
- (gf2
|^ 2))
* ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
|^ 2)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
ALGSTR_1: 7
.= (((
- (gf2
|^ 2))
* ((
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 ))))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
|^ 2)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
VECTSP_1:def 7
.= (((
- (gf2
|^ 2))
* ((((((P
`3_3 )
* (Q
`1_3 ))
|^ 2)
+ ((g2
* ((P
`3_3 )
* (Q
`1_3 )))
* ((P
`1_3 )
* (Q
`3_3 ))))
+ (((P
`1_3 )
* (Q
`3_3 ))
|^ 2))
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 )))))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
A1,
Th25
.= (((
- (gf2
|^ 2))
* ((((((P
`3_3 )
* (Q
`1_3 ))
|^ 2)
+ ((g2
* ((P
`3_3 )
* (Q
`1_3 )))
* ((P
`1_3 )
* (Q
`3_3 ))))
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 ))))
+ (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
ALGSTR_1: 8
.= (((
- (gf2
|^ 2))
* ((((((P
`3_3 )
* (Q
`1_3 ))
|^ 2)
+ (g2
* (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 )))))
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 ))))
+ (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
GROUP_1:def 3
.= (((
- (gf2
|^ 2))
* (((((P
`3_3 )
* (Q
`1_3 ))
|^ 2)
+ ((g2
* (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 ))))
- (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 )))))
+ (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
ALGSTR_1: 7
.= (((
- (gf2
|^ 2))
* (((((P
`3_3 )
* (Q
`1_3 ))
|^ 2)
+ (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 ))))
+ (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
A1,
Th24
.= (((
- (gf2
* gf2))
* (((((P
`3_3 )
* (Q
`1_3 ))
|^ 2)
+ (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 ))))
+ (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
EC_PF_1: 22
.= ((((
- gf2)
* gf2)
* (((((P
`3_3 )
* (Q
`1_3 ))
|^ 2)
+ (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 ))))
+ (((P
`1_3 )
* (Q
`3_3 ))
|^ 2)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
VECTSP_1: 9
.= (((
- gf2)
* ((((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 )))
* (((((P
`3_3 )
* (Q
`1_3 ))
|^ 2)
+ (((P
`3_3 )
* (Q
`1_3 ))
* ((P
`1_3 )
* (Q
`3_3 ))))
+ (((P
`1_3 )
* (Q
`3_3 ))
|^ 2))))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
A2,
GROUP_1:def 3
.= (((
- gf2)
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))) by
Th17;
then
A8: ((gf2
|^ 2)
* (((((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`1_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`1_3 ))))
= (((gf2
|^ 2)
* gf2)
* (((
- gf2)
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))) by
A6,
GROUP_1:def 3
.= ((gf2
|^ (2
+ 1))
* (((
- gf2)
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))) by
EC_PF_1: 24
.= ((gf2
|^ 3)
* (((
- gf2)
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))));
((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
Th35;
then (((Q
`3_3 )
|^ 3)
* (((P
`2_3 )
|^ 2)
* (P
`3_3 )))
= (((Q
`3_3 )
|^ 3)
* ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3)))) by
VECTSP_1: 19
.= ((((Q
`3_3 )
|^ 3)
* (((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))))
+ (((Q
`3_3 )
|^ 3)
* (b
* ((P
`3_3 )
|^ 3)))) by
VECTSP_1:def 7
.= (((((Q
`3_3 )
|^ 3)
* ((P
`1_3 )
|^ 3))
+ (((Q
`3_3 )
|^ 3)
* ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))))
+ (((Q
`3_3 )
|^ 3)
* (b
* ((P
`3_3 )
|^ 3)))) by
VECTSP_1:def 7
.= (((((Q
`3_3 )
* (P
`1_3 ))
|^ 3)
+ (((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
* ((Q
`3_3 )
|^ (2
+ 1))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
BINOM: 9
.= (((((P
`1_3 )
* (Q
`3_3 ))
|^ 3)
+ (((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))
* (((Q
`3_3 )
|^ 2)
* (Q
`3_3 ))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
EC_PF_1: 24
.= (((((P
`1_3 )
* (Q
`3_3 ))
|^ 3)
+ ((a
* (P
`1_3 ))
* (((P
`3_3 )
|^ 2)
* (((Q
`3_3 )
|^ 2)
* (Q
`3_3 )))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= (((((P
`1_3 )
* (Q
`3_3 ))
|^ 3)
+ ((a
* (P
`1_3 ))
* ((((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2))
* (Q
`3_3 ))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= (((((P
`1_3 )
* (Q
`3_3 ))
|^ 3)
+ ((a
* (P
`1_3 ))
* ((((P
`3_3 )
* (Q
`3_3 ))
|^ 2)
* (Q
`3_3 ))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
BINOM: 9
.= (((((P
`1_3 )
* (Q
`3_3 ))
|^ 3)
+ (a
* ((P
`1_3 )
* ((((P
`3_3 )
* (Q
`3_3 ))
|^ 2)
* (Q
`3_3 )))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= (((((P
`1_3 )
* (Q
`3_3 ))
|^ 3)
+ (a
* ((((P
`3_3 )
* (Q
`3_3 ))
|^ 2)
* ((Q
`3_3 )
* (P
`1_3 )))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= (((((P
`1_3 )
* (Q
`3_3 ))
|^ 3)
+ ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((P
`1_3 )
* (Q
`3_3 ))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
GROUP_1:def 3;
then
A9: (
- (((Q
`3_3 )
|^ 3)
* (((P
`2_3 )
|^ 2)
* (P
`3_3 ))))
= ((
- ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3)))
- ((((P
`1_3 )
* (Q
`3_3 ))
|^ 3)
+ ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((P
`1_3 )
* (Q
`3_3 ))))) by
VECTSP_1: 17
.= ((
- ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3)))
+ ((
- ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((P
`1_3 )
* (Q
`3_3 ))))
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))) by
VECTSP_1: 17;
((((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))
- ((((Q
`1_3 )
|^ 3)
+ ((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2)))
+ (b
* ((Q
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
Th35;
then
A10: (((P
`3_3 )
|^ 3)
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 )))
= (((P
`3_3 )
|^ 3)
* ((((Q
`1_3 )
|^ 3)
+ ((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2)))
+ (b
* ((Q
`3_3 )
|^ 3)))) by
VECTSP_1: 19
.= ((((P
`3_3 )
|^ 3)
* (((Q
`1_3 )
|^ 3)
+ ((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))))
+ (((P
`3_3 )
|^ 3)
* (b
* ((Q
`3_3 )
|^ 3)))) by
VECTSP_1:def 7
.= (((((P
`3_3 )
|^ 3)
* ((Q
`1_3 )
|^ 3))
+ (((P
`3_3 )
|^ 3)
* ((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))))
+ (((P
`3_3 )
|^ 3)
* (b
* ((Q
`3_3 )
|^ 3)))) by
VECTSP_1:def 7
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ (((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
* ((P
`3_3 )
|^ (2
+ 1))))
+ ((b
* ((Q
`3_3 )
|^ 3))
* ((P
`3_3 )
|^ 3))) by
BINOM: 9
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ (((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
* (((P
`3_3 )
|^ 2)
* (P
`3_3 ))))
+ ((b
* ((Q
`3_3 )
|^ 3))
* ((P
`3_3 )
|^ 3))) by
EC_PF_1: 24
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ (((a
* (Q
`1_3 ))
* ((Q
`3_3 )
|^ 2))
* (((P
`3_3 )
|^ 2)
* (P
`3_3 ))))
+ (b
* (((Q
`3_3 )
|^ 3)
* ((P
`3_3 )
|^ 3)))) by
GROUP_1:def 3
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((a
* (Q
`1_3 ))
* (((Q
`3_3 )
|^ 2)
* (((P
`3_3 )
|^ 2)
* (P
`3_3 )))))
+ (b
* (((P
`3_3 )
|^ 3)
* ((Q
`3_3 )
|^ 3)))) by
GROUP_1:def 3
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((a
* (Q
`1_3 ))
* ((((Q
`3_3 )
|^ 2)
* ((P
`3_3 )
|^ 2))
* (P
`3_3 ))))
+ (b
* (((P
`3_3 )
|^ 3)
* ((Q
`3_3 )
|^ 3)))) by
GROUP_1:def 3
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((a
* (Q
`1_3 ))
* ((((Q
`3_3 )
* (P
`3_3 ))
|^ 2)
* (P
`3_3 ))))
+ (b
* (((P
`3_3 )
|^ 3)
* ((Q
`3_3 )
|^ 3)))) by
BINOM: 9
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((a
* (Q
`1_3 ))
* ((((P
`3_3 )
* (Q
`3_3 ))
|^ 2)
* (P
`3_3 ))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ (a
* ((Q
`1_3 )
* ((((P
`3_3 )
* (Q
`3_3 ))
|^ 2)
* (P
`3_3 )))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ (a
* ((((P
`3_3 )
* (Q
`3_3 ))
|^ 2)
* ((P
`3_3 )
* (Q
`1_3 )))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((Q
`1_3 )
* (P
`3_3 ))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))) by
GROUP_1:def 3;
A11: ((gf1
* ((P
`3_3 )
* (Q
`3_3 )))
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
= (((P
`3_3 )
* (Q
`3_3 ))
* ((((Q
`2_3 )
* (P
`3_3 ))
- ((P
`2_3 )
* (Q
`3_3 )))
* (((Q
`2_3 )
* (P
`3_3 ))
+ ((P
`2_3 )
* (Q
`3_3 ))))) by
A2,
GROUP_1:def 3
.= (((P
`3_3 )
* (Q
`3_3 ))
* ((((Q
`2_3 )
* (P
`3_3 ))
|^ 2)
- (((P
`2_3 )
* (Q
`3_3 ))
|^ 2))) by
Th15
.= (((P
`3_3 )
* (Q
`3_3 ))
* ((((Q
`2_3 )
|^ 2)
* ((P
`3_3 )
|^ 2))
- (((P
`2_3 )
* (Q
`3_3 ))
|^ 2))) by
BINOM: 9
.= (((P
`3_3 )
* (Q
`3_3 ))
* ((((Q
`2_3 )
|^ 2)
* ((P
`3_3 )
|^ 2))
- (((P
`2_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2)))) by
BINOM: 9
.= ((((P
`3_3 )
* (Q
`3_3 ))
* (((Q
`2_3 )
|^ 2)
* ((P
`3_3 )
|^ 2)))
- (((P
`3_3 )
* (Q
`3_3 ))
* (((P
`2_3 )
|^ 2)
* ((Q
`3_3 )
|^ 2)))) by
VECTSP_1: 11
.= (((P
`3_3 )
* ((Q
`3_3 )
* (((Q
`2_3 )
|^ 2)
* ((P
`3_3 )
|^ 2))))
- (((P
`3_3 )
* (Q
`3_3 ))
* (((Q
`3_3 )
|^ 2)
* ((P
`2_3 )
|^ 2)))) by
GROUP_1:def 3
.= (((P
`3_3 )
* (((Q
`3_3 )
* ((Q
`2_3 )
|^ 2))
* ((P
`3_3 )
|^ 2)))
- (((P
`3_3 )
* (Q
`3_3 ))
* (((Q
`3_3 )
|^ 2)
* ((P
`2_3 )
|^ 2)))) by
GROUP_1:def 3
.= (((P
`3_3 )
* (((P
`3_3 )
|^ 2)
* ((Q
`3_3 )
* ((Q
`2_3 )
|^ 2))))
- ((P
`3_3 )
* ((Q
`3_3 )
* (((Q
`3_3 )
|^ 2)
* ((P
`2_3 )
|^ 2))))) by
GROUP_1:def 3
.= (((P
`3_3 )
* (((P
`3_3 )
|^ 2)
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 ))))
- ((P
`3_3 )
* (((Q
`3_3 )
* ((Q
`3_3 )
|^ 2))
* ((P
`2_3 )
|^ 2)))) by
GROUP_1:def 3
.= ((((P
`3_3 )
* ((P
`3_3 )
|^ 2))
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 )))
- ((P
`3_3 )
* ((((Q
`3_3 )
|^ 2)
* (Q
`3_3 ))
* ((P
`2_3 )
|^ 2)))) by
GROUP_1:def 3
.= (((((P
`3_3 )
|^ 2)
* (P
`3_3 ))
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 )))
- ((P
`3_3 )
* (((P
`2_3 )
|^ 2)
* ((Q
`3_3 )
|^ (2
+ 1))))) by
EC_PF_1: 24
.= ((((P
`3_3 )
|^ (2
+ 1))
* (((Q
`2_3 )
|^ 2)
* (Q
`3_3 )))
- ((P
`3_3 )
* (((P
`2_3 )
|^ 2)
* ((Q
`3_3 )
|^ 3)))) by
EC_PF_1: 24
.= ((((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((Q
`1_3 )
* (P
`3_3 ))))
+ ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3)))
- (((Q
`3_3 )
|^ 3)
* ((P
`3_3 )
* ((P
`2_3 )
|^ 2)))) by
A10,
GROUP_1:def 3
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((Q
`1_3 )
* (P
`3_3 ))))
+ (((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))
+ ((
- ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3)))
+ ((
- ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((P
`1_3 )
* (Q
`3_3 ))))
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))))) by
A9,
ALGSTR_1: 7
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((Q
`1_3 )
* (P
`3_3 ))))
+ ((((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3))
- ((b
* ((P
`3_3 )
|^ 3))
* ((Q
`3_3 )
|^ 3)))
+ ((
- ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((P
`1_3 )
* (Q
`3_3 ))))
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))) by
ALGSTR_1: 7
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((Q
`1_3 )
* (P
`3_3 ))))
+ ((
0. (
GF p))
+ ((
- ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((P
`1_3 )
* (Q
`3_3 ))))
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))) by
VECTSP_1: 19
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((Q
`1_3 )
* (P
`3_3 ))))
+ ((
- ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((P
`1_3 )
* (Q
`3_3 ))))
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))) by
ALGSTR_1: 7
.= ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ (((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((Q
`1_3 )
* (P
`3_3 )))
+ ((
- ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((P
`1_3 )
* (Q
`3_3 ))))
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))) by
ALGSTR_1: 7
.= ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((Q
`1_3 )
* (P
`3_3 )))
- ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* ((P
`1_3 )
* (Q
`3_3 ))))
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))) by
ALGSTR_1: 7
.= ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
+ ((
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))
+ ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* (((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 )))))) by
VECTSP_1: 11
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))
+ ((a
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))
* gf2)) by
A2,
ALGSTR_1: 7
.= (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))
+ ((a
* gf2)
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))) by
GROUP_1:def 3;
A12: ((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
= ((((g2
* (Q
`3_3 ))
* gf1)
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))) by
GROUP_1:def 3
.= (((g2
* (Q
`3_3 ))
* (gf1
* (R
`3_3 )))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))) by
GROUP_1:def 3
.= ((gf1
* (R
`3_3 ))
* ((g2
* (Q
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((gf1
* (R
`3_3 ))
* (g2
* ((Q
`3_3 )
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))) by
GROUP_1:def 3
.= ((gf1
* (R
`3_3 ))
* (((Q
`3_3 )
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
+ ((Q
`3_3 )
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))) by
A1,
Th20
.= ((gf1
* (R
`3_3 ))
* ((((Q
`3_3 )
* (gf2
* (P
`2_3 )))
- ((Q
`3_3 )
* (gf1
* (P
`1_3 ))))
+ (((gf2
* (Q
`2_3 ))
- (gf1
* (Q
`1_3 )))
* (P
`3_3 )))) by
A5,
VECTSP_1: 11
.= ((gf1
* (R
`3_3 ))
* ((((gf2
* (P
`2_3 ))
* (Q
`3_3 ))
- ((gf1
* (P
`1_3 ))
* (Q
`3_3 )))
+ (((gf2
* (Q
`2_3 ))
* (P
`3_3 ))
- ((gf1
* (Q
`1_3 ))
* (P
`3_3 ))))) by
VECTSP_1: 13
.= ((gf1
* (R
`3_3 ))
* (((gf2
* ((P
`2_3 )
* (Q
`3_3 )))
- ((gf1
* (P
`1_3 ))
* (Q
`3_3 )))
+ (((gf2
* (Q
`2_3 ))
* (P
`3_3 ))
- ((gf1
* (Q
`1_3 ))
* (P
`3_3 ))))) by
GROUP_1:def 3
.= ((gf1
* (R
`3_3 ))
* ((((gf2
* ((P
`2_3 )
* (Q
`3_3 )))
- ((gf1
* (P
`1_3 ))
* (Q
`3_3 )))
+ ((gf2
* (Q
`2_3 ))
* (P
`3_3 )))
- ((gf1
* (Q
`1_3 ))
* (P
`3_3 )))) by
ALGSTR_1: 7
.= ((gf1
* (R
`3_3 ))
* ((((gf2
* ((P
`2_3 )
* (Q
`3_3 )))
- (gf1
* ((P
`1_3 )
* (Q
`3_3 ))))
+ ((gf2
* (Q
`2_3 ))
* (P
`3_3 )))
- ((gf1
* (Q
`1_3 ))
* (P
`3_3 )))) by
GROUP_1:def 3
.= ((gf1
* (R
`3_3 ))
* ((((gf2
* ((P
`2_3 )
* (Q
`3_3 )))
- (gf1
* ((P
`1_3 )
* (Q
`3_3 ))))
+ (gf2
* ((Q
`2_3 )
* (P
`3_3 ))))
- ((gf1
* (Q
`1_3 ))
* (P
`3_3 )))) by
GROUP_1:def 3
.= ((gf1
* (R
`3_3 ))
* ((((gf2
* ((P
`2_3 )
* (Q
`3_3 )))
+ (gf2
* ((Q
`2_3 )
* (P
`3_3 ))))
- (gf1
* ((P
`1_3 )
* (Q
`3_3 ))))
- ((gf1
* (Q
`1_3 ))
* (P
`3_3 )))) by
ALGSTR_1: 8
.= ((gf1
* (R
`3_3 ))
* ((((gf2
* ((P
`2_3 )
* (Q
`3_3 )))
+ (gf2
* ((Q
`2_3 )
* (P
`3_3 ))))
- (gf1
* ((P
`1_3 )
* (Q
`3_3 ))))
- (gf1
* ((Q
`1_3 )
* (P
`3_3 ))))) by
GROUP_1:def 3
.= ((gf1
* (R
`3_3 ))
* (((gf2
* ((P
`2_3 )
* (Q
`3_3 )))
+ (gf2
* ((Q
`2_3 )
* (P
`3_3 ))))
+ ((
- (gf1
* ((P
`1_3 )
* (Q
`3_3 ))))
- (gf1
* ((Q
`1_3 )
* (P
`3_3 )))))) by
ALGSTR_1: 7
.= ((gf1
* (R
`3_3 ))
* ((gf2
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
+ ((
- (gf1
* ((P
`1_3 )
* (Q
`3_3 ))))
+ (
- (gf1
* ((Q
`1_3 )
* (P
`3_3 ))))))) by
VECTSP_1:def 7
.= ((gf1
* (R
`3_3 ))
* ((gf2
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
+ (((
- gf1)
* ((P
`1_3 )
* (Q
`3_3 )))
+ (
- (gf1
* ((Q
`1_3 )
* (P
`3_3 ))))))) by
VECTSP_1: 9
.= ((gf1
* (R
`3_3 ))
* ((gf2
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
+ (((
- gf1)
* ((P
`1_3 )
* (Q
`3_3 )))
+ ((
- gf1)
* ((Q
`1_3 )
* (P
`3_3 )))))) by
VECTSP_1: 9
.= ((gf1
* (R
`3_3 ))
* ((gf2
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
+ ((
- gf1)
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))))) by
VECTSP_1:def 7
.= ((gf1
* ((gf2
|^ 3)
* ((P
`3_3 )
* (Q
`3_3 ))))
* ((gf2
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
+ ((
- gf1)
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))))) by
GROUP_1:def 3,
A3
.= (((gf1
* (gf2
|^ 3))
* ((P
`3_3 )
* (Q
`3_3 )))
* ((gf2
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
+ ((
- gf1)
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))))) by
GROUP_1:def 3
.= ((((gf2
|^ 3)
* gf1)
* ((P
`3_3 )
* (Q
`3_3 )))
* ((gf2
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
- (gf1
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))))) by
VECTSP_1: 9
.= (((gf2
|^ 3)
* (gf1
* ((P
`3_3 )
* (Q
`3_3 ))))
* ((gf2
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
- (gf1
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))))) by
GROUP_1:def 3
.= ((gf2
|^ 3)
* ((gf1
* ((P
`3_3 )
* (Q
`3_3 )))
* ((gf2
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
- (gf1
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 ))))))) by
GROUP_1:def 3
.= ((gf2
|^ 3)
* (((gf1
* ((P
`3_3 )
* (Q
`3_3 )))
* (gf2
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 )))))
- ((gf1
* ((P
`3_3 )
* (Q
`3_3 )))
* (gf1
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 ))))))) by
VECTSP_1: 11
.= ((gf2
|^ 3)
* ((((gf1
* ((P
`3_3 )
* (Q
`3_3 )))
* gf2)
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
- ((gf1
* ((P
`3_3 )
* (Q
`3_3 )))
* (gf1
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 ))))))) by
GROUP_1:def 3
.= ((gf2
|^ 3)
* (((gf2
* (gf1
* ((P
`3_3 )
* (Q
`3_3 ))))
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 ))))
- (((gf1
* ((P
`3_3 )
* (Q
`3_3 )))
* gf1)
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))))) by
GROUP_1:def 3
.= ((gf2
|^ 3)
* ((gf2
* ((gf1
* ((P
`3_3 )
* (Q
`3_3 )))
* (((P
`2_3 )
* (Q
`3_3 ))
+ ((Q
`2_3 )
* (P
`3_3 )))))
- ((gf1
* (gf1
* ((P
`3_3 )
* (Q
`3_3 ))))
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))))) by
GROUP_1:def 3
.= ((gf2
|^ 3)
* ((gf2
* (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))
+ ((a
* gf2)
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))))
- (((gf1
* gf1)
* ((P
`3_3 )
* (Q
`3_3 )))
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))))) by
A11,
GROUP_1:def 3
.= ((gf2
|^ 3)
* ((gf2
* (((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))
+ ((a
* gf2)
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))))
- (((gf1
|^ 2)
* ((P
`3_3 )
* (Q
`3_3 )))
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))))) by
EC_PF_1: 22
.= ((gf2
|^ 3)
* ((
- (((gf1
|^ 2)
* ((P
`3_3 )
* (Q
`3_3 )))
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))))
+ ((gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ (gf2
* ((a
* gf2)
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2)))))) by
VECTSP_1:def 7
.= ((gf2
|^ 3)
* ((
- ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))))
+ ((gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ (gf2
* ((a
* gf2)
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2)))))) by
GROUP_1:def 3
.= ((gf2
|^ 3)
* ((
- ((((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))
+ ((gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((gf2
* (a
* gf2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))))) by
GROUP_1:def 3
.= ((gf2
|^ 3)
* ((
- ((((P
`1_3 )
* (Q
`3_3 ))
+ ((Q
`1_3 )
* (P
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))
+ ((gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((a
* (gf2
* gf2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))))) by
GROUP_1:def 3
.= ((gf2
|^ 3)
* ((
- ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))
+ ((gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((a
* (gf2
|^ 2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))))) by
EC_PF_1: 22;
thus (((gf2
|^ 2)
* (((((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`1_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`1_3 ))))
+ ((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))
= ((gf2
|^ 3)
* ((((
- gf2)
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))
+ ((
- ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))
+ ((gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((a
* (gf2
|^ 2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2)))))) by
A8,
A12,
VECTSP_1:def 7
.= ((gf2
|^ 3)
* (((((
- gf2)
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))
- ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))))
+ ((gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((a
* (gf2
|^ 2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))))) by
ALGSTR_1: 7
.= ((gf2
|^ 3)
* ((((
- gf2)
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ (((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))
- ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )))))
+ ((gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((a
* (gf2
|^ 2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))))) by
ALGSTR_1: 7
.= ((gf2
|^ 3)
* ((((
- gf2)
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ (
0. (
GF p)))
+ ((gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((a
* (gf2
|^ 2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))))) by
VECTSP_1: 19
.= ((gf2
|^ 3)
* (((
- gf2)
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ ((a
* (gf2
|^ 2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))))) by
ALGSTR_1: 7
.= ((gf2
|^ 3)
* ((((
- gf2)
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3)))
+ (gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))))
+ ((a
* (gf2
|^ 2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2)))) by
ALGSTR_1: 7
.= ((gf2
|^ 3)
* (((
- (gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))))
+ (gf2
* ((((P
`3_3 )
* (Q
`1_3 ))
|^ 3)
- (((P
`1_3 )
* (Q
`3_3 ))
|^ 3))))
+ ((a
* (gf2
|^ 2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2)))) by
VECTSP_1: 9
.= ((gf2
|^ 3)
* ((
0. (
GF p))
+ ((a
* (gf2
|^ 2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2)))) by
RLVECT_1: 5
.= ((gf2
|^ 3)
* ((a
* (gf2
|^ 2))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2))) by
ALGSTR_1: 7
.= (((gf2
|^ 3)
* (a
* (gf2
|^ 2)))
* (((P
`3_3 )
* (Q
`3_3 ))
|^ 2)) by
GROUP_1:def 3
.= (((a
* (gf2
|^ 2))
* (gf2
|^ 3))
* (((P
`3_3 )
* (Q
`3_3 ))
* ((P
`3_3 )
* (Q
`3_3 )))) by
EC_PF_1: 22
.= ((a
* (gf2
|^ 2))
* ((gf2
|^ 3)
* (((P
`3_3 )
* (Q
`3_3 ))
* ((P
`3_3 )
* (Q
`3_3 ))))) by
GROUP_1:def 3
.= ((a
* (gf2
|^ 2))
* (((gf2
|^ 3)
* ((P
`3_3 )
* (Q
`3_3 )))
* ((P
`3_3 )
* (Q
`3_3 )))) by
GROUP_1:def 3
.= ((a
* (gf2
|^ 2))
* ((((gf2
|^ 3)
* (P
`3_3 ))
* (Q
`3_3 ))
* ((P
`3_3 )
* (Q
`3_3 )))) by
GROUP_1:def 3
.= ((a
* (gf2
|^ 2))
* ((P
`3_3 )
* ((Q
`3_3 )
* (R
`3_3 )))) by
GROUP_1:def 3,
A3
.= (((a
* (gf2
|^ 2))
* (P
`3_3 ))
* ((Q
`3_3 )
* (R
`3_3 ))) by
GROUP_1:def 3
.= ((((a
* (gf2
|^ 2))
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 )) by
GROUP_1:def 3;
end;
theorem ::
EC_PF_2:58
Th58: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,gf1,gf2,gf3 be
Element of (
GF p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
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
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((((R
`2_3 )
|^ 2)
* (R
`3_3 ))
- ((((R
`1_3 )
|^ 3)
+ (((z
`1 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((z
`2 )
* ((R
`3_3 )
|^ 3)))))
= (
0. (
GF p))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,gf1,gf2,gf3 be
Element of (
GF p), P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):] such that
A1: g2
= (2
mod p) and
A2: 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
A3: 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 ))];
set a = (z
`1 );
set b = (z
`2 );
A4: ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
= (((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((R
`2_3 )
|^ 2))
* (R
`3_3 )) by
GROUP_1:def 3
.= (((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`2_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 )) by
GROUP_1:def 3
.= (((((gf2
* (P
`3_3 ))
* (R
`2_3 ))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )) by
Th13
.= ((((
- ((gf1
* (((R
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
+ ((gf2
* (P
`2_3 ))
* (R
`3_3 ))))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )) by
A1,
A2,
A3,
Th54
.= (((((gf1
* (((R
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
+ ((gf2
* (P
`2_3 ))
* (R
`3_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )) by
Th1
.= ((((((gf1
* ((R
`1_3 )
* (P
`3_3 )))
- (gf1
* ((P
`1_3 )
* (R
`3_3 ))))
+ ((gf2
* (P
`2_3 ))
* (R
`3_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )) by
VECTSP_1: 11
.= (((((((gf1
* (P
`3_3 ))
* (R
`1_3 ))
- (gf1
* ((P
`1_3 )
* (R
`3_3 ))))
+ ((gf2
* (P
`2_3 ))
* (R
`3_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )) by
GROUP_1:def 3
.= (((((((gf1
* (P
`3_3 ))
* (R
`1_3 ))
- ((gf1
* (P
`1_3 ))
* (R
`3_3 )))
+ ((gf2
* (P
`2_3 ))
* (R
`3_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )) by
GROUP_1:def 3
.= (((((((gf1
* (P
`3_3 ))
* (R
`1_3 ))
+ ((gf2
* (P
`2_3 ))
* (R
`3_3 )))
- ((gf1
* (P
`1_3 ))
* (R
`3_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )) by
ALGSTR_1: 8
.= ((((((gf1
* (P
`3_3 ))
* (R
`1_3 ))
+ (((gf2
* (P
`2_3 ))
* (R
`3_3 ))
- ((gf1
* (P
`1_3 ))
* (R
`3_3 ))))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )) by
ALGSTR_1: 7
.= ((((((gf1
* (P
`3_3 ))
* (R
`1_3 ))
+ (((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
* (R
`3_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )) by
VECTSP_1: 13
.= (((((((gf1
* (P
`3_3 ))
|^ 2)
* ((R
`1_3 )
|^ 2))
+ ((((g2
* (gf1
* (P
`3_3 )))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (R
`1_3 ))
* (R
`3_3 )))
+ ((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2)))
* (Q
`3_3 ))
* (R
`3_3 )) by
A1,
Th27
.= ((((((gf1
* (P
`3_3 ))
|^ 2)
* ((R
`1_3 )
|^ 2))
+ ((((g2
* (gf1
* (P
`3_3 )))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (R
`1_3 ))
* (R
`3_3 )))
+ ((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2)))
* ((Q
`3_3 )
* (R
`3_3 ))) by
GROUP_1:def 3
.= ((((((gf1
* (P
`3_3 ))
|^ 2)
* ((R
`1_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))
+ (((((g2
* (gf1
* (P
`3_3 )))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (R
`1_3 ))
* (R
`3_3 ))
* ((Q
`3_3 )
* (R
`3_3 ))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
Th14
.= ((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))
+ (((((g2
* (gf1
* (P
`3_3 )))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (R
`1_3 ))
* (R
`3_3 ))
* ((Q
`3_3 )
* (R
`3_3 ))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
BINOM: 9
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* (gf1
* (P
`3_3 )))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (R
`1_3 ))
* (R
`3_3 ))
* ((Q
`3_3 )
* (R
`3_3 ))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((((((g2
* gf1)
* (P
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (R
`1_3 ))
* (R
`3_3 ))
* ((Q
`3_3 )
* (R
`3_3 ))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((g2
* gf1)
* (((((P
`3_3 )
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (R
`1_3 ))
* (R
`3_3 ))
* ((Q
`3_3 )
* (R
`3_3 )))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
Th12
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((g2
* gf1)
* (((((Q
`3_3 )
* (R
`3_3 ))
* (R
`1_3 ))
* (R
`3_3 ))
* ((P
`3_3 )
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
Th10
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* gf1)
* ((Q
`3_3 )
* (R
`3_3 )))
* (R
`1_3 ))
* (R
`3_3 ))
* ((P
`3_3 )
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
Th11
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* (R
`1_3 ))
* (R
`3_3 ))
* ((P
`3_3 )
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* (((R
`1_3 )
* (R
`3_3 ))
* ((P
`3_3 )
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
Th11
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* (((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
* (((R
`1_3 )
* (R
`3_3 ))
* (P
`3_3 )))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (R
`3_3 ))
* (P
`3_3 ))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 )))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
+ ((((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* ((R
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
+ ((((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* ((R
`3_3 )
|^ 2))
* (R
`3_3 ))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* (((R
`3_3 )
|^ 2)
* (R
`3_3 )))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* ((R
`3_3 )
|^ (2
+ 1)))) by
EC_PF_1: 24;
A5: (
- ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))
= (
- ((((gf2
|^ 2)
* ((P
`3_3 )
* (P
`3_3 )))
* (Q
`3_3 ))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
EC_PF_1: 22
.= (
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`3_3 ))
* (Q
`3_3 ))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
GROUP_1:def 3
.= (
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (P
`3_3 ))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
GROUP_1:def 3
.= (
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (P
`3_3 ))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
* (R
`3_3 ))))) by
EC_PF_1: 22
.= (
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (P
`3_3 ))
* (((a
* (R
`1_3 ))
* (R
`3_3 ))
* (R
`3_3 )))) by
GROUP_1:def 3
.= (
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (P
`3_3 ))
* (((a
* (R
`3_3 ))
* (R
`1_3 ))
* (R
`3_3 )))) by
GROUP_1:def 3
.= (
- ((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* ((P
`3_3 )
* (((a
* (R
`3_3 ))
* (R
`1_3 ))
* (R
`3_3 ))))) by
GROUP_1:def 3
.= (
- ((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* ((((P
`3_3 )
* (a
* (R
`3_3 )))
* (R
`1_3 ))
* (R
`3_3 )))) by
Th11
.= (
- ((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* ((a
* (R
`3_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 ))))) by
Th11
.= (
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (a
* (R
`3_3 )))
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))) by
GROUP_1:def 3
.= (
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* a)
* (R
`3_3 ))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 )))) by
GROUP_1:def 3
.= (
- (((gf2
|^ 2)
* ((((P
`3_3 )
* (Q
`3_3 ))
* a)
* (R
`3_3 )))
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))) by
Th11
.= (
- (((gf2
|^ 2)
* (((a
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 )))
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))) by
GROUP_1:def 3
.= (
- ((((((gf2
|^ 2)
* a)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))) by
Th11
.= (
- ((((gf2
|^ 2)
* (((((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`1_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`1_3 ))))
+ ((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))) by
A1,
A2,
A3,
Th57
.= (
- ((((gf2
|^ 2)
* (((((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`1_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`1_3 ))))
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 ))))) by
VECTSP_1:def 7
.= (
- ((((gf2
|^ 2)
* ((((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ ((((P
`3_3 )
* (Q
`1_3 ))
* (R
`1_3 ))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`1_3 )))))
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 ))))) by
ALGSTR_1: 7
.= (
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* ((((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ ((((P
`3_3 )
* (Q
`1_3 ))
* (R
`1_3 ))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`1_3 )))))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))) by
GROUP_1:def 3
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- (((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* ((((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ ((((P
`3_3 )
* (Q
`1_3 ))
* (R
`1_3 ))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`1_3 )))))) by
VECTSP_1: 17
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ (((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* ((((P
`3_3 )
* (Q
`1_3 ))
* (R
`1_3 ))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`1_3 )))))) by
VECTSP_1:def 7
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ (((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (R
`1_3 ))))) by
VECTSP_1:def 7
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`1_3 )))) by
GROUP_1:def 3
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (R
`1_3 ))
* (R
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`1_3 )))) by
Th12
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((R
`1_3 )
* (R
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))))
* (R
`1_3 )))) by
Th11
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (R
`1_3 ))
* (R
`3_3 )))
* (R
`1_3 )))) by
GROUP_1:def 3
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`1_3 ))
* (R
`3_3 ))
* (R
`1_3 )))) by
Th11
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (((R
`1_3 )
* (R
`3_3 ))
* (R
`1_3 ))))) by
Th11
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (((R
`1_3 )
* (R
`1_3 ))
* (R
`3_3 ))))) by
GROUP_1:def 3
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (R
`3_3 )))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 ))))) by
EC_PF_1: 22
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (R
`1_3 ))
* (R
`3_3 ))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 ))))) by
Th11
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (P
`3_3 ))
* (((R
`1_3 )
* (R
`3_3 ))
* (((P
`1_3 )
* (Q
`1_3 ))
* (R
`3_3 ))))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 ))))) by
Th11
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`1_3 )
* (Q
`1_3 ))
* ((R
`3_3 )
* ((R
`1_3 )
* (R
`3_3 )))))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 ))))) by
GROUP_1:def 3
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`1_3 )
* (Q
`1_3 ))
* ((R
`1_3 )
* ((R
`3_3 )
* (R
`3_3 )))))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 ))))) by
GROUP_1:def 3
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`1_3 )
* (Q
`1_3 ))
* ((R
`1_3 )
* ((R
`3_3 )
|^ 2))))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 ))))) by
EC_PF_1: 22
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (P
`3_3 ))
* ((((P
`1_3 )
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 ))))) by
GROUP_1:def 3
.= ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- (((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 ))))) by
Th11
.= (((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((gf2
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 )))) by
VECTSP_1: 17
.= (((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (P
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 )))) by
GROUP_1:def 3
.= (((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* ((P
`3_3 )
* (((R
`1_3 )
|^ 2)
* (R
`3_3 ))))) by
GROUP_1:def 3
.= (((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* ((R
`3_3 )
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))) by
GROUP_1:def 3
.= (((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))) by
GROUP_1:def 3;
A6: (
- ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (b
* ((R
`3_3 )
|^ 3))))
= (
- (((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* b)
* ((R
`3_3 )
|^ (2
+ 1)))) by
GROUP_1:def 3
.= (
- (((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* b)
* (((R
`3_3 )
|^ 2)
* (R
`3_3 )))) by
EC_PF_1: 24
.= (
- (((b
* (((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 )))
* (R
`3_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (
- (((((b
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
* ((R
`3_3 )
|^ 2))) by
Th12
.= (
- (((
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 )))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )))
* ((R
`3_3 )
|^ 2))) by
A1,
A2,
A3,
Th56
.= ((
- ((
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 )))
+ (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 ))))
* ((R
`3_3 )
|^ 2)) by
VECTSP_1: 9
.= (((
- (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )))
+ (((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 )))
* ((R
`3_3 )
|^ 2)) by
VECTSP_1: 33
.= (((
- (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 )))
* ((R
`3_3 )
|^ 2))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
VECTSP_1:def 7
.= ((
- ((((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
VECTSP_1: 9
.= ((
- (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* ((R
`3_3 )
* ((R
`3_3 )
|^ 2))))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= ((
- (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* ((R
`3_3 )
|^ (2
+ 1))))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
EC_PF_1: 24;
A7: (((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (b
* ((R
`3_3 )
|^ 3))))
= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
+ ((((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* ((R
`3_3 )
|^ 3))
+ ((
- (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* ((R
`3_3 )
|^ 3)))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))) by
A4,
A6,
ALGSTR_1: 7
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
+ (((((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* ((R
`3_3 )
|^ 3))
- (((((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
* (Q
`3_3 ))
* ((R
`3_3 )
|^ 3)))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
ALGSTR_1: 7
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
+ ((
0. (
GF p))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
VECTSP_1: 19
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
RLVECT_1: 4
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 )))) by
ALGSTR_1: 8;
A8: ((((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (b
* ((R
`3_3 )
|^ 3))))
- ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))
= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 )))
+ (((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))))) by
A5,
A7,
ALGSTR_1: 7
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 )))
+ ((
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
+ ((
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))))) by
ALGSTR_1: 7
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 )))
- (((((g2
* gf1)
* (Q
`3_3 ))
* (R
`3_3 ))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
* (((R
`1_3 )
* (P
`3_3 ))
* (R
`3_3 ))))
+ ((
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))))) by
ALGSTR_1: 7
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((
0. (
GF p))
+ ((
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))))) by
VECTSP_1: 19
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))) by
RLVECT_1: 4
.= ((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ (((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ ((
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))))) by
ALGSTR_1: 7
.= ((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- ((((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`1_3 ))
* (Q
`1_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))) by
ALGSTR_1: 7
.= ((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
+ ((
0. (
GF p))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))) by
VECTSP_1: 19
.= ((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))) by
RLVECT_1: 14
.= ((((((gf1
|^ 2)
* ((P
`3_3 )
* (P
`3_3 )))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))) by
EC_PF_1: 22
.= (((((((gf1
|^ 2)
* (P
`3_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 ))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))) by
GROUP_1:def 3
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* ((((P
`3_3 )
* ((R
`1_3 )
|^ 2))
* (Q
`3_3 ))
* (R
`3_3 )))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))) by
Th11
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (((P
`3_3 )
* ((R
`1_3 )
|^ 2))
* ((Q
`3_3 )
* (R
`3_3 ))))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))) by
GROUP_1:def 3
.= (((((gf1
|^ 2)
* (P
`3_3 ))
* ((Q
`3_3 )
* (R
`3_3 )))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))) by
GROUP_1:def 3
.= ((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))) by
GROUP_1:def 3;
thus ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((((R
`2_3 )
|^ 2)
* (R
`3_3 ))
- ((((R
`1_3 )
|^ 3)
+ (((z
`1 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((z
`2 )
* ((R
`3_3 )
|^ 3)))))
= (((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((((R
`1_3 )
|^ 3)
+ ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (b
* ((R
`3_3 )
|^ 3))))) by
VECTSP_1: 11
.= (((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- ((((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((R
`1_3 )
|^ 3))
+ ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))
+ ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (b
* ((R
`3_3 )
|^ 3))))) by
Th14
.= (((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- (((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((R
`1_3 )
|^ 3))
+ (((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (b
* ((R
`3_3 )
|^ 3)))))) by
ALGSTR_1: 7
.= ((((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- (((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (b
* ((R
`3_3 )
|^ 3)))))
- ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((R
`1_3 )
|^ 3))) by
VECTSP_1: 17
.= (((((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* (b
* ((R
`3_3 )
|^ 3))))
- ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))
- ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((R
`1_3 )
|^ 3))) by
VECTSP_1: 17
.= (((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))
- ((((gf2
|^ 2)
* ((P
`3_3 )
* (P
`3_3 )))
* (Q
`3_3 ))
* ((R
`1_3 )
|^ 3))) by
A8,
EC_PF_1: 22
.= (((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (P
`3_3 ))
* (Q
`3_3 ))
* ((R
`1_3 )
|^ 3))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 3))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* ((R
`1_3 )
|^ (2
+ 1)))
* (P
`3_3 ))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (R
`1_3 )))
* (P
`3_3 ))) by
EC_PF_1: 24
.= (((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))
- ((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* ((((R
`1_3 )
|^ 2)
* (R
`1_3 ))
* (P
`3_3 )))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))
- ((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* ((R
`1_3 )
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))
- ((((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))))
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`1_3 ))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 )))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
- (((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 )))
- ((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`1_3 )))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))) by
Th14
.= ((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
- (((((gf2
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`1_3 ))
+ (((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))) by
VECTSP_1: 17
.= ((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
- (((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 )))
+ (((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 ))))
* (R
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))) by
Th11
.= ((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
- (((gf2
|^ 2)
* (((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 )))
+ ((gf2
|^ 2)
* ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (R
`3_3 )))))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))) by
GROUP_1:def 3
.= ((((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 ))
- ((gf2
|^ 2)
* ((((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 ))
+ ((((P
`3_3 )
* (Q
`1_3 ))
+ ((P
`1_3 )
* (Q
`3_3 )))
* (R
`3_3 )))))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))) by
VECTSP_1:def 7
.= (((
- ((gf2
|^ 2)
* ((((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 ))
+ ((((P
`3_3 )
* (Q
`1_3 ))
* (R
`3_3 ))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`3_3 ))))))
+ ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 )))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))) by
VECTSP_1:def 7
.= (((
- ((gf2
|^ 2)
* (((((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`3_3 )))))
+ ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
* (R
`3_3 )))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))) by
ALGSTR_1: 7
.= (((
- ((gf2
|^ 2)
* (((((P
`3_3 )
* (Q
`3_3 ))
* (R
`1_3 ))
+ (((P
`3_3 )
* (Q
`1_3 ))
* (R
`3_3 )))
+ (((P
`1_3 )
* (Q
`3_3 ))
* (R
`3_3 )))))
+ ((gf1
|^ 2)
* (((P
`3_3 )
* (Q
`3_3 ))
* (R
`3_3 ))))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))) by
Th11
.= ((
0. (
GF p))
* (((R
`1_3 )
|^ 2)
* (P
`3_3 ))) by
A1,
A2,
A3,
Th55
.= (
0. (
GF p));
end;
theorem ::
EC_PF_2:59
Th59: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
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)
* (P
`3_3 ))
* (R
`2_3 ))
= (
- ((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
+ (((g2
* gf2)
* (P
`2_3 ))
* (R
`3_3 ))))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):] such that
A1: g2
= (2
mod p) & g3
= (3
mod p) & g4
= (4
mod p) & g8
= (8
mod p) and
A2: 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)) and
A3: R
=
[((g2
* gf4)
* gf2), ((gf1
* ((g4
* gf3)
- gf4))
- ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2))), (g8
* (gf2
|^ 3))];
set a = (z
`1 );
set b = (z
`2 );
g8
= ((2
* 4)
mod p) by
A1
.= (g2
* g4) by
A1,
EC_PF_1: 18;
then
A4: (g8
+ (g2
* g8))
= (g2
* (g4
+ g8)) by
VECTSP_1:def 7;
A5: ((P
`3_3 )
* gf3)
= (((P
`3_3 )
* ((P
`1_3 )
* (P
`2_3 )))
* gf2) by
A2,
GROUP_1:def 3
.= (((P
`1_3 )
* gf2)
* gf2) by
A2,
GROUP_1:def 3
.= ((P
`1_3 )
* (gf2
* gf2)) by
GROUP_1:def 3
.= ((P
`1_3 )
* (gf2
|^ 2)) by
EC_PF_1: 22;
A6: (((P
`3_3 )
* gf2)
* gf3)
= (((P
`3_3 )
* gf3)
* gf2) by
GROUP_1:def 3
.= ((P
`1_3 )
* ((gf2
|^ 2)
* gf2)) by
A5,
GROUP_1:def 3
.= ((P
`1_3 )
* (gf2
|^ (2
+ 1))) by
EC_PF_1: 24
.= ((P
`1_3 )
* (gf2
|^ 3));
A7: ((g4
* gf3)
- gf4)
= ((g4
* gf3)
+ ((g8
* gf3)
- (gf1
|^ 2))) by
A2,
VECTSP_1: 17
.= (((g4
* gf3)
+ (g8
* gf3))
- (gf1
|^ 2)) by
ALGSTR_1: 7
.= (((g4
+ g8)
* gf3)
- (gf1
|^ 2)) by
VECTSP_1:def 7;
(((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
= (((P
`3_3 )
* ((g2
* gf4)
* gf2))
- ((P
`1_3 )
* (R
`3_3 ))) by
A3
.= (((P
`3_3 )
* ((g2
* gf2)
* ((gf1
|^ 2)
- (g8
* gf3))))
- ((P
`1_3 )
* (g8
* (gf2
|^ 3)))) by
A2,
A3,
GROUP_1:def 3
.= ((((P
`3_3 )
* (g2
* gf2))
* ((gf1
|^ 2)
- (g8
* gf3)))
- ((P
`1_3 )
* (g8
* (gf2
|^ 3)))) by
GROUP_1:def 3
.= (((((P
`3_3 )
* gf2)
* g2)
* ((gf1
|^ 2)
- (g8
* gf3)))
- ((P
`1_3 )
* (g8
* (gf2
|^ 3)))) by
GROUP_1:def 3
.= ((g2
* (((P
`3_3 )
* gf2)
* ((gf1
|^ 2)
- (g8
* gf3))))
- ((P
`1_3 )
* (g8
* (gf2
|^ 3)))) by
GROUP_1:def 3
.= ((g2
* ((((P
`3_3 )
* gf2)
* (gf1
|^ 2))
- (((P
`3_3 )
* gf2)
* (g8
* gf3))))
- ((P
`1_3 )
* (g8
* (gf2
|^ 3)))) by
VECTSP_1: 11
.= ((g2
* ((((P
`3_3 )
* gf2)
* (gf1
|^ 2))
- ((((P
`3_3 )
* gf2)
* gf3)
* g8)))
- ((P
`1_3 )
* (g8
* (gf2
|^ 3)))) by
GROUP_1:def 3
.= (((g2
* (((P
`3_3 )
* gf2)
* (gf1
|^ 2)))
- (g2
* (((P
`1_3 )
* (gf2
|^ 3))
* g8)))
- ((P
`1_3 )
* (g8
* (gf2
|^ 3)))) by
A6,
VECTSP_1: 11
.= (((g2
* (((P
`3_3 )
* gf2)
* (gf1
|^ 2)))
- ((g2
* g8)
* ((P
`1_3 )
* (gf2
|^ 3))))
- ((P
`1_3 )
* (g8
* (gf2
|^ 3)))) by
GROUP_1:def 3
.= (((g2
* (((P
`3_3 )
* gf2)
* (gf1
|^ 2)))
- ((g2
* g8)
* ((P
`1_3 )
* (gf2
|^ 3))))
- (g8
* ((P
`1_3 )
* (gf2
|^ 3)))) by
GROUP_1:def 3
.= ((g2
* (((P
`3_3 )
* gf2)
* (gf1
|^ 2)))
- ((g8
* ((P
`1_3 )
* (gf2
|^ 3)))
+ ((g2
* g8)
* ((P
`1_3 )
* (gf2
|^ 3))))) by
VECTSP_1: 17
.= ((g2
* (((P
`3_3 )
* gf2)
* (gf1
|^ 2)))
- ((g8
+ (g2
* g8))
* ((P
`1_3 )
* (gf2
|^ 3)))) by
VECTSP_1:def 7;
then
A8: (
- (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
= (((g8
+ (g2
* g8))
* ((P
`1_3 )
* (gf2
|^ 3)))
- (g2
* (((P
`3_3 )
* gf2)
* (gf1
|^ 2)))) by
VECTSP_1: 17
.= ((g2
* ((g4
+ g8)
* ((P
`1_3 )
* (gf2
|^ 3))))
- (g2
* (((P
`3_3 )
* gf2)
* (gf1
|^ 2)))) by
A4,
GROUP_1:def 3
.= (g2
* (((g4
+ g8)
* ((P
`1_3 )
* (gf2
|^ 3)))
- (((P
`3_3 )
* gf2)
* (gf1
|^ 2)))) by
VECTSP_1: 11;
A9: (((g2
* gf2)
* (P
`3_3 ))
* (R
`2_3 ))
= (((g2
* gf2)
* (P
`3_3 ))
* ((gf1
* ((g4
* gf3)
- gf4))
- ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2)))) by
A3
.= ((((g2
* gf2)
* (P
`3_3 ))
* (gf1
* (((g4
+ g8)
* gf3)
- (gf1
|^ 2))))
- (((g2
* gf2)
* (P
`3_3 ))
* ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2)))) by
A7,
VECTSP_1: 11;
A10: (
- (((g2
* gf2)
* (P
`3_3 ))
* ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2))))
= (
- ((g2
* (gf2
* (P
`3_3 )))
* ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2)))) by
GROUP_1:def 3
.= (
- ((g2
* ((P
`3_3 )
* gf2))
* (g8
* (((P
`2_3 )
|^ 2)
* (gf2
|^ 2))))) by
GROUP_1:def 3
.= (
- (((g2
* (P
`3_3 ))
* gf2)
* (g8
* ((gf2
|^ 2)
* ((P
`2_3 )
|^ 2))))) by
GROUP_1:def 3
.= (
- ((g2
* (P
`3_3 ))
* (gf2
* (g8
* ((gf2
|^ 2)
* ((P
`2_3 )
|^ 2)))))) by
GROUP_1:def 3
.= (
- ((g2
* (P
`3_3 ))
* (gf2
* ((g8
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))))) by
GROUP_1:def 3
.= (
- ((g2
* (P
`3_3 ))
* ((gf2
* (g8
* (gf2
|^ 2)))
* ((P
`2_3 )
|^ 2)))) by
GROUP_1:def 3
.= (
- ((g2
* (P
`3_3 ))
* (((gf2
* g8)
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2)))) by
GROUP_1:def 3
.= (
- (((g2
* (P
`3_3 ))
* ((P
`2_3 )
|^ 2))
* ((gf2
* g8)
* (gf2
|^ 2)))) by
GROUP_1:def 3
.= (
- ((g2
* ((P
`3_3 )
* ((P
`2_3 )
|^ 2)))
* ((g8
* gf2)
* (gf2
|^ 2)))) by
GROUP_1:def 3
.= (
- ((g2
* (((P
`2_3 )
|^ 2)
* (P
`3_3 )))
* (g8
* (gf2
* (gf2
|^ 2))))) by
GROUP_1:def 3
.= (
- ((g2
* (((P
`2_3 )
* (P
`2_3 ))
* (P
`3_3 )))
* (g8
* ((gf2
|^ 2)
* gf2)))) by
EC_PF_1: 22
.= (
- ((g2
* ((P
`2_3 )
* ((P
`2_3 )
* (P
`3_3 ))))
* (g8
* ((gf2
|^ 2)
* gf2)))) by
GROUP_1:def 3
.= (
- ((g2
* ((P
`2_3 )
* ((P
`2_3 )
* (P
`3_3 ))))
* (g8
* (gf2
|^ (2
+ 1))))) by
EC_PF_1: 24
.= (
- (g2
* (((P
`2_3 )
* gf2)
* (g8
* (gf2
|^ 3))))) by
A2,
GROUP_1:def 3
.= (
- (g2
* (gf2
* ((P
`2_3 )
* (R
`3_3 ))))) by
GROUP_1:def 3,
A3
.= (
- ((g2
* gf2)
* ((P
`2_3 )
* (R
`3_3 )))) by
GROUP_1:def 3
.= (
- (((g2
* gf2)
* (P
`2_3 ))
* (R
`3_3 ))) by
GROUP_1:def 3;
(((g2
* gf2)
* (P
`3_3 ))
* (gf1
* (((g4
+ g8)
* gf3)
- (gf1
|^ 2))))
= (gf1
* (((g2
* gf2)
* (P
`3_3 ))
* (((g4
+ g8)
* gf3)
- (gf1
|^ 2)))) by
GROUP_1:def 3
.= (gf1
* ((g2
* (gf2
* (P
`3_3 )))
* (((g4
+ g8)
* gf3)
- (gf1
|^ 2)))) by
GROUP_1:def 3
.= (gf1
* (g2
* ((gf2
* (P
`3_3 ))
* (((g4
+ g8)
* gf3)
- (gf1
|^ 2))))) by
GROUP_1:def 3
.= (gf1
* (g2
* ((((P
`3_3 )
* gf2)
* ((g4
+ g8)
* gf3))
- (((P
`3_3 )
* gf2)
* (gf1
|^ 2))))) by
VECTSP_1: 11
.= (gf1
* (g2
* (((((P
`3_3 )
* gf2)
* gf3)
* (g4
+ g8))
- (((P
`3_3 )
* gf2)
* (gf1
|^ 2))))) by
GROUP_1:def 3
.= (
- (gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))) by
A6,
A8,
VECTSP_1: 8;
hence (((g2
* gf2)
* (P
`3_3 ))
* (R
`2_3 ))
= ((
- (gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))))
- (((g2
* gf2)
* (P
`2_3 ))
* (R
`3_3 ))) by
A9,
A10
.= (
- ((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
+ (((g2
* gf2)
* (P
`2_3 ))
* (R
`3_3 )))) by
VECTSP_1: 17;
end;
theorem ::
EC_PF_2:60
Th60: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
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 (((g4
* (gf2
|^ 2))
* (P
`3_3 ))
* (R
`1_3 ))
= ((R
`3_3 )
* (((gf1
|^ 2)
* (P
`3_3 ))
- ((g8
* (gf2
|^ 2))
* (P
`1_3 ))))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):] such that
A1: g2
= (2
mod p) & g3
= (3
mod p) & g4
= (4
mod p) & g8
= (8
mod p) and
A2: 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)) and
A3: R
=
[((g2
* gf4)
* gf2), ((gf1
* ((g4
* gf3)
- gf4))
- ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2))), (g8
* (gf2
|^ 3))];
set a = (z
`1 );
set b = (z
`2 );
A4: g8
= ((2
* 4)
mod p) by
A1
.= (g2
* g4) by
A1,
EC_PF_1: 18;
A5: ((P
`3_3 )
* gf3)
= (((P
`3_3 )
* ((P
`1_3 )
* (P
`2_3 )))
* gf2) by
A2,
GROUP_1:def 3
.= (((P
`1_3 )
* ((P
`2_3 )
* (P
`3_3 )))
* gf2) by
GROUP_1:def 3
.= ((P
`1_3 )
* (gf2
* gf2)) by
A2,
GROUP_1:def 3
.= ((P
`1_3 )
* (gf2
|^ 2)) by
EC_PF_1: 22;
A6: (((g4
* (gf2
|^ 2))
* (P
`3_3 ))
* (R
`1_3 ))
= (((g4
* (gf2
|^ 2))
* (P
`3_3 ))
* ((g2
* gf4)
* gf2)) by
A3
.= (((g4
* (gf2
|^ 2))
* (P
`3_3 ))
* (g2
* (gf4
* gf2))) by
GROUP_1:def 3
.= ((g4
* ((gf2
|^ 2)
* (P
`3_3 )))
* (g2
* (gf2
* gf4))) by
GROUP_1:def 3
.= ((g4
* ((P
`3_3 )
* (gf2
|^ 2)))
* ((g2
* gf2)
* gf4)) by
GROUP_1:def 3
.= (((g4
* (P
`3_3 ))
* (gf2
|^ 2))
* ((gf2
* g2)
* gf4)) by
GROUP_1:def 3
.= ((((g4
* (P
`3_3 ))
* (gf2
|^ 2))
* (gf2
* g2))
* gf4) by
GROUP_1:def 3
.= (((g4
* (P
`3_3 ))
* ((gf2
|^ 2)
* (gf2
* g2)))
* gf4) by
GROUP_1:def 3
.= (((g4
* (P
`3_3 ))
* (((gf2
|^ 2)
* gf2)
* g2))
* gf4) by
GROUP_1:def 3
.= (((g4
* (P
`3_3 ))
* ((gf2
|^ (2
+ 1))
* g2))
* gf4) by
EC_PF_1: 24
.= ((((g4
* (P
`3_3 ))
* g2)
* (gf2
|^ 3))
* gf4) by
GROUP_1:def 3
.= ((((g2
* g4)
* (P
`3_3 ))
* (gf2
|^ 3))
* ((gf1
|^ 2)
- (g8
* gf3))) by
A2,
GROUP_1:def 3
.= (((g8
* (gf2
|^ 3))
* (P
`3_3 ))
* ((gf1
|^ 2)
- (g8
* gf3))) by
A4,
GROUP_1:def 3;
thus ((R
`3_3 )
* (((gf1
|^ 2)
* (P
`3_3 ))
- ((g8
* (gf2
|^ 2))
* (P
`1_3 ))))
= ((g8
* (gf2
|^ 3))
* (((gf1
|^ 2)
* (P
`3_3 ))
- ((g8
* (gf2
|^ 2))
* (P
`1_3 )))) by
A3
.= ((g8
* (gf2
|^ 3))
* (((P
`3_3 )
* (gf1
|^ 2))
- (g8
* ((gf2
|^ 2)
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((g8
* (gf2
|^ 3))
* (((P
`3_3 )
* (gf1
|^ 2))
- ((P
`3_3 )
* (g8
* gf3)))) by
A5,
GROUP_1:def 3
.= ((g8
* (gf2
|^ 3))
* ((P
`3_3 )
* ((gf1
|^ 2)
- (g8
* gf3)))) by
VECTSP_1: 11
.= (((g4
* (gf2
|^ 2))
* (P
`3_3 ))
* (R
`1_3 )) by
A6,
GROUP_1:def 3;
end;
theorem ::
EC_PF_2:61
Th61: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
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 (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((z
`2 )
* (R
`3_3 )))
= (((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2))
- (((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 )))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):] such that
A1: g2
= (2
mod p) & g3
= (3
mod p) & g4
= (4
mod p) & g8
= (8
mod p) and
A2: 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)) and
A3: R
=
[((g2
* gf4)
* gf2), ((gf1
* ((g4
* gf3)
- gf4))
- ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2))), (g8
* (gf2
|^ 3))];
set a = (z
`1 );
set b = (z
`2 );
A4: g4
= ((2
* 2)
mod p) by
A1
.= (g2
* g2) by
A1,
EC_PF_1: 18;
A5: g8
= ((2
* 4)
mod p) by
A1
.= (g2
* g4) by
A1,
EC_PF_1: 18;
A6: (
- (((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 )))
= (
- ((g4
* (gf2
|^ 2))
* (((P
`1_3 )
|^ 2)
* ((g2
* gf4)
* gf2)))) by
GROUP_1:def 3,
A3
.= (
- (g4
* ((gf2
|^ 2)
* (((P
`1_3 )
|^ 2)
* ((g2
* gf4)
* gf2))))) by
GROUP_1:def 3
.= (
- (g4
* (((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((g2
* gf4)
* gf2)))) by
GROUP_1:def 3
.= (
- (g4
* (((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))
* (g2
* (gf4
* gf2))))) by
GROUP_1:def 3
.= (
- (g4
* ((((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))
* (gf2
* gf4))
* g2))) by
GROUP_1:def 3
.= (
- ((g4
* g2)
* (((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))
* (gf2
* gf4)))) by
GROUP_1:def 3
.= (
- (g8
* (((P
`1_3 )
|^ 2)
* ((gf2
|^ 2)
* (gf2
* gf4))))) by
A5,
GROUP_1:def 3
.= (
- (g8
* (((P
`1_3 )
|^ 2)
* (((gf2
|^ 2)
* gf2)
* gf4)))) by
GROUP_1:def 3
.= (
- (g8
* (((P
`1_3 )
|^ 2)
* ((gf2
|^ (2
+ 1))
* gf4)))) by
EC_PF_1: 24
.= (
- (g8
* ((((P
`1_3 )
|^ 2)
* (gf2
|^ 3))
* gf4))) by
GROUP_1:def 3
.= (
- ((g8
* (((P
`1_3 )
|^ 2)
* (gf2
|^ 3)))
* gf4)) by
GROUP_1:def 3
.= (
- (((g8
* (gf2
|^ 3))
* ((P
`1_3 )
|^ 2))
* gf4)) by
GROUP_1:def 3
.= (
- ((((R
`3_3 )
* ((P
`1_3 )
|^ 2))
* (gf1
|^ 2))
- (((R
`3_3 )
* ((P
`1_3 )
|^ 2))
* (g8
* gf3)))) by
A2,
A3,
VECTSP_1: 11
.= ((((R
`3_3 )
* ((P
`1_3 )
|^ 2))
* (g8
* gf3))
- (((R
`3_3 )
* ((P
`1_3 )
|^ 2))
* (gf1
|^ 2))) by
VECTSP_1: 17
.= (((((R
`3_3 )
* ((P
`1_3 )
|^ 2))
* g8)
* gf3)
- (((R
`3_3 )
* ((P
`1_3 )
|^ 2))
* (gf1
|^ 2))) by
GROUP_1:def 3
.= (((g8
* ((R
`3_3 )
* ((P
`1_3 )
|^ 2)))
* gf3)
- ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* (gf1
|^ 2)))) by
GROUP_1:def 3
.= ((g8
* (((R
`3_3 )
* ((P
`1_3 )
|^ 2))
* gf3))
- ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* (gf1
|^ 2)))) by
GROUP_1:def 3
.= ((g8
* ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* (((P
`1_3 )
* (P
`2_3 ))
* gf2))))
- ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* (gf1
|^ 2)))) by
A2,
GROUP_1:def 3
.= ((g8
* ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* ((P
`1_3 )
* ((P
`2_3 )
* gf2)))))
- ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* (gf1
|^ 2)))) by
GROUP_1:def 3
.= ((g8
* ((R
`3_3 )
* ((((P
`1_3 )
|^ 2)
* (P
`1_3 ))
* ((P
`2_3 )
* gf2))))
- ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* (gf1
|^ 2)))) by
GROUP_1:def 3
.= ((g8
* ((R
`3_3 )
* (((P
`1_3 )
|^ (2
+ 1))
* ((P
`2_3 )
* gf2))))
- ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* (gf1
|^ 2)))) by
EC_PF_1: 24
.= ((g8
* (((R
`3_3 )
* ((P
`2_3 )
* gf2))
* ((P
`1_3 )
|^ 3)))
- ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* (gf1
|^ 2)))) by
GROUP_1:def 3
.= ((((g2
* g4)
* ((R
`3_3 )
* ((P
`2_3 )
* gf2)))
* ((P
`1_3 )
|^ 3))
- ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* (gf1
|^ 2)))) by
A5,
GROUP_1:def 3
.= (((g2
* (g4
* ((R
`3_3 )
* ((P
`2_3 )
* gf2))))
* ((P
`1_3 )
|^ 3))
- ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* (gf1
|^ 2)))) by
GROUP_1:def 3
.= (((g4
* ((R
`3_3 )
* ((P
`2_3 )
* gf2)))
* (g2
* ((P
`1_3 )
|^ 3)))
- ((R
`3_3 )
* (((P
`1_3 )
|^ 2)
* (gf1
|^ 2)))) by
GROUP_1:def 3
.= ((
- ((R
`3_3 )
* ((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))))
+ (((g4
* (R
`3_3 ))
* ((P
`2_3 )
* gf2))
* (g2
* ((P
`1_3 )
|^ 3)))) by
GROUP_1:def 3;
A7: (((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2))
- ((R
`3_3 )
* ((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))))
= ((R
`3_3 )
* (((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
- ((gf1
|^ 2)
* ((P
`1_3 )
|^ 2)))) by
VECTSP_1: 11
.= ((R
`3_3 )
* (((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)
- ((gf1
* (P
`1_3 ))
|^ 2))) by
BINOM: 9
.= ((R
`3_3 )
* (((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
+ (gf1
* (P
`1_3 )))
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
- (gf1
* (P
`1_3 ))))) by
Th15
.= ((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
- ((gf1
* (P
`1_3 ))
- (gf1
* (P
`1_3 ))))
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
- (gf1
* (P
`1_3 ))))) by
RLVECT_1: 29
.= ((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
- ((gf1
* (P
`1_3 ))
- (gf1
* (P
`1_3 ))))
* (((g2
* gf2)
* (P
`2_3 ))
- ((gf1
* (P
`1_3 ))
+ (gf1
* (P
`1_3 )))))) by
RLVECT_1: 27
.= ((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
- ((gf1
* (P
`1_3 ))
- (gf1
* (P
`1_3 ))))
* (((g2
* gf2)
* (P
`2_3 ))
- (g2
* (gf1
* (P
`1_3 )))))) by
A1,
Th20
.= ((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
- (
0. (
GF p)))
* (((g2
* gf2)
* (P
`2_3 ))
- (g2
* (gf1
* (P
`1_3 )))))) by
VECTSP_1: 19
.= ((R
`3_3 )
* (((g2
* gf2)
* (P
`2_3 ))
* (((g2
* gf2)
* (P
`2_3 ))
- (g2
* (gf1
* (P
`1_3 )))))) by
RLVECT_1: 13
.= ((R
`3_3 )
* (((g2
* gf2)
* (P
`2_3 ))
* ((g2
* (gf2
* (P
`2_3 )))
- (g2
* (gf1
* (P
`1_3 )))))) by
GROUP_1:def 3
.= ((R
`3_3 )
* (((g2
* gf2)
* (P
`2_3 ))
* (g2
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))) by
VECTSP_1: 11
.= ((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
* g2)
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((R
`3_3 )
* ((g2
* (g2
* (gf2
* (P
`2_3 ))))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((R
`3_3 )
* ((g4
* (gf2
* (P
`2_3 )))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
A4,
GROUP_1:def 3
.= (((R
`3_3 )
* (g4
* (gf2
* (P
`2_3 ))))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))) by
GROUP_1:def 3
.= (((g4
* (R
`3_3 ))
* ((P
`2_3 )
* gf2))
* ((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))) by
GROUP_1:def 3;
A8: 3
= (2
+ 1);
((((P
`2_3 )
|^ 2)
* (P
`3_3 ))
- ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3))))
= (
0. (
GF p)) by
Th35;
then
A9: ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3)))
= (((P
`2_3 )
|^ 2)
* (P
`3_3 )) by
VECTSP_1: 19
.= (((P
`2_3 )
* (P
`2_3 ))
* (P
`3_3 )) by
EC_PF_1: 22
.= (gf2
* (P
`2_3 )) by
A2,
GROUP_1:def 3;
thus (((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2))
- (((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 )))
= ((((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2))
- ((R
`3_3 )
* ((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))))
+ (((g4
* (R
`3_3 ))
* ((P
`2_3 )
* gf2))
* (g2
* ((P
`1_3 )
|^ 3)))) by
A6,
ALGSTR_1: 7
.= (((g4
* (R
`3_3 ))
* ((P
`2_3 )
* gf2))
* (((gf2
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
+ (g2
* ((P
`1_3 )
|^ 3)))) by
A7,
VECTSP_1:def 7
.= (((g4
* (R
`3_3 ))
* ((P
`2_3 )
* gf2))
* (((gf2
* (P
`2_3 ))
- (((a
* ((P
`3_3 )
|^ 2))
* (P
`1_3 ))
+ ((g3
* ((P
`1_3 )
|^ 2))
* (P
`1_3 ))))
+ (g2
* ((P
`1_3 )
|^ 3)))) by
A2,
VECTSP_1:def 7
.= (((g4
* (R
`3_3 ))
* ((P
`2_3 )
* gf2))
* ((gf2
* (P
`2_3 ))
- ((((a
* ((P
`3_3 )
|^ 2))
* (P
`1_3 ))
+ ((g3
* ((P
`1_3 )
|^ 2))
* (P
`1_3 )))
- (g2
* ((P
`1_3 )
|^ 3))))) by
RLVECT_1: 29
.= (((g4
* (R
`3_3 ))
* ((P
`2_3 )
* gf2))
* ((gf2
* (P
`2_3 ))
- ((((a
* ((P
`3_3 )
|^ 2))
* (P
`1_3 ))
+ (g3
* (((P
`1_3 )
|^ 2)
* (P
`1_3 ))))
- (g2
* ((P
`1_3 )
|^ 3))))) by
GROUP_1:def 3
.= (((g4
* (R
`3_3 ))
* ((P
`2_3 )
* gf2))
* ((gf2
* (P
`2_3 ))
- ((((a
* ((P
`3_3 )
|^ 2))
* (P
`1_3 ))
+ (g3
* ((P
`1_3 )
|^ (2
+ 1))))
- (g2
* ((P
`1_3 )
|^ 3))))) by
EC_PF_1: 24
.= (((g4
* (R
`3_3 ))
* ((P
`2_3 )
* gf2))
* ((gf2
* (P
`2_3 ))
- (((a
* ((P
`3_3 )
|^ 2))
* (P
`1_3 ))
+ ((g3
* ((P
`1_3 )
|^ 3))
- (g2
* ((P
`1_3 )
|^ 3)))))) by
ALGSTR_1: 7
.= (((g4
* (R
`3_3 ))
* ((P
`2_3 )
* gf2))
* ((gf2
* (P
`2_3 ))
- (((a
* ((P
`3_3 )
|^ 2))
* (P
`1_3 ))
+ ((P
`1_3 )
|^ 3)))) by
A1,
A8,
Th22
.= (((g4
* (R
`3_3 ))
* ((P
`2_3 )
* gf2))
* (((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
+ (b
* ((P
`3_3 )
|^ 3)))
- (((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2))))) by
A9,
GROUP_1:def 3
.= (((g4
* (R
`3_3 ))
* (gf2
* (P
`2_3 )))
* ((b
* ((P
`3_3 )
|^ 3))
+ ((((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))
- (((P
`1_3 )
|^ 3)
+ ((a
* (P
`1_3 ))
* ((P
`3_3 )
|^ 2)))))) by
ALGSTR_1: 7
.= (((g4
* (R
`3_3 ))
* (gf2
* (P
`2_3 )))
* ((b
* ((P
`3_3 )
|^ 3))
+ (
0. (
GF p)))) by
RLVECT_1: 5
.= (((g4
* (R
`3_3 ))
* (gf2
* (P
`2_3 )))
* (b
* ((P
`3_3 )
|^ 3))) by
ALGSTR_1: 7
.= ((((g4
* (R
`3_3 ))
* gf2)
* (P
`2_3 ))
* (b
* ((P
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= (((g4
* (R
`3_3 ))
* gf2)
* ((P
`2_3 )
* (b
* ((P
`3_3 )
|^ 3)))) by
GROUP_1:def 3
.= (((g4
* (R
`3_3 ))
* gf2)
* (b
* ((P
`2_3 )
* ((P
`3_3 )
|^ (2
+ 1))))) by
GROUP_1:def 3
.= (((g4
* (R
`3_3 ))
* gf2)
* (b
* ((P
`2_3 )
* (((P
`3_3 )
|^ 2)
* (P
`3_3 ))))) by
EC_PF_1: 24
.= ((((g4
* (R
`3_3 ))
* gf2)
* b)
* ((P
`2_3 )
* ((P
`3_3 )
* ((P
`3_3 )
|^ 2)))) by
GROUP_1:def 3
.= ((((g4
* (R
`3_3 ))
* gf2)
* b)
* (gf2
* ((P
`3_3 )
|^ 2))) by
A2,
GROUP_1:def 3
.= (((g4
* ((R
`3_3 )
* gf2))
* b)
* (gf2
* ((P
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= ((g4
* (((R
`3_3 )
* gf2)
* b))
* (gf2
* ((P
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (g4
* ((((R
`3_3 )
* gf2)
* b)
* (gf2
* ((P
`3_3 )
|^ 2)))) by
GROUP_1:def 3
.= (g4
* (((((R
`3_3 )
* gf2)
* b)
* gf2)
* ((P
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (g4
* ((gf2
* (gf2
* ((R
`3_3 )
* b)))
* ((P
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (g4
* (((gf2
* gf2)
* ((R
`3_3 )
* b))
* ((P
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (g4
* (((gf2
|^ 2)
* (b
* (R
`3_3 )))
* ((P
`3_3 )
|^ 2))) by
EC_PF_1: 22
.= (g4
* ((gf2
|^ 2)
* ((b
* (R
`3_3 ))
* ((P
`3_3 )
|^ 2)))) by
GROUP_1:def 3
.= ((g4
* (gf2
|^ 2))
* (((P
`3_3 )
|^ 2)
* (b
* (R
`3_3 )))) by
GROUP_1:def 3
.= (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (b
* (R
`3_3 ))) by
GROUP_1:def 3;
end;
theorem ::
EC_PF_2:62
Th62: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
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 )))))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):] such that
A1: g2
= (2
mod p) & g3
= (3
mod p) & g4
= (4
mod p) & g8
= (8
mod p) and
A2: 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)) and
A3: R
=
[((g2
* gf4)
* gf2), ((gf1
* ((g4
* gf3)
- gf4))
- ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2))), (g8
* (gf2
|^ 3))];
set a = (z
`1 );
set b = (z
`2 );
A4: g8
= ((2
* 4)
mod p) by
A1
.= (g2
* g4) by
A1,
EC_PF_1: 18;
A5: ((P
`3_3 )
* gf3)
= (((P
`3_3 )
* ((P
`1_3 )
* (P
`2_3 )))
* gf2) by
A2,
GROUP_1:def 3
.= (((P
`1_3 )
* ((P
`2_3 )
* (P
`3_3 )))
* gf2) by
GROUP_1:def 3
.= ((P
`1_3 )
* (gf2
* gf2)) by
A2,
GROUP_1:def 3
.= ((P
`1_3 )
* (gf2
|^ 2)) by
EC_PF_1: 22;
A6: (((g2
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (a
* (R
`3_3 )))
= ((((g2
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* a)
* (R
`3_3 )) by
GROUP_1:def 3
.= ((R
`3_3 )
* ((g2
* (gf2
|^ 2))
* (a
* ((P
`3_3 )
|^ 2)))) by
GROUP_1:def 3;
A7: (((gf1
* (P
`3_3 ))
* (R
`3_3 ))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))
= ((R
`3_3 )
* ((gf1
* (P
`3_3 ))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((R
`3_3 )
* (gf1
* ((P
`3_3 )
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))) by
GROUP_1:def 3
.= ((R
`3_3 )
* (gf1
* (((P
`3_3 )
* ((g2
* gf2)
* (P
`2_3 )))
- ((P
`3_3 )
* (gf1
* (P
`1_3 )))))) by
VECTSP_1: 11
.= ((R
`3_3 )
* (gf1
* (((P
`3_3 )
* ((P
`2_3 )
* (g2
* gf2)))
- (((P
`3_3 )
* gf1)
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((R
`3_3 )
* (gf1
* ((((P
`3_3 )
* (P
`2_3 ))
* (g2
* gf2))
- ((P
`1_3 )
* ((P
`3_3 )
* gf1))))) by
GROUP_1:def 3
.= ((R
`3_3 )
* (gf1
* ((((P
`2_3 )
* (P
`3_3 ))
* (g2
* gf2))
- (((P
`1_3 )
* (P
`3_3 ))
* gf1)))) by
GROUP_1:def 3
.= ((R
`3_3 )
* (gf1
* ((g2
* (gf2
* gf2))
- (gf1
* ((P
`1_3 )
* (P
`3_3 )))))) by
A2,
GROUP_1:def 3
.= ((R
`3_3 )
* (gf1
* ((g2
* (gf2
|^ 2))
- (gf1
* ((P
`1_3 )
* (P
`3_3 )))))) by
EC_PF_1: 22
.= ((R
`3_3 )
* ((gf1
* (g2
* (gf2
|^ 2)))
- (gf1
* (gf1
* ((P
`1_3 )
* (P
`3_3 )))))) by
VECTSP_1: 11
.= ((R
`3_3 )
* ((gf1
* (g2
* (gf2
|^ 2)))
- ((gf1
* gf1)
* ((P
`1_3 )
* (P
`3_3 ))))) by
GROUP_1:def 3
.= ((R
`3_3 )
* (((g2
* (gf2
|^ 2))
* gf1)
- ((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 ))))) by
EC_PF_1: 22;
A8: ((gf2
|^ 2)
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 ))))
= (((gf2
|^ 2)
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 )))
+ ((gf2
|^ 2)
* ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))) by
VECTSP_1:def 7
.= (((gf2
|^ 2)
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 )))
+ (((gf2
|^ 2)
* (g2
* ((P
`1_3 )
|^ 2)))
* (R
`3_3 ))) by
GROUP_1:def 3
.= (((gf2
|^ 2)
* ((R
`1_3 )
* ((g4
* (P
`1_3 ))
* (P
`3_3 ))))
+ ((((gf2
|^ 2)
* g2)
* ((P
`1_3 )
|^ 2))
* (R
`3_3 ))) by
GROUP_1:def 3
.= ((((gf2
|^ 2)
* ((g2
* gf4)
* gf2))
* ((g4
* (P
`1_3 ))
* (P
`3_3 )))
+ ((R
`3_3 )
* (((gf2
|^ 2)
* g2)
* ((P
`1_3 )
|^ 2)))) by
A3,
GROUP_1:def 3
.= ((((gf2
|^ 2)
* (gf2
* (g2
* gf4)))
* ((g4
* (P
`1_3 ))
* (P
`3_3 )))
+ ((R
`3_3 )
* (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
GROUP_1:def 3
.= (((((gf2
|^ 2)
* gf2)
* (g2
* gf4))
* ((g4
* (P
`1_3 ))
* (P
`3_3 )))
+ ((R
`3_3 )
* (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
GROUP_1:def 3
.= ((((gf2
|^ (2
+ 1))
* (g2
* gf4))
* ((g4
* (P
`1_3 ))
* (P
`3_3 )))
+ ((R
`3_3 )
* (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
EC_PF_1: 24
.= (((gf2
|^ 3)
* ((g2
* gf4)
* ((g4
* (P
`1_3 ))
* (P
`3_3 ))))
+ ((R
`3_3 )
* (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
GROUP_1:def 3
.= (((gf2
|^ 3)
* ((g2
* gf4)
* (g4
* ((P
`1_3 )
* (P
`3_3 )))))
+ ((R
`3_3 )
* (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
GROUP_1:def 3
.= (((gf2
|^ 3)
* (((g2
* gf4)
* g4)
* ((P
`1_3 )
* (P
`3_3 ))))
+ ((R
`3_3 )
* (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
GROUP_1:def 3
.= (((gf2
|^ 3)
* (((g2
* g4)
* gf4)
* ((P
`1_3 )
* (P
`3_3 ))))
+ ((R
`3_3 )
* (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
GROUP_1:def 3
.= (((gf2
|^ 3)
* (g8
* (gf4
* ((P
`1_3 )
* (P
`3_3 )))))
+ ((R
`3_3 )
* (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
A4,
GROUP_1:def 3
.= ((((gf2
|^ 3)
* g8)
* (gf4
* ((P
`1_3 )
* (P
`3_3 ))))
+ ((R
`3_3 )
* (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
GROUP_1:def 3
.= ((R
`3_3 )
* ((((P
`1_3 )
* (P
`3_3 ))
* gf4)
+ (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
VECTSP_1:def 7,
A3
.= ((R
`3_3 )
* (((P
`1_3 )
* ((P
`3_3 )
* gf4))
+ (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
GROUP_1:def 3;
A9: 4
= (3
+ 1);
((P
`3_3 )
* gf4)
= (((P
`3_3 )
* (gf1
|^ 2))
- ((P
`3_3 )
* (g8
* gf3))) by
A2,
VECTSP_1: 11
.= (((P
`3_3 )
* (gf1
|^ 2))
- (g8
* ((P
`3_3 )
* gf3))) by
GROUP_1:def 3
.= (((P
`3_3 )
* (gf1
|^ 2))
- (g4
* (g2
* ((P
`1_3 )
* (gf2
|^ 2))))) by
A4,
A5,
GROUP_1:def 3
.= (((P
`3_3 )
* (gf1
|^ 2))
- ((g3
* (g2
* ((P
`1_3 )
* (gf2
|^ 2))))
+ (g2
* ((P
`1_3 )
* (gf2
|^ 2))))) by
A1,
A9,
Th19
.= (((P
`3_3 )
* (gf1
|^ 2))
- ((g3
* ((g2
* (gf2
|^ 2))
* (P
`1_3 )))
+ (g2
* ((gf2
|^ 2)
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((P
`3_3 )
* (gf1
|^ 2))
- (((g2
* (gf2
|^ 2))
* (g3
* (P
`1_3 )))
+ (g2
* ((gf2
|^ 2)
* (P
`1_3 ))))) by
GROUP_1:def 3;
then ((P
`1_3 )
* ((P
`3_3 )
* gf4))
= (((P
`1_3 )
* ((P
`3_3 )
* (gf1
|^ 2)))
- ((P
`1_3 )
* (((g2
* (gf2
|^ 2))
* (g3
* (P
`1_3 )))
+ (g2
* ((gf2
|^ 2)
* (P
`1_3 )))))) by
VECTSP_1: 11
.= ((((P
`1_3 )
* (P
`3_3 ))
* (gf1
|^ 2))
- ((P
`1_3 )
* (((g2
* (gf2
|^ 2))
* (g3
* (P
`1_3 )))
+ (g2
* ((gf2
|^ 2)
* (P
`1_3 )))))) by
GROUP_1:def 3
.= ((((P
`1_3 )
* (P
`3_3 ))
* (gf1
|^ 2))
- (((P
`1_3 )
* ((g2
* (gf2
|^ 2))
* (g3
* (P
`1_3 ))))
+ ((P
`1_3 )
* (g2
* ((gf2
|^ 2)
* (P
`1_3 )))))) by
VECTSP_1:def 7
.= ((((P
`1_3 )
* (P
`3_3 ))
* (gf1
|^ 2))
- (((g2
* (gf2
|^ 2))
* ((g3
* (P
`1_3 ))
* (P
`1_3 )))
+ ((g2
* ((gf2
|^ 2)
* (P
`1_3 )))
* (P
`1_3 )))) by
GROUP_1:def 3
.= ((((P
`1_3 )
* (P
`3_3 ))
* (gf1
|^ 2))
- (((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
* (P
`1_3 ))))
+ ((g2
* ((gf2
|^ 2)
* (P
`1_3 )))
* (P
`1_3 )))) by
GROUP_1:def 3
.= ((((P
`1_3 )
* (P
`3_3 ))
* (gf1
|^ 2))
- (((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2)))
+ ((g2
* ((gf2
|^ 2)
* (P
`1_3 )))
* (P
`1_3 )))) by
EC_PF_1: 22
.= ((((P
`1_3 )
* (P
`3_3 ))
* (gf1
|^ 2))
- (((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2)))
+ (g2
* (((gf2
|^ 2)
* (P
`1_3 ))
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((((P
`1_3 )
* (P
`3_3 ))
* (gf1
|^ 2))
- (((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2)))
+ (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
* (P
`1_3 )))))) by
GROUP_1:def 3
.= (((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 )))
- (((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2)))
+ (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
EC_PF_1: 22
.= ((((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 )))
- ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2))))
- (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2)))) by
VECTSP_1: 17;
then (((P
`1_3 )
* ((P
`3_3 )
* gf4))
+ (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))
= ((((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 )))
- ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2))))
+ ((
- (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))
+ (g2
* ((gf2
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
ALGSTR_1: 7
.= ((((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 )))
- ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2))))
+ (
0. (
GF p))) by
VECTSP_1: 19
.= (((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 )))
- ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2)))) by
ALGSTR_1: 7;
hence ((((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 )))))
= ((R
`3_3 )
* ((((g2
* (gf2
|^ 2))
* gf1)
- ((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 ))))
+ (((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 )))
- ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2)))))) by
A7,
A8,
VECTSP_1:def 7
.= ((R
`3_3 )
* (((g2
* (gf2
|^ 2))
* gf1)
+ ((
- ((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 ))))
+ (((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 )))
- ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2))))))) by
ALGSTR_1: 7
.= ((R
`3_3 )
* (((g2
* (gf2
|^ 2))
* gf1)
+ (((
- ((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 ))))
+ ((gf1
|^ 2)
* ((P
`1_3 )
* (P
`3_3 ))))
- ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2)))))) by
ALGSTR_1: 7
.= ((R
`3_3 )
* (((g2
* (gf2
|^ 2))
* gf1)
+ ((
0. (
GF p))
- ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2)))))) by
VECTSP_1: 19
.= ((R
`3_3 )
* (((g2
* (gf2
|^ 2))
* gf1)
- ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2))))) by
VECTSP_1: 18
.= ((R
`3_3 )
* ((((g2
* (gf2
|^ 2))
* (a
* ((P
`3_3 )
|^ 2)))
+ ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2))))
- ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2))))) by
A2,
VECTSP_1:def 7
.= ((R
`3_3 )
* (((g2
* (gf2
|^ 2))
* (a
* ((P
`3_3 )
|^ 2)))
+ (((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2)))
- ((g2
* (gf2
|^ 2))
* (g3
* ((P
`1_3 )
|^ 2)))))) by
ALGSTR_1: 7
.= ((R
`3_3 )
* (((g2
* (gf2
|^ 2))
* (a
* ((P
`3_3 )
|^ 2)))
+ (
0. (
GF p)))) by
VECTSP_1: 19
.= (((g2
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (a
* (R
`3_3 ))) by
A6,
ALGSTR_1: 7;
end;
theorem ::
EC_PF_2:63
Th63: for p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
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 (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((((R
`2_3 )
|^ 2)
* (R
`3_3 ))
- ((((R
`1_3 )
|^ 3)
+ (((z
`1 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((z
`2 )
* ((R
`3_3 )
|^ 3)))))
= (
0. (
GF p))
proof
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p), g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p), P be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), R be
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):] such that
A1: g2
= (2
mod p) & g3
= (3
mod p) & g4
= (4
mod p) & g8
= (8
mod p) and
A2: 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)) and
A3: R
=
[((g2
* gf4)
* gf2), ((gf1
* ((g4
* gf3)
- gf4))
- ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2))), (g8
* (gf2
|^ 3))];
set a = (z
`1 );
set b = (z
`2 );
A4: g4
= ((2
* 2)
mod p) by
A1
.= (g2
* g2) by
A1,
EC_PF_1: 18;
then
A5: g4
= (g2
|^ 2) by
EC_PF_1: 22;
A6: g8
= ((2
* 4)
mod p) by
A1
.= (g2
* g4) by
A1,
EC_PF_1: 18;
A7: (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
= (((((g2
|^ 2)
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((R
`2_3 )
|^ 2))
* (R
`3_3 )) by
A5,
GROUP_1:def 3
.= (((((g2
* gf2)
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`2_3 )
|^ 2))
* (R
`3_3 )) by
BINOM: 9
.= (((((g2
* gf2)
* (P
`3_3 ))
* (R
`2_3 ))
|^ 2)
* (R
`3_3 )) by
Th13
.= (((
- ((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
+ (((g2
* gf2)
* (P
`2_3 ))
* (R
`3_3 ))))
|^ 2)
* (R
`3_3 )) by
A1,
A2,
A3,
Th59
.= ((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
+ (((g2
* gf2)
* (P
`2_3 ))
* (R
`3_3 )))
|^ 2)
* (R
`3_3 )) by
Th1
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ ((g2
* (gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))))
* (((g2
* gf2)
* (P
`2_3 ))
* (R
`3_3 ))))
+ ((((g2
* gf2)
* (P
`2_3 ))
* (R
`3_3 ))
|^ 2))
* (R
`3_3 )) by
A1,
Th25
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ ((g2
* (gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))))
* (g2
* ((gf2
* (P
`2_3 ))
* (R
`3_3 )))))
+ ((((g2
* gf2)
* (P
`2_3 ))
* (R
`3_3 ))
|^ 2))
* (R
`3_3 )) by
Th11
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ (((g2
* (gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))))
* g2)
* ((gf2
* (P
`2_3 ))
* (R
`3_3 ))))
+ ((((g2
* gf2)
* (P
`2_3 ))
* (R
`3_3 ))
|^ 2))
* (R
`3_3 )) by
GROUP_1:def 3
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ (((g2
* g2)
* (gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))))
* ((gf2
* (P
`2_3 ))
* (R
`3_3 ))))
+ ((((g2
* gf2)
* (P
`2_3 ))
* (R
`3_3 ))
|^ 2))
* (R
`3_3 )) by
GROUP_1:def 3
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ ((g4
* (gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))))
* ((gf2
* (P
`2_3 ))
* (R
`3_3 ))))
+ (((g2
* gf2)
* ((P
`2_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 )) by
A4,
GROUP_1:def 3
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ ((g4
* (gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))))
* ((gf2
* (P
`2_3 ))
* (R
`3_3 ))))
+ (((g2
|^ 2)
* (gf2
|^ 2))
* (((P
`2_3 )
* (R
`3_3 ))
|^ 2)))
* (R
`3_3 )) by
Th13
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ ((g4
* (gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))))
* ((gf2
* (P
`2_3 ))
* (R
`3_3 ))))
+ ((g4
* (gf2
|^ 2))
* (((P
`2_3 )
|^ 2)
* ((R
`3_3 )
|^ 2))))
* (R
`3_3 )) by
A5,
BINOM: 9
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ (((g4
* gf1)
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
* ((gf2
* (P
`2_3 ))
* (R
`3_3 ))))
+ ((g4
* (gf2
|^ 2))
* (((P
`2_3 )
|^ 2)
* ((R
`3_3 )
|^ 2))))
* (R
`3_3 )) by
GROUP_1:def 3
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ (((g4
* gf1)
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
* ((gf2
* (P
`2_3 ))
* (R
`3_3 ))))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 2)))
* (R
`3_3 )) by
GROUP_1:def 3
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ (((((g4
* gf1)
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
* gf2)
* (P
`2_3 ))
* (R
`3_3 )))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 2)))
* (R
`3_3 )) by
Th11
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ (((g4
* gf1)
* (((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
* gf2)
* (P
`2_3 )))
* (R
`3_3 )))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 2)))
* (R
`3_3 )) by
Th11
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ (((g4
* gf1)
* ((gf2
* (P
`2_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))))
* (R
`3_3 )))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 2)))
* (R
`3_3 )) by
GROUP_1:def 3
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
+ (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
* (R
`3_3 )))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 2)))
* (R
`3_3 )) by
Th11
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
* (R
`3_3 ))
* (R
`3_3 )))
+ ((((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 2))
* (R
`3_3 ))) by
Th14
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
* (R
`3_3 ))
+ (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
* ((R
`3_3 )
* (R
`3_3 ))))
+ ((((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 2))
* (R
`3_3 ))) by
GROUP_1:def 3
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
* (R
`3_3 ))
+ (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
* ((R
`3_3 )
* (R
`3_3 ))))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* (((R
`3_3 )
|^ 2)
* (R
`3_3 )))) by
GROUP_1:def 3
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
* (R
`3_3 ))
+ (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
* ((R
`3_3 )
* (R
`3_3 ))))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ (2
+ 1)))) by
EC_PF_1: 24
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
* (R
`3_3 ))
+ (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
* ((R
`3_3 )
|^ 2)))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
EC_PF_1: 22
.= (((((gf1
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 ))))
|^ 2)
* (R
`3_3 ))
+ (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* ((R
`3_3 )
|^ 2))
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= (((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* ((R
`3_3 )
|^ 2))
* (((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
BINOM: 9
.= (((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* ((R
`3_3 )
|^ 2))
* ((P
`3_3 )
* (R
`1_3 )))
- (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* ((R
`3_3 )
|^ 2))
* ((P
`1_3 )
* (R
`3_3 )))))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
VECTSP_1: 11
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* ((R
`3_3 )
|^ 2))
* ((P
`3_3 )
* (R
`1_3 ))))
- (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* ((R
`3_3 )
|^ 2))
* ((P
`1_3 )
* (R
`3_3 ))))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
ALGSTR_1: 7
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* ((P
`3_3 )
* (R
`1_3 )))
* ((R
`3_3 )
|^ 2)))
- (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* ((R
`3_3 )
|^ 2))
* ((P
`1_3 )
* (R
`3_3 ))))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* ((R
`3_3 )
|^ 2))
* ((P
`1_3 )
* (R
`3_3 ))))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* ((R
`3_3 )
|^ 2))
* (P
`1_3 ))
* (R
`3_3 )))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (R
`3_3 )))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`1_3 ))
* (((R
`3_3 )
|^ 2)
* (R
`3_3 ))))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
GROUP_1:def 3
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`1_3 ))
* ((R
`3_3 )
|^ (2
+ 1))))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
EC_PF_1: 24
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))) by
GROUP_1:def 3;
A8: (
- (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))
= (
- ((((((g2
* g2)
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (R
`1_3 ))
* a)
* ((R
`3_3 )
|^ 2))) by
A4,
Th11
.= (
- ((((g2
* ((g2
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2)))
* (R
`1_3 ))
* a)
* ((R
`3_3 )
|^ 2))) by
Th11
.= (
- ((((g2
* (R
`1_3 ))
* ((g2
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2)))
* a)
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (
- ((((g2
* (R
`1_3 ))
* ((g2
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2)))
* a)
* ((R
`3_3 )
* (R
`3_3 )))) by
EC_PF_1: 22
.= (
- (((((g2
* (R
`1_3 ))
* ((g2
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2)))
* a)
* (R
`3_3 ))
* (R
`3_3 ))) by
GROUP_1:def 3
.= (
- (((((g2
* (R
`1_3 ))
* ((g2
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2)))
* (R
`3_3 ))
* a)
* (R
`3_3 ))) by
GROUP_1:def 3
.= (
- (((((g2
* (R
`1_3 ))
* (R
`3_3 ))
* ((g2
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2)))
* a)
* (R
`3_3 ))) by
GROUP_1:def 3
.= (
- ((((g2
* (R
`1_3 ))
* (R
`3_3 ))
* ((g2
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2)))
* (a
* (R
`3_3 )))) by
GROUP_1:def 3
.= (
- (((g2
* (R
`1_3 ))
* (R
`3_3 ))
* (((g2
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (a
* (R
`3_3 ))))) by
GROUP_1:def 3
.= (
- (((g2
* (R
`1_3 ))
* (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 ))))))) by
A1,
A2,
A3,
Th62
.= (
- ((((g2
* (R
`1_3 ))
* (R
`3_3 ))
* (((gf1
* (P
`3_3 ))
* (R
`3_3 ))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))
+ (((g2
* (R
`1_3 ))
* (R
`3_3 ))
* ((gf2
|^ 2)
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 ))))))) by
VECTSP_1:def 7
.= ((
- (((g2
* (R
`1_3 ))
* (R
`3_3 ))
* ((gf2
|^ 2)
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 ))))))
- (((g2
* (R
`1_3 ))
* (R
`3_3 ))
* (((gf1
* (P
`3_3 ))
* (R
`3_3 ))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))) by
VECTSP_1: 17
.= ((
- ((((g2
* (R
`1_3 ))
* (R
`3_3 ))
* (gf2
|^ 2))
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- (((g2
* (R
`1_3 ))
* (R
`3_3 ))
* (((gf1
* (P
`3_3 ))
* (R
`3_3 ))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))))) by
GROUP_1:def 3
.= ((
- ((((g2
* (R
`1_3 ))
* (R
`3_3 ))
* (gf2
|^ 2))
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- ((((g2
* (R
`1_3 ))
* (R
`3_3 ))
* ((gf1
* (P
`3_3 ))
* (R
`3_3 )))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((
- ((((g2
* (R
`1_3 ))
* (gf2
|^ 2))
* (R
`3_3 ))
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- ((((g2
* (R
`1_3 ))
* (R
`3_3 ))
* ((gf1
* (P
`3_3 ))
* (R
`3_3 )))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- ((((g2
* (R
`1_3 ))
* (R
`3_3 ))
* ((gf1
* (P
`3_3 ))
* (R
`3_3 )))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- ((((((g2
* (R
`1_3 ))
* (R
`3_3 ))
* gf1)
* (P
`3_3 ))
* (R
`3_3 ))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
Th11
.= ((
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- (((((g2
* ((R
`1_3 )
* (R
`3_3 )))
* gf1)
* (P
`3_3 ))
* (R
`3_3 ))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- (((((g2
* gf1)
* ((R
`1_3 )
* (R
`3_3 )))
* (P
`3_3 ))
* (R
`3_3 ))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- (((((g2
* gf1)
* (P
`3_3 ))
* ((R
`1_3 )
* (R
`3_3 )))
* (R
`3_3 ))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- ((((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* (R
`3_3 ))
* (R
`3_3 ))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* ((((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
+ ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
* (R
`3_3 )))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= ((
- (((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 )))
+ ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
* (R
`3_3 )))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
VECTSP_1:def 7
.= ((
- (((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 )))
+ ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 )))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
EC_PF_1: 22
.= (((
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* ((g2
* ((P
`1_3 )
|^ 2))
* (R
`3_3 ))))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
VECTSP_1: 17
.= (((
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (g2
* (((P
`1_3 )
|^ 2)
* (R
`3_3 )))))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- (((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* g2)
* (((P
`1_3 )
|^ 2)
* (R
`3_3 ))))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- ((((g2
* (gf2
|^ 2))
* ((R
`1_3 )
* (R
`3_3 )))
* g2)
* (((P
`1_3 )
|^ 2)
* (R
`3_3 ))))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- ((((g2
* (gf2
|^ 2))
* g2)
* ((R
`1_3 )
* (R
`3_3 )))
* (((P
`1_3 )
|^ 2)
* (R
`3_3 ))))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- (((g2
* (gf2
|^ 2))
* g2)
* (((R
`1_3 )
* (R
`3_3 ))
* (((P
`1_3 )
|^ 2)
* (R
`3_3 )))))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- (((g2
* g2)
* (gf2
|^ 2))
* (((R
`1_3 )
* (R
`3_3 ))
* (((P
`1_3 )
|^ 2)
* (R
`3_3 )))))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- ((g4
* (gf2
|^ 2))
* ((((R
`1_3 )
* (R
`3_3 ))
* ((P
`1_3 )
|^ 2))
* (R
`3_3 ))))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
A4,
GROUP_1:def 3
.= (((
- ((g4
* (gf2
|^ 2))
* ((((R
`1_3 )
* ((P
`1_3 )
|^ 2))
* (R
`3_3 ))
* (R
`3_3 ))))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- ((g4
* (gf2
|^ 2))
* (((R
`1_3 )
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
* (R
`3_3 )))))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- ((g4
* (gf2
|^ 2))
* (((R
`1_3 )
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 2))))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
EC_PF_1: 22
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (((g4
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
Th11
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* (g4
* (((P
`1_3 )
* (P
`3_3 ))
* (R
`1_3 )))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
Th11
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g2
* (gf2
|^ 2))
* (R
`1_3 ))
* (R
`3_3 ))
* g4)
* (((P
`1_3 )
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((g2
* (gf2
|^ 2))
* ((R
`1_3 )
* (R
`3_3 )))
* g4)
* (((P
`1_3 )
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((g2
* (gf2
|^ 2))
* g4)
* ((R
`1_3 )
* (R
`3_3 )))
* (((P
`1_3 )
* (P
`3_3 ))
* (R
`1_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((g2
* (gf2
|^ 2))
* g4)
* (((R
`1_3 )
* (R
`3_3 ))
* (((P
`1_3 )
* (P
`3_3 ))
* (R
`1_3 )))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((g2
* g4)
* (gf2
|^ 2))
* (((R
`1_3 )
* (R
`3_3 ))
* (((P
`1_3 )
* (P
`3_3 ))
* (R
`1_3 )))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((g8
* (gf2
|^ 2))
* (((((P
`1_3 )
* (P
`3_3 ))
* (R
`1_3 ))
* (R
`1_3 ))
* (R
`3_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
A6,
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((g8
* (gf2
|^ 2))
* ((((P
`1_3 )
* (P
`3_3 ))
* ((R
`1_3 )
* (R
`1_3 )))
* (R
`3_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((g8
* (gf2
|^ 2))
* ((((P
`1_3 )
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
EC_PF_1: 22
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 ))))) by
Th11
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
- ((((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* ((g2
* gf2)
* (P
`2_3 )))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (gf1
* (P
`1_3 ))))) by
VECTSP_1: 11
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ ((((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* (gf1
* (P
`1_3 )))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* ((g2
* gf2)
* (P
`2_3 ))))) by
VECTSP_1: 17
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ ((((g2
* gf1)
* (((P
`3_3 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
* (gf1
* (P
`1_3 )))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* ((g2
* gf2)
* (P
`2_3 ))))) by
Th11
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ ((((g2
* gf1)
* (gf1
* (P
`1_3 )))
* (((P
`3_3 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* ((g2
* gf2)
* (P
`2_3 ))))) by
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ (((((g2
* gf1)
* gf1)
* (P
`1_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* ((g2
* gf2)
* (P
`2_3 ))))) by
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ ((((g2
* (gf1
* gf1))
* (P
`1_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* ((g2
* gf2)
* (P
`2_3 ))))) by
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ ((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* ((g2
* gf2)
* (P
`2_3 ))))) by
EC_PF_1: 22
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ ((((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- (((((g2
* gf1)
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
* ((g2
* gf2)
* (P
`2_3 ))))) by
Th11
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ ((((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- (((g2
* gf1)
* (((P
`3_3 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
* ((g2
* gf2)
* (P
`2_3 ))))) by
Th11
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ ((((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- (((g2
* gf1)
* ((g2
* gf2)
* (P
`2_3 )))
* (((P
`3_3 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))) by
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ ((((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- (((((g2
* gf1)
* g2)
* gf2)
* (P
`2_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))) by
Th11
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ ((((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- (((((g2
* g2)
* gf1)
* gf2)
* (P
`2_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))) by
GROUP_1:def 3
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )))
+ ((((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
A4,
Th11
.= (((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
ALGSTR_1: 8
.= ((((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
ALGSTR_1: 7
.= ((((
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
ALGSTR_1: 8;
A9: (
- (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (b
* ((R
`3_3 )
|^ 3))))
= (
- ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* b)
* ((R
`3_3 )
|^ (2
+ 1)))) by
GROUP_1:def 3
.= (
- ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* b)
* (((R
`3_3 )
|^ 2)
* (R
`3_3 )))) by
EC_PF_1: 24
.= (
- (((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* b)
* (R
`3_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (
- (((R
`3_3 )
|^ 2)
* (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (b
* (R
`3_3 ))))) by
GROUP_1:def 3
.= (
- (((R
`3_3 )
|^ 2)
* (((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2))
- (((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))))) by
A1,
A2,
A3,
Th61
.= (
- ((((R
`3_3 )
|^ 2)
* ((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)))
- (((R
`3_3 )
|^ 2)
* (((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))))) by
VECTSP_1: 11
.= ((((R
`3_3 )
|^ 2)
* (((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 )))
- (((R
`3_3 )
|^ 2)
* ((R
`3_3 )
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2)))) by
VECTSP_1: 17
.= (((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- ((((R
`3_3 )
|^ 2)
* (R
`3_3 ))
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2))) by
GROUP_1:def 3
.= (((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- (((R
`3_3 )
|^ (2
+ 1))
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2))) by
EC_PF_1: 24
.= (((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ ((
- ((R
`3_3 )
|^ 3))
* ((((g2
* gf2)
* (P
`2_3 ))
- (gf1
* (P
`1_3 )))
|^ 2))) by
VECTSP_1: 9
.= (((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ ((
- ((R
`3_3 )
|^ 3))
* (((((g2
* gf2)
* (P
`2_3 ))
|^ 2)
- ((g2
* ((g2
* gf2)
* (P
`2_3 )))
* (gf1
* (P
`1_3 ))))
+ ((gf1
* (P
`1_3 ))
|^ 2)))) by
A1,
Th26
.= (((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ ((
- ((R
`3_3 )
|^ 3))
* (((((g2
|^ 2)
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
- ((g2
* ((g2
* gf2)
* (P
`2_3 )))
* (gf1
* (P
`1_3 ))))
+ ((gf1
* (P
`1_3 ))
|^ 2)))) by
Th13
.= (((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ ((
- ((R
`3_3 )
|^ 3))
* ((((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
- ((g2
* ((g2
* gf2)
* (P
`2_3 )))
* (gf1
* (P
`1_3 ))))
+ ((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
A5,
BINOM: 9
.= (((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ ((
- ((R
`3_3 )
|^ 3))
* ((((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
- ((((g2
* g2)
* gf2)
* (P
`2_3 ))
* (gf1
* (P
`1_3 ))))
+ ((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
Th11
.= (((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ ((
- ((R
`3_3 )
|^ 3))
* ((((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
- (((g4
* gf2)
* (gf1
* (P
`1_3 )))
* (P
`2_3 )))
+ ((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
A4,
GROUP_1:def 3
.= (((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ ((
- ((R
`3_3 )
|^ 3))
* ((((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
- ((((g4
* gf2)
* gf1)
* (P
`1_3 ))
* (P
`2_3 )))
+ ((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
GROUP_1:def 3
.= (((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ ((
- ((R
`3_3 )
|^ 3))
* ((((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
- ((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 )))
+ ((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))))) by
GROUP_1:def 3
.= ((((((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* (
- ((R
`3_3 )
|^ 3)))
- (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* (
- ((R
`3_3 )
|^ 3))))
+ (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* (
- ((R
`3_3 )
|^ 3))))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
Th14
.= ((((
- (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
- (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* (
- ((R
`3_3 )
|^ 3))))
+ (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* (
- ((R
`3_3 )
|^ 3))))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
VECTSP_1: 8
.= ((((
- (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
- (
- (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3))))
+ (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* (
- ((R
`3_3 )
|^ 3))))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
VECTSP_1: 8
.= ((((
- (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* (
- ((R
`3_3 )
|^ 3))))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
RLVECT_1: 17
.= ((((
- (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
VECTSP_1: 8;
A10: ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (b
* ((R
`3_3 )
|^ 3))))
= (((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ (((
- (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ ((
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))) by
A7,
A9,
ALGSTR_1: 7
.= (((((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
- (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ ((
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
Th8
.= ((((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))
- (((g4
* (gf2
|^ 2))
* ((P
`2_3 )
|^ 2))
* ((R
`3_3 )
|^ 3))))
+ (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ ((
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
ALGSTR_1: 7
.= ((((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ (
0. (
GF p)))
+ (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ ((
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
VECTSP_1: 19
.= (((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ ((
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
ALGSTR_1: 7
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((
- (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3)))
+ (((((g4
* gf1)
* gf2)
* (P
`1_3 ))
* (P
`2_3 ))
* ((R
`3_3 )
|^ 3))))
+ ((
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
ALGSTR_1: 7
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (
0. (
GF p)))
+ ((
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
RLVECT_1: 5
.= (((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
ALGSTR_1: 7
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 3)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
ALGSTR_1: 7
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ (2
+ 1))))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
ALGSTR_1: 8
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* (((R
`3_3 )
|^ 2)
* (R
`3_3 ))))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
EC_PF_1: 24
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
* (R
`3_3 ))
- ((((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 2))
* (R
`3_3 )))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
- (((gf1
|^ 2)
* ((P
`1_3 )
|^ 2))
* ((R
`3_3 )
|^ 2)))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
VECTSP_1: 13
.= ((((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2))
- ((gf1
|^ 2)
* (((P
`1_3 )
|^ 2)
* ((R
`3_3 )
|^ 2))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (((((gf1
|^ 2)
* (((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2)
- (((P
`1_3 )
|^ 2)
* ((R
`3_3 )
|^ 2))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
VECTSP_1: 11
.= (((((gf1
|^ 2)
* (((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
|^ 2)
- (((P
`1_3 )
* (R
`3_3 ))
|^ 2)))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
BINOM: 9
.= (((((gf1
|^ 2)
* (((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
+ ((P
`1_3 )
* (R
`3_3 )))
* ((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
- ((P
`1_3 )
* (R
`3_3 )))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
Th15
.= (((((gf1
|^ 2)
* (((((P
`3_3 )
* (R
`1_3 ))
- ((P
`1_3 )
* (R
`3_3 )))
+ ((P
`1_3 )
* (R
`3_3 )))
* (((P
`3_3 )
* (R
`1_3 ))
- (((P
`1_3 )
* (R
`3_3 ))
+ ((P
`1_3 )
* (R
`3_3 ))))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
VECTSP_1: 17
.= (((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
+ ((
- ((P
`1_3 )
* (R
`3_3 )))
+ ((P
`1_3 )
* (R
`3_3 ))))
* (((P
`3_3 )
* (R
`1_3 ))
- (((P
`1_3 )
* (R
`3_3 ))
+ ((P
`1_3 )
* (R
`3_3 ))))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
ALGSTR_1: 7
.= (((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
+ (
0. (
GF p)))
* (((P
`3_3 )
* (R
`1_3 ))
- (((P
`1_3 )
* (R
`3_3 ))
+ ((P
`1_3 )
* (R
`3_3 ))))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
RLVECT_1: 5
.= (((((gf1
|^ 2)
* ((((P
`3_3 )
* (R
`1_3 ))
+ (
0. (
GF p)))
* (((P
`3_3 )
* (R
`1_3 ))
- (g2
* ((P
`1_3 )
* (R
`3_3 ))))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
A1,
Th20
.= (((((gf1
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* (((P
`3_3 )
* (R
`1_3 ))
- (g2
* ((P
`1_3 )
* (R
`3_3 ))))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
RLVECT_1: 4
.= ((((((gf1
|^ 2)
* ((P
`3_3 )
* (R
`1_3 )))
* (((P
`3_3 )
* (R
`1_3 ))
- (g2
* ((P
`1_3 )
* (R
`3_3 )))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
* (R
`1_3 )))
* ((P
`3_3 )
* (R
`1_3 )))
- (((gf1
|^ 2)
* ((P
`3_3 )
* (R
`1_3 )))
* (g2
* ((P
`1_3 )
* (R
`3_3 )))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
VECTSP_1: 11
.= ((((((gf1
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
* ((P
`3_3 )
* (R
`1_3 ))))
- (((gf1
|^ 2)
* ((P
`3_3 )
* (R
`1_3 )))
* (g2
* ((P
`1_3 )
* (R
`3_3 )))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= ((((((gf1
|^ 2)
* (((P
`3_3 )
* (R
`1_3 ))
|^ 2))
- (((gf1
|^ 2)
* ((P
`3_3 )
* (R
`1_3 )))
* (g2
* ((P
`1_3 )
* (R
`3_3 )))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
EC_PF_1: 22
.= ((((((gf1
|^ 2)
* (((P
`3_3 )
|^ 2)
* ((R
`1_3 )
|^ 2)))
- (((gf1
|^ 2)
* ((P
`3_3 )
* (R
`1_3 )))
* (g2
* ((P
`1_3 )
* (R
`3_3 )))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
BINOM: 9
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
- (((gf1
|^ 2)
* ((P
`3_3 )
* (R
`1_3 )))
* (g2
* ((P
`1_3 )
* (R
`3_3 )))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
- ((((gf1
|^ 2)
* ((P
`3_3 )
* (R
`1_3 )))
* g2)
* ((P
`1_3 )
* (R
`3_3 ))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
- ((((gf1
|^ 2)
* g2)
* ((P
`3_3 )
* (R
`1_3 )))
* ((P
`1_3 )
* (R
`3_3 ))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
- ((g2
* (gf1
|^ 2))
* (((P
`3_3 )
* (R
`1_3 ))
* ((P
`1_3 )
* (R
`3_3 )))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
- ((g2
* (gf1
|^ 2))
* ((((P
`3_3 )
* (R
`1_3 ))
* (P
`1_3 ))
* (R
`3_3 ))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
- ((g2
* (gf1
|^ 2))
* ((((P
`1_3 )
* (P
`3_3 ))
* (R
`1_3 ))
* (R
`3_3 ))))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
- (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* (R
`3_3 )))
* (R
`3_3 ))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
Th11
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
- ((((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* (R
`3_3 ))
* (R
`3_3 )))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
VECTSP_1: 13
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
- (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
* (R
`3_3 ))))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
GROUP_1:def 3
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
- (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))) by
EC_PF_1: 22;
A11: ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ (b
* ((R
`3_3 )
|^ 3)))))
= ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (b
* ((R
`3_3 )
|^ 3))))) by
VECTSP_1:def 7
.= (((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (b
* ((R
`3_3 )
|^ 3))))
- (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))) by
VECTSP_1: 17
.= (((((((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
- (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
A8,
A10,
Th8
.= ((((((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
- (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- ((((g4
* (gf2
|^ 2))
* ((P
`1_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))
- ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
ALGSTR_1: 7
.= ((((((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
- (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (
0. (
GF p)))
- ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
RLVECT_1: 5
.= (((((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
- (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
RLVECT_1: 4
.= ((((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
- (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
- ((((((g4
* gf1)
* gf2)
* (P
`2_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))
+ (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
ALGSTR_1: 7
.= ((((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
- (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (
0. (
GF p)))
+ (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
RLVECT_1: 5
.= (((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
- (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
RLVECT_1: 4
.= ((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
+ ((
- (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (((((g2
* (gf1
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
ALGSTR_1: 7
.= ((((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
+ (
0. (
GF p)))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
RLVECT_1: 5
.= (((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
RLVECT_1: 4
.= ((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 )))
- (((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 ))) by
GROUP_1:def 3
.= ((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 )))
- ((((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 ))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 )))) by
GROUP_1:def 3
.= ((((gf1
|^ 2)
* ((P
`3_3 )
|^ 2))
- (((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 )))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 ))) by
VECTSP_1: 13
.= ((((gf1
|^ 2)
* ((P
`3_3 )
* (P
`3_3 )))
- (((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 )))
* (((R
`1_3 )
|^ 2)
* (R
`3_3 ))) by
EC_PF_1: 22
.= (((((gf1
|^ 2)
* ((P
`3_3 )
* (P
`3_3 )))
- (((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 )))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )) by
GROUP_1:def 3
.= ((((((gf1
|^ 2)
* (P
`3_3 ))
* (P
`3_3 ))
- (((g8
* (gf2
|^ 2))
* (P
`1_3 ))
* (P
`3_3 )))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )) by
GROUP_1:def 3
.= ((((((gf1
|^ 2)
* (P
`3_3 ))
- ((g8
* (gf2
|^ 2))
* (P
`1_3 )))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2))
* (R
`3_3 )) by
VECTSP_1: 13
.= ((((R
`3_3 )
* (((gf1
|^ 2)
* (P
`3_3 ))
- ((g8
* (gf2
|^ 2))
* (P
`1_3 ))))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2)) by
Th11
.= (((((g4
* (gf2
|^ 2))
* (P
`3_3 ))
* (R
`1_3 ))
* (P
`3_3 ))
* ((R
`1_3 )
|^ 2)) by
A1,
A2,
A3,
Th60
.= (((((g4
* (gf2
|^ 2))
* (P
`3_3 ))
* (P
`3_3 ))
* (R
`1_3 ))
* ((R
`1_3 )
|^ 2)) by
GROUP_1:def 3
.= ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
* (P
`3_3 )))
* (R
`1_3 ))
* ((R
`1_3 )
|^ 2)) by
GROUP_1:def 3
.= ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (R
`1_3 ))
* ((R
`1_3 )
|^ 2)) by
EC_PF_1: 22
.= (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((R
`1_3 )
|^ 2)
* (R
`1_3 ))) by
GROUP_1:def 3
.= (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ (2
+ 1))) by
EC_PF_1: 24
.= (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 3));
thus (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((((R
`2_3 )
|^ 2)
* (R
`3_3 ))
- ((((R
`1_3 )
|^ 3)
+ ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (b
* ((R
`3_3 )
|^ 3)))))
= ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((((R
`1_3 )
|^ 3)
+ ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (b
* ((R
`3_3 )
|^ 3))))) by
VECTSP_1: 11
.= ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((R
`1_3 )
|^ 3)
+ (((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ (b
* ((R
`3_3 )
|^ 3)))))) by
ALGSTR_1: 7
.= ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- ((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 3))
+ (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ (b
* ((R
`3_3 )
|^ 3)))))) by
VECTSP_1:def 7
.= (((((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((R
`2_3 )
|^ 2)
* (R
`3_3 )))
- (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* (((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2))
+ (b
* ((R
`3_3 )
|^ 3)))))
- (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((R
`1_3 )
|^ 3))) by
VECTSP_1: 17
.= (
0. (
GF p)) by
A11,
RLVECT_1: 5;
end;
definition
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p);
::
EC_PF_2:def9
func
addell_ProjCo (z,p) ->
Function of
[:(
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)):], (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) means for P,Q,O be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st O
=
[
0 , 1,
0 ] holds (P
_EQ_ O implies (it
. (P,Q))
= Q) & ((Q
_EQ_ O & not P
_EQ_ O) implies (it
. (P,Q))
= P) & (( not P
_EQ_ O & not Q
_EQ_ O & not P
_EQ_ Q) implies for g2,gf1,gf2,gf3 be
Element 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 ))) holds (it
. (P,Q))
=
[(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 ))]) & (( not P
_EQ_ O & not Q
_EQ_ O & P
_EQ_ Q) implies for g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element 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)) holds (it
. (P,Q))
=
[((g2
* gf4)
* gf2), ((gf1
* ((g4
* gf3)
- gf4))
- ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2))), (g8
* (gf2
|^ 3))]);
existence
proof
set a = (z
`1 );
set b = (z
`2 );
A1: p
> (2
+ 1) & (
Disc (a,b,p))
<> (
0. (
GF p)) by
Th30;
defpred
P[
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)),
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)),
set] means for O be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st O
=
[
0 , 1,
0 ] holds ($1
_EQ_ O implies $3
= $2) & (($2
_EQ_ O & not $1
_EQ_ O) implies $3
= $1) & (( not $1
_EQ_ O & not $2
_EQ_ O & not $1
_EQ_ $2) implies for g2,gf1,gf2,gf3 be
Element of (
GF p) st g2
= (2
mod p) & gf1
= ((($2
`2_3 )
* ($1
`3_3 ))
- (($1
`2_3 )
* ($2
`3_3 ))) & gf2
= ((($2
`1_3 )
* ($1
`3_3 ))
- (($1
`1_3 )
* ($2
`3_3 ))) & gf3
= (((((gf1
|^ 2)
* ($1
`3_3 ))
* ($2
`3_3 ))
- (gf2
|^ 3))
- (((g2
* (gf2
|^ 2))
* ($1
`1_3 ))
* ($2
`3_3 ))) holds $3
=
[(gf2
* gf3), ((gf1
* ((((gf2
|^ 2)
* ($1
`1_3 ))
* ($2
`3_3 ))
- gf3))
- (((gf2
|^ 3)
* ($1
`2_3 ))
* ($2
`3_3 ))), (((gf2
|^ 3)
* ($1
`3_3 ))
* ($2
`3_3 ))]) & (( not $1
_EQ_ O & not $2
_EQ_ O & $1
_EQ_ $2) implies for g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p) st g2
= (2
mod p) & g3
= (3
mod p) & g4
= (4
mod p) & g8
= (8
mod p) & gf1
= ((a
* (($1
`3_3 )
|^ 2))
+ (g3
* (($1
`1_3 )
|^ 2))) & gf2
= (($1
`2_3 )
* ($1
`3_3 )) & gf3
= ((($1
`1_3 )
* ($1
`2_3 ))
* gf2) & gf4
= ((gf1
|^ 2)
- (g8
* gf3)) holds $3
=
[((g2
* gf4)
* gf2), ((gf1
* ((g4
* gf3)
- gf4))
- ((g8
* (($1
`2_3 )
|^ 2))
* (gf2
|^ 2))), (g8
* (gf2
|^ 3))]);
A2: for P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds ex R be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st
P[P, Q, R]
proof
let P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
consider PP be
Element of (
ProjCo (
GF p)) such that
A3: PP
= P & PP
in (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
A4: (PP
`1_3 )
= (P
`1_3 ) & (PP
`2_3 )
= (P
`2_3 ) & (PP
`3_3 )
= (P
`3_3 ) by
A3,
Th32;
consider QQ be
Element of (
ProjCo (
GF p)) such that
A5: QQ
= Q & QQ
in (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
A6: (QQ
`1_3 )
= (Q
`1_3 ) & (QQ
`2_3 )
= (Q
`2_3 ) & (QQ
`3_3 )
= (Q
`3_3 ) by
A5,
Th32;
set O =
[
0 , 1,
0 ];
reconsider O as
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) by
EC_PF_1: 42;
consider OO be
Element of (
ProjCo (
GF p)) such that
A7: OO
= O & OO
in (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
per cases ;
suppose
A8: P
_EQ_ O;
take Q;
thus thesis by
A8;
end;
suppose
A9: Q
_EQ_ O & not P
_EQ_ O;
take P;
thus thesis by
A9;
end;
suppose
A10: not P
_EQ_ O & not Q
_EQ_ O & not P
_EQ_ Q;
A11: (OO
`3_3 )
=
0 by
A7;
(
rep_pt PP)
<> (
rep_pt OO) by
A3,
A7,
A10,
Th39;
then (
rep_pt PP)
<>
[
0 , 1,
0 ] by
A11,
Def7;
then
A12: ((
rep_pt PP)
`3_3 )
<>
0 by
Th37;
then
A13: (P
`3_3 )
<>
0 by
A4,
Th38;
(
rep_pt QQ)
<> (
rep_pt OO) by
A5,
A7,
A10,
Th39;
then (
rep_pt QQ)
<>
[
0 , 1,
0 ] by
A11,
Def7;
then ((
rep_pt QQ)
`3_3 )
<>
0 by
Th37;
then
A14: (Q
`3_3 )
<>
0 by
A6,
Th38;
reconsider g2 = (2
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
set gf1 = (((Q
`2_3 )
* (P
`3_3 ))
- ((P
`2_3 )
* (Q
`3_3 )));
set gf2 = (((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 )));
set gf3 = (((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- (gf2
|^ 3))
- (((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 )));
reconsider gf1, gf2, gf3 as
Element of (
GF p);
set 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 ))];
reconsider R as
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):];
per cases ;
suppose
A15: P
_EQ_ ((
compell_ProjCo (z,p))
. Q);
then ((P
`1_3 )
* (Q
`3_3 ))
= ((Q
`1_3 )
* (P
`3_3 )) by
A13,
A14,
Th50;
then
A16: gf2
= (
0. (
GF p)) by
VECTSP_1: 19;
then
A17: (gf2
|^ 2)
= (
0. (
GF p)) & (gf2
|^ 3)
= (
0. (
GF p)) by
Th5;
((P
`2_3 )
* (Q
`3_3 ))
<> ((Q
`2_3 )
* (P
`3_3 )) by
A10,
A15,
Th52;
then gf1
<> (
0. (
GF p)) by
VECTSP_1: 19;
then
A18: gf1
<>
0 by
EC_PF_1: 11;
then
A19: (gf1
|^ 2)
<>
0 by
EC_PF_1: 25;
A20: gf3
= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- ((
0. (
GF p))
+ (((g2
* (
0. (
GF p)))
* (P
`1_3 ))
* (Q
`3_3 )))) by
A17,
VECTSP_1: 17
.= ((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- (
0. (
GF p))) by
ALGSTR_1: 7
.= (((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 )) by
VECTSP_1: 18
.= ((gf1
|^ 2)
* ((P
`3_3 )
* (Q
`3_3 ))) by
GROUP_1:def 3;
((P
`3_3 )
* (Q
`3_3 ))
<>
0 by
A13,
A14,
EC_PF_1: 20;
then gf3
<>
0 by
A19,
A20,
EC_PF_1: 20;
then (gf1
* gf3)
<>
0 by
A18,
EC_PF_1: 20;
then
A21: (gf1
* gf3)
<> (
0. (
GF p)) by
EC_PF_1: 11;
((gf1
* ((((gf2
|^ 2)
* (P
`1_3 ))
* (Q
`3_3 ))
- gf3))
- (((gf2
|^ 3)
* (P
`2_3 ))
* (Q
`3_3 )))
= ((gf1
* (((
0. (
GF p))
* ((P
`1_3 )
* (Q
`3_3 )))
- gf3))
- (((
0. (
GF p))
* (P
`2_3 ))
* (Q
`3_3 ))) by
A17
.= (gf1
* ((
0. (
GF p))
- gf3)) by
VECTSP_1: 18
.= (gf1
* (
- gf3)) by
VECTSP_1: 18
.= (
- (gf1
* gf3)) by
VECTSP_1: 8;
then ((gf1
* ((((gf2
|^ 2)
* (P
`1_3 ))
* (Q
`3_3 ))
- gf3))
- (((gf2
|^ 3)
* (P
`2_3 ))
* (Q
`3_3 )))
<> (
0. (
GF p)) by
A21,
VECTSP_2: 3;
then
A22: (R
`2_3 )
<> (
0. (
GF p));
then R
<>
[
0 ,
0 ,
0 ] by
EC_PF_1: 11;
then not R
in
{
[
0 ,
0 ,
0 ]} by
TARSKI:def 1;
then R
in (
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):]
\
{
[
0 ,
0 ,
0 ]}) by
XBOOLE_0:def 5;
then
reconsider R as
Element of (
ProjCo (
GF p)) by
EC_PF_1: 40;
R
in (
EC_SetProjCo (a,b,p))
proof
A23: (R
`1_3 )
= (
0. (
GF p)) by
A16;
A24: (R
`3_3 )
= (
0. (
GF p)) by
A17;
set d = (R
`2_3 );
reconsider d as
Element of (
GF p);
A25: (OO
`1_3 )
=
0 & (OO
`2_3 )
= 1 & (OO
`3_3 )
=
0 by
A7;
A26: (d
* (OO
`1_3 ))
=
0 by
A25,
EC_PF_1: 20
.= (R
`1_3 ) by
A23,
EC_PF_1: 11;
A27: (d
* (OO
`3_3 ))
=
0 by
A25,
EC_PF_1: 20
.= (R
`3_3 ) by
A24,
EC_PF_1: 11;
(d
* (OO
`2_3 ))
= (d
* (
1. (
GF p))) by
A25,
EC_PF_1: 12
.= (R
`2_3 );
hence R
in (
EC_SetProjCo (a,b,p)) by
A1,
A7,
A22,
A26,
A27,
EC_PF_1: 45;
end;
then
reconsider R as
Element of (
EC_SetProjCo (a,b,p));
take R;
thus thesis by
A10;
end;
suppose not P
_EQ_ ((
compell_ProjCo (z,p))
. Q);
then ((P
`1_3 )
* (Q
`3_3 ))
<> ((Q
`1_3 )
* (P
`3_3 )) by
A10,
A13,
A14,
Th50;
then
A28: gf2
<> (
0. (
GF p)) by
VECTSP_1: 19;
then gf2
<>
0 by
EC_PF_1: 11;
then
A29: (gf2
|^ 3)
<>
0 by
EC_PF_1: 25;
((P
`3_3 )
* (Q
`3_3 ))
<>
0 by
A13,
A14,
EC_PF_1: 20;
then ((gf2
|^ 3)
* ((P
`3_3 )
* (Q
`3_3 )))
<>
0 by
A29,
EC_PF_1: 20;
then ((gf2
|^ 3)
* ((P
`3_3 )
* (Q
`3_3 )))
<> (
0. (
GF p)) by
EC_PF_1: 11;
then
A30: (((gf2
|^ 3)
* (P
`3_3 ))
* (Q
`3_3 ))
<> (
0. (
GF p)) by
GROUP_1:def 3;
(R
`3_3 )
<> (
0. (
GF p)) by
A30;
then R
<>
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))];
then not R
in
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]} by
TARSKI:def 1;
then R
in (
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):]
\
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]}) by
XBOOLE_0:def 5;
then
reconsider R as
Element of (
ProjCo (
GF p)) by
EC_PF_1:def 6;
(gf2
* gf2)
<> (
0. (
GF p)) by
A28,
VECTSP_1: 12;
then
A31: (gf2
|^ 2)
<> (
0. (
GF p)) by
EC_PF_1: 22;
((P
`3_3 )
|^ 2)
<>
0 by
A4,
A12,
Th38,
EC_PF_1: 25;
then
A32: ((P
`3_3 )
|^ 2)
<> (
0. (
GF p)) by
EC_PF_1: 11;
A33: (Q
`3_3 )
<> (
0. (
GF p)) by
A14,
EC_PF_1: 11;
((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
<> (
0. (
GF p)) by
A31,
A32,
VECTSP_1: 12;
then
A34: (((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
<> (
0. (
GF p)) by
A33,
VECTSP_1: 12;
((
EC_WEqProjCo (a,b,p))
. R)
= (
0. (
GF p))
proof
((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((((R
`2_3 )
|^ 2)
* (R
`3_3 ))
- ((((R
`1_3 )
|^ 3)
+ ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (b
* ((R
`3_3 )
|^ 3)))))
= (
0. (
GF p)) by
Th58;
then ((((gf2
|^ 2)
* ((P
`3_3 )
|^ 2))
* (Q
`3_3 ))
* ((
EC_WEqProjCo (a,b,p))
. R))
= (
0. (
GF p)) by
EC_PF_1:def 8;
hence thesis by
A34,
VECTSP_1: 12;
end;
then R
is_on_curve (
EC_WEqProjCo (a,b,p));
then
reconsider R as
Element of (
EC_SetProjCo (a,b,p)) by
Th34;
take R;
thus thesis by
A10;
end;
end;
suppose
A35: not P
_EQ_ O & not Q
_EQ_ O & P
_EQ_ Q;
A36: (OO
`3_3 )
=
0 by
A7;
(
rep_pt PP)
<> (
rep_pt OO) by
A3,
A7,
A35,
Th39;
then (
rep_pt PP)
<>
[
0 , 1,
0 ] by
A36,
Def7;
then
A37: ((
rep_pt PP)
`3_3 )
<>
0 by
Th37;
then
A38: (P
`3_3 )
<>
0 by
A4,
Th38;
reconsider g2 = (2
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
reconsider g3 = (3
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
reconsider g4 = (4
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
reconsider g8 = (8
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
set gf1 = ((a
* ((P
`3_3 )
|^ 2))
+ (g3
* ((P
`1_3 )
|^ 2)));
set gf2 = ((P
`2_3 )
* (P
`3_3 ));
set gf3 = (((P
`1_3 )
* (P
`2_3 ))
* gf2);
set gf4 = ((gf1
|^ 2)
- (g8
* gf3));
reconsider gf1, gf2, gf3, gf4 as
Element of (
GF p);
set R =
[((g2
* gf4)
* gf2), ((gf1
* ((g4
* gf3)
- gf4))
- ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2))), (g8
* (gf2
|^ 3))];
reconsider R as
Element of
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):];
per cases ;
suppose
A39: (P
`2_3 )
=
0 ;
A40: gf2
=
0 by
A39,
EC_PF_1: 20;
then
A41: gf3
=
0 by
EC_PF_1: 20;
A42: gf2
= (
0. (
GF p)) by
A40,
EC_PF_1: 11;
A43: gf3
= (
0. (
GF p)) by
A41,
EC_PF_1: 11;
A44: gf1
<>
0 by
A4,
A37,
A39,
Th38,
Th53;
then
A45: gf1
<> (
0. (
GF p)) by
EC_PF_1: 11;
gf4
= ((gf1
|^ 2)
- (
0. (
GF p))) by
A43
.= (gf1
|^ 2) by
VECTSP_1: 18;
then gf4
<>
0 by
A44,
EC_PF_1: 25;
then
A46: gf4
<> (
0. (
GF p)) by
EC_PF_1: 11;
(R
`2_3 )
= ((gf1
* ((g4
* (
0. (
GF p)))
- gf4))
- ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
* gf2))) by
A43,
EC_PF_1: 22
.= ((gf1
* (
- gf4))
- ((g8
* ((P
`2_3 )
|^ 2))
* ((
0. (
GF p))
* (
0. (
GF p))))) by
VECTSP_1: 18,
A42
.= (gf1
* (
- gf4)) by
VECTSP_1: 18
.= (
- (gf1
* gf4)) by
VECTSP_1: 8;
then (
- (R
`2_3 ))
= (gf1
* gf4) by
RLVECT_1: 17;
then (
- (R
`2_3 ))
<> (
0. (
GF p)) by
A45,
A46,
VECTSP_1: 12;
then
A47: (R
`2_3 )
<> (
0. (
GF p)) by
VECTSP_2: 3;
then R
<>
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))];
then not R
in
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]} by
TARSKI:def 1;
then R
in (
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):]
\
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]}) by
XBOOLE_0:def 5;
then
reconsider R as
Element of (
ProjCo (
GF p)) by
EC_PF_1:def 6;
R
in (
EC_SetProjCo (a,b,p))
proof
A48: (R
`1_3 )
= (
0. (
GF p)) by
A42;
0
= (
0. (
GF p)) by
EC_PF_1: 11;
then
A49: (gf2
|^ 3)
= (
0. (
GF p)) by
A40,
Th5;
A50: (R
`3_3 )
= (
0. (
GF p)) by
A49;
set d = (R
`2_3 );
reconsider d as
Element of (
GF p);
A51: (OO
`1_3 )
=
0 & (OO
`2_3 )
= 1 & (OO
`3_3 )
=
0 by
A7;
A52: (d
* (OO
`1_3 ))
=
0 by
A51,
EC_PF_1: 20
.= (R
`1_3 ) by
A48,
EC_PF_1: 11;
A53: (d
* (OO
`3_3 ))
=
0 by
A51,
EC_PF_1: 20
.= (R
`3_3 ) by
A50,
EC_PF_1: 11;
(d
* (OO
`2_3 ))
= (d
* (
1. (
GF p))) by
A51,
EC_PF_1: 12
.= (R
`2_3 );
hence R
in (
EC_SetProjCo (a,b,p)) by
A1,
A7,
A47,
A52,
A53,
EC_PF_1: 45;
end;
then
reconsider R as
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
take R;
thus thesis by
A35;
end;
suppose
A54: (P
`2_3 )
<>
0 ;
A55: gf2
<>
0 by
A38,
A54,
EC_PF_1: 20;
then (gf2
|^ 3)
<>
0 by
EC_PF_1: 25;
then
A56: (gf2
|^ 3)
<> (
0. (
GF p)) by
EC_PF_1: 11;
A57: g4
= ((2
* 2)
mod p)
.= (g2
* g2) by
EC_PF_1: 18;
A58: p
> 2 by
A1,
NAT_1: 13;
g8
= ((4
* 2)
mod p)
.= (g4
* g2) by
EC_PF_1: 18
.= ((g2
|^ 2)
* g2) by
A57,
EC_PF_1: 22
.= (g2
|^ (2
+ 1)) by
EC_PF_1: 24
.= (g2
|^ 3);
then
A59: g8
<> (
0. (
GF p)) by
Th28,
A58;
(R
`3_3 )
<> (
0. (
GF p)) by
A56,
A59,
VECTSP_1: 12;
then R
<>
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))];
then not R
in
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]} by
TARSKI:def 1;
then R
in (
[:the
carrier of (
GF p), the
carrier of (
GF p), the
carrier of (
GF p):]
\
{
[(
0. (
GF p)), (
0. (
GF p)), (
0. (
GF p))]}) by
XBOOLE_0:def 5;
then
reconsider R as
Element of (
ProjCo (
GF p)) by
EC_PF_1:def 6;
g4
= (g2
|^ 2) by
A57,
EC_PF_1: 22;
then
A60: g4
<> (
0. (
GF p)) by
Th28,
A58;
(gf2
|^ 2)
<>
0 by
A55,
EC_PF_1: 25;
then
A61: (gf2
|^ 2)
<> (
0. (
GF p)) by
EC_PF_1: 11;
((P
`3_3 )
|^ 2)
<>
0 by
A4,
A37,
Th38,
EC_PF_1: 25;
then
A62: ((P
`3_3 )
|^ 2)
<> (
0. (
GF p)) by
EC_PF_1: 11;
(g4
* (gf2
|^ 2))
<> (
0. (
GF p)) by
A60,
A61,
VECTSP_1: 12;
then
A63: ((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
<> (
0. (
GF p)) by
A62,
VECTSP_1: 12;
((
EC_WEqProjCo (a,b,p))
. R)
= (
0. (
GF p))
proof
(((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((((R
`2_3 )
|^ 2)
* (R
`3_3 ))
- ((((R
`1_3 )
|^ 3)
+ ((a
* (R
`1_3 ))
* ((R
`3_3 )
|^ 2)))
+ (b
* ((R
`3_3 )
|^ 3)))))
= (
0. (
GF p)) by
Th63;
then (((g4
* (gf2
|^ 2))
* ((P
`3_3 )
|^ 2))
* ((
EC_WEqProjCo (a,b,p))
. R))
= (
0. (
GF p)) by
EC_PF_1:def 8;
hence thesis by
A63,
VECTSP_1: 12;
end;
then R
is_on_curve (
EC_WEqProjCo (a,b,p));
then
reconsider R as
Element of (
EC_SetProjCo (a,b,p)) by
Th34;
take R;
thus thesis by
A35;
end;
end;
end;
consider f be
Function of
[:(
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)):], (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) such that
A64: for P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds
P[P, Q, (f
. (P,Q))] from
BINOP_1:sch 3(
A2);
take f;
thus thesis by
A64;
end;
uniqueness
proof
set a = (z
`1 );
set b = (z
`2 );
defpred
P[
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)),
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)),
set] means for O be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st O
=
[
0 , 1,
0 ] holds ($1
_EQ_ O implies $3
= $2) & (($2
_EQ_ O & not $1
_EQ_ O) implies $3
= $1) & (( not $1
_EQ_ O & not $2
_EQ_ O & not $1
_EQ_ $2) implies for g2,gf1,gf2,gf3 be
Element of (
GF p) st g2
= (2
mod p) & gf1
= ((($2
`2_3 )
* ($1
`3_3 ))
- (($1
`2_3 )
* ($2
`3_3 ))) & gf2
= ((($2
`1_3 )
* ($1
`3_3 ))
- (($1
`1_3 )
* ($2
`3_3 ))) & gf3
= (((((gf1
|^ 2)
* ($1
`3_3 ))
* ($2
`3_3 ))
- (gf2
|^ 3))
- (((g2
* (gf2
|^ 2))
* ($1
`1_3 ))
* ($2
`3_3 ))) holds $3
=
[(gf2
* gf3), ((gf1
* ((((gf2
|^ 2)
* ($1
`1_3 ))
* ($2
`3_3 ))
- gf3))
- (((gf2
|^ 3)
* ($1
`2_3 ))
* ($2
`3_3 ))), (((gf2
|^ 3)
* ($1
`3_3 ))
* ($2
`3_3 ))]) & (( not $1
_EQ_ O & not $2
_EQ_ O & $1
_EQ_ $2) implies for g2,g3,g4,g8,gf1,gf2,gf3,gf4 be
Element of (
GF p) st g2
= (2
mod p) & g3
= (3
mod p) & g4
= (4
mod p) & g8
= (8
mod p) & gf1
= ((a
* (($1
`3_3 )
|^ 2))
+ (g3
* (($1
`1_3 )
|^ 2))) & gf2
= (($1
`2_3 )
* ($1
`3_3 )) & gf3
= ((($1
`1_3 )
* ($1
`2_3 ))
* gf2) & gf4
= ((gf1
|^ 2)
- (g8
* gf3)) holds $3
=
[((g2
* gf4)
* gf2), ((gf1
* ((g4
* gf3)
- gf4))
- ((g8
* (($1
`2_3 )
|^ 2))
* (gf2
|^ 2))), (g8
* (gf2
|^ 3))]);
for o1,o2 be
Function of
[:(
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)):], (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) st (for P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds
P[P, Q, (o1
. (P,Q))]) & (for P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds
P[P, Q, (o2
. (P,Q))]) holds o1
= o2
proof
let o1,o2 be
Function of
[:(
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)):], (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) such that
A65: for P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds
P[P, Q, (o1
. (P,Q))] and
A66: for P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds
P[P, Q, (o2
. (P,Q))];
set O =
[
0 , 1,
0 ];
reconsider O as
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) by
EC_PF_1: 42;
for P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) holds (o1
. (P,Q))
= (o2
. (P,Q))
proof
let P,Q be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
per cases ;
suppose
A67: P
_EQ_ O;
(o1
. (P,Q))
= Q by
A65,
A67;
hence thesis by
A66,
A67;
end;
suppose
A68: Q
_EQ_ O & not P
_EQ_ O;
(o1
. (P,Q))
= P by
A65,
A68;
hence thesis by
A66,
A68;
end;
suppose
A69: not P
_EQ_ O & not Q
_EQ_ O & not P
_EQ_ Q;
reconsider g2 = (2
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
set gf1 = (((Q
`2_3 )
* (P
`3_3 ))
- ((P
`2_3 )
* (Q
`3_3 )));
set gf2 = (((Q
`1_3 )
* (P
`3_3 ))
- ((P
`1_3 )
* (Q
`3_3 )));
set gf3 = (((((gf1
|^ 2)
* (P
`3_3 ))
* (Q
`3_3 ))
- (gf2
|^ 3))
- (((g2
* (gf2
|^ 2))
* (P
`1_3 ))
* (Q
`3_3 )));
reconsider gf1, gf2, gf3 as
Element of (
GF p);
(o1
. (P,Q))
=
[(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 ))] by
A65,
A69;
hence thesis by
A66,
A69;
end;
suppose
A70: not P
_EQ_ O & not Q
_EQ_ O & P
_EQ_ Q;
reconsider g2 = (2
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
reconsider g3 = (3
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
reconsider g4 = (4
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
reconsider g8 = (8
mod p) as
Element of (
GF p) by
EC_PF_1: 14;
set gf1 = ((a
* ((P
`3_3 )
|^ 2))
+ (g3
* ((P
`1_3 )
|^ 2)));
set gf2 = ((P
`2_3 )
* (P
`3_3 ));
set gf3 = (((P
`1_3 )
* (P
`2_3 ))
* gf2);
set gf4 = ((gf1
|^ 2)
- (g8
* gf3));
reconsider gf1, gf2, gf3, gf4 as
Element of (
GF p);
(o1
. (P,Q))
=
[((g2
* gf4)
* gf2), ((gf1
* ((g4
* gf3)
- gf4))
- ((g8
* ((P
`2_3 )
|^ 2))
* (gf2
|^ 2))), (g8
* (gf2
|^ 3))] by
A65,
A70;
hence thesis by
A66,
A70;
end;
end;
hence thesis by
BINOP_1: 2;
end;
hence thesis;
end;
end
definition
let p be 5
_or_greater
Prime, z be
Element of (
EC_WParam p);
let F be
Function of
[:(
EC_SetProjCo ((z
`1 ),(z
`2 ),p)), (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)):], (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
let Q,R be
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
:: original:
.
redefine
func F
. (Q,R) ->
Element of (
EC_SetProjCo ((z
`1 ),(z
`2 ),p)) ;
correctness
proof
(F
. (Q,R))
in (
EC_SetProjCo ((z
`1 ),(z
`2 ),p));
hence thesis;
end;
end