K122() is Element of bool K118()
K118() is set
bool K118() is non empty set
K117() is set
bool K117() is non empty set
bool K122() is non empty set
{} is empty set
1 is non empty set
X is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the non empty set is non empty set
[: the non empty set , the non empty set :] is non empty set
[:[: the non empty set , the non empty set :], the non empty set :] is non empty set
bool [:[: the non empty set , the non empty set :], the non empty set :] is non empty set
the V1() V4([: the non empty set , the non empty set :]) V5( the non empty set ) V6() V18([: the non empty set , the non empty set :], the non empty set ) Element of bool [:[: the non empty set , the non empty set :], the non empty set :] is V1() V4([: the non empty set , the non empty set :]) V5( the non empty set ) V6() V18([: the non empty set , the non empty set :], the non empty set ) Element of bool [:[: the non empty set , the non empty set :], the non empty set :]
the Element of the non empty set is Element of the non empty set
the carrier of X is non empty non trivial set
[: the carrier of X, the non empty set :] is non empty set
[:[: the carrier of X, the non empty set :], the non empty set :] is non empty set
bool [:[: the carrier of X, the non empty set :], the non empty set :] is non empty set
the V1() V4([: the carrier of X, the non empty set :]) V5( the non empty set ) V6() V18([: the carrier of X, the non empty set :], the non empty set ) Element of bool [:[: the carrier of X, the non empty set :], the non empty set :] is V1() V4([: the carrier of X, the non empty set :]) V5( the non empty set ) V6() V18([: the carrier of X, the non empty set :], the non empty set ) Element of bool [:[: the carrier of X, the non empty set :], the non empty set :]
bool [: the non empty set , the non empty set :] is non empty set
the V1() V4( the non empty set ) V5( the non empty set ) Element of bool [: the non empty set , the non empty set :] is V1() V4( the non empty set ) V5( the non empty set ) Element of bool [: the non empty set , the non empty set :]
(X, the non empty set , the V1() V4([: the non empty set , the non empty set :]) V5( the non empty set ) V6() V18([: the non empty set , the non empty set :], the non empty set ) Element of bool [:[: the non empty set , the non empty set :], the non empty set :], the Element of the non empty set , the V1() V4([: the carrier of X, the non empty set :]) V5( the non empty set ) V6() V18([: the carrier of X, the non empty set :], the non empty set ) Element of bool [:[: the carrier of X, the non empty set :], the non empty set :], the V1() V4( the non empty set ) V5( the non empty set ) Element of bool [: the non empty set , the non empty set :]) is (X) (X)
the carrier of (X, the non empty set , the V1() V4([: the non empty set , the non empty set :]) V5( the non empty set ) V6() V18([: the non empty set , the non empty set :], the non empty set ) Element of bool [:[: the non empty set , the non empty set :], the non empty set :], the Element of the non empty set , the V1() V4([: the carrier of X, the non empty set :]) V5( the non empty set ) V6() V18([: the carrier of X, the non empty set :], the non empty set ) Element of bool [:[: the carrier of X, the non empty set :], the non empty set :], the V1() V4( the non empty set ) V5( the non empty set ) Element of bool [: the non empty set , the non empty set :]) is set
X is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
o is (X)
the carrier of o is set
0 is empty Element of K122()
{0} is non empty set
o is Element of {0}
[:{0},{0}:] is non empty set
[:[:{0},{0}:],{0}:] is non empty set
bool [:[:{0},{0}:],{0}:] is non empty set
md is V1() V4([:{0},{0}:]) V5({0}) V6() V18([:{0},{0}:],{0}) Element of bool [:[:{0},{0}:],{0}:]
b is set
a is set
bool [:{0},{0}:] is non empty set
a is V1() V4({0}) V5({0}) Element of bool [:{0},{0}:]
x is V1() V4({0}) V5({0}) Element of bool [:{0},{0}:]
y is set
z is set
0F is Element of {0}
p is Element of {0}
[0F,p] is Element of [:{0},{0}:]
{0F,p} is non empty set
{0F} is non empty set
{{0F,p},{0F}} is non empty set
S is non empty set
[:S,S:] is non empty set
[:[:S,S:],S:] is non empty set
bool [:[:S,S:],S:] is non empty set
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the carrier of F is non empty non trivial set
[: the carrier of F,S:] is non empty set
[:[: the carrier of F,S:],S:] is non empty set
bool [:[: the carrier of F,S:],S:] is non empty set
bool [:S,S:] is non empty set
b is V1() V4([:S,S:]) V5(S) V6() V18([:S,S:],S) Element of bool [:[:S,S:],S:]
a is Element of S
x is V1() V4([: the carrier of F,S:]) V5(S) V6() V18([: the carrier of F,S:],S) Element of bool [:[: the carrier of F,S:],S:]
y is V1() V4(S) V5(S) Element of bool [:S,S:]
(F,S,b,a,x,y) is (F) (F)
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the carrier of F is non empty non trivial set
[: the carrier of F,{0}:] is non empty set
[:[: the carrier of F,{0}:],{0}:] is non empty set
bool [:[: the carrier of F,{0}:],{0}:] is non empty set
S is V1() V4([: the carrier of F,{0}:]) V5({0}) V6() V18([: the carrier of F,{0}:],{0}) Element of bool [:[: the carrier of F,{0}:],{0}:]
b is V1() V4({0}) V5({0}) Element of bool [:{0},{0}:]
(F,{0},md,o,S,b) is non empty (F) (F)
the carrier of (F,{0},md,o,S,b) is non empty set
x is Element of the carrier of (F,{0},md,o,S,b)
y is Element of the carrier of (F,{0},md,o,S,b)
x + y is Element of the carrier of (F,{0},md,o,S,b)
the addF of (F,{0},md,o,S,b) is V1() V4([: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):]) V5( the carrier of (F,{0},md,o,S,b)) V6() V18([: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b)) Element of bool [:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):]
[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):] is non empty set
[:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):] is non empty set
bool [:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):] is non empty set
the addF of (F,{0},md,o,S,b) . (x,y) is Element of the carrier of (F,{0},md,o,S,b)
y + x is Element of the carrier of (F,{0},md,o,S,b)
the addF of (F,{0},md,o,S,b) . (y,x) is Element of the carrier of (F,{0},md,o,S,b)
z is Element of {0}
0F is Element of {0}
md . (z,0F) is Element of {0}
the carrier of (F,{0},md,o,S,b) is non empty set
x is Element of the carrier of (F,{0},md,o,S,b)
y is Element of the carrier of (F,{0},md,o,S,b)
x + y is Element of the carrier of (F,{0},md,o,S,b)
the addF of (F,{0},md,o,S,b) is V1() V4([: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):]) V5( the carrier of (F,{0},md,o,S,b)) V6() V18([: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b)) Element of bool [:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):]
[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):] is non empty set
[:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):] is non empty set
bool [:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):] is non empty set
the addF of (F,{0},md,o,S,b) . (x,y) is Element of the carrier of (F,{0},md,o,S,b)
z is Element of the carrier of (F,{0},md,o,S,b)
(x + y) + z is Element of the carrier of (F,{0},md,o,S,b)
the addF of (F,{0},md,o,S,b) . ((x + y),z) is Element of the carrier of (F,{0},md,o,S,b)
y + z is Element of the carrier of (F,{0},md,o,S,b)
the addF of (F,{0},md,o,S,b) . (y,z) is Element of the carrier of (F,{0},md,o,S,b)
x + (y + z) is Element of the carrier of (F,{0},md,o,S,b)
the addF of (F,{0},md,o,S,b) . (x,(y + z)) is Element of the carrier of (F,{0},md,o,S,b)
0F is Element of {0}
p is Element of {0}
r is Element of {0}
md . (p,r) is Element of {0}
md . (0F,(md . (p,r))) is Element of {0}
the carrier of (F,{0},md,o,S,b) is non empty set
x is Element of the carrier of (F,{0},md,o,S,b)
0. (F,{0},md,o,S,b) is V51((F,{0},md,o,S,b)) Element of the carrier of (F,{0},md,o,S,b)
the ZeroF of (F,{0},md,o,S,b) is Element of the carrier of (F,{0},md,o,S,b)
x + (0. (F,{0},md,o,S,b)) is Element of the carrier of (F,{0},md,o,S,b)
the addF of (F,{0},md,o,S,b) is V1() V4([: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):]) V5( the carrier of (F,{0},md,o,S,b)) V6() V18([: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b)) Element of bool [:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):]
[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):] is non empty set
[:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):] is non empty set
bool [:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):] is non empty set
the addF of (F,{0},md,o,S,b) . (x,(0. (F,{0},md,o,S,b))) is Element of the carrier of (F,{0},md,o,S,b)
y is Element of {0}
md . (y,(0. (F,{0},md,o,S,b))) is set
the carrier of (F,{0},md,o,S,b) is non empty set
x is Element of the carrier of (F,{0},md,o,S,b)
- x is Element of the carrier of (F,{0},md,o,S,b)
x + (- x) is Element of the carrier of (F,{0},md,o,S,b)
the addF of (F,{0},md,o,S,b) is V1() V4([: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):]) V5( the carrier of (F,{0},md,o,S,b)) V6() V18([: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b)) Element of bool [:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):]
[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):] is non empty set
[:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):] is non empty set
bool [:[: the carrier of (F,{0},md,o,S,b), the carrier of (F,{0},md,o,S,b):], the carrier of (F,{0},md,o,S,b):] is non empty set
the addF of (F,{0},md,o,S,b) . (x,(- x)) is Element of the carrier of (F,{0},md,o,S,b)
0. (F,{0},md,o,S,b) is V51((F,{0},md,o,S,b)) Element of the carrier of (F,{0},md,o,S,b)
the ZeroF of (F,{0},md,o,S,b) is Element of the carrier of (F,{0},md,o,S,b)
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the V1() V4({0}) V5({0}) Element of bool [:{0},{0}:] is V1() V4({0}) V5({0}) Element of bool [:{0},{0}:]
the carrier of F is non empty non trivial set
[: the carrier of F,{0}:] is non empty set
[:[: the carrier of F,{0}:],{0}:] is non empty set
bool [:[: the carrier of F,{0}:],{0}:] is non empty set
the V1() V4([: the carrier of F,{0}:]) V5({0}) V6() V18([: the carrier of F,{0}:],{0}) Element of bool [:[: the carrier of F,{0}:],{0}:] is V1() V4([: the carrier of F,{0}:]) V5({0}) V6() V18([: the carrier of F,{0}:],{0}) Element of bool [:[: the carrier of F,{0}:],{0}:]
(F,{0},md,o, the V1() V4([: the carrier of F,{0}:]) V5({0}) V6() V18([: the carrier of F,{0}:],{0}) Element of bool [:[: the carrier of F,{0}:],{0}:], the V1() V4({0}) V5({0}) Element of bool [:{0},{0}:]) is non empty (F) (F)
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the carrier of F is non empty non trivial set
[: the carrier of F,{0}:] is non empty set
[:[: the carrier of F,{0}:],{0}:] is non empty set
bool [:[: the carrier of F,{0}:],{0}:] is non empty set
S is V1() V4([: the carrier of F,{0}:]) V5({0}) V6() V18([: the carrier of F,{0}:],{0}) Element of bool [:[: the carrier of F,{0}:],{0}:]
a is non empty right_complementable Abelian add-associative right_zeroed (F)
b is V1() V4({0}) V5({0}) Element of bool [:{0},{0}:]
(F,{0},md,o,S,b) is non empty (F) (F)
the carrier of a is non empty set
1_ F is Element of the carrier of F
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
x is Element of the carrier of F
y is Element of the carrier of F
x + y is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . (x,y) is Element of the carrier of F
x * y is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . (x,y) is Element of the carrier of F
z is Element of the carrier of a
0F is Element of the carrier of a
z + 0F is Element of the carrier of a
the addF of a is V1() V4([: the carrier of a, the carrier of a:]) V5( the carrier of a) V6() V18([: the carrier of a, the carrier of a:], the carrier of a) Element of bool [:[: the carrier of a, the carrier of a:], the carrier of a:]
[: the carrier of a, the carrier of a:] is non empty set
[:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
bool [:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
the addF of a . (z,0F) is Element of the carrier of a
x * (z + 0F) is Element of the carrier of a
x * z is Element of the carrier of a
x * 0F is Element of the carrier of a
(x * z) + (x * 0F) is Element of the carrier of a
the addF of a . ((x * z),(x * 0F)) is Element of the carrier of a
(x + y) * z is Element of the carrier of a
y * z is Element of the carrier of a
(x * z) + (y * z) is Element of the carrier of a
the addF of a . ((x * z),(y * z)) is Element of the carrier of a
(x * y) * z is Element of the carrier of a
x * (y * z) is Element of the carrier of a
(1_ F) * z is Element of the carrier of a
S . ((x * y),z) is set
r is Element of the carrier of a
v is Element of the carrier of a
(x * y) * v is Element of the carrier of a
v is Element of the carrier of a
S . (y,v) is set
v is Element of the carrier of a
y * v is Element of the carrier of a
x * (y * v) is Element of the carrier of a
S . (x,o) is Element of {0}
p is Element of {0}
r is Element of {0}
md . (p,r) is Element of {0}
v is Element of the carrier of a
v is Element of the carrier of a
v + v is Element of the carrier of a
the addF of a . (v,v) is Element of the carrier of a
x * (v + v) is Element of the carrier of a
S . (x,o) is Element of {0}
S . (x,v) is set
x * v is Element of the carrier of a
S . (x,v) is set
x * v is Element of the carrier of a
0. a is V51(a) Element of the carrier of a
the ZeroF of a is Element of the carrier of a
S . ((x + y),z) is set
r is Element of the carrier of a
v is Element of the carrier of a
(x + y) * v is Element of the carrier of a
v is Element of the carrier of a
S . (x,v) is set
v is Element of the carrier of a
x * v is Element of the carrier of a
v is Element of the carrier of a
S . (y,v) is set
0. a is V51(a) Element of the carrier of a
the ZeroF of a is Element of the carrier of a
v is Element of the carrier of a
y * v is Element of the carrier of a
S . ((1_ F),z) is set
r is Element of the carrier of a
S . ((1_ F),r) is set
S is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the carrier of S is non empty non trivial set
[: the carrier of S,{0}:] is non empty set
[:[: the carrier of S,{0}:],{0}:] is non empty set
bool [:[: the carrier of S,{0}:],{0}:] is non empty set
a is V1() V4({0}) V5({0}) Element of bool [:{0},{0}:]
x is non empty right_complementable Abelian add-associative right_zeroed (S)
b is V1() V4([: the carrier of S,{0}:]) V5({0}) V6() V18([: the carrier of S,{0}:],{0}) Element of bool [:[: the carrier of S,{0}:],{0}:]
(S,{0},md,o,b,a) is non empty (S) (S)
the carrier of x is non empty set
0. x is V51(x) Element of the carrier of x
the ZeroF of x is Element of the carrier of x
y is Element of the carrier of x
y is Element of the carrier of x
z is Element of the carrier of x
[y,z] is Element of [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
{y,z} is non empty set
{y} is non empty set
{{y,z},{y}} is non empty set
0F is Element of {0}
p is Element of {0}
[0F,p] is Element of [:{0},{0}:]
{0F,p} is non empty set
{0F} is non empty set
{{0F,p},{0F}} is non empty set
y is Element of the carrier of x
z is Element of the carrier of x
0F is Element of the carrier of x
p is Element of the carrier of x
[y,z] is Element of [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
{y,z} is non empty set
{y} is non empty set
{{y,z},{y}} is non empty set
[0F,p] is Element of [: the carrier of x, the carrier of x:]
{0F,p} is non empty set
{0F} is non empty set
{{0F,p},{0F}} is non empty set
[y,z] is Element of [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
{y,z} is non empty set
{y} is non empty set
{{y,z},{y}} is non empty set
0F is Element of {0}
p is Element of {0}
[0F,p] is Element of [:{0},{0}:]
{0F,p} is non empty set
{0F} is non empty set
{{0F,p},{0F}} is non empty set
y is Element of the carrier of x
z is Element of the carrier of x
0F is Element of the carrier of S
0F * y is Element of the carrier of x
z is Element of the carrier of x
y is Element of the carrier of x
0F is Element of the carrier of x
z + 0F is Element of the carrier of x
the addF of x is V1() V4([: the carrier of x, the carrier of x:]) V5( the carrier of x) V6() V18([: the carrier of x, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
the addF of x . (z,0F) is Element of the carrier of x
z is Element of the carrier of x
y is Element of the carrier of x
0F is Element of the carrier of x
y is Element of the carrier of x
z is Element of the carrier of x
0F is Element of the carrier of x
z + 0F is Element of the carrier of x
the addF of x is V1() V4([: the carrier of x, the carrier of x:]) V5( the carrier of x) V6() V18([: the carrier of x, the carrier of x:], the carrier of x) Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
the addF of x . (z,0F) is Element of the carrier of x
0F + y is Element of the carrier of x
the addF of x . (0F,y) is Element of the carrier of x
y + z is Element of the carrier of x
the addF of x . (y,z) is Element of the carrier of x
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the carrier of F is non empty non trivial set
[: the carrier of F,{0}:] is non empty set
[:[: the carrier of F,{0}:],{0}:] is non empty set
bool [:[: the carrier of F,{0}:],{0}:] is non empty set
S is V1() V4([: the carrier of F,{0}:]) V5({0}) V6() V18([: the carrier of F,{0}:],{0}) Element of bool [:[: the carrier of F,{0}:],{0}:]
b is V1() V4({0}) V5({0}) Element of bool [:{0},{0}:]
(F,{0},md,o,S,b) is non empty (F) (F)
a is non empty right_complementable Abelian add-associative right_zeroed (F)
the carrier of a is non empty set
0. a is V51(a) Element of the carrier of a
the ZeroF of a is Element of the carrier of a
x is Element of the carrier of a
y is Element of the carrier of a
z is Element of the carrier of a
0F is Element of the carrier of F
0F * y is Element of the carrier of a
r is Element of the carrier of a
p is Element of the carrier of a
v is Element of the carrier of a
r + v is Element of the carrier of a
the addF of a is V1() V4([: the carrier of a, the carrier of a:]) V5( the carrier of a) V6() V18([: the carrier of a, the carrier of a:], the carrier of a) Element of bool [:[: the carrier of a, the carrier of a:], the carrier of a:]
[: the carrier of a, the carrier of a:] is non empty set
[:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
bool [:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
the addF of a . (r,v) is Element of the carrier of a
v is Element of the carrier of a
v is Element of the carrier of a
v is Element of the carrier of a
v is Element of the carrier of a
c19 is Element of the carrier of a
c20 is Element of the carrier of a
c19 + c20 is Element of the carrier of a
the addF of a . (c19,c20) is Element of the carrier of a
c20 + v is Element of the carrier of a
the addF of a . (c20,v) is Element of the carrier of a
v + c19 is Element of the carrier of a
the addF of a . (v,c19) is Element of the carrier of a
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
b is Element of the carrier of S
0. F is V51(F) Element of the carrier of F
the carrier of F is non empty non trivial set
the ZeroF of F is Element of the carrier of F
y is Element of the carrier of F
y * b is Element of the carrier of S
(0. S) - (y * b) is Element of the carrier of S
- (y * b) is Element of the carrier of S
(0. S) + (- (y * b)) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . ((0. S),(- (y * b))) is Element of the carrier of S
(0. F) * ((0. S) - (y * b)) is Element of the carrier of S
(0. F) * b is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
(0. S) + a is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . ((0. S),a) is Element of the carrier of S
a + b is Element of the carrier of S
the addF of S . (a,b) is Element of the carrier of S
b + (0. S) is Element of the carrier of S
the addF of S . (b,(0. S)) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
x + b is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,b) is Element of the carrier of S
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
- (1_ F) is Element of the carrier of F
(- (1_ F)) * x is Element of the carrier of S
- x is Element of the carrier of S
(- x) + (x + b) is Element of the carrier of S
the addF of S . ((- x),(x + b)) is Element of the carrier of S
x + (- x) is Element of the carrier of S
the addF of S . (x,(- x)) is Element of the carrier of S
(x + (- x)) + b is Element of the carrier of S
the addF of S . ((x + (- x)),b) is Element of the carrier of S
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
(0. S) + b is Element of the carrier of S
the addF of S . ((0. S),b) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
b + x is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (b,x) is Element of the carrier of S
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
- (1_ F) is Element of the carrier of F
(- (1_ F)) * x is Element of the carrier of S
- x is Element of the carrier of S
(b + x) + (- x) is Element of the carrier of S
the addF of S . ((b + x),(- x)) is Element of the carrier of S
x + (- x) is Element of the carrier of S
the addF of S . (x,(- x)) is Element of the carrier of S
b + (x + (- x)) is Element of the carrier of S
the addF of S . (b,(x + (- x))) is Element of the carrier of S
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
b + (0. S) is Element of the carrier of S
the addF of S . (b,(0. S)) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the carrier of F is non empty non trivial set
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of F
x * b is Element of the carrier of S
x * a is Element of the carrier of S
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
z is Element of the carrier of F
z * x is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the multF of F . (z,x) is Element of the carrier of F
z * (x * a) is Element of the carrier of S
(1. F) * a is Element of the carrier of S
z is Element of the carrier of F
z * x is Element of the carrier of F
the multF of F . (z,x) is Element of the carrier of F
z * (x * b) is Element of the carrier of S
(1. F) * b is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
- b is Element of the carrier of S
a is Element of the carrier of S
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
- (1_ F) is Element of the carrier of F
(- (1_ F)) * b is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
((1_ F) + (1_ F)) * b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
b + x is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (b,x) is Element of the carrier of S
(((1_ F) + (1_ F)) * b) + x is Element of the carrier of S
the addF of S . ((((1_ F) + (1_ F)) * b),x) is Element of the carrier of S
(1_ F) * b is Element of the carrier of S
((1_ F) * b) + ((1_ F) * b) is Element of the carrier of S
the addF of S . (((1_ F) * b),((1_ F) * b)) is Element of the carrier of S
(((1_ F) * b) + ((1_ F) * b)) + x is Element of the carrier of S
the addF of S . ((((1_ F) * b) + ((1_ F) * b)),x) is Element of the carrier of S
b + ((1_ F) * b) is Element of the carrier of S
the addF of S . (b,((1_ F) * b)) is Element of the carrier of S
(b + ((1_ F) * b)) + x is Element of the carrier of S
the addF of S . ((b + ((1_ F) * b)),x) is Element of the carrier of S
b + b is Element of the carrier of S
the addF of S . (b,b) is Element of the carrier of S
(b + b) + x is Element of the carrier of S
the addF of S . ((b + b),x) is Element of the carrier of S
b + (b + x) is Element of the carrier of S
the addF of S . (b,(b + x)) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
y is Element of the carrier of S
b + y is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (b,y) is Element of the carrier of S
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
- (1_ F) is Element of the carrier of F
(- (1_ F)) * y is Element of the carrier of S
(- (1_ F)) * b is Element of the carrier of S
- y is Element of the carrier of S
- b is Element of the carrier of S
(b + y) + (- y) is Element of the carrier of S
the addF of S . ((b + y),(- y)) is Element of the carrier of S
(- b) + (b + y) is Element of the carrier of S
the addF of S . ((- b),(b + y)) is Element of the carrier of S
y + (- y) is Element of the carrier of S
the addF of S . (y,(- y)) is Element of the carrier of S
b + (y + (- y)) is Element of the carrier of S
the addF of S . (b,(y + (- y))) is Element of the carrier of S
(- b) + b is Element of the carrier of S
the addF of S . ((- b),b) is Element of the carrier of S
((- b) + b) + y is Element of the carrier of S
the addF of S . (((- b) + b),y) is Element of the carrier of S
b + (0. S) is Element of the carrier of S
the addF of S . (b,(0. S)) is Element of the carrier of S
(0. S) + y is Element of the carrier of S
the addF of S . ((0. S),y) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
y is Element of the carrier of S
x + y is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,y) is Element of the carrier of S
z is Element of the carrier of S
y is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
y is Element of the carrier of S
z is Element of the carrier of S
((1_ F) + (1_ F)) * y is Element of the carrier of S
(((1_ F) + (1_ F)) * y) + z is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . ((((1_ F) + (1_ F)) * y),z) is Element of the carrier of S
y + z is Element of the carrier of S
the addF of S . (y,z) is Element of the carrier of S
0F is Element of the carrier of S
p is Element of the carrier of S
0F is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
b - a is Element of the carrier of S
- a is Element of the carrier of S
b + (- a) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (b,(- a)) is Element of the carrier of S
x is Element of the carrier of S
y is Element of the carrier of S
b - y is Element of the carrier of S
- y is Element of the carrier of S
b + (- y) is Element of the carrier of S
the addF of S . (b,(- y)) is Element of the carrier of S
a - y is Element of the carrier of S
a + (- y) is Element of the carrier of S
the addF of S . (a,(- y)) is Element of the carrier of S
- (b - a) is Element of the carrier of S
- b is Element of the carrier of S
(- b) + a is Element of the carrier of S
the addF of S . ((- b),a) is Element of the carrier of S
a + (- b) is Element of the carrier of S
the addF of S . (a,(- b)) is Element of the carrier of S
(a + (- b)) + (b - y) is Element of the carrier of S
the addF of S . ((a + (- b)),(b - y)) is Element of the carrier of S
(- b) + (b - y) is Element of the carrier of S
the addF of S . ((- b),(b - y)) is Element of the carrier of S
a + ((- b) + (b - y)) is Element of the carrier of S
the addF of S . (a,((- b) + (b - y))) is Element of the carrier of S
(- b) + b is Element of the carrier of S
the addF of S . ((- b),b) is Element of the carrier of S
((- b) + b) + (- y) is Element of the carrier of S
the addF of S . (((- b) + b),(- y)) is Element of the carrier of S
a + (((- b) + b) + (- y)) is Element of the carrier of S
the addF of S . (a,(((- b) + b) + (- y))) is Element of the carrier of S
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
(0. S) + (- y) is Element of the carrier of S
the addF of S . ((0. S),(- y)) is Element of the carrier of S
a + ((0. S) + (- y)) is Element of the carrier of S
the addF of S . (a,((0. S) + (- y))) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the carrier of F is non empty non trivial set
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
y is Element of the carrier of F
y * b is Element of the carrier of S
x - (y * b) is Element of the carrier of S
- (y * b) is Element of the carrier of S
x + (- (y * b)) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,(- (y * b))) is Element of the carrier of S
z is Element of the carrier of F
z * b is Element of the carrier of S
x - (z * b) is Element of the carrier of S
- (z * b) is Element of the carrier of S
x + (- (z * b)) is Element of the carrier of S
the addF of S . (x,(- (z * b))) is Element of the carrier of S
1_ F is Element of the carrier of F
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(y * b) - (z * b) is Element of the carrier of S
(y * b) + (- (z * b)) is Element of the carrier of S
the addF of S . ((y * b),(- (z * b))) is Element of the carrier of S
- (1_ F) is Element of the carrier of F
(- (1_ F)) * (z * b) is Element of the carrier of S
(y * b) + ((- (1_ F)) * (z * b)) is Element of the carrier of S
the addF of S . ((y * b),((- (1_ F)) * (z * b))) is Element of the carrier of S
(- (1_ F)) * z is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the multF of F . ((- (1_ F)),z) is Element of the carrier of F
((- (1_ F)) * z) * b is Element of the carrier of S
(y * b) + (((- (1_ F)) * z) * b) is Element of the carrier of S
the addF of S . ((y * b),(((- (1_ F)) * z) * b)) is Element of the carrier of S
z * (1_ F) is Element of the carrier of F
the multF of F . (z,(1_ F)) is Element of the carrier of F
- (z * (1_ F)) is Element of the carrier of F
(- (z * (1_ F))) * b is Element of the carrier of S
(y * b) + ((- (z * (1_ F))) * b) is Element of the carrier of S
the addF of S . ((y * b),((- (z * (1_ F))) * b)) is Element of the carrier of S
- z is Element of the carrier of F
(- z) * b is Element of the carrier of S
(y * b) + ((- z) * b) is Element of the carrier of S
the addF of S . ((y * b),((- z) * b)) is Element of the carrier of S
y + (- z) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the addF of F . (y,(- z)) is Element of the carrier of F
(y + (- z)) * b is Element of the carrier of S
(y + (- z)) " is Element of the carrier of F
((y + (- z)) ") * ((y + (- z)) * b) is Element of the carrier of S
((y + (- z)) ") * (y + (- z)) is Element of the carrier of F
the multF of F . (((y + (- z)) "),(y + (- z))) is Element of the carrier of F
(((y + (- z)) ") * (y + (- z))) * b is Element of the carrier of S
y - z is Element of the carrier of F
y + (- z) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
(1_ F) * b is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
x is Element of the carrier of S
y is Element of the carrier of F
y * x is Element of the carrier of S
b - (y * x) is Element of the carrier of S
- (y * x) is Element of the carrier of S
b + (- (y * x)) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (b,(- (y * x))) is Element of the carrier of S
b + (- (y * x)) is Element of the carrier of S
b + b is Element of the carrier of S
the addF of S . (b,b) is Element of the carrier of S
(1_ F) * b is Element of the carrier of S
b + ((1_ F) * b) is Element of the carrier of S
the addF of S . (b,((1_ F) * b)) is Element of the carrier of S
((1_ F) * b) + ((1_ F) * b) is Element of the carrier of S
the addF of S . (((1_ F) * b),((1_ F) * b)) is Element of the carrier of S
- (1_ F) is Element of the carrier of F
(- (1_ F)) * (y * x) is Element of the carrier of S
((1_ F) + (1_ F)) * b is Element of the carrier of S
(- (1_ F)) * y is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . ((- (1_ F)),y) is Element of the carrier of F
((- (1_ F)) * y) * x is Element of the carrier of S
y * (1_ F) is Element of the carrier of F
the multF of F . (y,(1_ F)) is Element of the carrier of F
- (y * (1_ F)) is Element of the carrier of F
(- (y * (1_ F))) * x is Element of the carrier of S
- y is Element of the carrier of F
(- y) * x is Element of the carrier of S
((1_ F) + (1_ F)) " is Element of the carrier of F
(((1_ F) + (1_ F)) ") * (((1_ F) + (1_ F)) * b) is Element of the carrier of S
b + ((- y) * x) is Element of the carrier of S
the addF of S . (b,((- y) * x)) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
a is Element of the carrier of S
b is Element of the carrier of S
the carrier of F is non empty non trivial set
x is Element of the carrier of S
y is Element of the carrier of F
y * a is Element of the carrier of S
x - (y * a) is Element of the carrier of S
- (y * a) is Element of the carrier of S
x + (- (y * a)) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,(- (y * a))) is Element of the carrier of S
z is Element of the carrier of F
z * a is Element of the carrier of S
x - (z * a) is Element of the carrier of S
- (z * a) is Element of the carrier of S
x + (- (z * a)) is Element of the carrier of S
the addF of S . (x,(- (z * a))) is Element of the carrier of S
y is Element of the carrier of F
z is Element of the carrier of F
0F is Element of the carrier of F
0F * a is Element of the carrier of S
x - (0F * a) is Element of the carrier of S
- (0F * a) is Element of the carrier of S
x + (- (0F * a)) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,(- (0F * a))) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,a,b,x) is Element of the carrier of F
the carrier of F is non empty non trivial set
(F,S,a,b,x) * b is Element of the carrier of S
x - ((F,S,a,b,x) * b) is Element of the carrier of S
- ((F,S,a,b,x) * b) is Element of the carrier of S
x + (- ((F,S,a,b,x) * b)) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,(- ((F,S,a,b,x) * b))) is Element of the carrier of S
y is Element of the carrier of F
y * b is Element of the carrier of S
x - (y * b) is Element of the carrier of S
- (y * b) is Element of the carrier of S
x + (- (y * b)) is Element of the carrier of S
the addF of S . (x,(- (y * b))) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the carrier of F is non empty non trivial set
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,a,b,x) is Element of the carrier of F
y is Element of the carrier of F
y * x is Element of the carrier of S
(F,S,a,b,(y * x)) is Element of the carrier of F
y * (F,S,a,b,x) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the multF of F . (y,(F,S,a,b,x)) is Element of the carrier of F
(F,S,a,b,x) * b is Element of the carrier of S
x - ((F,S,a,b,x) * b) is Element of the carrier of S
- ((F,S,a,b,x) * b) is Element of the carrier of S
x + (- ((F,S,a,b,x) * b)) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,(- ((F,S,a,b,x) * b))) is Element of the carrier of S
y * (x - ((F,S,a,b,x) * b)) is Element of the carrier of S
y * ((F,S,a,b,x) * b) is Element of the carrier of S
(y * x) - (y * ((F,S,a,b,x) * b)) is Element of the carrier of S
- (y * ((F,S,a,b,x) * b)) is Element of the carrier of S
(y * x) + (- (y * ((F,S,a,b,x) * b))) is Element of the carrier of S
the addF of S . ((y * x),(- (y * ((F,S,a,b,x) * b)))) is Element of the carrier of S
(y * (F,S,a,b,x)) * b is Element of the carrier of S
(y * x) - ((y * (F,S,a,b,x)) * b) is Element of the carrier of S
- ((y * (F,S,a,b,x)) * b) is Element of the carrier of S
(y * x) + (- ((y * (F,S,a,b,x)) * b)) is Element of the carrier of S
the addF of S . ((y * x),(- ((y * (F,S,a,b,x)) * b))) is Element of the carrier of S
(F,S,a,b,(y * x)) * b is Element of the carrier of S
(y * x) - ((F,S,a,b,(y * x)) * b) is Element of the carrier of S
- ((F,S,a,b,(y * x)) * b) is Element of the carrier of S
(y * x) + (- ((F,S,a,b,(y * x)) * b)) is Element of the carrier of S
the addF of S . ((y * x),(- ((F,S,a,b,(y * x)) * b))) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,a,b,x) is Element of the carrier of F
the carrier of F is non empty non trivial set
y is Element of the carrier of S
x + y is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,y) is Element of the carrier of S
(F,S,a,b,(x + y)) is Element of the carrier of F
(F,S,a,b,y) is Element of the carrier of F
(F,S,a,b,x) + (F,S,a,b,y) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((F,S,a,b,x),(F,S,a,b,y)) is Element of the carrier of F
1_ F is Element of the carrier of F
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(F,S,a,b,x) * b is Element of the carrier of S
x - ((F,S,a,b,x) * b) is Element of the carrier of S
- ((F,S,a,b,x) * b) is Element of the carrier of S
x + (- ((F,S,a,b,x) * b)) is Element of the carrier of S
the addF of S . (x,(- ((F,S,a,b,x) * b))) is Element of the carrier of S
(F,S,a,b,y) * b is Element of the carrier of S
y - ((F,S,a,b,y) * b) is Element of the carrier of S
- ((F,S,a,b,y) * b) is Element of the carrier of S
y + (- ((F,S,a,b,y) * b)) is Element of the carrier of S
the addF of S . (y,(- ((F,S,a,b,y) * b))) is Element of the carrier of S
(x - ((F,S,a,b,x) * b)) + (y - ((F,S,a,b,y) * b)) is Element of the carrier of S
the addF of S . ((x - ((F,S,a,b,x) * b)),(y - ((F,S,a,b,y) * b))) is Element of the carrier of S
(- ((F,S,a,b,x) * b)) + x is Element of the carrier of S
the addF of S . ((- ((F,S,a,b,x) * b)),x) is Element of the carrier of S
((- ((F,S,a,b,x) * b)) + x) + y is Element of the carrier of S
the addF of S . (((- ((F,S,a,b,x) * b)) + x),y) is Element of the carrier of S
(((- ((F,S,a,b,x) * b)) + x) + y) + (- ((F,S,a,b,y) * b)) is Element of the carrier of S
the addF of S . ((((- ((F,S,a,b,x) * b)) + x) + y),(- ((F,S,a,b,y) * b))) is Element of the carrier of S
(x + y) + (- ((F,S,a,b,x) * b)) is Element of the carrier of S
the addF of S . ((x + y),(- ((F,S,a,b,x) * b))) is Element of the carrier of S
((x + y) + (- ((F,S,a,b,x) * b))) + (- ((F,S,a,b,y) * b)) is Element of the carrier of S
the addF of S . (((x + y) + (- ((F,S,a,b,x) * b))),(- ((F,S,a,b,y) * b))) is Element of the carrier of S
(- ((F,S,a,b,x) * b)) + (- ((F,S,a,b,y) * b)) is Element of the carrier of S
the addF of S . ((- ((F,S,a,b,x) * b)),(- ((F,S,a,b,y) * b))) is Element of the carrier of S
(x + y) + ((- ((F,S,a,b,x) * b)) + (- ((F,S,a,b,y) * b))) is Element of the carrier of S
the addF of S . ((x + y),((- ((F,S,a,b,x) * b)) + (- ((F,S,a,b,y) * b)))) is Element of the carrier of S
(1_ F) * (- ((F,S,a,b,x) * b)) is Element of the carrier of S
((1_ F) * (- ((F,S,a,b,x) * b))) + (- ((F,S,a,b,y) * b)) is Element of the carrier of S
the addF of S . (((1_ F) * (- ((F,S,a,b,x) * b))),(- ((F,S,a,b,y) * b))) is Element of the carrier of S
(x + y) + (((1_ F) * (- ((F,S,a,b,x) * b))) + (- ((F,S,a,b,y) * b))) is Element of the carrier of S
the addF of S . ((x + y),(((1_ F) * (- ((F,S,a,b,x) * b))) + (- ((F,S,a,b,y) * b)))) is Element of the carrier of S
(1_ F) * (- ((F,S,a,b,y) * b)) is Element of the carrier of S
((1_ F) * (- ((F,S,a,b,x) * b))) + ((1_ F) * (- ((F,S,a,b,y) * b))) is Element of the carrier of S
the addF of S . (((1_ F) * (- ((F,S,a,b,x) * b))),((1_ F) * (- ((F,S,a,b,y) * b)))) is Element of the carrier of S
(x + y) + (((1_ F) * (- ((F,S,a,b,x) * b))) + ((1_ F) * (- ((F,S,a,b,y) * b)))) is Element of the carrier of S
the addF of S . ((x + y),(((1_ F) * (- ((F,S,a,b,x) * b))) + ((1_ F) * (- ((F,S,a,b,y) * b))))) is Element of the carrier of S
- (F,S,a,b,x) is Element of the carrier of F
(- (F,S,a,b,x)) * b is Element of the carrier of S
(1_ F) * ((- (F,S,a,b,x)) * b) is Element of the carrier of S
((1_ F) * ((- (F,S,a,b,x)) * b)) + ((1_ F) * (- ((F,S,a,b,y) * b))) is Element of the carrier of S
the addF of S . (((1_ F) * ((- (F,S,a,b,x)) * b)),((1_ F) * (- ((F,S,a,b,y) * b)))) is Element of the carrier of S
(x + y) + (((1_ F) * ((- (F,S,a,b,x)) * b)) + ((1_ F) * (- ((F,S,a,b,y) * b)))) is Element of the carrier of S
the addF of S . ((x + y),(((1_ F) * ((- (F,S,a,b,x)) * b)) + ((1_ F) * (- ((F,S,a,b,y) * b))))) is Element of the carrier of S
- (F,S,a,b,y) is Element of the carrier of F
(- (F,S,a,b,y)) * b is Element of the carrier of S
(1_ F) * ((- (F,S,a,b,y)) * b) is Element of the carrier of S
((1_ F) * ((- (F,S,a,b,x)) * b)) + ((1_ F) * ((- (F,S,a,b,y)) * b)) is Element of the carrier of S
the addF of S . (((1_ F) * ((- (F,S,a,b,x)) * b)),((1_ F) * ((- (F,S,a,b,y)) * b))) is Element of the carrier of S
(x + y) + (((1_ F) * ((- (F,S,a,b,x)) * b)) + ((1_ F) * ((- (F,S,a,b,y)) * b))) is Element of the carrier of S
the addF of S . ((x + y),(((1_ F) * ((- (F,S,a,b,x)) * b)) + ((1_ F) * ((- (F,S,a,b,y)) * b)))) is Element of the carrier of S
(1_ F) * (- (F,S,a,b,x)) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . ((1_ F),(- (F,S,a,b,x))) is Element of the carrier of F
((1_ F) * (- (F,S,a,b,x))) * b is Element of the carrier of S
(((1_ F) * (- (F,S,a,b,x))) * b) + ((1_ F) * ((- (F,S,a,b,y)) * b)) is Element of the carrier of S
the addF of S . ((((1_ F) * (- (F,S,a,b,x))) * b),((1_ F) * ((- (F,S,a,b,y)) * b))) is Element of the carrier of S
(x + y) + ((((1_ F) * (- (F,S,a,b,x))) * b) + ((1_ F) * ((- (F,S,a,b,y)) * b))) is Element of the carrier of S
the addF of S . ((x + y),((((1_ F) * (- (F,S,a,b,x))) * b) + ((1_ F) * ((- (F,S,a,b,y)) * b)))) is Element of the carrier of S
(- (F,S,a,b,x)) * (1_ F) is Element of the carrier of F
the multF of F . ((- (F,S,a,b,x)),(1_ F)) is Element of the carrier of F
((- (F,S,a,b,x)) * (1_ F)) * b is Element of the carrier of S
(1_ F) * (- (F,S,a,b,y)) is Element of the carrier of F
the multF of F . ((1_ F),(- (F,S,a,b,y))) is Element of the carrier of F
((1_ F) * (- (F,S,a,b,y))) * b is Element of the carrier of S
(((- (F,S,a,b,x)) * (1_ F)) * b) + (((1_ F) * (- (F,S,a,b,y))) * b) is Element of the carrier of S
the addF of S . ((((- (F,S,a,b,x)) * (1_ F)) * b),(((1_ F) * (- (F,S,a,b,y))) * b)) is Element of the carrier of S
(x + y) + ((((- (F,S,a,b,x)) * (1_ F)) * b) + (((1_ F) * (- (F,S,a,b,y))) * b)) is Element of the carrier of S
the addF of S . ((x + y),((((- (F,S,a,b,x)) * (1_ F)) * b) + (((1_ F) * (- (F,S,a,b,y))) * b))) is Element of the carrier of S
(- (F,S,a,b,y)) * (1_ F) is Element of the carrier of F
the multF of F . ((- (F,S,a,b,y)),(1_ F)) is Element of the carrier of F
((- (F,S,a,b,y)) * (1_ F)) * b is Element of the carrier of S
((- (F,S,a,b,x)) * b) + (((- (F,S,a,b,y)) * (1_ F)) * b) is Element of the carrier of S
the addF of S . (((- (F,S,a,b,x)) * b),(((- (F,S,a,b,y)) * (1_ F)) * b)) is Element of the carrier of S
(x + y) + (((- (F,S,a,b,x)) * b) + (((- (F,S,a,b,y)) * (1_ F)) * b)) is Element of the carrier of S
the addF of S . ((x + y),(((- (F,S,a,b,x)) * b) + (((- (F,S,a,b,y)) * (1_ F)) * b))) is Element of the carrier of S
((- (F,S,a,b,x)) * b) + ((- (F,S,a,b,y)) * b) is Element of the carrier of S
the addF of S . (((- (F,S,a,b,x)) * b),((- (F,S,a,b,y)) * b)) is Element of the carrier of S
(x + y) + (((- (F,S,a,b,x)) * b) + ((- (F,S,a,b,y)) * b)) is Element of the carrier of S
the addF of S . ((x + y),(((- (F,S,a,b,x)) * b) + ((- (F,S,a,b,y)) * b))) is Element of the carrier of S
(- (F,S,a,b,x)) + (- (F,S,a,b,y)) is Element of the carrier of F
the addF of F . ((- (F,S,a,b,x)),(- (F,S,a,b,y))) is Element of the carrier of F
((- (F,S,a,b,x)) + (- (F,S,a,b,y))) * b is Element of the carrier of S
(x + y) + (((- (F,S,a,b,x)) + (- (F,S,a,b,y))) * b) is Element of the carrier of S
the addF of S . ((x + y),(((- (F,S,a,b,x)) + (- (F,S,a,b,y))) * b)) is Element of the carrier of S
- ((F,S,a,b,x) + (F,S,a,b,y)) is Element of the carrier of F
(- ((F,S,a,b,x) + (F,S,a,b,y))) * b is Element of the carrier of S
(x + y) + ((- ((F,S,a,b,x) + (F,S,a,b,y))) * b) is Element of the carrier of S
the addF of S . ((x + y),((- ((F,S,a,b,x) + (F,S,a,b,y))) * b)) is Element of the carrier of S
((F,S,a,b,x) + (F,S,a,b,y)) * b is Element of the carrier of S
(x + y) - (((F,S,a,b,x) + (F,S,a,b,y)) * b) is Element of the carrier of S
- (((F,S,a,b,x) + (F,S,a,b,y)) * b) is Element of the carrier of S
(x + y) + (- (((F,S,a,b,x) + (F,S,a,b,y)) * b)) is Element of the carrier of S
the addF of S . ((x + y),(- (((F,S,a,b,x) + (F,S,a,b,y)) * b))) is Element of the carrier of S
(F,S,a,b,(x + y)) * b is Element of the carrier of S
(x + y) - ((F,S,a,b,(x + y)) * b) is Element of the carrier of S
- ((F,S,a,b,(x + y)) * b) is Element of the carrier of S
(x + y) + (- ((F,S,a,b,(x + y)) * b)) is Element of the carrier of S
the addF of S . ((x + y),(- ((F,S,a,b,(x + y)) * b))) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the carrier of F is non empty non trivial set
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,a,b,x) is Element of the carrier of F
y is Element of the carrier of F
y * b is Element of the carrier of S
(F,S,a,(y * b),x) is Element of the carrier of F
y " is Element of the carrier of F
(y ") * (F,S,a,b,x) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the multF of F . ((y "),(F,S,a,b,x)) is Element of the carrier of F
(F,S,a,(y * b),x) * (y * b) is Element of the carrier of S
x - ((F,S,a,(y * b),x) * (y * b)) is Element of the carrier of S
- ((F,S,a,(y * b),x) * (y * b)) is Element of the carrier of S
x + (- ((F,S,a,(y * b),x) * (y * b))) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,(- ((F,S,a,(y * b),x) * (y * b)))) is Element of the carrier of S
(F,S,a,(y * b),x) * y is Element of the carrier of F
the multF of F . ((F,S,a,(y * b),x),y) is Element of the carrier of F
((F,S,a,(y * b),x) * y) * b is Element of the carrier of S
x - (((F,S,a,(y * b),x) * y) * b) is Element of the carrier of S
- (((F,S,a,(y * b),x) * y) * b) is Element of the carrier of S
x + (- (((F,S,a,(y * b),x) * y) * b)) is Element of the carrier of S
the addF of S . (x,(- (((F,S,a,(y * b),x) * y) * b))) is Element of the carrier of S
(F,S,a,b,x) * b is Element of the carrier of S
x - ((F,S,a,b,x) * b) is Element of the carrier of S
- ((F,S,a,b,x) * b) is Element of the carrier of S
x + (- ((F,S,a,b,x) * b)) is Element of the carrier of S
the addF of S . (x,(- ((F,S,a,b,x) * b))) is Element of the carrier of S
(F,S,a,b,x) * (y ") is Element of the carrier of F
the multF of F . ((F,S,a,b,x),(y ")) is Element of the carrier of F
((F,S,a,(y * b),x) * y) * (y ") is Element of the carrier of F
the multF of F . (((F,S,a,(y * b),x) * y),(y ")) is Element of the carrier of F
y * (y ") is Element of the carrier of F
the multF of F . (y,(y ")) is Element of the carrier of F
(F,S,a,(y * b),x) * (y * (y ")) is Element of the carrier of F
the multF of F . ((F,S,a,(y * b),x),(y * (y "))) is Element of the carrier of F
1_ F is Element of the carrier of F
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(F,S,a,(y * b),x) * (1_ F) is Element of the carrier of F
the multF of F . ((F,S,a,(y * b),x),(1_ F)) is Element of the carrier of F
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the carrier of F is non empty non trivial set
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,a,b,x) is Element of the carrier of F
y is Element of the carrier of F
y * a is Element of the carrier of S
(F,S,(y * a),b,x) is Element of the carrier of F
(F,S,(y * a),b,x) * b is Element of the carrier of S
x - ((F,S,(y * a),b,x) * b) is Element of the carrier of S
- ((F,S,(y * a),b,x) * b) is Element of the carrier of S
x + (- ((F,S,(y * a),b,x) * b)) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,(- ((F,S,(y * a),b,x) * b))) is Element of the carrier of S
y " is Element of the carrier of F
(y ") * (y * a) is Element of the carrier of S
(y ") * y is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the multF of F . ((y "),y) is Element of the carrier of F
((y ") * y) * a is Element of the carrier of S
1_ F is Element of the carrier of F
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) * a is Element of the carrier of S
(F,S,a,b,x) * b is Element of the carrier of S
x - ((F,S,a,b,x) * b) is Element of the carrier of S
- ((F,S,a,b,x) * b) is Element of the carrier of S
x + (- ((F,S,a,b,x) * b)) is Element of the carrier of S
the addF of S . (x,(- ((F,S,a,b,x) * b))) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
b + x is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (b,x) is Element of the carrier of S
y is Element of the carrier of S
(F,S,a,(b + x),y) is Element of the carrier of F
the carrier of F is non empty non trivial set
(F,S,a,b,y) is Element of the carrier of F
y + x is Element of the carrier of S
the addF of S . (y,x) is Element of the carrier of S
(F,S,a,b,(y + x)) is Element of the carrier of F
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
(F,S,a,(b + x),y) * (b + x) is Element of the carrier of S
y - ((F,S,a,(b + x),y) * (b + x)) is Element of the carrier of S
- ((F,S,a,(b + x),y) * (b + x)) is Element of the carrier of S
y + (- ((F,S,a,(b + x),y) * (b + x))) is Element of the carrier of S
the addF of S . (y,(- ((F,S,a,(b + x),y) * (b + x)))) is Element of the carrier of S
(F,S,a,(b + x),y) * b is Element of the carrier of S
(F,S,a,(b + x),y) * x is Element of the carrier of S
((F,S,a,(b + x),y) * b) + ((F,S,a,(b + x),y) * x) is Element of the carrier of S
the addF of S . (((F,S,a,(b + x),y) * b),((F,S,a,(b + x),y) * x)) is Element of the carrier of S
y - (((F,S,a,(b + x),y) * b) + ((F,S,a,(b + x),y) * x)) is Element of the carrier of S
- (((F,S,a,(b + x),y) * b) + ((F,S,a,(b + x),y) * x)) is Element of the carrier of S
y + (- (((F,S,a,(b + x),y) * b) + ((F,S,a,(b + x),y) * x))) is Element of the carrier of S
the addF of S . (y,(- (((F,S,a,(b + x),y) * b) + ((F,S,a,(b + x),y) * x)))) is Element of the carrier of S
y - ((F,S,a,(b + x),y) * b) is Element of the carrier of S
- ((F,S,a,(b + x),y) * b) is Element of the carrier of S
y + (- ((F,S,a,(b + x),y) * b)) is Element of the carrier of S
the addF of S . (y,(- ((F,S,a,(b + x),y) * b))) is Element of the carrier of S
(y - ((F,S,a,(b + x),y) * b)) - ((F,S,a,(b + x),y) * x) is Element of the carrier of S
- ((F,S,a,(b + x),y) * x) is Element of the carrier of S
(y - ((F,S,a,(b + x),y) * b)) + (- ((F,S,a,(b + x),y) * x)) is Element of the carrier of S
the addF of S . ((y - ((F,S,a,(b + x),y) * b)),(- ((F,S,a,(b + x),y) * x))) is Element of the carrier of S
(F,S,a,b,(y + x)) * b is Element of the carrier of S
(y + x) - ((F,S,a,b,(y + x)) * b) is Element of the carrier of S
- ((F,S,a,b,(y + x)) * b) is Element of the carrier of S
(y + x) + (- ((F,S,a,b,(y + x)) * b)) is Element of the carrier of S
the addF of S . ((y + x),(- ((F,S,a,b,(y + x)) * b))) is Element of the carrier of S
- x is Element of the carrier of S
x + y is Element of the carrier of S
the addF of S . (x,y) is Element of the carrier of S
(x + y) - ((F,S,a,b,(y + x)) * b) is Element of the carrier of S
(x + y) + (- ((F,S,a,b,(y + x)) * b)) is Element of the carrier of S
the addF of S . ((x + y),(- ((F,S,a,b,(y + x)) * b))) is Element of the carrier of S
(- x) + ((x + y) - ((F,S,a,b,(y + x)) * b)) is Element of the carrier of S
the addF of S . ((- x),((x + y) - ((F,S,a,b,(y + x)) * b))) is Element of the carrier of S
y + (- ((F,S,a,b,(y + x)) * b)) is Element of the carrier of S
the addF of S . (y,(- ((F,S,a,b,(y + x)) * b))) is Element of the carrier of S
x + (y + (- ((F,S,a,b,(y + x)) * b))) is Element of the carrier of S
the addF of S . (x,(y + (- ((F,S,a,b,(y + x)) * b)))) is Element of the carrier of S
(- x) + (x + (y + (- ((F,S,a,b,(y + x)) * b)))) is Element of the carrier of S
the addF of S . ((- x),(x + (y + (- ((F,S,a,b,(y + x)) * b))))) is Element of the carrier of S
(- x) + x is Element of the carrier of S
the addF of S . ((- x),x) is Element of the carrier of S
((- x) + x) + (y + (- ((F,S,a,b,(y + x)) * b))) is Element of the carrier of S
the addF of S . (((- x) + x),(y + (- ((F,S,a,b,(y + x)) * b)))) is Element of the carrier of S
(0. S) + (y + (- ((F,S,a,b,(y + x)) * b))) is Element of the carrier of S
the addF of S . ((0. S),(y + (- ((F,S,a,b,(y + x)) * b)))) is Element of the carrier of S
y - ((F,S,a,b,(y + x)) * b) is Element of the carrier of S
y + (- ((F,S,a,b,(y + x)) * b)) is Element of the carrier of S
y + (- ((F,S,a,(b + x),y) * b)) is Element of the carrier of S
(y + (- ((F,S,a,(b + x),y) * b))) - ((F,S,a,(b + x),y) * x) is Element of the carrier of S
(y + (- ((F,S,a,(b + x),y) * b))) + (- ((F,S,a,(b + x),y) * x)) is Element of the carrier of S
the addF of S . ((y + (- ((F,S,a,(b + x),y) * b))),(- ((F,S,a,(b + x),y) * x))) is Element of the carrier of S
((y + (- ((F,S,a,(b + x),y) * b))) - ((F,S,a,(b + x),y) * x)) + ((F,S,a,(b + x),y) * x) is Element of the carrier of S
the addF of S . (((y + (- ((F,S,a,(b + x),y) * b))) - ((F,S,a,(b + x),y) * x)),((F,S,a,(b + x),y) * x)) is Element of the carrier of S
(- ((F,S,a,(b + x),y) * x)) + ((F,S,a,(b + x),y) * x) is Element of the carrier of S
the addF of S . ((- ((F,S,a,(b + x),y) * x)),((F,S,a,(b + x),y) * x)) is Element of the carrier of S
(y + (- ((F,S,a,(b + x),y) * b))) + ((- ((F,S,a,(b + x),y) * x)) + ((F,S,a,(b + x),y) * x)) is Element of the carrier of S
the addF of S . ((y + (- ((F,S,a,(b + x),y) * b))),((- ((F,S,a,(b + x),y) * x)) + ((F,S,a,(b + x),y) * x))) is Element of the carrier of S
(y + (- ((F,S,a,(b + x),y) * b))) + (0. S) is Element of the carrier of S
the addF of S . ((y + (- ((F,S,a,(b + x),y) * b))),(0. S)) is Element of the carrier of S
(F,S,a,b,y) * b is Element of the carrier of S
y - ((F,S,a,b,y) * b) is Element of the carrier of S
- ((F,S,a,b,y) * b) is Element of the carrier of S
y + (- ((F,S,a,b,y) * b)) is Element of the carrier of S
the addF of S . (y,(- ((F,S,a,b,y) * b))) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
a + x is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (a,x) is Element of the carrier of S
y is Element of the carrier of S
(F,S,(a + x),b,y) is Element of the carrier of F
the carrier of F is non empty non trivial set
(F,S,a,b,y) is Element of the carrier of F
(F,S,a,b,y) * b is Element of the carrier of S
- ((F,S,a,b,y) * b) is Element of the carrier of S
y + (- ((F,S,a,b,y) * b)) is Element of the carrier of S
the addF of S . (y,(- ((F,S,a,b,y) * b))) is Element of the carrier of S
y - ((F,S,a,b,y) * b) is Element of the carrier of S
y + (- ((F,S,a,b,y) * b)) is Element of the carrier of S
(F,S,(a + x),b,y) * b is Element of the carrier of S
y - ((F,S,(a + x),b,y) * b) is Element of the carrier of S
- ((F,S,(a + x),b,y) * b) is Element of the carrier of S
y + (- ((F,S,(a + x),b,y) * b)) is Element of the carrier of S
the addF of S . (y,(- ((F,S,(a + x),b,y) * b))) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
x - b is Element of the carrier of S
- b is Element of the carrier of S
x + (- b) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,(- b)) is Element of the carrier of S
(F,S,a,b,x) is Element of the carrier of F
(1_ F) * b is Element of the carrier of S
x - ((1_ F) * b) is Element of the carrier of S
- ((1_ F) * b) is Element of the carrier of S
x + (- ((1_ F) * b)) is Element of the carrier of S
the addF of S . (x,(- ((1_ F) * b))) is Element of the carrier of S
(F,S,a,b,x) * b is Element of the carrier of S
x - ((F,S,a,b,x) * b) is Element of the carrier of S
- ((F,S,a,b,x) * b) is Element of the carrier of S
x + (- ((F,S,a,b,x) * b)) is Element of the carrier of S
the addF of S . (x,(- ((F,S,a,b,x) * b))) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
(F,S,a,b,b) is Element of the carrier of F
b - b is Element of the carrier of S
- b is Element of the carrier of S
b + (- b) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (b,(- b)) is Element of the carrier of S
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
0. F is V51(F) Element of the carrier of F
the carrier of F is non empty non trivial set
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,a,b,x) is Element of the carrier of F
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
x + (0. S) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,(0. S)) is Element of the carrier of S
- (0. S) is Element of the carrier of S
x + (- (0. S)) is Element of the carrier of S
the addF of S . (x,(- (0. S))) is Element of the carrier of S
(0. F) * b is Element of the carrier of S
x - ((0. F) * b) is Element of the carrier of S
- ((0. F) * b) is Element of the carrier of S
x + (- ((0. F) * b)) is Element of the carrier of S
the addF of S . (x,(- ((0. F) * b))) is Element of the carrier of S
(F,S,a,b,x) * b is Element of the carrier of S
x - ((F,S,a,b,x) * b) is Element of the carrier of S
- ((F,S,a,b,x) * b) is Element of the carrier of S
x + (- ((F,S,a,b,x) * b)) is Element of the carrier of S
the addF of S . (x,(- ((F,S,a,b,x) * b))) is Element of the carrier of S
x - (0. S) is Element of the carrier of S
x + (- (0. S)) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,a,b,x) is Element of the carrier of F
the carrier of F is non empty non trivial set
(F,S,a,b,x) " is Element of the carrier of F
y is Element of the carrier of S
(F,S,a,b,y) is Element of the carrier of F
(F,S,a,b,y) * ((F,S,a,b,x) ") is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the multF of F . ((F,S,a,b,y),((F,S,a,b,x) ")) is Element of the carrier of F
(F,S,a,x,y) is Element of the carrier of F
(F,S,a,x,y) * x is Element of the carrier of S
(F,S,a,b,((F,S,a,x,y) * x)) is Element of the carrier of F
(F,S,a,b,((F,S,a,x,y) * x)) * b is Element of the carrier of S
((F,S,a,x,y) * x) - ((F,S,a,b,((F,S,a,x,y) * x)) * b) is Element of the carrier of S
- ((F,S,a,b,((F,S,a,x,y) * x)) * b) is Element of the carrier of S
((F,S,a,x,y) * x) + (- ((F,S,a,b,((F,S,a,x,y) * x)) * b)) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (((F,S,a,x,y) * x),(- ((F,S,a,b,((F,S,a,x,y) * x)) * b))) is Element of the carrier of S
y - ((F,S,a,x,y) * x) is Element of the carrier of S
- ((F,S,a,x,y) * x) is Element of the carrier of S
y + (- ((F,S,a,x,y) * x)) is Element of the carrier of S
the addF of S . (y,(- ((F,S,a,x,y) * x))) is Element of the carrier of S
y + (- ((F,S,a,x,y) * x)) is Element of the carrier of S
(y + (- ((F,S,a,x,y) * x))) + (((F,S,a,x,y) * x) - ((F,S,a,b,((F,S,a,x,y) * x)) * b)) is Element of the carrier of S
the addF of S . ((y + (- ((F,S,a,x,y) * x))),(((F,S,a,x,y) * x) - ((F,S,a,b,((F,S,a,x,y) * x)) * b))) is Element of the carrier of S
(y + (- ((F,S,a,x,y) * x))) + ((F,S,a,x,y) * x) is Element of the carrier of S
the addF of S . ((y + (- ((F,S,a,x,y) * x))),((F,S,a,x,y) * x)) is Element of the carrier of S
((y + (- ((F,S,a,x,y) * x))) + ((F,S,a,x,y) * x)) + (- ((F,S,a,b,((F,S,a,x,y) * x)) * b)) is Element of the carrier of S
the addF of S . (((y + (- ((F,S,a,x,y) * x))) + ((F,S,a,x,y) * x)),(- ((F,S,a,b,((F,S,a,x,y) * x)) * b))) is Element of the carrier of S
(- ((F,S,a,x,y) * x)) + ((F,S,a,x,y) * x) is Element of the carrier of S
the addF of S . ((- ((F,S,a,x,y) * x)),((F,S,a,x,y) * x)) is Element of the carrier of S
y + ((- ((F,S,a,x,y) * x)) + ((F,S,a,x,y) * x)) is Element of the carrier of S
the addF of S . (y,((- ((F,S,a,x,y) * x)) + ((F,S,a,x,y) * x))) is Element of the carrier of S
(y + ((- ((F,S,a,x,y) * x)) + ((F,S,a,x,y) * x))) + (- ((F,S,a,b,((F,S,a,x,y) * x)) * b)) is Element of the carrier of S
the addF of S . ((y + ((- ((F,S,a,x,y) * x)) + ((F,S,a,x,y) * x))),(- ((F,S,a,b,((F,S,a,x,y) * x)) * b))) is Element of the carrier of S
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
y + (0. S) is Element of the carrier of S
the addF of S . (y,(0. S)) is Element of the carrier of S
(y + (0. S)) + (- ((F,S,a,b,((F,S,a,x,y) * x)) * b)) is Element of the carrier of S
the addF of S . ((y + (0. S)),(- ((F,S,a,b,((F,S,a,x,y) * x)) * b))) is Element of the carrier of S
y + (- ((F,S,a,b,((F,S,a,x,y) * x)) * b)) is Element of the carrier of S
the addF of S . (y,(- ((F,S,a,b,((F,S,a,x,y) * x)) * b))) is Element of the carrier of S
(F,S,a,x,y) * (F,S,a,b,x) is Element of the carrier of F
the multF of F . ((F,S,a,x,y),(F,S,a,b,x)) is Element of the carrier of F
((F,S,a,x,y) * (F,S,a,b,x)) * b is Element of the carrier of S
y - (((F,S,a,x,y) * (F,S,a,b,x)) * b) is Element of the carrier of S
- (((F,S,a,x,y) * (F,S,a,b,x)) * b) is Element of the carrier of S
y + (- (((F,S,a,x,y) * (F,S,a,b,x)) * b)) is Element of the carrier of S
the addF of S . (y,(- (((F,S,a,x,y) * (F,S,a,b,x)) * b))) is Element of the carrier of S
(F,S,a,b,y) * b is Element of the carrier of S
y - ((F,S,a,b,y) * b) is Element of the carrier of S
- ((F,S,a,b,y) * b) is Element of the carrier of S
y + (- ((F,S,a,b,y) * b)) is Element of the carrier of S
the addF of S . (y,(- ((F,S,a,b,y) * b))) is Element of the carrier of S
(F,S,a,b,x) * ((F,S,a,b,x) ") is Element of the carrier of F
the multF of F . ((F,S,a,b,x),((F,S,a,b,x) ")) is Element of the carrier of F
(F,S,a,x,y) * ((F,S,a,b,x) * ((F,S,a,b,x) ")) is Element of the carrier of F
the multF of F . ((F,S,a,x,y),((F,S,a,b,x) * ((F,S,a,b,x) "))) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
1_ F is Element of the carrier of F
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(F,S,a,x,y) * (1_ F) is Element of the carrier of F
the multF of F . ((F,S,a,x,y),(1_ F)) is Element of the carrier of F
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,a,b,x) is Element of the carrier of F
the carrier of F is non empty non trivial set
(F,S,a,x,b) is Element of the carrier of F
(F,S,a,x,b) " is Element of the carrier of F
1_ F is Element of the carrier of F
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
(F,S,a,b,b) is Element of the carrier of F
(F,S,a,b,x) " is Element of the carrier of F
(F,S,a,b,b) * ((F,S,a,b,x) ") is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the multF of F . ((F,S,a,b,b),((F,S,a,b,x) ")) is Element of the carrier of F
((F,S,a,b,x) ") * (1_ F) is Element of the carrier of F
the multF of F . (((F,S,a,b,x) "),(1_ F)) is Element of the carrier of F
(F,S,a,x,b) * (F,S,a,b,x) is Element of the carrier of F
the multF of F . ((F,S,a,x,b),(F,S,a,b,x)) is Element of the carrier of F
((F,S,a,x,b) ") * ((F,S,a,x,b) * (F,S,a,b,x)) is Element of the carrier of F
the multF of F . (((F,S,a,x,b) "),((F,S,a,x,b) * (F,S,a,b,x))) is Element of the carrier of F
((F,S,a,x,b) ") * (F,S,a,x,b) is Element of the carrier of F
the multF of F . (((F,S,a,x,b) "),(F,S,a,x,b)) is Element of the carrier of F
(((F,S,a,x,b) ") * (F,S,a,x,b)) * (F,S,a,b,x) is Element of the carrier of F
the multF of F . ((((F,S,a,x,b) ") * (F,S,a,x,b)),(F,S,a,b,x)) is Element of the carrier of F
(F,S,a,b,x) * (1_ F) is Element of the carrier of F
the multF of F . ((F,S,a,b,x),(1_ F)) is Element of the carrier of F
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
x + a is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (x,a) is Element of the carrier of S
(F,S,a,b,x) is Element of the carrier of F
the carrier of F is non empty non trivial set
(F,S,x,b,a) is Element of the carrier of F
(F,S,a,b,x) * b is Element of the carrier of S
- ((F,S,a,b,x) * b) is Element of the carrier of S
x - ((F,S,a,b,x) * b) is Element of the carrier of S
x + (- ((F,S,a,b,x) * b)) is Element of the carrier of S
the addF of S . (x,(- ((F,S,a,b,x) * b))) is Element of the carrier of S
(- ((F,S,a,b,x) * b)) + x is Element of the carrier of S
the addF of S . ((- ((F,S,a,b,x) * b)),x) is Element of the carrier of S
a + (- ((F,S,a,b,x) * b)) is Element of the carrier of S
the addF of S . (a,(- ((F,S,a,b,x) * b))) is Element of the carrier of S
a - ((F,S,a,b,x) * b) is Element of the carrier of S
a + (- ((F,S,a,b,x) * b)) is Element of the carrier of S
(F,S,x,b,a) * b is Element of the carrier of S
a - ((F,S,x,b,a) * b) is Element of the carrier of S
- ((F,S,x,b,a) * b) is Element of the carrier of S
a + (- ((F,S,x,b,a) * b)) is Element of the carrier of S
the addF of S . (a,(- ((F,S,x,b,a) * b))) is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,x,a,b) is Element of the carrier of F
the carrier of F is non empty non trivial set
(F,S,a,b,x) is Element of the carrier of F
(F,S,a,b,x) " is Element of the carrier of F
- ((F,S,a,b,x) ") is Element of the carrier of F
(F,S,b,a,x) is Element of the carrier of F
(- ((F,S,a,b,x) ")) * (F,S,b,a,x) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the multF of F . ((- ((F,S,a,b,x) ")),(F,S,b,a,x)) is Element of the carrier of F
1_ F is Element of the carrier of F
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
- (1_ F) is Element of the carrier of F
((F,S,a,b,x) ") * (F,S,a,b,x) is Element of the carrier of F
the multF of F . (((F,S,a,b,x) "),(F,S,a,b,x)) is Element of the carrier of F
(- (1_ F)) * (((F,S,a,b,x) ") * (F,S,a,b,x)) is Element of the carrier of F
the multF of F . ((- (1_ F)),(((F,S,a,b,x) ") * (F,S,a,b,x))) is Element of the carrier of F
(- (1_ F)) * (1_ F) is Element of the carrier of F
the multF of F . ((- (1_ F)),(1_ F)) is Element of the carrier of F
(- (1_ F)) * ((F,S,a,b,x) ") is Element of the carrier of F
the multF of F . ((- (1_ F)),((F,S,a,b,x) ")) is Element of the carrier of F
((- (1_ F)) * ((F,S,a,b,x) ")) * (F,S,a,b,x) is Element of the carrier of F
the multF of F . (((- (1_ F)) * ((F,S,a,b,x) ")),(F,S,a,b,x)) is Element of the carrier of F
((F,S,a,b,x) ") * (1_ F) is Element of the carrier of F
the multF of F . (((F,S,a,b,x) "),(1_ F)) is Element of the carrier of F
- (((F,S,a,b,x) ") * (1_ F)) is Element of the carrier of F
(- (((F,S,a,b,x) ") * (1_ F))) * (F,S,a,b,x) is Element of the carrier of F
the multF of F . ((- (((F,S,a,b,x) ") * (1_ F))),(F,S,a,b,x)) is Element of the carrier of F
(- ((F,S,a,b,x) ")) * (F,S,a,b,x) is Element of the carrier of F
the multF of F . ((- ((F,S,a,b,x) ")),(F,S,a,b,x)) is Element of the carrier of F
(- ((F,S,a,b,x) ")) * x is Element of the carrier of S
(F,S,a,b,((- ((F,S,a,b,x) ")) * x)) is Element of the carrier of F
(- (1_ F)) * b is Element of the carrier of S
((- ((F,S,a,b,x) ")) * x) - ((- (1_ F)) * b) is Element of the carrier of S
- ((- (1_ F)) * b) is Element of the carrier of S
((- ((F,S,a,b,x) ")) * x) + (- ((- (1_ F)) * b)) is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (((- ((F,S,a,b,x) ")) * x),(- ((- (1_ F)) * b))) is Element of the carrier of S
- b is Element of the carrier of S
((- ((F,S,a,b,x) ")) * x) - (- b) is Element of the carrier of S
- (- b) is Element of the carrier of S
((- ((F,S,a,b,x) ")) * x) + (- (- b)) is Element of the carrier of S
the addF of S . (((- ((F,S,a,b,x) ")) * x),(- (- b))) is Element of the carrier of S
((- ((F,S,a,b,x) ")) * x) + b is Element of the carrier of S
the addF of S . (((- ((F,S,a,b,x) ")) * x),b) is Element of the carrier of S
(F,S,b,a,((- ((F,S,a,b,x) ")) * x)) is Element of the carrier of F
(F,S,((- ((F,S,a,b,x) ")) * x),a,b) is Element of the carrier of F
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,b,a,x) is Element of the carrier of F
y is Element of the carrier of S
(F,S,y,x,a) is Element of the carrier of F
(F,S,b,a,x) * (F,S,y,x,a) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . ((F,S,b,a,x),(F,S,y,x,a)) is Element of the carrier of F
(F,S,a,b,y) is Element of the carrier of F
(F,S,x,y,b) is Element of the carrier of F
(F,S,a,b,y) * (F,S,x,y,b) is Element of the carrier of F
the multF of F . ((F,S,a,b,y),(F,S,x,y,b)) is Element of the carrier of F
a + b is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (a,b) is Element of the carrier of S
(F,S,y,x,(a + b)) is Element of the carrier of F
(F,S,x,(a + b),y) is Element of the carrier of F
(F,S,x,(a + b),y) " is Element of the carrier of F
- ((F,S,x,(a + b),y) ") is Element of the carrier of F
(F,S,(a + b),x,y) is Element of the carrier of F
(- ((F,S,x,(a + b),y) ")) * (F,S,(a + b),x,y) is Element of the carrier of F
the multF of F . ((- ((F,S,x,(a + b),y) ")),(F,S,(a + b),x,y)) is Element of the carrier of F
(F,S,b,(a + b),x) is Element of the carrier of F
(F,S,(a + b),x,b) is Element of the carrier of F
(F,S,(a + b),x,b) " is Element of the carrier of F
- ((F,S,(a + b),x,b) ") is Element of the carrier of F
(F,S,x,(a + b),b) is Element of the carrier of F
(- ((F,S,(a + b),x,b) ")) * (F,S,x,(a + b),b) is Element of the carrier of F
the multF of F . ((- ((F,S,(a + b),x,b) ")),(F,S,x,(a + b),b)) is Element of the carrier of F
(F,S,x,(a + b),b) * (- ((F,S,(a + b),x,b) ")) is Element of the carrier of F
the multF of F . ((F,S,x,(a + b),b),(- ((F,S,(a + b),x,b) "))) is Element of the carrier of F
((F,S,x,(a + b),b) * (- ((F,S,(a + b),x,b) "))) * (- ((F,S,x,(a + b),y) ")) is Element of the carrier of F
the multF of F . (((F,S,x,(a + b),b) * (- ((F,S,(a + b),x,b) "))),(- ((F,S,x,(a + b),y) "))) is Element of the carrier of F
(((F,S,x,(a + b),b) * (- ((F,S,(a + b),x,b) "))) * (- ((F,S,x,(a + b),y) "))) * (F,S,(a + b),x,y) is Element of the carrier of F
the multF of F . ((((F,S,x,(a + b),b) * (- ((F,S,(a + b),x,b) "))) * (- ((F,S,x,(a + b),y) "))),(F,S,(a + b),x,y)) is Element of the carrier of F
(- ((F,S,(a + b),x,b) ")) * (- ((F,S,x,(a + b),y) ")) is Element of the carrier of F
the multF of F . ((- ((F,S,(a + b),x,b) ")),(- ((F,S,x,(a + b),y) "))) is Element of the carrier of F
(F,S,x,(a + b),b) * ((- ((F,S,(a + b),x,b) ")) * (- ((F,S,x,(a + b),y) "))) is Element of the carrier of F
the multF of F . ((F,S,x,(a + b),b),((- ((F,S,(a + b),x,b) ")) * (- ((F,S,x,(a + b),y) ")))) is Element of the carrier of F
((F,S,x,(a + b),b) * ((- ((F,S,(a + b),x,b) ")) * (- ((F,S,x,(a + b),y) ")))) * (F,S,(a + b),x,y) is Element of the carrier of F
the multF of F . (((F,S,x,(a + b),b) * ((- ((F,S,(a + b),x,b) ")) * (- ((F,S,x,(a + b),y) ")))),(F,S,(a + b),x,y)) is Element of the carrier of F
((F,S,x,(a + b),y) ") * ((F,S,(a + b),x,b) ") is Element of the carrier of F
the multF of F . (((F,S,x,(a + b),y) "),((F,S,(a + b),x,b) ")) is Element of the carrier of F
(F,S,x,(a + b),b) * (((F,S,x,(a + b),y) ") * ((F,S,(a + b),x,b) ")) is Element of the carrier of F
the multF of F . ((F,S,x,(a + b),b),(((F,S,x,(a + b),y) ") * ((F,S,(a + b),x,b) "))) is Element of the carrier of F
((F,S,x,(a + b),b) * (((F,S,x,(a + b),y) ") * ((F,S,(a + b),x,b) "))) * (F,S,(a + b),x,y) is Element of the carrier of F
the multF of F . (((F,S,x,(a + b),b) * (((F,S,x,(a + b),y) ") * ((F,S,(a + b),x,b) "))),(F,S,(a + b),x,y)) is Element of the carrier of F
(F,S,x,(a + b),b) * ((F,S,x,(a + b),y) ") is Element of the carrier of F
the multF of F . ((F,S,x,(a + b),b),((F,S,x,(a + b),y) ")) is Element of the carrier of F
((F,S,x,(a + b),b) * ((F,S,x,(a + b),y) ")) * ((F,S,(a + b),x,b) ") is Element of the carrier of F
the multF of F . (((F,S,x,(a + b),b) * ((F,S,x,(a + b),y) ")),((F,S,(a + b),x,b) ")) is Element of the carrier of F
(((F,S,x,(a + b),b) * ((F,S,x,(a + b),y) ")) * ((F,S,(a + b),x,b) ")) * (F,S,(a + b),x,y) is Element of the carrier of F
the multF of F . ((((F,S,x,(a + b),b) * ((F,S,x,(a + b),y) ")) * ((F,S,(a + b),x,b) ")),(F,S,(a + b),x,y)) is Element of the carrier of F
(F,S,x,y,b) * ((F,S,(a + b),x,b) ") is Element of the carrier of F
the multF of F . ((F,S,x,y,b),((F,S,(a + b),x,b) ")) is Element of the carrier of F
((F,S,x,y,b) * ((F,S,(a + b),x,b) ")) * (F,S,(a + b),x,y) is Element of the carrier of F
the multF of F . (((F,S,x,y,b) * ((F,S,(a + b),x,b) ")),(F,S,(a + b),x,y)) is Element of the carrier of F
(F,S,(a + b),x,y) * ((F,S,(a + b),x,b) ") is Element of the carrier of F
the multF of F . ((F,S,(a + b),x,y),((F,S,(a + b),x,b) ")) is Element of the carrier of F
(F,S,x,y,b) * ((F,S,(a + b),x,y) * ((F,S,(a + b),x,b) ")) is Element of the carrier of F
the multF of F . ((F,S,x,y,b),((F,S,(a + b),x,y) * ((F,S,(a + b),x,b) "))) is Element of the carrier of F
(F,S,(a + b),b,y) is Element of the carrier of F
(F,S,x,y,b) * (F,S,(a + b),b,y) is Element of the carrier of F
the multF of F . ((F,S,x,y,b),(F,S,(a + b),b,y)) is Element of the carrier of F
(F,S,x,y,b) * (F,S,a,b,y) is Element of the carrier of F
the multF of F . ((F,S,x,y,b),(F,S,a,b,y)) is Element of the carrier of F
(F,S,y,b,x) is Element of the carrier of F
(F,S,y,b,x) " is Element of the carrier of F
- ((F,S,y,b,x) ") is Element of the carrier of F
(F,S,b,y,x) is Element of the carrier of F
(- ((F,S,y,b,x) ")) * (F,S,b,y,x) is Element of the carrier of F
the multF of F . ((- ((F,S,y,b,x) ")),(F,S,b,y,x)) is Element of the carrier of F
(F,S,b,y,a) is Element of the carrier of F
(F,S,b,y,a) " is Element of the carrier of F
- ((F,S,b,y,a) ") is Element of the carrier of F
(F,S,y,b,a) is Element of the carrier of F
(- ((F,S,b,y,a) ")) * (F,S,y,b,a) is Element of the carrier of F
the multF of F . ((- ((F,S,b,y,a) ")),(F,S,y,b,a)) is Element of the carrier of F
(F,S,y,b,a) * (- ((F,S,b,y,a) ")) is Element of the carrier of F
the multF of F . ((F,S,y,b,a),(- ((F,S,b,y,a) "))) is Element of the carrier of F
((F,S,y,b,a) * (- ((F,S,b,y,a) "))) * (- ((F,S,y,b,x) ")) is Element of the carrier of F
the multF of F . (((F,S,y,b,a) * (- ((F,S,b,y,a) "))),(- ((F,S,y,b,x) "))) is Element of the carrier of F
(((F,S,y,b,a) * (- ((F,S,b,y,a) "))) * (- ((F,S,y,b,x) "))) * (F,S,b,y,x) is Element of the carrier of F
the multF of F . ((((F,S,y,b,a) * (- ((F,S,b,y,a) "))) * (- ((F,S,y,b,x) "))),(F,S,b,y,x)) is Element of the carrier of F
(- ((F,S,b,y,a) ")) * (- ((F,S,y,b,x) ")) is Element of the carrier of F
the multF of F . ((- ((F,S,b,y,a) ")),(- ((F,S,y,b,x) "))) is Element of the carrier of F
(F,S,y,b,a) * ((- ((F,S,b,y,a) ")) * (- ((F,S,y,b,x) "))) is Element of the carrier of F
the multF of F . ((F,S,y,b,a),((- ((F,S,b,y,a) ")) * (- ((F,S,y,b,x) ")))) is Element of the carrier of F
((F,S,y,b,a) * ((- ((F,S,b,y,a) ")) * (- ((F,S,y,b,x) ")))) * (F,S,b,y,x) is Element of the carrier of F
the multF of F . (((F,S,y,b,a) * ((- ((F,S,b,y,a) ")) * (- ((F,S,y,b,x) ")))),(F,S,b,y,x)) is Element of the carrier of F
((F,S,y,b,x) ") * ((F,S,b,y,a) ") is Element of the carrier of F
the multF of F . (((F,S,y,b,x) "),((F,S,b,y,a) ")) is Element of the carrier of F
(F,S,y,b,a) * (((F,S,y,b,x) ") * ((F,S,b,y,a) ")) is Element of the carrier of F
the multF of F . ((F,S,y,b,a),(((F,S,y,b,x) ") * ((F,S,b,y,a) "))) is Element of the carrier of F
((F,S,y,b,a) * (((F,S,y,b,x) ") * ((F,S,b,y,a) "))) * (F,S,b,y,x) is Element of the carrier of F
the multF of F . (((F,S,y,b,a) * (((F,S,y,b,x) ") * ((F,S,b,y,a) "))),(F,S,b,y,x)) is Element of the carrier of F
(F,S,y,b,a) * ((F,S,y,b,x) ") is Element of the carrier of F
the multF of F . ((F,S,y,b,a),((F,S,y,b,x) ")) is Element of the carrier of F
((F,S,y,b,a) * ((F,S,y,b,x) ")) * ((F,S,b,y,a) ") is Element of the carrier of F
the multF of F . (((F,S,y,b,a) * ((F,S,y,b,x) ")),((F,S,b,y,a) ")) is Element of the carrier of F
(((F,S,y,b,a) * ((F,S,y,b,x) ")) * ((F,S,b,y,a) ")) * (F,S,b,y,x) is Element of the carrier of F
the multF of F . ((((F,S,y,b,a) * ((F,S,y,b,x) ")) * ((F,S,b,y,a) ")),(F,S,b,y,x)) is Element of the carrier of F
(F,S,y,x,a) * ((F,S,b,y,a) ") is Element of the carrier of F
the multF of F . ((F,S,y,x,a),((F,S,b,y,a) ")) is Element of the carrier of F
((F,S,y,x,a) * ((F,S,b,y,a) ")) * (F,S,b,y,x) is Element of the carrier of F
the multF of F . (((F,S,y,x,a) * ((F,S,b,y,a) ")),(F,S,b,y,x)) is Element of the carrier of F
(F,S,b,y,x) * ((F,S,b,y,a) ") is Element of the carrier of F
the multF of F . ((F,S,b,y,x),((F,S,b,y,a) ")) is Element of the carrier of F
(F,S,y,x,a) * ((F,S,b,y,x) * ((F,S,b,y,a) ")) is Element of the carrier of F
the multF of F . ((F,S,y,x,a),((F,S,b,y,x) * ((F,S,b,y,a) "))) is Element of the carrier of F
(F,S,y,x,a) * (F,S,b,a,x) is Element of the carrier of F
the multF of F . ((F,S,y,x,a),(F,S,b,a,x)) is Element of the carrier of F
(F,S,x,a,y) is Element of the carrier of F
(F,S,x,a,y) " is Element of the carrier of F
- ((F,S,x,a,y) ") is Element of the carrier of F
(F,S,a,x,y) is Element of the carrier of F
(- ((F,S,x,a,y) ")) * (F,S,a,x,y) is Element of the carrier of F
the multF of F . ((- ((F,S,x,a,y) ")),(F,S,a,x,y)) is Element of the carrier of F
(F,S,a,x,b) is Element of the carrier of F
(F,S,a,x,b) " is Element of the carrier of F
- ((F,S,a,x,b) ") is Element of the carrier of F
(F,S,x,a,b) is Element of the carrier of F
(- ((F,S,a,x,b) ")) * (F,S,x,a,b) is Element of the carrier of F
the multF of F . ((- ((F,S,a,x,b) ")),(F,S,x,a,b)) is Element of the carrier of F
(F,S,x,a,b) * (- ((F,S,a,x,b) ")) is Element of the carrier of F
the multF of F . ((F,S,x,a,b),(- ((F,S,a,x,b) "))) is Element of the carrier of F
((F,S,x,a,b) * (- ((F,S,a,x,b) "))) * (- ((F,S,x,a,y) ")) is Element of the carrier of F
the multF of F . (((F,S,x,a,b) * (- ((F,S,a,x,b) "))),(- ((F,S,x,a,y) "))) is Element of the carrier of F
(((F,S,x,a,b) * (- ((F,S,a,x,b) "))) * (- ((F,S,x,a,y) "))) * (F,S,a,x,y) is Element of the carrier of F
the multF of F . ((((F,S,x,a,b) * (- ((F,S,a,x,b) "))) * (- ((F,S,x,a,y) "))),(F,S,a,x,y)) is Element of the carrier of F
(- ((F,S,a,x,b) ")) * (- ((F,S,x,a,y) ")) is Element of the carrier of F
the multF of F . ((- ((F,S,a,x,b) ")),(- ((F,S,x,a,y) "))) is Element of the carrier of F
(F,S,x,a,b) * ((- ((F,S,a,x,b) ")) * (- ((F,S,x,a,y) "))) is Element of the carrier of F
the multF of F . ((F,S,x,a,b),((- ((F,S,a,x,b) ")) * (- ((F,S,x,a,y) ")))) is Element of the carrier of F
((F,S,x,a,b) * ((- ((F,S,a,x,b) ")) * (- ((F,S,x,a,y) ")))) * (F,S,a,x,y) is Element of the carrier of F
the multF of F . (((F,S,x,a,b) * ((- ((F,S,a,x,b) ")) * (- ((F,S,x,a,y) ")))),(F,S,a,x,y)) is Element of the carrier of F
((F,S,x,a,y) ") * ((F,S,a,x,b) ") is Element of the carrier of F
the multF of F . (((F,S,x,a,y) "),((F,S,a,x,b) ")) is Element of the carrier of F
(F,S,x,a,b) * (((F,S,x,a,y) ") * ((F,S,a,x,b) ")) is Element of the carrier of F
the multF of F . ((F,S,x,a,b),(((F,S,x,a,y) ") * ((F,S,a,x,b) "))) is Element of the carrier of F
((F,S,x,a,b) * (((F,S,x,a,y) ") * ((F,S,a,x,b) "))) * (F,S,a,x,y) is Element of the carrier of F
the multF of F . (((F,S,x,a,b) * (((F,S,x,a,y) ") * ((F,S,a,x,b) "))),(F,S,a,x,y)) is Element of the carrier of F
(F,S,x,a,b) * ((F,S,x,a,y) ") is Element of the carrier of F
the multF of F . ((F,S,x,a,b),((F,S,x,a,y) ")) is Element of the carrier of F
((F,S,x,a,b) * ((F,S,x,a,y) ")) * ((F,S,a,x,b) ") is Element of the carrier of F
the multF of F . (((F,S,x,a,b) * ((F,S,x,a,y) ")),((F,S,a,x,b) ")) is Element of the carrier of F
(((F,S,x,a,b) * ((F,S,x,a,y) ")) * ((F,S,a,x,b) ")) * (F,S,a,x,y) is Element of the carrier of F
the multF of F . ((((F,S,x,a,b) * ((F,S,x,a,y) ")) * ((F,S,a,x,b) ")),(F,S,a,x,y)) is Element of the carrier of F
(F,S,x,y,b) * ((F,S,a,x,b) ") is Element of the carrier of F
the multF of F . ((F,S,x,y,b),((F,S,a,x,b) ")) is Element of the carrier of F
((F,S,x,y,b) * ((F,S,a,x,b) ")) * (F,S,a,x,y) is Element of the carrier of F
the multF of F . (((F,S,x,y,b) * ((F,S,a,x,b) ")),(F,S,a,x,y)) is Element of the carrier of F
(F,S,a,x,y) * ((F,S,a,x,b) ") is Element of the carrier of F
the multF of F . ((F,S,a,x,y),((F,S,a,x,b) ")) is Element of the carrier of F
(F,S,x,y,b) * ((F,S,a,x,y) * ((F,S,a,x,b) ")) is Element of the carrier of F
the multF of F . ((F,S,x,y,b),((F,S,a,x,y) * ((F,S,a,x,b) "))) is Element of the carrier of F
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,b,a,x) is Element of the carrier of F
y is Element of the carrier of S
(F,S,a,y,b) is Element of the carrier of F
(F,S,a,y,b) * (F,S,b,a,x) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . ((F,S,a,y,b),(F,S,b,a,x)) is Element of the carrier of F
(F,S,x,y,b) is Element of the carrier of F
(F,S,y,a,x) is Element of the carrier of F
(F,S,x,y,b) * (F,S,y,a,x) is Element of the carrier of F
the multF of F . ((F,S,x,y,b),(F,S,y,a,x)) is Element of the carrier of F
(F,S,y,x,a) is Element of the carrier of F
(F,S,b,a,x) * (F,S,y,x,a) is Element of the carrier of F
the multF of F . ((F,S,b,a,x),(F,S,y,x,a)) is Element of the carrier of F
(F,S,a,b,y) is Element of the carrier of F
(F,S,a,b,y) * (F,S,x,y,b) is Element of the carrier of F
the multF of F . ((F,S,a,b,y),(F,S,x,y,b)) is Element of the carrier of F
(F,S,a,y,b) " is Element of the carrier of F
((F,S,a,y,b) ") * (F,S,x,y,b) is Element of the carrier of F
the multF of F . (((F,S,a,y,b) "),(F,S,x,y,b)) is Element of the carrier of F
(F,S,a,y,b) * ((F,S,b,a,x) * (F,S,y,x,a)) is Element of the carrier of F
the multF of F . ((F,S,a,y,b),((F,S,b,a,x) * (F,S,y,x,a))) is Element of the carrier of F
(F,S,a,y,b) * ((F,S,a,y,b) ") is Element of the carrier of F
the multF of F . ((F,S,a,y,b),((F,S,a,y,b) ")) is Element of the carrier of F
((F,S,a,y,b) * ((F,S,a,y,b) ")) * (F,S,x,y,b) is Element of the carrier of F
the multF of F . (((F,S,a,y,b) * ((F,S,a,y,b) ")),(F,S,x,y,b)) is Element of the carrier of F
(F,S,x,y,b) * (1_ F) is Element of the carrier of F
the multF of F . ((F,S,x,y,b),(1_ F)) is Element of the carrier of F
((F,S,a,y,b) * (F,S,b,a,x)) * (F,S,y,x,a) is Element of the carrier of F
the multF of F . (((F,S,a,y,b) * (F,S,b,a,x)),(F,S,y,x,a)) is Element of the carrier of F
(F,S,y,a,x) " is Element of the carrier of F
((F,S,a,y,b) * (F,S,b,a,x)) * ((F,S,y,a,x) ") is Element of the carrier of F
the multF of F . (((F,S,a,y,b) * (F,S,b,a,x)),((F,S,y,a,x) ")) is Element of the carrier of F
((F,S,y,a,x) ") * (F,S,y,a,x) is Element of the carrier of F
the multF of F . (((F,S,y,a,x) "),(F,S,y,a,x)) is Element of the carrier of F
((F,S,a,y,b) * (F,S,b,a,x)) * (((F,S,y,a,x) ") * (F,S,y,a,x)) is Element of the carrier of F
the multF of F . (((F,S,a,y,b) * (F,S,b,a,x)),(((F,S,y,a,x) ") * (F,S,y,a,x))) is Element of the carrier of F
((F,S,a,y,b) * (F,S,b,a,x)) * (1_ F) is Element of the carrier of F
the multF of F . (((F,S,a,y,b) * (F,S,b,a,x)),(1_ F)) is Element of the carrier of F
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,b,a,x) is Element of the carrier of F
y is Element of the carrier of S
(F,S,y,a,x) is Element of the carrier of F
z is Element of the carrier of S
(F,S,a,z,b) is Element of the carrier of F
(F,S,a,z,b) * (F,S,b,a,x) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . ((F,S,a,z,b),(F,S,b,a,x)) is Element of the carrier of F
(F,S,a,z,y) is Element of the carrier of F
(F,S,a,z,y) * (F,S,y,a,x) is Element of the carrier of F
the multF of F . ((F,S,a,z,y),(F,S,y,a,x)) is Element of the carrier of F
0F is Element of the carrier of S
(F,S,x,b,0F) is Element of the carrier of F
((F,S,a,z,b) * (F,S,b,a,x)) * (F,S,x,b,0F) is Element of the carrier of F
the multF of F . (((F,S,a,z,b) * (F,S,b,a,x)),(F,S,x,b,0F)) is Element of the carrier of F
(F,S,x,y,0F) is Element of the carrier of F
((F,S,a,z,y) * (F,S,y,a,x)) * (F,S,x,y,0F) is Element of the carrier of F
the multF of F . (((F,S,a,z,y) * (F,S,y,a,x)),(F,S,x,y,0F)) is Element of the carrier of F
(F,S,a,y,b) is Element of the carrier of F
(F,S,a,y,b) * (F,S,b,a,x) is Element of the carrier of F
the multF of F . ((F,S,a,y,b),(F,S,b,a,x)) is Element of the carrier of F
(F,S,x,y,b) is Element of the carrier of F
(F,S,x,y,b) * (F,S,y,a,x) is Element of the carrier of F
the multF of F . ((F,S,x,y,b),(F,S,y,a,x)) is Element of the carrier of F
(F,S,a,z,y) " is Element of the carrier of F
(F,S,a,z,b) * ((F,S,a,z,y) ") is Element of the carrier of F
the multF of F . ((F,S,a,z,b),((F,S,a,z,y) ")) is Element of the carrier of F
((F,S,a,z,b) * ((F,S,a,z,y) ")) * (F,S,b,a,x) is Element of the carrier of F
the multF of F . (((F,S,a,z,b) * ((F,S,a,z,y) ")),(F,S,b,a,x)) is Element of the carrier of F
((F,S,a,z,y) ") * (F,S,a,z,b) is Element of the carrier of F
the multF of F . (((F,S,a,z,y) "),(F,S,a,z,b)) is Element of the carrier of F
(((F,S,a,z,y) ") * (F,S,a,z,b)) * (F,S,b,a,x) is Element of the carrier of F
the multF of F . ((((F,S,a,z,y) ") * (F,S,a,z,b)),(F,S,b,a,x)) is Element of the carrier of F
(F,S,x,0F,b) is Element of the carrier of F
(F,S,x,0F,y) is Element of the carrier of F
(F,S,x,0F,y) " is Element of the carrier of F
(F,S,x,0F,b) * ((F,S,x,0F,y) ") is Element of the carrier of F
the multF of F . ((F,S,x,0F,b),((F,S,x,0F,y) ")) is Element of the carrier of F
((F,S,x,0F,b) * ((F,S,x,0F,y) ")) * (F,S,y,a,x) is Element of the carrier of F
the multF of F . (((F,S,x,0F,b) * ((F,S,x,0F,y) ")),(F,S,y,a,x)) is Element of the carrier of F
((F,S,a,z,y) ") * ((F,S,a,z,b) * (F,S,b,a,x)) is Element of the carrier of F
the multF of F . (((F,S,a,z,y) "),((F,S,a,z,b) * (F,S,b,a,x))) is Element of the carrier of F
(F,S,a,z,y) * (((F,S,a,z,y) ") * ((F,S,a,z,b) * (F,S,b,a,x))) is Element of the carrier of F
the multF of F . ((F,S,a,z,y),(((F,S,a,z,y) ") * ((F,S,a,z,b) * (F,S,b,a,x)))) is Element of the carrier of F
(F,S,a,z,y) * (((F,S,x,0F,b) * ((F,S,x,0F,y) ")) * (F,S,y,a,x)) is Element of the carrier of F
the multF of F . ((F,S,a,z,y),(((F,S,x,0F,b) * ((F,S,x,0F,y) ")) * (F,S,y,a,x))) is Element of the carrier of F
(F,S,a,z,y) * ((F,S,a,z,y) ") is Element of the carrier of F
the multF of F . ((F,S,a,z,y),((F,S,a,z,y) ")) is Element of the carrier of F
((F,S,a,z,y) * ((F,S,a,z,y) ")) * ((F,S,a,z,b) * (F,S,b,a,x)) is Element of the carrier of F
the multF of F . (((F,S,a,z,y) * ((F,S,a,z,y) ")),((F,S,a,z,b) * (F,S,b,a,x))) is Element of the carrier of F
((F,S,a,z,b) * (F,S,b,a,x)) * (1_ F) is Element of the carrier of F
the multF of F . (((F,S,a,z,b) * (F,S,b,a,x)),(1_ F)) is Element of the carrier of F
(F,S,x,b,0F) " is Element of the carrier of F
((F,S,x,0F,y) ") * ((F,S,x,b,0F) ") is Element of the carrier of F
the multF of F . (((F,S,x,0F,y) "),((F,S,x,b,0F) ")) is Element of the carrier of F
(((F,S,x,0F,y) ") * ((F,S,x,b,0F) ")) * (F,S,y,a,x) is Element of the carrier of F
the multF of F . ((((F,S,x,0F,y) ") * ((F,S,x,b,0F) ")),(F,S,y,a,x)) is Element of the carrier of F
(F,S,a,z,y) * ((((F,S,x,0F,y) ") * ((F,S,x,b,0F) ")) * (F,S,y,a,x)) is Element of the carrier of F
the multF of F . ((F,S,a,z,y),((((F,S,x,0F,y) ") * ((F,S,x,b,0F) ")) * (F,S,y,a,x))) is Element of the carrier of F
(F,S,a,z,y) * (((F,S,x,0F,y) ") * ((F,S,x,b,0F) ")) is Element of the carrier of F
the multF of F . ((F,S,a,z,y),(((F,S,x,0F,y) ") * ((F,S,x,b,0F) "))) is Element of the carrier of F
((F,S,a,z,y) * (((F,S,x,0F,y) ") * ((F,S,x,b,0F) "))) * (F,S,y,a,x) is Element of the carrier of F
the multF of F . (((F,S,a,z,y) * (((F,S,x,0F,y) ") * ((F,S,x,b,0F) "))),(F,S,y,a,x)) is Element of the carrier of F
(F,S,a,z,y) * ((F,S,x,0F,y) ") is Element of the carrier of F
the multF of F . ((F,S,a,z,y),((F,S,x,0F,y) ")) is Element of the carrier of F
((F,S,a,z,y) * ((F,S,x,0F,y) ")) * ((F,S,x,b,0F) ") is Element of the carrier of F
the multF of F . (((F,S,a,z,y) * ((F,S,x,0F,y) ")),((F,S,x,b,0F) ")) is Element of the carrier of F
(F,S,y,a,x) * (((F,S,a,z,y) * ((F,S,x,0F,y) ")) * ((F,S,x,b,0F) ")) is Element of the carrier of F
the multF of F . ((F,S,y,a,x),(((F,S,a,z,y) * ((F,S,x,0F,y) ")) * ((F,S,x,b,0F) "))) is Element of the carrier of F
(F,S,y,a,x) * ((F,S,a,z,y) * ((F,S,x,0F,y) ")) is Element of the carrier of F
the multF of F . ((F,S,y,a,x),((F,S,a,z,y) * ((F,S,x,0F,y) "))) is Element of the carrier of F
((F,S,y,a,x) * ((F,S,a,z,y) * ((F,S,x,0F,y) "))) * ((F,S,x,b,0F) ") is Element of the carrier of F
the multF of F . (((F,S,y,a,x) * ((F,S,a,z,y) * ((F,S,x,0F,y) "))),((F,S,x,b,0F) ")) is Element of the carrier of F
(((F,S,y,a,x) * ((F,S,a,z,y) * ((F,S,x,0F,y) "))) * ((F,S,x,b,0F) ")) * (F,S,x,b,0F) is Element of the carrier of F
the multF of F . ((((F,S,y,a,x) * ((F,S,a,z,y) * ((F,S,x,0F,y) "))) * ((F,S,x,b,0F) ")),(F,S,x,b,0F)) is Element of the carrier of F
((F,S,x,b,0F) ") * (F,S,x,b,0F) is Element of the carrier of F
the multF of F . (((F,S,x,b,0F) "),(F,S,x,b,0F)) is Element of the carrier of F
((F,S,y,a,x) * ((F,S,a,z,y) * ((F,S,x,0F,y) "))) * (((F,S,x,b,0F) ") * (F,S,x,b,0F)) is Element of the carrier of F
the multF of F . (((F,S,y,a,x) * ((F,S,a,z,y) * ((F,S,x,0F,y) "))),(((F,S,x,b,0F) ") * (F,S,x,b,0F))) is Element of the carrier of F
((F,S,y,a,x) * ((F,S,a,z,y) * ((F,S,x,0F,y) "))) * (1_ F) is Element of the carrier of F
the multF of F . (((F,S,y,a,x) * ((F,S,a,z,y) * ((F,S,x,0F,y) "))),(1_ F)) is Element of the carrier of F
(F,S,a,z,y) * (F,S,x,y,0F) is Element of the carrier of F
the multF of F . ((F,S,a,z,y),(F,S,x,y,0F)) is Element of the carrier of F
(F,S,y,a,x) * ((F,S,a,z,y) * (F,S,x,y,0F)) is Element of the carrier of F
the multF of F . ((F,S,y,a,x),((F,S,a,z,y) * (F,S,x,y,0F))) is Element of the carrier of F
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
(F,S,a,b,x) is Element of the carrier of F
the carrier of F is non empty non trivial set
y is Element of the carrier of S
(F,S,x,a,y) is Element of the carrier of F
(F,S,a,b,x) * (F,S,x,a,y) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the multF of F . ((F,S,a,b,x),(F,S,x,a,y)) is Element of the carrier of F
(F,S,a,b,y) is Element of the carrier of F
- (F,S,a,b,y) is Element of the carrier of F
(F,S,y,a,x) is Element of the carrier of F
(- (F,S,a,b,y)) * (F,S,y,a,x) is Element of the carrier of F
the multF of F . ((- (F,S,a,b,y)),(F,S,y,a,x)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
1_ F is Element of the carrier of F
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(F,S,a,b,x) " is Element of the carrier of F
(F,S,a,b,y) * ((F,S,a,b,x) ") is Element of the carrier of F
the multF of F . ((F,S,a,b,y),((F,S,a,b,x) ")) is Element of the carrier of F
(F,S,a,x,y) is Element of the carrier of F
((F,S,a,b,y) * ((F,S,a,b,x) ")) * (F,S,a,b,x) is Element of the carrier of F
the multF of F . (((F,S,a,b,y) * ((F,S,a,b,x) ")),(F,S,a,b,x)) is Element of the carrier of F
(F,S,x,y,a) is Element of the carrier of F
(F,S,x,y,a) " is Element of the carrier of F
- ((F,S,x,y,a) ") is Element of the carrier of F
(F,S,y,x,a) is Element of the carrier of F
(- ((F,S,x,y,a) ")) * (F,S,y,x,a) is Element of the carrier of F
the multF of F . ((- ((F,S,x,y,a) ")),(F,S,y,x,a)) is Element of the carrier of F
((- ((F,S,x,y,a) ")) * (F,S,y,x,a)) * (F,S,a,b,x) is Element of the carrier of F
the multF of F . (((- ((F,S,x,y,a) ")) * (F,S,y,x,a)),(F,S,a,b,x)) is Element of the carrier of F
((F,S,a,b,x) ") * (F,S,a,b,x) is Element of the carrier of F
the multF of F . (((F,S,a,b,x) "),(F,S,a,b,x)) is Element of the carrier of F
(F,S,a,b,y) * (((F,S,a,b,x) ") * (F,S,a,b,x)) is Element of the carrier of F
the multF of F . ((F,S,a,b,y),(((F,S,a,b,x) ") * (F,S,a,b,x))) is Element of the carrier of F
(F,S,a,b,y) * (1_ F) is Element of the carrier of F
the multF of F . ((F,S,a,b,y),(1_ F)) is Element of the carrier of F
(F,S,y,x,a) * (- ((F,S,x,y,a) ")) is Element of the carrier of F
the multF of F . ((F,S,y,x,a),(- ((F,S,x,y,a) "))) is Element of the carrier of F
((F,S,y,x,a) * (- ((F,S,x,y,a) "))) * (F,S,a,b,x) is Element of the carrier of F
the multF of F . (((F,S,y,x,a) * (- ((F,S,x,y,a) "))),(F,S,a,b,x)) is Element of the carrier of F
(- ((F,S,x,y,a) ")) * (F,S,a,b,x) is Element of the carrier of F
the multF of F . ((- ((F,S,x,y,a) ")),(F,S,a,b,x)) is Element of the carrier of F
(F,S,y,x,a) * ((- ((F,S,x,y,a) ")) * (F,S,a,b,x)) is Element of the carrier of F
the multF of F . ((F,S,y,x,a),((- ((F,S,x,y,a) ")) * (F,S,a,b,x))) is Element of the carrier of F
(F,S,y,a,x) * (F,S,a,b,y) is Element of the carrier of F
the multF of F . ((F,S,y,a,x),(F,S,a,b,y)) is Element of the carrier of F
(F,S,y,a,x) " is Element of the carrier of F
((F,S,y,a,x) ") * ((- ((F,S,x,y,a) ")) * (F,S,a,b,x)) is Element of the carrier of F
the multF of F . (((F,S,y,a,x) "),((- ((F,S,x,y,a) ")) * (F,S,a,b,x))) is Element of the carrier of F
(F,S,y,a,x) * (((F,S,y,a,x) ") * ((- ((F,S,x,y,a) ")) * (F,S,a,b,x))) is Element of the carrier of F
the multF of F . ((F,S,y,a,x),(((F,S,y,a,x) ") * ((- ((F,S,x,y,a) ")) * (F,S,a,b,x)))) is Element of the carrier of F
(F,S,y,a,x) * ((F,S,y,a,x) ") is Element of the carrier of F
the multF of F . ((F,S,y,a,x),((F,S,y,a,x) ")) is Element of the carrier of F
((F,S,y,a,x) * ((F,S,y,a,x) ")) * ((- ((F,S,x,y,a) ")) * (F,S,a,b,x)) is Element of the carrier of F
the multF of F . (((F,S,y,a,x) * ((F,S,y,a,x) ")),((- ((F,S,x,y,a) ")) * (F,S,a,b,x))) is Element of the carrier of F
((- ((F,S,x,y,a) ")) * (F,S,a,b,x)) * (1_ F) is Element of the carrier of F
the multF of F . (((- ((F,S,x,y,a) ")) * (F,S,a,b,x)),(1_ F)) is Element of the carrier of F
(F,S,a,b,y) * (F,S,y,a,x) is Element of the carrier of F
the multF of F . ((F,S,a,b,y),(F,S,y,a,x)) is Element of the carrier of F
- ((F,S,a,b,y) * (F,S,y,a,x)) is Element of the carrier of F
((F,S,x,y,a) ") * (F,S,a,b,x) is Element of the carrier of F
the multF of F . (((F,S,x,y,a) "),(F,S,a,b,x)) is Element of the carrier of F
- (((F,S,x,y,a) ") * (F,S,a,b,x)) is Element of the carrier of F
- (- (((F,S,x,y,a) ") * (F,S,a,b,x))) is Element of the carrier of F
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
y is Element of the carrier of S
x is Element of the carrier of S
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
b is Element of the carrier of S
a is Element of the carrier of S
z is Element of the carrier of S
(F,S,x,y,z) is Element of the carrier of F
(F,S,z,x,b) is Element of the carrier of F
(F,S,x,y,z) * (F,S,z,x,b) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . ((F,S,x,y,z),(F,S,z,x,b)) is Element of the carrier of F
(F,S,b,z,a) is Element of the carrier of F
((F,S,x,y,z) * (F,S,z,x,b)) * (F,S,b,z,a) is Element of the carrier of F
the multF of F . (((F,S,x,y,z) * (F,S,z,x,b)),(F,S,b,z,a)) is Element of the carrier of F
0F is Element of the carrier of S
(F,S,x,y,0F) is Element of the carrier of F
(F,S,0F,x,b) is Element of the carrier of F
(F,S,x,y,0F) * (F,S,0F,x,b) is Element of the carrier of F
the multF of F . ((F,S,x,y,0F),(F,S,0F,x,b)) is Element of the carrier of F
(F,S,b,0F,a) is Element of the carrier of F
((F,S,x,y,0F) * (F,S,0F,x,b)) * (F,S,b,0F,a) is Element of the carrier of F
the multF of F . (((F,S,x,y,0F) * (F,S,0F,x,b)),(F,S,b,0F,a)) is Element of the carrier of F
z is Element of the carrier of F
0F is Element of the carrier of F
p is Element of the carrier of S
r is Element of the carrier of S
(F,S,x,y,r) is Element of the carrier of F
(F,S,r,x,b) is Element of the carrier of F
(F,S,x,y,r) * (F,S,r,x,b) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . ((F,S,x,y,r),(F,S,r,x,b)) is Element of the carrier of F
(F,S,b,r,a) is Element of the carrier of F
((F,S,x,y,r) * (F,S,r,x,b)) * (F,S,b,r,a) is Element of the carrier of F
the multF of F . (((F,S,x,y,r) * (F,S,r,x,b)),(F,S,b,r,a)) is Element of the carrier of F
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
y is Element of the carrier of S
(F,S,x,y,a,b) is Element of the carrier of F
z is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
y is Element of the carrier of S
(F,S,x,y,a,b) is Element of the carrier of F
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
0F is Element of the carrier of S
(F,S,0F,a,x) is Element of the carrier of F
(F,S,a,b,0F) is Element of the carrier of F
(F,S,a,b,0F) * (F,S,0F,a,x) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . ((F,S,a,b,0F),(F,S,0F,a,x)) is Element of the carrier of F
(F,S,x,0F,y) is Element of the carrier of F
((F,S,a,b,0F) * (F,S,0F,a,x)) * (F,S,x,0F,y) is Element of the carrier of F
the multF of F . (((F,S,a,b,0F) * (F,S,0F,a,x)),(F,S,x,0F,y)) is Element of the carrier of F
0F is Element of the carrier of S
0F is Element of the carrier of S
(F,S,x,0F,y) is Element of the carrier of F
(F,S,a,b,0F) is Element of the carrier of F
(F,S,0F,a,x) is Element of the carrier of F
(F,S,a,b,0F) * (F,S,0F,a,x) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . ((F,S,a,b,0F),(F,S,0F,a,x)) is Element of the carrier of F
((F,S,a,b,0F) * (F,S,0F,a,x)) * (F,S,x,0F,y) is Element of the carrier of F
the multF of F . (((F,S,a,b,0F) * (F,S,0F,a,x)),(F,S,x,0F,y)) is Element of the carrier of F
0F is Element of the carrier of S
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
y is Element of the carrier of S
(F,S,x,y,a,b) is Element of the carrier of F
(F,S,y,x,a,b) is Element of the carrier of F
- (F,S,y,x,a,b) is Element of the carrier of F
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
p is Element of the carrier of S
(F,S,a,b,p) is Element of the carrier of F
(F,S,p,a,y) is Element of the carrier of F
(F,S,a,b,p) * (F,S,p,a,y) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . ((F,S,a,b,p),(F,S,p,a,y)) is Element of the carrier of F
(F,S,y,p,x) is Element of the carrier of F
((F,S,a,b,p) * (F,S,p,a,y)) * (F,S,y,p,x) is Element of the carrier of F
the multF of F . (((F,S,a,b,p) * (F,S,p,a,y)),(F,S,y,p,x)) is Element of the carrier of F
(F,S,p,a,y) * (F,S,y,p,x) is Element of the carrier of F
the multF of F . ((F,S,p,a,y),(F,S,y,p,x)) is Element of the carrier of F
(F,S,a,b,p) * ((F,S,p,a,y) * (F,S,y,p,x)) is Element of the carrier of F
the multF of F . ((F,S,a,b,p),((F,S,p,a,y) * (F,S,y,p,x))) is Element of the carrier of F
(F,S,p,a,x) is Element of the carrier of F
- (F,S,p,a,x) is Element of the carrier of F
(F,S,x,p,y) is Element of the carrier of F
(- (F,S,p,a,x)) * (F,S,x,p,y) is Element of the carrier of F
the multF of F . ((- (F,S,p,a,x)),(F,S,x,p,y)) is Element of the carrier of F
(F,S,a,b,p) * ((- (F,S,p,a,x)) * (F,S,x,p,y)) is Element of the carrier of F
the multF of F . ((F,S,a,b,p),((- (F,S,p,a,x)) * (F,S,x,p,y))) is Element of the carrier of F
(- (F,S,p,a,x)) * (F,S,a,b,p) is Element of the carrier of F
the multF of F . ((- (F,S,p,a,x)),(F,S,a,b,p)) is Element of the carrier of F
((- (F,S,p,a,x)) * (F,S,a,b,p)) * (F,S,x,p,y) is Element of the carrier of F
the multF of F . (((- (F,S,p,a,x)) * (F,S,a,b,p)),(F,S,x,p,y)) is Element of the carrier of F
(F,S,p,a,x) * (F,S,a,b,p) is Element of the carrier of F
the multF of F . ((F,S,p,a,x),(F,S,a,b,p)) is Element of the carrier of F
- ((F,S,p,a,x) * (F,S,a,b,p)) is Element of the carrier of F
(- ((F,S,p,a,x) * (F,S,a,b,p))) * (F,S,x,p,y) is Element of the carrier of F
the multF of F . ((- ((F,S,p,a,x) * (F,S,a,b,p))),(F,S,x,p,y)) is Element of the carrier of F
- (1_ F) is Element of the carrier of F
(- (1_ F)) * (F,S,y,x,a,b) is Element of the carrier of F
the multF of F . ((- (1_ F)),(F,S,y,x,a,b)) is Element of the carrier of F
(F,S,a,b,p) * (F,S,p,a,x) is Element of the carrier of F
the multF of F . ((F,S,a,b,p),(F,S,p,a,x)) is Element of the carrier of F
((F,S,a,b,p) * (F,S,p,a,x)) * (F,S,x,p,y) is Element of the carrier of F
the multF of F . (((F,S,a,b,p) * (F,S,p,a,x)),(F,S,x,p,y)) is Element of the carrier of F
- (((F,S,a,b,p) * (F,S,p,a,x)) * (F,S,x,p,y)) is Element of the carrier of F
(- (1_ F)) * (- (((F,S,a,b,p) * (F,S,p,a,x)) * (F,S,x,p,y))) is Element of the carrier of F
the multF of F . ((- (1_ F)),(- (((F,S,a,b,p) * (F,S,p,a,x)) * (F,S,x,p,y)))) is Element of the carrier of F
(F,S,y,x,a,b) * (1_ F) is Element of the carrier of F
the multF of F . ((F,S,y,x,a,b),(1_ F)) is Element of the carrier of F
- ((F,S,y,x,a,b) * (1_ F)) is Element of the carrier of F
(((F,S,a,b,p) * (F,S,p,a,x)) * (F,S,x,p,y)) * (1_ F) is Element of the carrier of F
the multF of F . ((((F,S,a,b,p) * (F,S,p,a,x)) * (F,S,x,p,y)),(1_ F)) is Element of the carrier of F
(- (1_ F)) * (0. F) is Element of the carrier of F
the multF of F . ((- (1_ F)),(0. F)) is Element of the carrier of F
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
the carrier of F is non empty non trivial set
1_ F is Element of the carrier of F
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
y is Element of the carrier of S
(F,S,x,y,a,b) is Element of the carrier of F
z is Element of the carrier of F
z * y is Element of the carrier of S
(F,S,x,(z * y),a,b) is Element of the carrier of F
z * (F,S,x,y,a,b) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . (z,(F,S,x,y,a,b)) is Element of the carrier of F
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
p is Element of the carrier of S
(F,S,a,b,p) is Element of the carrier of F
(F,S,p,a,x) is Element of the carrier of F
(F,S,a,b,p) * (F,S,p,a,x) is Element of the carrier of F
the multF of F . ((F,S,a,b,p),(F,S,p,a,x)) is Element of the carrier of F
(F,S,x,p,(z * y)) is Element of the carrier of F
((F,S,a,b,p) * (F,S,p,a,x)) * (F,S,x,p,(z * y)) is Element of the carrier of F
the multF of F . (((F,S,a,b,p) * (F,S,p,a,x)),(F,S,x,p,(z * y))) is Element of the carrier of F
(F,S,x,p,y) is Element of the carrier of F
z * (F,S,x,p,y) is Element of the carrier of F
the multF of F . (z,(F,S,x,p,y)) is Element of the carrier of F
(z * (F,S,x,p,y)) * ((F,S,a,b,p) * (F,S,p,a,x)) is Element of the carrier of F
the multF of F . ((z * (F,S,x,p,y)),((F,S,a,b,p) * (F,S,p,a,x))) is Element of the carrier of F
((F,S,a,b,p) * (F,S,p,a,x)) * (F,S,x,p,y) is Element of the carrier of F
the multF of F . (((F,S,a,b,p) * (F,S,p,a,x)),(F,S,x,p,y)) is Element of the carrier of F
z * (0. F) is Element of the carrier of F
the multF of F . (z,(0. F)) is Element of the carrier of F
F is non empty non degenerated non trivial right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital V114() left_unital doubleLoopStr
1_ F is Element of the carrier of F
the carrier of F is non empty non trivial set
1. F is V51(F) Element of the carrier of F
the OneF of F is Element of the carrier of F
(1_ F) + (1_ F) is Element of the carrier of F
the addF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
[: the carrier of F, the carrier of F:] is non empty set
[:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
bool [:[: the carrier of F, the carrier of F:], the carrier of F:] is non empty set
the addF of F . ((1_ F),(1_ F)) is Element of the carrier of F
0. F is V51(F) Element of the carrier of F
the ZeroF of F is Element of the carrier of F
S is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital (F) (F)
the carrier of S is non empty set
b is Element of the carrier of S
a is Element of the carrier of S
x is Element of the carrier of S
y is Element of the carrier of S
(F,S,x,y,a,b) is Element of the carrier of F
z is Element of the carrier of S
y + z is Element of the carrier of S
the addF of S is V1() V4([: the carrier of S, the carrier of S:]) V5( the carrier of S) V6() V18([: the carrier of S, the carrier of S:], the carrier of S) Element of bool [:[: the carrier of S, the carrier of S:], the carrier of S:]
[: the carrier of S, the carrier of S:] is non empty set
[:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
bool [:[: the carrier of S, the carrier of S:], the carrier of S:] is non empty set
the addF of S . (y,z) is Element of the carrier of S
(F,S,x,(y + z),a,b) is Element of the carrier of F
(F,S,x,z,a,b) is Element of the carrier of F
(F,S,x,y,a,b) + (F,S,x,z,a,b) is Element of the carrier of F
the addF of F . ((F,S,x,y,a,b),(F,S,x,z,a,b)) is Element of the carrier of F
0. S is V51(S) Element of the carrier of S
the ZeroF of S is Element of the carrier of S
0F is Element of the carrier of S
(F,S,a,b,0F) is Element of the carrier of F
(F,S,0F,a,x) is Element of the carrier of F
(F,S,a,b,0F) * (F,S,0F,a,x) is Element of the carrier of F
the multF of F is V1() V4([: the carrier of F, the carrier of F:]) V5( the carrier of F) V6() V18([: the carrier of F, the carrier of F:], the carrier of F) Element of bool [:[: the carrier of F, the carrier of F:], the carrier of F:]
the multF of F . ((F,S,a,b,0F),(F,S,0F,a,x)) is Element of the carrier of F
(F,S,x,0F,(y + z)) is Element of the carrier of F
((F,S,a,b,0F) * (F,S,0F,a,x)) * (F,S,x,0F,(y + z)) is Element of the carrier of F
the multF of F . (((F,S,a,b,0F) * (F,S,0F,a,x)),(F,S,x,0F,(y + z))) is Element of the carrier of F
(F,S,x,0F,y) is Element of the carrier of F
((F,S,a,b,0F) * (F,S,0F,a,x)) * (F,S,x,0F,y) is Element of the carrier of F
the multF of F . (((F,S,a,b,0F) * (F,S,0F,a,x)),(F,S,x,0F,y)) is Element of the carrier of F
(F,S,x,0F,z) is Element of the carrier of F
((F,S,a,b,0F) * (F,S,0F,a,x)) * (F,S,x,0F,z) is Element of the carrier of F
the multF of F . (((F,S,a,b,0F) * (F,S,0F,a,x)),(F,S,x,0F,z)) is Element of the carrier of F
(F,S,x,0F,y) + (F,S,x,0F,z) is Element of the carrier of F
the addF of F . ((F,S,x,0F,y),(F,S,x,0F,z)) is Element of the carrier of F