REAL is non empty V37() set
NAT is non empty V24() V25() V26() Element of bool REAL
bool REAL is non empty set
COMPLEX is non empty V37() set
RAT is non empty V37() set
INT is non empty V37() set
[:COMPLEX,COMPLEX:] is non empty set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:REAL,REAL:] is non empty set
bool [:REAL,REAL:] is non empty set
[:[:REAL,REAL:],REAL:] is non empty set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is non empty set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is non empty set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is non empty set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is non empty set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is non empty set
[:[:NAT,NAT:],NAT:] is non empty set
bool [:[:NAT,NAT:],NAT:] is non empty set
NAT is non empty V24() V25() V26() set
bool NAT is non empty set
bool NAT is non empty set
K243() is set
1 is non empty V24() V25() V26() V30() V31() V32() V33() ext-real positive non negative Element of NAT
K502() is non empty strict unital Group-like associative commutative cyclic multMagma
the carrier of K502() is non empty V171() set
{} is functional empty V24() V25() V26() V28() V29() V30() V31() V32() V33() FinSequence-membered ext-real non positive non negative set
2 is non empty V24() V25() V26() V30() V31() V32() V33() ext-real positive non negative Element of NAT
3 is non empty V24() V25() V26() V30() V31() V32() V33() ext-real positive non negative Element of NAT
Seg 1 is non empty V37() V44(1) Element of bool NAT
{ b1 where b1 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT : ( 1 <= b1 & b1 <= 1 ) } is set
{1} is non empty set
Seg 2 is non empty V37() V44(2) Element of bool NAT
{ b1 where b1 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT : ( 1 <= b1 & b1 <= 2 ) } is set
{1,2} is non empty set
0 is functional empty V24() V25() V26() V28() V29() V30() V31() V32() V33() FinSequence-membered ext-real non positive non negative Element of NAT
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
F is Element of the carrier of G
R is Element of the carrier of G
F |^ R is Element of the carrier of G
R " is Element of the carrier of G
(R ") * F is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,(R "),F) is Element of the carrier of G
((R ") * F) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * F),R) is Element of the carrier of G
[.F,R.] is Element of the carrier of G
F " is Element of the carrier of G
(F ") * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),(R ")) is Element of the carrier of G
((F ") * (R ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * (R ")),F) is Element of the carrier of G
(((F ") * (R ")) * F) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * (R ")) * F),R) is Element of the carrier of G
F * [.F,R.] is Element of the carrier of G
K84( the carrier of G, the multF of G,F,[.F,R.]) is Element of the carrier of G
F * R is Element of the carrier of G
K84( the carrier of G, the multF of G,F,R) is Element of the carrier of G
((F ") * (R ")) * (F * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * (R ")),(F * R)) is Element of the carrier of G
F * (((F ") * (R ")) * (F * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,F,(((F ") * (R ")) * (F * R))) is Element of the carrier of G
(R ") * (F * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(F * R)) is Element of the carrier of G
(F ") * ((R ") * (F * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),((R ") * (F * R))) is Element of the carrier of G
F * ((F ") * ((R ") * (F * R))) is Element of the carrier of G
K84( the carrier of G, the multF of G,F,((F ") * ((R ") * (F * R)))) is Element of the carrier of G
F * (F ") is Element of the carrier of G
K84( the carrier of G, the multF of G,F,(F ")) is Element of the carrier of G
(F * (F ")) * ((R ") * (F * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(F * (F ")),((R ") * (F * R))) is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * ((R ") * (F * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(1_ G),((R ") * (F * R))) is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
F is Element of the carrier of G
R is Element of the carrier of G
[.F,R.] is Element of the carrier of G
F " is Element of the carrier of G
R " is Element of the carrier of G
(F ") * (R ") is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,(F "),(R ")) is Element of the carrier of G
((F ") * (R ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * (R ")),F) is Element of the carrier of G
(((F ") * (R ")) * F) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * (R ")) * F),R) is Element of the carrier of G
[.F,R.] " is Element of the carrier of G
[.F,(R ").] is Element of the carrier of G
(R ") " is Element of the carrier of G
(F ") * ((R ") ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),((R ") ")) is Element of the carrier of G
((F ") * ((R ") ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * ((R ") ")),F) is Element of the carrier of G
(((F ") * ((R ") ")) * F) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * ((R ") ")) * F),(R ")) is Element of the carrier of G
[.F,(R ").] |^ R is Element of the carrier of G
(R ") * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),[.F,(R ").]) is Element of the carrier of G
((R ") * [.F,(R ").]) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.F,(R ").]),R) is Element of the carrier of G
F * R is Element of the carrier of G
K84( the carrier of G, the multF of G,F,R) is Element of the carrier of G
((F ") * (R ")) * (F * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * (R ")),(F * R)) is Element of the carrier of G
(((F ") * (R ")) * (F * R)) " is Element of the carrier of G
(F * R) " is Element of the carrier of G
((F ") * (R ")) " is Element of the carrier of G
((F * R) ") * (((F ") * (R ")) ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((F * R) "),(((F ") * (R ")) ")) is Element of the carrier of G
(R ") * (F ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(F ")) is Element of the carrier of G
((R ") * (F ")) * (((F ") * (R ")) ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (F ")),(((F ") * (R ")) ")) is Element of the carrier of G
(F ") " is Element of the carrier of G
((R ") ") * ((F ") ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") "),((F ") ")) is Element of the carrier of G
((R ") * (F ")) * (((R ") ") * ((F ") ")) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (F ")),(((R ") ") * ((F ") "))) is Element of the carrier of G
R * F is Element of the carrier of G
K84( the carrier of G, the multF of G,R,F) is Element of the carrier of G
(F ") * (R * F) is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),(R * F)) is Element of the carrier of G
(R ") * ((F ") * (R * F)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),((F ") * (R * F))) is Element of the carrier of G
(F ") * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),R) is Element of the carrier of G
((F ") * R) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * R),F) is Element of the carrier of G
(R ") * (((F ") * R) * F) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(((F ") * R) * F)) is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
((R ") * (((F ") * R) * F)) * (1_ G) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (((F ") * R) * F)),(1_ G)) is Element of the carrier of G
(R ") * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),R) is Element of the carrier of G
((R ") * (((F ") * R) * F)) * ((R ") * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (((F ") * R) * F)),((R ") * R)) is Element of the carrier of G
((R ") * (((F ") * R) * F)) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (((F ") * R) * F)),(R ")) is Element of the carrier of G
(((R ") * (((F ") * R) * F)) * (R ")) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") * (((F ") * R) * F)) * (R ")),R) is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
F is Element of the carrier of G
F " is Element of the carrier of G
R is Element of the carrier of G
[.F,R.] is Element of the carrier of G
R " is Element of the carrier of G
(F ") * (R ") is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,(F "),(R ")) is Element of the carrier of G
((F ") * (R ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * (R ")),F) is Element of the carrier of G
(((F ") * (R ")) * F) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * (R ")) * F),R) is Element of the carrier of G
[.F,R.] " is Element of the carrier of G
[.(F "),R.] is Element of the carrier of G
(F ") " is Element of the carrier of G
((F ") ") * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") "),(R ")) is Element of the carrier of G
(((F ") ") * (R ")) * (F ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") ") * (R ")),(F ")) is Element of the carrier of G
((((F ") ") * (R ")) * (F ")) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((((F ") ") * (R ")) * (F ")),R) is Element of the carrier of G
[.(F "),R.] |^ F is Element of the carrier of G
(F ") * [.(F "),R.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),[.(F "),R.]) is Element of the carrier of G
((F ") * [.(F "),R.]) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * [.(F "),R.]),F) is Element of the carrier of G
F * R is Element of the carrier of G
K84( the carrier of G, the multF of G,F,R) is Element of the carrier of G
((F ") * (R ")) * (F * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * (R ")),(F * R)) is Element of the carrier of G
(((F ") * (R ")) * (F * R)) " is Element of the carrier of G
(F * R) " is Element of the carrier of G
((F ") * (R ")) " is Element of the carrier of G
((F * R) ") * (((F ") * (R ")) ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((F * R) "),(((F ") * (R ")) ")) is Element of the carrier of G
(R ") * (F ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(F ")) is Element of the carrier of G
((R ") * (F ")) * (((F ") * (R ")) ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (F ")),(((F ") * (R ")) ")) is Element of the carrier of G
(R ") " is Element of the carrier of G
((R ") ") * ((F ") ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") "),((F ") ")) is Element of the carrier of G
((R ") * (F ")) * (((R ") ") * ((F ") ")) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (F ")),(((R ") ") * ((F ") "))) is Element of the carrier of G
R * F is Element of the carrier of G
K84( the carrier of G, the multF of G,R,F) is Element of the carrier of G
(F ") * (R * F) is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),(R * F)) is Element of the carrier of G
(R ") * ((F ") * (R * F)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),((F ") * (R * F))) is Element of the carrier of G
(F ") * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),R) is Element of the carrier of G
((F ") * R) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * R),F) is Element of the carrier of G
(R ") * (((F ") * R) * F) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(((F ") * R) * F)) is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * ((R ") * (((F ") * R) * F)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(1_ G),((R ") * (((F ") * R) * F))) is Element of the carrier of G
(F ") * F is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),F) is Element of the carrier of G
((F ") * F) * ((R ") * (((F ") * R) * F)) is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * F),((R ") * (((F ") * R) * F))) is Element of the carrier of G
F * ((R ") * (((F ") * R) * F)) is Element of the carrier of G
K84( the carrier of G, the multF of G,F,((R ") * (((F ") * R) * F))) is Element of the carrier of G
(F ") * (F * ((R ") * (((F ") * R) * F))) is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),(F * ((R ") * (((F ") * R) * F)))) is Element of the carrier of G
F * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,F,(R ")) is Element of the carrier of G
(F * (R ")) * (((F ") * R) * F) is Element of the carrier of G
K84( the carrier of G, the multF of G,(F * (R ")),(((F ") * R) * F)) is Element of the carrier of G
(F ") * ((F * (R ")) * (((F ") * R) * F)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),((F * (R ")) * (((F ") * R) * F))) is Element of the carrier of G
(F * (R ")) * ((F ") * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(F * (R ")),((F ") * R)) is Element of the carrier of G
((F * (R ")) * ((F ") * R)) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F * (R ")) * ((F ") * R)),F) is Element of the carrier of G
(F ") * (((F * (R ")) * ((F ") * R)) * F) is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),(((F * (R ")) * ((F ") * R)) * F)) is Element of the carrier of G
(F ") * ((F * (R ")) * ((F ") * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),((F * (R ")) * ((F ") * R))) is Element of the carrier of G
((F ") * ((F * (R ")) * ((F ") * R))) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * ((F * (R ")) * ((F ") * R))),F) is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
F is Element of the carrier of G
R is Element of the carrier of G
R " is Element of the carrier of G
[.F,(R ").] is Element of the carrier of G
F " is Element of the carrier of G
(R ") " is Element of the carrier of G
(F ") * ((R ") ") is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,(F "),((R ") ")) is Element of the carrier of G
((F ") * ((R ") ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * ((R ") ")),F) is Element of the carrier of G
(((F ") * ((R ") ")) * F) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * ((R ") ")) * F),(R ")) is Element of the carrier of G
[.F,(R ").] |^ R is Element of the carrier of G
(R ") * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),[.F,(R ").]) is Element of the carrier of G
((R ") * [.F,(R ").]) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.F,(R ").]),R) is Element of the carrier of G
([.F,(R ").] |^ R) " is Element of the carrier of G
[.(R "),F.] is Element of the carrier of G
((R ") ") * (F ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") "),(F ")) is Element of the carrier of G
(((R ") ") * (F ")) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") ") * (F ")),(R ")) is Element of the carrier of G
((((R ") ") * (F ")) * (R ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((((R ") ") * (F ")) * (R ")),F) is Element of the carrier of G
[.(R "),F.] |^ R is Element of the carrier of G
(R ") * [.(R "),F.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),[.(R "),F.]) is Element of the carrier of G
((R ") * [.(R "),F.]) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.(R "),F.]),R) is Element of the carrier of G
[.F,(R ").] " is Element of the carrier of G
([.F,(R ").] ") |^ R is Element of the carrier of G
(R ") * ([.F,(R ").] ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),([.F,(R ").] ")) is Element of the carrier of G
((R ") * ([.F,(R ").] ")) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * ([.F,(R ").] ")),R) is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
F is Element of the carrier of G
R is Element of the carrier of G
R " is Element of the carrier of G
[.F,(R ").] is Element of the carrier of G
F " is Element of the carrier of G
(R ") " is Element of the carrier of G
(F ") * ((R ") ") is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,(F "),((R ") ")) is Element of the carrier of G
((F ") * ((R ") ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * ((R ") ")),F) is Element of the carrier of G
(((F ") * ((R ") ")) * F) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * ((R ") ")) * F),(R ")) is Element of the carrier of G
[.F,(R ").] |^ R is Element of the carrier of G
(R ") * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),[.F,(R ").]) is Element of the carrier of G
((R ") * [.F,(R ").]) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.F,(R ").]),R) is Element of the carrier of G
c4 is Element of the carrier of G
[.F,(R "),c4.] is Element of the carrier of G
[.[.F,(R ").],c4.] is Element of the carrier of G
[.F,(R ").] " is Element of the carrier of G
c4 " is Element of the carrier of G
([.F,(R ").] ") * (c4 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,([.F,(R ").] "),(c4 ")) is Element of the carrier of G
(([.F,(R ").] ") * (c4 ")) * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(R ").] ") * (c4 ")),[.F,(R ").]) is Element of the carrier of G
((([.F,(R ").] ") * (c4 ")) * [.F,(R ").]) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.F,(R ").] ") * (c4 ")) * [.F,(R ").]),c4) is Element of the carrier of G
[.F,(R "),c4.] |^ R is Element of the carrier of G
(R ") * [.F,(R "),c4.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),[.F,(R "),c4.]) is Element of the carrier of G
((R ") * [.F,(R "),c4.]) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.F,(R "),c4.]),R) is Element of the carrier of G
c4 |^ R is Element of the carrier of G
(R ") * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),c4) is Element of the carrier of G
((R ") * c4) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * c4),R) is Element of the carrier of G
[.([.F,(R ").] |^ R),(c4 |^ R).] is Element of the carrier of G
([.F,(R ").] |^ R) " is Element of the carrier of G
(c4 |^ R) " is Element of the carrier of G
(([.F,(R ").] |^ R) ") * ((c4 |^ R) ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(R ").] |^ R) "),((c4 |^ R) ")) is Element of the carrier of G
((([.F,(R ").] |^ R) ") * ((c4 |^ R) ")) * ([.F,(R ").] |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.F,(R ").] |^ R) ") * ((c4 |^ R) ")),([.F,(R ").] |^ R)) is Element of the carrier of G
(((([.F,(R ").] |^ R) ") * ((c4 |^ R) ")) * ([.F,(R ").] |^ R)) * (c4 |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(((([.F,(R ").] |^ R) ") * ((c4 |^ R) ")) * ([.F,(R ").] |^ R)),(c4 |^ R)) is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
([.F,(R ").] ") * (1_ G) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.F,(R ").] "),(1_ G)) is Element of the carrier of G
(([.F,(R ").] ") * (1_ G)) * (c4 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(R ").] ") * (1_ G)),(c4 ")) is Element of the carrier of G
((([.F,(R ").] ") * (1_ G)) * (c4 ")) * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.F,(R ").] ") * (1_ G)) * (c4 ")),[.F,(R ").]) is Element of the carrier of G
(((([.F,(R ").] ") * (1_ G)) * (c4 ")) * [.F,(R ").]) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,(((([.F,(R ").] ") * (1_ G)) * (c4 ")) * [.F,(R ").]),c4) is Element of the carrier of G
(R ") * ((((([.F,(R ").] ") * (1_ G)) * (c4 ")) * [.F,(R ").]) * c4) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),((((([.F,(R ").] ") * (1_ G)) * (c4 ")) * [.F,(R ").]) * c4)) is Element of the carrier of G
((R ") * ((((([.F,(R ").] ") * (1_ G)) * (c4 ")) * [.F,(R ").]) * c4)) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * ((((([.F,(R ").] ") * (1_ G)) * (c4 ")) * [.F,(R ").]) * c4)),R) is Element of the carrier of G
R * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,R,(R ")) is Element of the carrier of G
([.F,(R ").] ") * (R * (R ")) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.F,(R ").] "),(R * (R "))) is Element of the carrier of G
(([.F,(R ").] ") * (R * (R "))) * (c4 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(R ").] ") * (R * (R "))),(c4 ")) is Element of the carrier of G
((([.F,(R ").] ") * (R * (R "))) * (c4 ")) * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.F,(R ").] ") * (R * (R "))) * (c4 ")),[.F,(R ").]) is Element of the carrier of G
(((([.F,(R ").] ") * (R * (R "))) * (c4 ")) * [.F,(R ").]) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,(((([.F,(R ").] ") * (R * (R "))) * (c4 ")) * [.F,(R ").]),c4) is Element of the carrier of G
(R ") * ((((([.F,(R ").] ") * (R * (R "))) * (c4 ")) * [.F,(R ").]) * c4) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),((((([.F,(R ").] ") * (R * (R "))) * (c4 ")) * [.F,(R ").]) * c4)) is Element of the carrier of G
((R ") * ((((([.F,(R ").] ") * (R * (R "))) * (c4 ")) * [.F,(R ").]) * c4)) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * ((((([.F,(R ").] ") * (R * (R "))) * (c4 ")) * [.F,(R ").]) * c4)),R) is Element of the carrier of G
([.F,(R ").] ") * R is Element of the carrier of G
K84( the carrier of G, the multF of G,([.F,(R ").] "),R) is Element of the carrier of G
(([.F,(R ").] ") * R) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(R ").] ") * R),(R ")) is Element of the carrier of G
((([.F,(R ").] ") * R) * (R ")) * (c4 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.F,(R ").] ") * R) * (R ")),(c4 ")) is Element of the carrier of G
(((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(((([.F,(R ").] ") * R) * (R ")) * (c4 ")),[.F,(R ").]) is Element of the carrier of G
((((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * [.F,(R ").]) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,((((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * [.F,(R ").]),c4) is Element of the carrier of G
(R ") * (((((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * [.F,(R ").]) * c4) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(((((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * [.F,(R ").]) * c4)) is Element of the carrier of G
((R ") * (((((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * [.F,(R ").]) * c4)) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (((((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * [.F,(R ").]) * c4)),R) is Element of the carrier of G
[.F,(R ").] * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,[.F,(R ").],c4) is Element of the carrier of G
(((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * ([.F,(R ").] * c4) is Element of the carrier of G
K84( the carrier of G, the multF of G,(((([.F,(R ").] ") * R) * (R ")) * (c4 ")),([.F,(R ").] * c4)) is Element of the carrier of G
(R ") * ((((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * ([.F,(R ").] * c4)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),((((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * ([.F,(R ").] * c4))) is Element of the carrier of G
((R ") * ((((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * ([.F,(R ").] * c4))) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * ((((([.F,(R ").] ") * R) * (R ")) * (c4 ")) * ([.F,(R ").] * c4))),R) is Element of the carrier of G
(c4 ") * ([.F,(R ").] * c4) is Element of the carrier of G
K84( the carrier of G, the multF of G,(c4 "),([.F,(R ").] * c4)) is Element of the carrier of G
((([.F,(R ").] ") * R) * (R ")) * ((c4 ") * ([.F,(R ").] * c4)) is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.F,(R ").] ") * R) * (R ")),((c4 ") * ([.F,(R ").] * c4))) is Element of the carrier of G
(R ") * (((([.F,(R ").] ") * R) * (R ")) * ((c4 ") * ([.F,(R ").] * c4))) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(((([.F,(R ").] ") * R) * (R ")) * ((c4 ") * ([.F,(R ").] * c4)))) is Element of the carrier of G
((R ") * (((([.F,(R ").] ") * R) * (R ")) * ((c4 ") * ([.F,(R ").] * c4)))) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (((([.F,(R ").] ") * R) * (R ")) * ((c4 ") * ([.F,(R ").] * c4)))),R) is Element of the carrier of G
(R ") * ((c4 ") * ([.F,(R ").] * c4)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),((c4 ") * ([.F,(R ").] * c4))) is Element of the carrier of G
(([.F,(R ").] ") * R) * ((R ") * ((c4 ") * ([.F,(R ").] * c4))) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(R ").] ") * R),((R ") * ((c4 ") * ([.F,(R ").] * c4)))) is Element of the carrier of G
(R ") * ((([.F,(R ").] ") * R) * ((R ") * ((c4 ") * ([.F,(R ").] * c4)))) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),((([.F,(R ").] ") * R) * ((R ") * ((c4 ") * ([.F,(R ").] * c4))))) is Element of the carrier of G
((R ") * ((([.F,(R ").] ") * R) * ((R ") * ((c4 ") * ([.F,(R ").] * c4))))) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * ((([.F,(R ").] ") * R) * ((R ") * ((c4 ") * ([.F,(R ").] * c4))))),R) is Element of the carrier of G
(R ") * (([.F,(R ").] ") * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(([.F,(R ").] ") * R)) is Element of the carrier of G
((R ") * (([.F,(R ").] ") * R)) * ((R ") * ((c4 ") * ([.F,(R ").] * c4))) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (([.F,(R ").] ") * R)),((R ") * ((c4 ") * ([.F,(R ").] * c4)))) is Element of the carrier of G
(((R ") * (([.F,(R ").] ") * R)) * ((R ") * ((c4 ") * ([.F,(R ").] * c4)))) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") * (([.F,(R ").] ") * R)) * ((R ") * ((c4 ") * ([.F,(R ").] * c4)))),R) is Element of the carrier of G
([.F,(R ").] ") |^ R is Element of the carrier of G
(R ") * ([.F,(R ").] ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),([.F,(R ").] ")) is Element of the carrier of G
((R ") * ([.F,(R ").] ")) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * ([.F,(R ").] ")),R) is Element of the carrier of G
(([.F,(R ").] ") |^ R) * ((R ") * ((c4 ") * ([.F,(R ").] * c4))) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(R ").] ") |^ R),((R ") * ((c4 ") * ([.F,(R ").] * c4)))) is Element of the carrier of G
((([.F,(R ").] ") |^ R) * ((R ") * ((c4 ") * ([.F,(R ").] * c4)))) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.F,(R ").] ") |^ R) * ((R ") * ((c4 ") * ([.F,(R ").] * c4)))),R) is Element of the carrier of G
((R ") * ((c4 ") * ([.F,(R ").] * c4))) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * ((c4 ") * ([.F,(R ").] * c4))),R) is Element of the carrier of G
(([.F,(R ").] ") |^ R) * (((R ") * ((c4 ") * ([.F,(R ").] * c4))) * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(R ").] ") |^ R),(((R ") * ((c4 ") * ([.F,(R ").] * c4))) * R)) is Element of the carrier of G
[.F,(R ").] |^ c4 is Element of the carrier of G
(c4 ") * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(c4 "),[.F,(R ").]) is Element of the carrier of G
((c4 ") * [.F,(R ").]) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,((c4 ") * [.F,(R ").]),c4) is Element of the carrier of G
([.F,(R ").] |^ c4) |^ R is Element of the carrier of G
(R ") * ([.F,(R ").] |^ c4) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),([.F,(R ").] |^ c4)) is Element of the carrier of G
((R ") * ([.F,(R ").] |^ c4)) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * ([.F,(R ").] |^ c4)),R) is Element of the carrier of G
(([.F,(R ").] ") |^ R) * (([.F,(R ").] |^ c4) |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(R ").] ") |^ R),(([.F,(R ").] |^ c4) |^ R)) is Element of the carrier of G
[.(R "),F.] is Element of the carrier of G
((R ") ") * (F ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") "),(F ")) is Element of the carrier of G
(((R ") ") * (F ")) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") ") * (F ")),(R ")) is Element of the carrier of G
((((R ") ") * (F ")) * (R ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((((R ") ") * (F ")) * (R ")),F) is Element of the carrier of G
[.(R "),F.] |^ R is Element of the carrier of G
(R ") * [.(R "),F.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),[.(R "),F.]) is Element of the carrier of G
((R ") * [.(R "),F.]) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.(R "),F.]),R) is Element of the carrier of G
([.(R "),F.] |^ R) * (([.F,(R ").] |^ c4) |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),(([.F,(R ").] |^ c4) |^ R)) is Element of the carrier of G
c4 * R is Element of the carrier of G
K84( the carrier of G, the multF of G,c4,R) is Element of the carrier of G
[.F,(R ").] |^ (c4 * R) is Element of the carrier of G
(c4 * R) " is Element of the carrier of G
((c4 * R) ") * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,((c4 * R) "),[.F,(R ").]) is Element of the carrier of G
(((c4 * R) ") * [.F,(R ").]) * (c4 * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(((c4 * R) ") * [.F,(R ").]),(c4 * R)) is Element of the carrier of G
([.(R "),F.] |^ R) * ([.F,(R ").] |^ (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),([.F,(R ").] |^ (c4 * R))) is Element of the carrier of G
([.(R "),F.] |^ R) * ((c4 |^ R) ") is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),((c4 |^ R) ")) is Element of the carrier of G
(([.(R "),F.] |^ R) * ((c4 |^ R) ")) * ([.F,(R ").] |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.(R "),F.] |^ R) * ((c4 |^ R) ")),([.F,(R ").] |^ R)) is Element of the carrier of G
((([.(R "),F.] |^ R) * ((c4 |^ R) ")) * ([.F,(R ").] |^ R)) * (c4 |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.(R "),F.] |^ R) * ((c4 |^ R) ")) * ([.F,(R ").] |^ R)),(c4 |^ R)) is Element of the carrier of G
(c4 ") |^ R is Element of the carrier of G
(R ") * (c4 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(c4 ")) is Element of the carrier of G
((R ") * (c4 ")) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (c4 ")),R) is Element of the carrier of G
([.(R "),F.] |^ R) * ((c4 ") |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),((c4 ") |^ R)) is Element of the carrier of G
(([.(R "),F.] |^ R) * ((c4 ") |^ R)) * ([.F,(R ").] |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.(R "),F.] |^ R) * ((c4 ") |^ R)),([.F,(R ").] |^ R)) is Element of the carrier of G
((([.(R "),F.] |^ R) * ((c4 ") |^ R)) * ([.F,(R ").] |^ R)) * (c4 |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.(R "),F.] |^ R) * ((c4 ") |^ R)) * ([.F,(R ").] |^ R)),(c4 |^ R)) is Element of the carrier of G
([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),(((R ") * (c4 ")) * R)) is Element of the carrier of G
(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)) * (((R ") * [.F,(R ").]) * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)),(((R ") * [.F,(R ").]) * R)) is Element of the carrier of G
(R ") * (c4 * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(c4 * R)) is Element of the carrier of G
((([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)) * (((R ") * [.F,(R ").]) * R)) * ((R ") * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)) * (((R ") * [.F,(R ").]) * R)),((R ") * (c4 * R))) is Element of the carrier of G
(((R ") * [.F,(R ").]) * R) * ((R ") * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") * [.F,(R ").]) * R),((R ") * (c4 * R))) is Element of the carrier of G
(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)) * ((((R ") * [.F,(R ").]) * R) * ((R ") * (c4 * R))) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)),((((R ") * [.F,(R ").]) * R) * ((R ") * (c4 * R)))) is Element of the carrier of G
R * ((R ") * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,R,((R ") * (c4 * R))) is Element of the carrier of G
((R ") * [.F,(R ").]) * (R * ((R ") * (c4 * R))) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.F,(R ").]),(R * ((R ") * (c4 * R)))) is Element of the carrier of G
(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)) * (((R ") * [.F,(R ").]) * (R * ((R ") * (c4 * R)))) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)),(((R ") * [.F,(R ").]) * (R * ((R ") * (c4 * R))))) is Element of the carrier of G
(R * (R ")) * (c4 * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R * (R ")),(c4 * R)) is Element of the carrier of G
((R ") * [.F,(R ").]) * ((R * (R ")) * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.F,(R ").]),((R * (R ")) * (c4 * R))) is Element of the carrier of G
(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)) * (((R ") * [.F,(R ").]) * ((R * (R ")) * (c4 * R))) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)),(((R ") * [.F,(R ").]) * ((R * (R ")) * (c4 * R)))) is Element of the carrier of G
(1_ G) * (c4 * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(1_ G),(c4 * R)) is Element of the carrier of G
((R ") * [.F,(R ").]) * ((1_ G) * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.F,(R ").]),((1_ G) * (c4 * R))) is Element of the carrier of G
(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)) * (((R ") * [.F,(R ").]) * ((1_ G) * (c4 * R))) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)),(((R ") * [.F,(R ").]) * ((1_ G) * (c4 * R)))) is Element of the carrier of G
((R ") * [.F,(R ").]) * (c4 * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.F,(R ").]),(c4 * R)) is Element of the carrier of G
(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)) * (((R ") * [.F,(R ").]) * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.(R "),F.] |^ R) * (((R ") * (c4 ")) * R)),(((R ") * [.F,(R ").]) * (c4 * R))) is Element of the carrier of G
(((R ") * (c4 ")) * R) * (((R ") * [.F,(R ").]) * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") * (c4 ")) * R),(((R ") * [.F,(R ").]) * (c4 * R))) is Element of the carrier of G
([.(R "),F.] |^ R) * ((((R ") * (c4 ")) * R) * (((R ") * [.F,(R ").]) * (c4 * R))) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),((((R ") * (c4 ")) * R) * (((R ") * [.F,(R ").]) * (c4 * R)))) is Element of the carrier of G
R * (((R ") * [.F,(R ").]) * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,R,(((R ") * [.F,(R ").]) * (c4 * R))) is Element of the carrier of G
((R ") * (c4 ")) * (R * (((R ") * [.F,(R ").]) * (c4 * R))) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (c4 ")),(R * (((R ") * [.F,(R ").]) * (c4 * R)))) is Element of the carrier of G
([.(R "),F.] |^ R) * (((R ") * (c4 ")) * (R * (((R ") * [.F,(R ").]) * (c4 * R)))) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),(((R ") * (c4 ")) * (R * (((R ") * [.F,(R ").]) * (c4 * R))))) is Element of the carrier of G
R * ((R ") * [.F,(R ").]) is Element of the carrier of G
K84( the carrier of G, the multF of G,R,((R ") * [.F,(R ").])) is Element of the carrier of G
(R * ((R ") * [.F,(R ").])) * (c4 * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R * ((R ") * [.F,(R ").])),(c4 * R)) is Element of the carrier of G
((R ") * (c4 ")) * ((R * ((R ") * [.F,(R ").])) * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (c4 ")),((R * ((R ") * [.F,(R ").])) * (c4 * R))) is Element of the carrier of G
([.(R "),F.] |^ R) * (((R ") * (c4 ")) * ((R * ((R ") * [.F,(R ").])) * (c4 * R))) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),(((R ") * (c4 ")) * ((R * ((R ") * [.F,(R ").])) * (c4 * R)))) is Element of the carrier of G
(R * (R ")) * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(R * (R ")),[.F,(R ").]) is Element of the carrier of G
((R * (R ")) * [.F,(R ").]) * (c4 * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R * (R ")) * [.F,(R ").]),(c4 * R)) is Element of the carrier of G
((R ") * (c4 ")) * (((R * (R ")) * [.F,(R ").]) * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (c4 ")),(((R * (R ")) * [.F,(R ").]) * (c4 * R))) is Element of the carrier of G
([.(R "),F.] |^ R) * (((R ") * (c4 ")) * (((R * (R ")) * [.F,(R ").]) * (c4 * R))) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),(((R ") * (c4 ")) * (((R * (R ")) * [.F,(R ").]) * (c4 * R)))) is Element of the carrier of G
(1_ G) * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(1_ G),[.F,(R ").]) is Element of the carrier of G
((1_ G) * [.F,(R ").]) * (c4 * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((1_ G) * [.F,(R ").]),(c4 * R)) is Element of the carrier of G
((R ") * (c4 ")) * (((1_ G) * [.F,(R ").]) * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (c4 ")),(((1_ G) * [.F,(R ").]) * (c4 * R))) is Element of the carrier of G
([.(R "),F.] |^ R) * (((R ") * (c4 ")) * (((1_ G) * [.F,(R ").]) * (c4 * R))) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),(((R ") * (c4 ")) * (((1_ G) * [.F,(R ").]) * (c4 * R)))) is Element of the carrier of G
[.F,(R ").] * (c4 * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,[.F,(R ").],(c4 * R)) is Element of the carrier of G
((R ") * (c4 ")) * ([.F,(R ").] * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (c4 ")),([.F,(R ").] * (c4 * R))) is Element of the carrier of G
([.(R "),F.] |^ R) * (((R ") * (c4 ")) * ([.F,(R ").] * (c4 * R))) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),(((R ") * (c4 ")) * ([.F,(R ").] * (c4 * R)))) is Element of the carrier of G
((R ") * (c4 ")) * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (c4 ")),[.F,(R ").]) is Element of the carrier of G
(((R ") * (c4 ")) * [.F,(R ").]) * (c4 * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") * (c4 ")) * [.F,(R ").]),(c4 * R)) is Element of the carrier of G
([.(R "),F.] |^ R) * ((((R ") * (c4 ")) * [.F,(R ").]) * (c4 * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,([.(R "),F.] |^ R),((((R ") * (c4 ")) * [.F,(R ").]) * (c4 * R))) is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
F is Element of the carrier of G
R is Element of the carrier of G
R " is Element of the carrier of G
[.F,(R ").] is Element of the carrier of G
F " is Element of the carrier of G
(R ") " is Element of the carrier of G
(F ") * ((R ") ") is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,(F "),((R ") ")) is Element of the carrier of G
((F ") * ((R ") ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * ((R ") ")),F) is Element of the carrier of G
(((F ") * ((R ") ")) * F) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * ((R ") ")) * F),(R ")) is Element of the carrier of G
[.F,(R ").] |^ R is Element of the carrier of G
(R ") * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),[.F,(R ").]) is Element of the carrier of G
((R ") * [.F,(R ").]) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.F,(R ").]),R) is Element of the carrier of G
[.R,F.] is Element of the carrier of G
(R ") * (F ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(F ")) is Element of the carrier of G
((R ") * (F ")) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (F ")),R) is Element of the carrier of G
(((R ") * (F ")) * R) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") * (F ")) * R),F) is Element of the carrier of G
((((F ") * ((R ") ")) * F) * (R ")) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((((F ") * ((R ") ")) * F) * (R ")),R) is Element of the carrier of G
(R ") * (((((F ") * ((R ") ")) * F) * (R ")) * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(((((F ") * ((R ") ")) * F) * (R ")) * R)) is Element of the carrier of G
(R ") * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),R) is Element of the carrier of G
(((F ") * ((R ") ")) * F) * ((R ") * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * ((R ") ")) * F),((R ") * R)) is Element of the carrier of G
(R ") * ((((F ") * ((R ") ")) * F) * ((R ") * R)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),((((F ") * ((R ") ")) * F) * ((R ") * R))) is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
(((F ") * ((R ") ")) * F) * (1_ G) is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * ((R ") ")) * F),(1_ G)) is Element of the carrier of G
(R ") * ((((F ") * ((R ") ")) * F) * (1_ G)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),((((F ") * ((R ") ")) * F) * (1_ G))) is Element of the carrier of G
(R ") * (((F ") * ((R ") ")) * F) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(((F ") * ((R ") ")) * F)) is Element of the carrier of G
R * F is Element of the carrier of G
K84( the carrier of G, the multF of G,R,F) is Element of the carrier of G
(F ") * (R * F) is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),(R * F)) is Element of the carrier of G
(R ") * ((F ") * (R * F)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),((F ") * (R * F))) is Element of the carrier of G
((R ") * (F ")) * (R * F) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (F ")),(R * F)) is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
F is Element of the carrier of G
R is Element of the carrier of G
R " is Element of the carrier of G
c4 is Element of the carrier of G
[.F,(R "),c4.] is Element of the carrier of G
[.F,(R ").] is Element of the carrier of G
F " is Element of the carrier of G
(R ") " is Element of the carrier of G
(F ") * ((R ") ") is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,(F "),((R ") ")) is Element of the carrier of G
((F ") * ((R ") ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * ((R ") ")),F) is Element of the carrier of G
(((F ") * ((R ") ")) * F) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * ((R ") ")) * F),(R ")) is Element of the carrier of G
[.[.F,(R ").],c4.] is Element of the carrier of G
[.F,(R ").] " is Element of the carrier of G
c4 " is Element of the carrier of G
([.F,(R ").] ") * (c4 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,([.F,(R ").] "),(c4 ")) is Element of the carrier of G
(([.F,(R ").] ") * (c4 ")) * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(R ").] ") * (c4 ")),[.F,(R ").]) is Element of the carrier of G
((([.F,(R ").] ") * (c4 ")) * [.F,(R ").]) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.F,(R ").] ") * (c4 ")) * [.F,(R ").]),c4) is Element of the carrier of G
[.F,(R "),c4.] |^ R is Element of the carrier of G
(R ") * [.F,(R "),c4.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),[.F,(R "),c4.]) is Element of the carrier of G
((R ") * [.F,(R "),c4.]) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.F,(R "),c4.]),R) is Element of the carrier of G
c4 |^ R is Element of the carrier of G
(R ") * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),c4) is Element of the carrier of G
((R ") * c4) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * c4),R) is Element of the carrier of G
[.R,F,(c4 |^ R).] is Element of the carrier of G
[.R,F.] is Element of the carrier of G
(R ") * (F ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(F ")) is Element of the carrier of G
((R ") * (F ")) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (F ")),R) is Element of the carrier of G
(((R ") * (F ")) * R) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") * (F ")) * R),F) is Element of the carrier of G
[.[.R,F.],(c4 |^ R).] is Element of the carrier of G
[.R,F.] " is Element of the carrier of G
(c4 |^ R) " is Element of the carrier of G
([.R,F.] ") * ((c4 |^ R) ") is Element of the carrier of G
K84( the carrier of G, the multF of G,([.R,F.] "),((c4 |^ R) ")) is Element of the carrier of G
(([.R,F.] ") * ((c4 |^ R) ")) * [.R,F.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.R,F.] ") * ((c4 |^ R) ")),[.R,F.]) is Element of the carrier of G
((([.R,F.] ") * ((c4 |^ R) ")) * [.R,F.]) * (c4 |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.R,F.] ") * ((c4 |^ R) ")) * [.R,F.]),(c4 |^ R)) is Element of the carrier of G
[.F,(R ").] |^ R is Element of the carrier of G
(R ") * [.F,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),[.F,(R ").]) is Element of the carrier of G
((R ") * [.F,(R ").]) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.F,(R ").]),R) is Element of the carrier of G
[.([.F,(R ").] |^ R),(c4 |^ R).] is Element of the carrier of G
([.F,(R ").] |^ R) " is Element of the carrier of G
(([.F,(R ").] |^ R) ") * ((c4 |^ R) ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(R ").] |^ R) "),((c4 |^ R) ")) is Element of the carrier of G
((([.F,(R ").] |^ R) ") * ((c4 |^ R) ")) * ([.F,(R ").] |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.F,(R ").] |^ R) ") * ((c4 |^ R) ")),([.F,(R ").] |^ R)) is Element of the carrier of G
(((([.F,(R ").] |^ R) ") * ((c4 |^ R) ")) * ([.F,(R ").] |^ R)) * (c4 |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(((([.F,(R ").] |^ R) ") * ((c4 |^ R) ")) * ([.F,(R ").] |^ R)),(c4 |^ R)) is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
F is Element of the carrier of G
R is Element of the carrier of G
F |^ R is Element of the carrier of G
R " is Element of the carrier of G
(R ") * F is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,(R "),F) is Element of the carrier of G
((R ") * F) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * F),R) is Element of the carrier of G
c4 is Element of the carrier of G
c4 |^ F is Element of the carrier of G
F " is Element of the carrier of G
(F ") * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),c4) is Element of the carrier of G
((F ") * c4) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * c4),F) is Element of the carrier of G
[.F,R,(c4 |^ F).] is Element of the carrier of G
[.F,R.] is Element of the carrier of G
(F ") * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),(R ")) is Element of the carrier of G
((F ") * (R ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * (R ")),F) is Element of the carrier of G
(((F ") * (R ")) * F) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * (R ")) * F),R) is Element of the carrier of G
[.[.F,R.],(c4 |^ F).] is Element of the carrier of G
[.F,R.] " is Element of the carrier of G
(c4 |^ F) " is Element of the carrier of G
([.F,R.] ") * ((c4 |^ F) ") is Element of the carrier of G
K84( the carrier of G, the multF of G,([.F,R.] "),((c4 |^ F) ")) is Element of the carrier of G
(([.F,R.] ") * ((c4 |^ F) ")) * [.F,R.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,R.] ") * ((c4 |^ F) ")),[.F,R.]) is Element of the carrier of G
((([.F,R.] ") * ((c4 |^ F) ")) * [.F,R.]) * (c4 |^ F) is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.F,R.] ") * ((c4 |^ F) ")) * [.F,R.]),(c4 |^ F)) is Element of the carrier of G
R |^ c4 is Element of the carrier of G
c4 " is Element of the carrier of G
(c4 ") * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(c4 "),R) is Element of the carrier of G
((c4 ") * R) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,((c4 ") * R),c4) is Element of the carrier of G
[.c4,F,(R |^ c4).] is Element of the carrier of G
[.c4,F.] is Element of the carrier of G
(c4 ") * (F ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(c4 "),(F ")) is Element of the carrier of G
((c4 ") * (F ")) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,((c4 ") * (F ")),c4) is Element of the carrier of G
(((c4 ") * (F ")) * c4) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,(((c4 ") * (F ")) * c4),F) is Element of the carrier of G
[.[.c4,F.],(R |^ c4).] is Element of the carrier of G
[.c4,F.] " is Element of the carrier of G
(R |^ c4) " is Element of the carrier of G
([.c4,F.] ") * ((R |^ c4) ") is Element of the carrier of G
K84( the carrier of G, the multF of G,([.c4,F.] "),((R |^ c4) ")) is Element of the carrier of G
(([.c4,F.] ") * ((R |^ c4) ")) * [.c4,F.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.c4,F.] ") * ((R |^ c4) ")),[.c4,F.]) is Element of the carrier of G
((([.c4,F.] ") * ((R |^ c4) ")) * [.c4,F.]) * (R |^ c4) is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.c4,F.] ") * ((R |^ c4) ")) * [.c4,F.]),(R |^ c4)) is Element of the carrier of G
[.F,R,(c4 |^ F).] * [.c4,F,(R |^ c4).] is Element of the carrier of G
K84( the carrier of G, the multF of G,[.F,R,(c4 |^ F).],[.c4,F,(R |^ c4).]) is Element of the carrier of G
[.R,c4,(F |^ R).] is Element of the carrier of G
[.R,c4.] is Element of the carrier of G
(R ") * (c4 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),(c4 ")) is Element of the carrier of G
((R ") * (c4 ")) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * (c4 ")),R) is Element of the carrier of G
(((R ") * (c4 ")) * R) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") * (c4 ")) * R),c4) is Element of the carrier of G
[.[.R,c4.],(F |^ R).] is Element of the carrier of G
[.R,c4.] " is Element of the carrier of G
(F |^ R) " is Element of the carrier of G
([.R,c4.] ") * ((F |^ R) ") is Element of the carrier of G
K84( the carrier of G, the multF of G,([.R,c4.] "),((F |^ R) ")) is Element of the carrier of G
(([.R,c4.] ") * ((F |^ R) ")) * [.R,c4.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.R,c4.] ") * ((F |^ R) ")),[.R,c4.]) is Element of the carrier of G
((([.R,c4.] ") * ((F |^ R) ")) * [.R,c4.]) * (F |^ R) is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.R,c4.] ") * ((F |^ R) ")) * [.R,c4.]),(F |^ R)) is Element of the carrier of G
([.F,R,(c4 |^ F).] * [.c4,F,(R |^ c4).]) * [.R,c4,(F |^ R).] is Element of the carrier of G
K84( the carrier of G, the multF of G,([.F,R,(c4 |^ F).] * [.c4,F,(R |^ c4).]),[.R,c4,(F |^ R).]) is Element of the carrier of G
[.R,(F "),c4.] is Element of the carrier of G
[.R,(F ").] is Element of the carrier of G
(F ") " is Element of the carrier of G
(R ") * ((F ") ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),((F ") ")) is Element of the carrier of G
((R ") * ((F ") ")) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * ((F ") ")),R) is Element of the carrier of G
(((R ") * ((F ") ")) * R) * (F ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") * ((F ") ")) * R),(F ")) is Element of the carrier of G
[.[.R,(F ").],c4.] is Element of the carrier of G
[.R,(F ").] " is Element of the carrier of G
([.R,(F ").] ") * (c4 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,([.R,(F ").] "),(c4 ")) is Element of the carrier of G
(([.R,(F ").] ") * (c4 ")) * [.R,(F ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.R,(F ").] ") * (c4 ")),[.R,(F ").]) is Element of the carrier of G
((([.R,(F ").] ") * (c4 ")) * [.R,(F ").]) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.R,(F ").] ") * (c4 ")) * [.R,(F ").]),c4) is Element of the carrier of G
[.R,(F "),c4.] |^ F is Element of the carrier of G
(F ") * [.R,(F "),c4.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),[.R,(F "),c4.]) is Element of the carrier of G
((F ") * [.R,(F "),c4.]) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * [.R,(F "),c4.]),F) is Element of the carrier of G
[.F,(c4 "),R.] is Element of the carrier of G
[.F,(c4 ").] is Element of the carrier of G
(c4 ") " is Element of the carrier of G
(F ") * ((c4 ") ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(F "),((c4 ") ")) is Element of the carrier of G
((F ") * ((c4 ") ")) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((F ") * ((c4 ") ")),F) is Element of the carrier of G
(((F ") * ((c4 ") ")) * F) * (c4 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((F ") * ((c4 ") ")) * F),(c4 ")) is Element of the carrier of G
[.[.F,(c4 ").],R.] is Element of the carrier of G
[.F,(c4 ").] " is Element of the carrier of G
([.F,(c4 ").] ") * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,([.F,(c4 ").] "),(R ")) is Element of the carrier of G
(([.F,(c4 ").] ") * (R ")) * [.F,(c4 ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.F,(c4 ").] ") * (R ")),[.F,(c4 ").]) is Element of the carrier of G
((([.F,(c4 ").] ") * (R ")) * [.F,(c4 ").]) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.F,(c4 ").] ") * (R ")) * [.F,(c4 ").]),R) is Element of the carrier of G
[.F,(c4 "),R.] |^ c4 is Element of the carrier of G
(c4 ") * [.F,(c4 "),R.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(c4 "),[.F,(c4 "),R.]) is Element of the carrier of G
((c4 ") * [.F,(c4 "),R.]) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,((c4 ") * [.F,(c4 "),R.]),c4) is Element of the carrier of G
[.c4,(R "),F.] is Element of the carrier of G
[.c4,(R ").] is Element of the carrier of G
(R ") " is Element of the carrier of G
(c4 ") * ((R ") ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(c4 "),((R ") ")) is Element of the carrier of G
((c4 ") * ((R ") ")) * c4 is Element of the carrier of G
K84( the carrier of G, the multF of G,((c4 ") * ((R ") ")),c4) is Element of the carrier of G
(((c4 ") * ((R ") ")) * c4) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((c4 ") * ((R ") ")) * c4),(R ")) is Element of the carrier of G
[.[.c4,(R ").],F.] is Element of the carrier of G
[.c4,(R ").] " is Element of the carrier of G
([.c4,(R ").] ") * (F ") is Element of the carrier of G
K84( the carrier of G, the multF of G,([.c4,(R ").] "),(F ")) is Element of the carrier of G
(([.c4,(R ").] ") * (F ")) * [.c4,(R ").] is Element of the carrier of G
K84( the carrier of G, the multF of G,(([.c4,(R ").] ") * (F ")),[.c4,(R ").]) is Element of the carrier of G
((([.c4,(R ").] ") * (F ")) * [.c4,(R ").]) * F is Element of the carrier of G
K84( the carrier of G, the multF of G,((([.c4,(R ").] ") * (F ")) * [.c4,(R ").]),F) is Element of the carrier of G
[.c4,(R "),F.] |^ R is Element of the carrier of G
(R ") * [.c4,(R "),F.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),[.c4,(R "),F.]) is Element of the carrier of G
((R ") * [.c4,(R "),F.]) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") * [.c4,(R "),F.]),R) is Element of the carrier of G
G is non empty unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
R is non empty unital Group-like associative Subgroup of G
[.F,R.] is non empty strict unital Group-like associative Subgroup of G
carr F is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of F is non empty set
carr R is Element of bool the carrier of G
the carrier of R is non empty set
[.(carr F),(carr R).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr F),(carr R)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr F & b2 in carr R ) } is set
gr (commutators ((carr F),(carr R))) is non empty strict unital Group-like associative Subgroup of G
[.R,F.] is non empty strict unital Group-like associative Subgroup of G
[.(carr R),(carr F).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr R),(carr F)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in carr F ) } is set
gr (commutators ((carr R),(carr F))) is non empty strict unital Group-like associative Subgroup of G
c4 is Element of the carrier of G
commutators (F,R) is Element of bool the carrier of G
H1 is Relation-like NAT -defined the carrier of G -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len H1 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
H2 is Relation-like NAT -defined INT -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of INT
len H2 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
rng H1 is Element of bool the carrier of G
H1 |^ H2 is Relation-like NAT -defined the carrier of G -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of the carrier of G
Product (H1 |^ H2) is Element of the carrier of G
N is Relation-like NAT -defined the carrier of G -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len N is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
dom N is Element of bool NAT
Seg (len H1) is V37() V44( len H1) Element of bool NAT
{ b1 where b1 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT : ( 1 <= b1 & b1 <= len H1 ) } is set
N9 is Relation-like NAT -defined INT -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of INT
len N9 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
dom N9 is Element of bool NAT
dom H1 is Element of bool NAT
N |^ N9 is Relation-like NAT -defined the carrier of G -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of the carrier of G
len (N |^ N9) is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
len (H1 |^ H2) is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
dom (H1 |^ H2) is Element of bool NAT
G2 is V24() V25() V26() V30() V31() V32() V33() ext-real set
(N |^ N9) . G2 is set
N9 /. G2 is V31() V32() V33() ext-real Element of INT
@ (N9 /. G2) is V31() V32() V33() ext-real Element of INT
N /. G2 is Element of the carrier of G
(N /. G2) |^ (@ (N9 /. G2)) is Element of the carrier of G
N . G2 is set
H1 /. G2 is Element of the carrier of G
(H1 /. G2) " is Element of the carrier of G
N9 . G2 is set
H2 /. G2 is V31() V32() V33() ext-real Element of INT
@ (H2 /. G2) is V31() V32() V33() ext-real Element of INT
- (@ (H2 /. G2)) is V31() V32() V33() ext-real set
@ (- (@ (H2 /. G2))) is V31() V32() V33() ext-real Element of INT
((H1 /. G2) ") |^ (@ (H2 /. G2)) is Element of the carrier of G
(((H1 /. G2) ") |^ (@ (H2 /. G2))) " is Element of the carrier of G
(H1 /. G2) |^ (@ (H2 /. G2)) is Element of the carrier of G
((H1 /. G2) |^ (@ (H2 /. G2))) " is Element of the carrier of G
(((H1 /. G2) |^ (@ (H2 /. G2))) ") " is Element of the carrier of G
(H1 |^ H2) . G2 is set
rng N is Element of bool the carrier of G
commutators (R,F) is Element of bool the carrier of G
G2 is set
g1 is set
N . g1 is set
g is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
H1 . g is set
h1 is Element of the carrier of G
h2 is Element of the carrier of G
[.h1,h2.] is Element of the carrier of G
h1 " is Element of the carrier of G
h2 " is Element of the carrier of G
(h1 ") * (h2 ") is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,(h1 "),(h2 ")) is Element of the carrier of G
((h1 ") * (h2 ")) * h1 is Element of the carrier of G
K84( the carrier of G, the multF of G,((h1 ") * (h2 ")),h1) is Element of the carrier of G
(((h1 ") * (h2 ")) * h1) * h2 is Element of the carrier of G
K84( the carrier of G, the multF of G,(((h1 ") * (h2 ")) * h1),h2) is Element of the carrier of G
H1 /. g is Element of the carrier of G
(H1 /. g) " is Element of the carrier of G
[.h2,h1.] is Element of the carrier of G
(h2 ") * (h1 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(h2 "),(h1 ")) is Element of the carrier of G
((h2 ") * (h1 ")) * h2 is Element of the carrier of G
K84( the carrier of G, the multF of G,((h2 ") * (h1 ")),h2) is Element of the carrier of G
(((h2 ") * (h1 ")) * h2) * h1 is Element of the carrier of G
K84( the carrier of G, the multF of G,(((h2 ") * (h1 ")) * h2),h1) is Element of the carrier of G
G is non empty unital Group-like associative multMagma
c4 is non empty unital Group-like associative Subgroup of G
H1 is non empty unital Group-like associative Subgroup of G
[.c4,H1.] is non empty strict unital Group-like associative Subgroup of G
carr c4 is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of c4 is non empty set
carr H1 is Element of bool the carrier of G
the carrier of H1 is non empty set
[.(carr c4),(carr H1).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr c4),(carr H1)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr c4 & b2 in carr H1 ) } is set
gr (commutators ((carr c4),(carr H1))) is non empty strict unital Group-like associative Subgroup of G
[.H1,c4.] is non empty strict unital Group-like associative Subgroup of G
[.(carr H1),(carr c4).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr H1),(carr c4)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr H1 & b2 in carr c4 ) } is set
gr (commutators ((carr H1),(carr c4))) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
R is non empty unital Group-like associative Subgroup of G
commutators (R,F) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
carr R is Element of bool the carrier of G
the carrier of R is non empty set
carr F is Element of bool the carrier of G
the carrier of F is non empty set
commutators ((carr R),(carr F)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in carr F ) } is set
c4 is set
H1 is Element of the carrier of G
H2 is Element of the carrier of G
[.H1,H2.] is Element of the carrier of G
H1 " is Element of the carrier of G
H2 " is Element of the carrier of G
(H1 ") * (H2 ") is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,(H1 "),(H2 ")) is Element of the carrier of G
((H1 ") * (H2 ")) * H1 is Element of the carrier of G
K84( the carrier of G, the multF of G,((H1 ") * (H2 ")),H1) is Element of the carrier of G
(((H1 ") * (H2 ")) * H1) * H2 is Element of the carrier of G
K84( the carrier of G, the multF of G,(((H1 ") * (H2 ")) * H1),H2) is Element of the carrier of G
H1 * H2 is Element of the carrier of G
K84( the carrier of G, the multF of G,H1,H2) is Element of the carrier of G
((H1 ") * (H2 ")) * (H1 * H2) is Element of the carrier of G
K84( the carrier of G, the multF of G,((H1 ") * (H2 ")),(H1 * H2)) is Element of the carrier of G
G is non empty unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
carr F is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of F is non empty set
gr (carr F) is non empty strict unital Group-like associative Subgroup of G
the multF of F is Relation-like [: the carrier of F, the carrier of F:] -defined the carrier of F -valued Function-like V18([: the carrier of F, the carrier of F:], the carrier of F) V22( 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
multMagma(# the carrier of F, the multF of F #) is non empty strict unital Group-like associative multMagma
the carrier of multMagma(# the carrier of F, the multF of F #) is non empty set
the multF of multMagma(# the carrier of F, the multF of F #) is Relation-like [: the carrier of multMagma(# the carrier of F, the multF of F #), the carrier of multMagma(# the carrier of F, the multF of F #):] -defined the carrier of multMagma(# the carrier of F, the multF of F #) -valued Function-like V18([: the carrier of multMagma(# the carrier of F, the multF of F #), the carrier of multMagma(# the carrier of F, the multF of F #):], the carrier of multMagma(# the carrier of F, the multF of F #)) V22( the carrier of multMagma(# the carrier of F, the multF of F #)) Element of bool [:[: the carrier of multMagma(# the carrier of F, the multF of F #), the carrier of multMagma(# the carrier of F, the multF of F #):], the carrier of multMagma(# the carrier of F, the multF of F #):]
[: the carrier of multMagma(# the carrier of F, the multF of F #), the carrier of multMagma(# the carrier of F, the multF of F #):] is non empty set
[:[: the carrier of multMagma(# the carrier of F, the multF of F #), the carrier of multMagma(# the carrier of F, the multF of F #):], the carrier of multMagma(# the carrier of F, the multF of F #):] is non empty set
bool [:[: the carrier of multMagma(# the carrier of F, the multF of F #), the carrier of multMagma(# the carrier of F, the multF of F #):], the carrier of multMagma(# the carrier of F, the multF of F #):] is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G || the carrier of F is set
the multF of G | [: the carrier of F, the carrier of F:] is Relation-like set
c4 is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
R is non empty unital Group-like associative Subgroup of G
(G,R,F) is non empty strict unital Group-like associative Subgroup of G
carr R is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of R is non empty set
carr F is Element of bool the carrier of G
the carrier of F is non empty set
[.(carr R),(carr F).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr R),(carr F)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in carr F ) } is set
gr (commutators ((carr R),(carr F))) is non empty strict unital Group-like associative Subgroup of G
gr (carr R) is non empty strict unital Group-like associative Subgroup of G
commutators (R,F) is Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
R is non empty unital Group-like associative Subgroup of G
(G,F,R) is non empty strict unital Group-like associative Subgroup of G
carr F is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of F is non empty set
carr R is Element of bool the carrier of G
the carrier of R is non empty set
[.(carr F),(carr R).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr F),(carr R)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr F & b2 in carr R ) } is set
gr (commutators ((carr F),(carr R))) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
dom the multF of G is Relation-like the carrier of G -defined the carrier of G -valued Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty set
the multF of ((Omega). G) is Relation-like [: the carrier of ((Omega). G), the carrier of ((Omega). G):] -defined the carrier of ((Omega). G) -valued Function-like V18([: the carrier of ((Omega). G), the carrier of ((Omega). G):], the carrier of ((Omega). G)) V22( the carrier of ((Omega). G)) Element of bool [:[: the carrier of ((Omega). G), the carrier of ((Omega). G):], the carrier of ((Omega). G):]
the carrier of ((Omega). G) is non empty set
[: the carrier of ((Omega). G), the carrier of ((Omega). G):] is non empty set
[:[: the carrier of ((Omega). G), the carrier of ((Omega). G):], the carrier of ((Omega). G):] is non empty set
bool [:[: the carrier of ((Omega). G), the carrier of ((Omega). G):], the carrier of ((Omega). G):] is non empty set
the multF of ((Omega). G) || the carrier of ((Omega). G) is set
the multF of ((Omega). G) | [: the carrier of ((Omega). G), the carrier of ((Omega). G):] is Relation-like set
G is non empty unital Group-like associative multMagma
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
(G,F,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr F is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of F is non empty set
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
[.(carr F),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr F),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr F & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr F),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
R is non empty unital Group-like associative Subgroup of G
c4 is non empty unital Group-like associative Subgroup of G
F /\ c4 is non empty strict unital Group-like associative Subgroup of G
(G,(F /\ c4),c4) is non empty strict unital Group-like associative Subgroup of G
carr (F /\ c4) is Element of bool the carrier of G
the carrier of (F /\ c4) is non empty set
carr c4 is Element of bool the carrier of G
the carrier of c4 is non empty set
[.(carr (F /\ c4)),(carr c4).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr (F /\ c4)),(carr c4)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr (F /\ c4) & b2 in carr c4 ) } is set
gr (commutators ((carr (F /\ c4)),(carr c4))) is non empty strict unital Group-like associative Subgroup of G
R /\ c4 is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
(G,F,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr F is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of F is non empty set
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
[.(carr F),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr F),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr F & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr F),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
R is non empty unital Group-like associative Subgroup of G
(G,F,R) is non empty strict unital Group-like associative Subgroup of G
carr R is Element of bool the carrier of G
the carrier of R is non empty set
[.(carr F),(carr R).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr F),(carr R)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr F & b2 in carr R ) } is set
gr (commutators ((carr F),(carr R))) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
R is non empty unital Group-like associative normal Subgroup of G
(G,R,F) is non empty strict unital Group-like associative Subgroup of G
carr R is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of R is non empty set
carr F is Element of bool the carrier of G
the carrier of F is non empty set
[.(carr R),(carr F).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr R),(carr F)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in carr F ) } is set
gr (commutators ((carr R),(carr F))) is non empty strict unital Group-like associative Subgroup of G
the multF of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like V18([: the carrier of R, the carrier of R:], the carrier of R) V22( the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G || the carrier of R is set
the multF of G | [: the carrier of R, the carrier of R:] is Relation-like set
multMagma(# the carrier of R, the multF of R #) is non empty strict unital Group-like associative multMagma
H1 is Element of the carrier of G
R |^ H1 is non empty strict unital Group-like associative Subgroup of G
the carrier of (R |^ H1) is non empty set
(carr R) |^ H1 is Element of bool the carrier of G
{H1} is non empty Element of bool the carrier of G
(carr R) |^ {H1} is Element of bool the carrier of G
{ (b1 |^ b2) where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in {H1} ) } is set
c4 is non empty strict unital Group-like associative Subgroup of G
carr c4 is Element of bool the carrier of G
the carrier of c4 is non empty set
(carr c4) |^ H1 is Element of bool the carrier of G
(carr c4) |^ {H1} is Element of bool the carrier of G
{ (b1 |^ b2) where b1, b2 is Element of the carrier of G : ( b1 in carr c4 & b2 in {H1} ) } is set
c4 |^ H1 is non empty strict unital Group-like associative Subgroup of G
the carrier of (c4 |^ H1) is non empty set
the multF of c4 is Relation-like [: the carrier of c4, the carrier of c4:] -defined the carrier of c4 -valued Function-like V18([: the carrier of c4, the carrier of c4:], the carrier of c4) V22( the carrier of c4) Element of bool [:[: the carrier of c4, the carrier of c4:], the carrier of c4:]
[: the carrier of c4, the carrier of c4:] is non empty set
[:[: the carrier of c4, the carrier of c4:], the carrier of c4:] is non empty set
bool [:[: the carrier of c4, the carrier of c4:], the carrier of c4:] is non empty set
multMagma(# the carrier of c4, the multF of c4 #) is non empty strict unital Group-like associative multMagma
H1 is non empty strict unital Group-like associative normal Subgroup of G
(G,H1,F) is non empty strict unital Group-like associative Subgroup of G
carr H1 is Element of bool the carrier of G
the carrier of H1 is non empty set
[.(carr H1),(carr F).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr H1),(carr F)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr H1 & b2 in carr F ) } is set
gr (commutators ((carr H1),(carr F))) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
(G,F,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr F is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of F is non empty set
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
[.(carr F),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr F),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr F & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr F),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
R is Element of the carrier of G
R * F is Element of bool the carrier of G
R * (carr F) is Element of bool the carrier of G
{R} is non empty Element of bool the carrier of G
{R} * (carr F) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {R} & b2 in carr F ) } is set
F * R is Element of bool the carrier of G
(carr F) * R is Element of bool the carrier of G
(carr F) * {R} is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in carr F & b2 in {R} ) } is set
c4 is set
H1 is Element of the carrier of G
H2 is Element of the carrier of G
R * H2 is Element of the carrier of G
K84( the carrier of G, the multF of G,R,H2) is Element of the carrier of G
H2 " is Element of the carrier of G
R " is Element of the carrier of G
(G,((Omega). G),F) is non empty strict unital Group-like associative Subgroup of G
[.(carr ((Omega). G)),(carr F).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr ((Omega). G)),(carr F)) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr ((Omega). G) & b2 in carr F ) } is set
gr (commutators ((carr ((Omega). G)),(carr F))) is non empty strict unital Group-like associative Subgroup of G
[.(R "),(H2 ").] is Element of the carrier of G
(R ") " is Element of the carrier of G
(H2 ") " is Element of the carrier of G
((R ") ") * ((H2 ") ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((R ") "),((H2 ") ")) is Element of the carrier of G
(((R ") ") * ((H2 ") ")) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R ") ") * ((H2 ") ")),(R ")) is Element of the carrier of G
((((R ") ") * ((H2 ") ")) * (R ")) * (H2 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((((R ") ") * ((H2 ") ")) * (R ")),(H2 ")) is Element of the carrier of G
(R * H2) * (R ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(R * H2),(R ")) is Element of the carrier of G
((R * H2) * (R ")) * (H2 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,((R * H2) * (R ")),(H2 ")) is Element of the carrier of G
(((R * H2) * (R ")) * (H2 ")) * H2 is Element of the carrier of G
K84( the carrier of G, the multF of G,(((R * H2) * (R ")) * (H2 ")),H2) is Element of the carrier of G
(H2 ") * H2 is Element of the carrier of G
K84( the carrier of G, the multF of G,(H2 "),H2) is Element of the carrier of G
((R * H2) * (R ")) * ((H2 ") * H2) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R * H2) * (R ")),((H2 ") * H2)) is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
((R * H2) * (R ")) * (1_ G) is Element of the carrier of G
K84( the carrier of G, the multF of G,((R * H2) * (R ")),(1_ G)) is Element of the carrier of G
((R * H2) * (R ")) * R is Element of the carrier of G
K84( the carrier of G, the multF of G,((R * H2) * (R ")),R) is Element of the carrier of G
(R ") * R is Element of the carrier of G
K84( the carrier of G, the multF of G,(R "),R) is Element of the carrier of G
(R * H2) * ((R ") * R) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R * H2),((R ") * R)) is Element of the carrier of G
(R * H2) * (1_ G) is Element of the carrier of G
K84( the carrier of G, the multF of G,(R * H2),(1_ G)) is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
F is set
c4 is set
H2 is non empty strict unital Group-like associative normal Subgroup of G
R is set
the carrier of H2 is non empty set
H1 is set
N is non empty strict unital Group-like associative normal Subgroup of G
the carrier of N is non empty set
R is Relation-like Function-like set
c4 is set
H1 is non empty strict unital Group-like associative normal Subgroup of G
the carrier of H1 is non empty set
H2 is set
[c4,H2] is set
{c4,H2} is non empty set
{c4} is non empty set
{{c4,H2},{c4}} is non empty set
H1 is set
[c4,H1] is set
{c4,H1} is non empty set
{c4} is non empty set
{{c4,H1},{c4}} is non empty set
dom R is set
rng R is set
c4 is set
H1 is set
R . H1 is set
[H1,c4] is set
{H1,c4} is non empty set
{H1} is non empty set
{{H1,c4},{H1}} is non empty set
H2 is non empty strict unital Group-like associative normal Subgroup of G
the carrier of H2 is non empty set
H1 is non empty strict unital Group-like associative normal Subgroup of G
the carrier of H1 is non empty set
H2 is set
[H2,c4] is set
{H2,c4} is non empty set
{H2} is non empty set
{{H2,c4},{H2}} is non empty set
R . H2 is set
F is set
R is set
G is non empty unital Group-like associative multMagma
(G) is set
the non empty strict unital Group-like associative normal Subgroup of G is non empty strict unital Group-like associative normal Subgroup of G
G is non empty unital Group-like associative multMagma
(G) is non empty set
F is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
dom F is Element of bool NAT
R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
F . R is set
rng F is Element of bool (G)
bool (G) is non empty set
G is non empty unital Group-like associative multMagma
(G) is non empty set
Subgroups G is non empty set
F is set
G is non empty unital Group-like associative multMagma
(G) is non empty set
Subgroups G is non empty set
F is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
rng F is Element of bool (G)
bool (G) is non empty set
the non empty unital Group-like associative multMagma is non empty unital Group-like associative multMagma
(1). the non empty unital Group-like associative multMagma is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic Subgroup of the non empty unital Group-like associative multMagma
F is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic Subgroup of the non empty unital Group-like associative multMagma
<*F*> is Relation-like NAT -defined Function-like non empty V37() V44(1) FinSequence-like FinSubsequence-like set
[1,F] is set
{1,F} is non empty set
{{1,F},{1}} is non empty set
{[1,F]} is non empty set
rng <*F*> is set
(F) is non empty set
R is set
{F} is non empty set
R is Relation-like NAT -defined (F) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (F)
len R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R . 1 is set
(Omega). F is non empty V78() strict unital Group-like associative commutative normal Subgroup of F
the carrier of F is non empty trivial V37() set
the multF of F is Relation-like [: the carrier of F, the carrier of F:] -defined the carrier of F -valued Function-like V18([: the carrier of F, the carrier of F:], the carrier of F) V22( 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
multMagma(# the carrier of F, the multF of F #) is non empty trivial V78() 1 -element strict unital Group-like associative multMagma
R . (len R) is set
(1). F is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic Subgroup of F
dom R is Element of bool NAT
c4 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
c4 + 1 is V31() V32() V33() ext-real set
R . c4 is set
R . (c4 + 1) is set
H1 is non empty V78() strict unital Group-like associative commutative normal Subgroup of F
H2 is non empty V78() strict unital Group-like associative commutative normal Subgroup of F
(H1,H2) `*` is non empty V78() strict unital Group-like associative commutative normal Subgroup of H1
H1 ./. ((H1,H2) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((H1,H2) `*`) is non empty Element of bool (bool the carrier of H1)
the carrier of H1 is non empty V37() set
bool the carrier of H1 is non empty set
bool (bool the carrier of H1) is non empty set
CosOp ((H1,H2) `*`) is Relation-like [:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):] -defined Left_Cosets ((H1,H2) `*`) -valued Function-like V18([:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):], Left_Cosets ((H1,H2) `*`)) Element of bool [:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):]
[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):] is non empty set
[:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):] is non empty set
bool [:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):] is non empty set
multMagma(# (Left_Cosets ((H1,H2) `*`)),(CosOp ((H1,H2) `*`)) #) is non empty strict multMagma
F ./. H2 is non empty strict unital Group-like associative multMagma
Left_Cosets H2 is non empty Element of bool (bool the carrier of F)
bool the carrier of F is non empty set
bool (bool the carrier of F) is non empty set
CosOp H2 is Relation-like [:(Left_Cosets H2),(Left_Cosets H2):] -defined Left_Cosets H2 -valued Function-like V18([:(Left_Cosets H2),(Left_Cosets H2):], Left_Cosets H2) Element of bool [:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):]
[:(Left_Cosets H2),(Left_Cosets H2):] is non empty set
[:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):] is non empty set
bool [:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):] is non empty set
multMagma(# (Left_Cosets H2),(CosOp H2) #) is non empty strict multMagma
center (F ./. H2) is non empty strict unital Group-like associative Subgroup of F ./. H2
{1} is non empty Element of bool NAT
c4 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
c4 + 1 is V31() V32() V33() ext-real set
H1 is non empty V78() strict unital Group-like associative commutative normal Subgroup of F
R . c4 is set
H2 is non empty V78() strict unital Group-like associative commutative normal Subgroup of F
R . (c4 + 1) is set
(H1,H2) `*` is non empty V78() strict unital Group-like associative commutative normal Subgroup of H1
H1 ./. ((H1,H2) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((H1,H2) `*`) is non empty Element of bool (bool the carrier of H1)
the carrier of H1 is non empty V37() set
bool the carrier of H1 is non empty set
bool (bool the carrier of H1) is non empty set
CosOp ((H1,H2) `*`) is Relation-like [:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):] -defined Left_Cosets ((H1,H2) `*`) -valued Function-like V18([:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):], Left_Cosets ((H1,H2) `*`)) Element of bool [:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):]
[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):] is non empty set
[:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):] is non empty set
bool [:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):] is non empty set
multMagma(# (Left_Cosets ((H1,H2) `*`)),(CosOp ((H1,H2) `*`)) #) is non empty strict multMagma
F ./. H2 is non empty strict unital Group-like associative multMagma
Left_Cosets H2 is non empty Element of bool (bool the carrier of F)
bool the carrier of F is non empty set
bool (bool the carrier of F) is non empty set
CosOp H2 is Relation-like [:(Left_Cosets H2),(Left_Cosets H2):] -defined Left_Cosets H2 -valued Function-like V18([:(Left_Cosets H2),(Left_Cosets H2):], Left_Cosets H2) Element of bool [:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):]
[:(Left_Cosets H2),(Left_Cosets H2):] is non empty set
[:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):] is non empty set
bool [:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):] is non empty set
multMagma(# (Left_Cosets H2),(CosOp H2) #) is non empty strict multMagma
center (F ./. H2) is non empty strict unital Group-like associative Subgroup of F ./. H2
G is non empty unital Group-like associative multMagma
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
(G,F,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr F is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of F is non empty set
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
[.(carr F),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr F),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr F & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr F),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
R is non empty strict unital Group-like associative normal Subgroup of G
(F,R) `*` is non empty strict unital Group-like associative normal Subgroup of F
F ./. ((F,R) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((F,R) `*`) is non empty Element of bool (bool the carrier of F)
bool the carrier of F is non empty set
bool (bool the carrier of F) is non empty set
CosOp ((F,R) `*`) is Relation-like [:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):] -defined Left_Cosets ((F,R) `*`) -valued Function-like V18([:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):], Left_Cosets ((F,R) `*`)) Element of bool [:[:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):],(Left_Cosets ((F,R) `*`)):]
[:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):] is non empty set
[:[:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):],(Left_Cosets ((F,R) `*`)):] is non empty set
bool [:[:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):],(Left_Cosets ((F,R) `*`)):] is non empty set
multMagma(# (Left_Cosets ((F,R) `*`)),(CosOp ((F,R) `*`)) #) is non empty strict multMagma
G ./. R is non empty strict unital Group-like associative multMagma
Left_Cosets R is non empty Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
CosOp R is Relation-like [:(Left_Cosets R),(Left_Cosets R):] -defined Left_Cosets R -valued Function-like V18([:(Left_Cosets R),(Left_Cosets R):], Left_Cosets R) Element of bool [:[:(Left_Cosets R),(Left_Cosets R):],(Left_Cosets R):]
[:(Left_Cosets R),(Left_Cosets R):] is non empty set
[:[:(Left_Cosets R),(Left_Cosets R):],(Left_Cosets R):] is non empty set
bool [:[:(Left_Cosets R),(Left_Cosets R):],(Left_Cosets R):] is non empty set
multMagma(# (Left_Cosets R),(CosOp R) #) is non empty strict multMagma
center (G ./. R) is non empty strict unital Group-like associative Subgroup of G ./. R
commutators (F,((Omega). G)) is Element of bool the carrier of G
carr R is Element of bool the carrier of G
the carrier of R is non empty set
H2 is Element of the carrier of G
N is Element of the carrier of G
N9 is Element of the carrier of G
[.N,N9.] is Element of the carrier of G
N " is Element of the carrier of G
N9 " is Element of the carrier of G
(N ") * (N9 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(N "),(N9 ")) is Element of the carrier of G
((N ") * (N9 ")) * N is Element of the carrier of G
K84( the carrier of G, the multF of G,((N ") * (N9 ")),N) is Element of the carrier of G
(((N ") * (N9 ")) * N) * N9 is Element of the carrier of G
K84( the carrier of G, the multF of G,(((N ") * (N9 ")) * N),N9) is Element of the carrier of G
c4 is non empty unital Group-like associative Subgroup of G ./. R
the carrier of c4 is non empty set
H1 is non empty unital Group-like associative normal Subgroup of F
G1 is Element of the carrier of F
G1 * H1 is Element of bool the carrier of F
carr H1 is Element of bool the carrier of F
the carrier of H1 is non empty set
G1 * (carr H1) is Element of bool the carrier of F
{G1} is non empty Element of bool the carrier of F
{G1} * (carr H1) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {G1} & b2 in carr H1 ) } is set
G2 is Element of the carrier of c4
the carrier of (G ./. R) is non empty set
N9 * R is Element of bool the carrier of G
N9 * (carr R) is Element of bool the carrier of G
{N9} is non empty Element of bool the carrier of G
{N9} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {N9} & b2 in carr R ) } is set
g is Element of the carrier of G
g * R is Element of bool the carrier of G
g * (carr R) is Element of bool the carrier of G
{g} is non empty Element of bool the carrier of G
{g} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {g} & b2 in carr R ) } is set
h1 is Element of the carrier of (G ./. R)
g1 is Element of the carrier of (G ./. R)
h1 * g1 is Element of the carrier of (G ./. R)
the multF of (G ./. R) is Relation-like [: the carrier of (G ./. R), the carrier of (G ./. R):] -defined the carrier of (G ./. R) -valued Function-like V18([: the carrier of (G ./. R), the carrier of (G ./. R):], the carrier of (G ./. R)) V22( the carrier of (G ./. R)) Element of bool [:[: the carrier of (G ./. R), the carrier of (G ./. R):], the carrier of (G ./. R):]
[: the carrier of (G ./. R), the carrier of (G ./. R):] is non empty set
[:[: the carrier of (G ./. R), the carrier of (G ./. R):], the carrier of (G ./. R):] is non empty set
bool [:[: the carrier of (G ./. R), the carrier of (G ./. R):], the carrier of (G ./. R):] is non empty set
K84( the carrier of (G ./. R), the multF of (G ./. R),h1,g1) is Element of the carrier of (G ./. R)
g1 * h1 is Element of the carrier of (G ./. R)
K84( the carrier of (G ./. R), the multF of (G ./. R),g1,h1) is Element of the carrier of (G ./. R)
@ h1 is Element of bool the carrier of G
@ g1 is Element of bool the carrier of G
(g * R) * (N9 * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in g * R & b2 in N9 * R ) } is set
R * (N9 * R) is Element of bool the carrier of G
(carr R) * (N9 * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in N9 * R ) } is set
g * (R * (N9 * R)) is Element of bool the carrier of G
{g} * (R * (N9 * R)) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {g} & b2 in R * (N9 * R) ) } is set
R * N9 is Element of bool the carrier of G
(carr R) * N9 is Element of bool the carrier of G
(carr R) * {N9} is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in {N9} ) } is set
(R * N9) * R is Element of bool the carrier of G
(R * N9) * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in R * N9 & b2 in carr R ) } is set
g * ((R * N9) * R) is Element of bool the carrier of G
{g} * ((R * N9) * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {g} & b2 in (R * N9) * R ) } is set
(N9 * R) * R is Element of bool the carrier of G
(N9 * R) * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in N9 * R & b2 in carr R ) } is set
g * ((N9 * R) * R) is Element of bool the carrier of G
{g} * ((N9 * R) * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {g} & b2 in (N9 * R) * R ) } is set
g * (N9 * R) is Element of bool the carrier of G
{g} * (N9 * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {g} & b2 in N9 * R ) } is set
g * N9 is Element of the carrier of G
K84( the carrier of G, the multF of G,g,N9) is Element of the carrier of G
(g * N9) * R is Element of bool the carrier of G
(g * N9) * (carr R) is Element of bool the carrier of G
{(g * N9)} is non empty Element of bool the carrier of G
{(g * N9)} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(g * N9)} & b2 in carr R ) } is set
(N9 * R) * (g * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in N9 * R & b2 in g * R ) } is set
R * (g * R) is Element of bool the carrier of G
(carr R) * (g * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in g * R ) } is set
N9 * (R * (g * R)) is Element of bool the carrier of G
{N9} * (R * (g * R)) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {N9} & b2 in R * (g * R) ) } is set
R * g is Element of bool the carrier of G
(carr R) * g is Element of bool the carrier of G
(carr R) * {g} is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in {g} ) } is set
(R * g) * R is Element of bool the carrier of G
(R * g) * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in R * g & b2 in carr R ) } is set
N9 * ((R * g) * R) is Element of bool the carrier of G
{N9} * ((R * g) * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {N9} & b2 in (R * g) * R ) } is set
(g * R) * R is Element of bool the carrier of G
(g * R) * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in g * R & b2 in carr R ) } is set
N9 * ((g * R) * R) is Element of bool the carrier of G
{N9} * ((g * R) * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {N9} & b2 in (g * R) * R ) } is set
N9 * (g * R) is Element of bool the carrier of G
{N9} * (g * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {N9} & b2 in g * R ) } is set
N9 * g is Element of the carrier of G
K84( the carrier of G, the multF of G,N9,g) is Element of the carrier of G
(N9 * g) * R is Element of bool the carrier of G
(N9 * g) * (carr R) is Element of bool the carrier of G
{(N9 * g)} is non empty Element of bool the carrier of G
{(N9 * g)} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(N9 * g)} & b2 in carr R ) } is set
g " is Element of the carrier of G
(g ") * (N9 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(g "),(N9 ")) is Element of the carrier of G
((g ") * (N9 ")) * ((g * N9) * R) is Element of bool the carrier of G
{((g ") * (N9 "))} is non empty Element of bool the carrier of G
{((g ") * (N9 "))} * ((g * N9) * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((g ") * (N9 "))} & b2 in (g * N9) * R ) } is set
((g ") * (N9 ")) * (N9 * g) is Element of the carrier of G
K84( the carrier of G, the multF of G,((g ") * (N9 ")),(N9 * g)) is Element of the carrier of G
(((g ") * (N9 ")) * (N9 * g)) * R is Element of bool the carrier of G
(((g ") * (N9 ")) * (N9 * g)) * (carr R) is Element of bool the carrier of G
{(((g ") * (N9 ")) * (N9 * g))} is non empty Element of bool the carrier of G
{(((g ") * (N9 ")) * (N9 * g))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(((g ") * (N9 ")) * (N9 * g))} & b2 in carr R ) } is set
(N9 ") * (N9 * g) is Element of the carrier of G
K84( the carrier of G, the multF of G,(N9 "),(N9 * g)) is Element of the carrier of G
(g ") * ((N9 ") * (N9 * g)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(g "),((N9 ") * (N9 * g))) is Element of the carrier of G
((g ") * ((N9 ") * (N9 * g))) * R is Element of bool the carrier of G
((g ") * ((N9 ") * (N9 * g))) * (carr R) is Element of bool the carrier of G
{((g ") * ((N9 ") * (N9 * g)))} is non empty Element of bool the carrier of G
{((g ") * ((N9 ") * (N9 * g)))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((g ") * ((N9 ") * (N9 * g)))} & b2 in carr R ) } is set
(N9 ") * N9 is Element of the carrier of G
K84( the carrier of G, the multF of G,(N9 "),N9) is Element of the carrier of G
((N9 ") * N9) * g is Element of the carrier of G
K84( the carrier of G, the multF of G,((N9 ") * N9),g) is Element of the carrier of G
(g ") * (((N9 ") * N9) * g) is Element of the carrier of G
K84( the carrier of G, the multF of G,(g "),(((N9 ") * N9) * g)) is Element of the carrier of G
((g ") * (((N9 ") * N9) * g)) * R is Element of bool the carrier of G
((g ") * (((N9 ") * N9) * g)) * (carr R) is Element of bool the carrier of G
{((g ") * (((N9 ") * N9) * g))} is non empty Element of bool the carrier of G
{((g ") * (((N9 ") * N9) * g))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((g ") * (((N9 ") * N9) * g))} & b2 in carr R ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * g is Element of the carrier of G
K84( the carrier of G, the multF of G,(1_ G),g) is Element of the carrier of G
(g ") * ((1_ G) * g) is Element of the carrier of G
K84( the carrier of G, the multF of G,(g "),((1_ G) * g)) is Element of the carrier of G
((g ") * ((1_ G) * g)) * R is Element of bool the carrier of G
((g ") * ((1_ G) * g)) * (carr R) is Element of bool the carrier of G
{((g ") * ((1_ G) * g))} is non empty Element of bool the carrier of G
{((g ") * ((1_ G) * g))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((g ") * ((1_ G) * g))} & b2 in carr R ) } is set
(g ") * g is Element of the carrier of G
K84( the carrier of G, the multF of G,(g "),g) is Element of the carrier of G
((g ") * g) * R is Element of bool the carrier of G
((g ") * g) * (carr R) is Element of bool the carrier of G
{((g ") * g)} is non empty Element of bool the carrier of G
{((g ") * g)} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((g ") * g)} & b2 in carr R ) } is set
(1_ G) * R is Element of bool the carrier of G
(1_ G) * (carr R) is Element of bool the carrier of G
{(1_ G)} is non empty Element of bool the carrier of G
{(1_ G)} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(1_ G)} & b2 in carr R ) } is set
((g ") * (N9 ")) * (g * N9) is Element of the carrier of G
K84( the carrier of G, the multF of G,((g ") * (N9 ")),(g * N9)) is Element of the carrier of G
(((g ") * (N9 ")) * (g * N9)) * R is Element of bool the carrier of G
(((g ") * (N9 ")) * (g * N9)) * (carr R) is Element of bool the carrier of G
{(((g ") * (N9 ")) * (g * N9))} is non empty Element of bool the carrier of G
{(((g ") * (N9 ")) * (g * N9))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(((g ") * (N9 ")) * (g * N9))} & b2 in carr R ) } is set
[.g,N9.] is Element of the carrier of G
((g ") * (N9 ")) * g is Element of the carrier of G
K84( the carrier of G, the multF of G,((g ") * (N9 ")),g) is Element of the carrier of G
(((g ") * (N9 ")) * g) * N9 is Element of the carrier of G
K84( the carrier of G, the multF of G,(((g ") * (N9 ")) * g),N9) is Element of the carrier of G
[.g,N9.] * R is Element of bool the carrier of G
[.g,N9.] * (carr R) is Element of bool the carrier of G
{[.g,N9.]} is non empty Element of bool the carrier of G
{[.g,N9.]} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {[.g,N9.]} & b2 in carr R ) } is set
gr (carr R) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
(G,F,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr F is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of F is non empty set
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
[.(carr F),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr F),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr F & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr F),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
R is non empty unital Group-like associative normal Subgroup of G
(F,R) `*` is non empty strict unital Group-like associative normal Subgroup of F
F ./. ((F,R) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((F,R) `*`) is non empty Element of bool (bool the carrier of F)
bool the carrier of F is non empty set
bool (bool the carrier of F) is non empty set
CosOp ((F,R) `*`) is Relation-like [:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):] -defined Left_Cosets ((F,R) `*`) -valued Function-like V18([:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):], Left_Cosets ((F,R) `*`)) Element of bool [:[:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):],(Left_Cosets ((F,R) `*`)):]
[:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):] is non empty set
[:[:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):],(Left_Cosets ((F,R) `*`)):] is non empty set
bool [:[:(Left_Cosets ((F,R) `*`)),(Left_Cosets ((F,R) `*`)):],(Left_Cosets ((F,R) `*`)):] is non empty set
multMagma(# (Left_Cosets ((F,R) `*`)),(CosOp ((F,R) `*`)) #) is non empty strict multMagma
G ./. R is non empty strict unital Group-like associative multMagma
Left_Cosets R is non empty Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
CosOp R is Relation-like [:(Left_Cosets R),(Left_Cosets R):] -defined Left_Cosets R -valued Function-like V18([:(Left_Cosets R),(Left_Cosets R):], Left_Cosets R) Element of bool [:[:(Left_Cosets R),(Left_Cosets R):],(Left_Cosets R):]
[:(Left_Cosets R),(Left_Cosets R):] is non empty set
[:[:(Left_Cosets R),(Left_Cosets R):],(Left_Cosets R):] is non empty set
bool [:[:(Left_Cosets R),(Left_Cosets R):],(Left_Cosets R):] is non empty set
multMagma(# (Left_Cosets R),(CosOp R) #) is non empty strict multMagma
center (G ./. R) is non empty strict unital Group-like associative Subgroup of G ./. R
the carrier of (G ./. R) is non empty set
c4 is non empty unital Group-like associative Subgroup of G ./. R
H2 is Element of the carrier of (G ./. R)
N is Element of the carrier of (G ./. R)
H2 * N is Element of the carrier of (G ./. R)
the multF of (G ./. R) is Relation-like [: the carrier of (G ./. R), the carrier of (G ./. R):] -defined the carrier of (G ./. R) -valued Function-like V18([: the carrier of (G ./. R), the carrier of (G ./. R):], the carrier of (G ./. R)) V22( the carrier of (G ./. R)) Element of bool [:[: the carrier of (G ./. R), the carrier of (G ./. R):], the carrier of (G ./. R):]
[: the carrier of (G ./. R), the carrier of (G ./. R):] is non empty set
[:[: the carrier of (G ./. R), the carrier of (G ./. R):], the carrier of (G ./. R):] is non empty set
bool [:[: the carrier of (G ./. R), the carrier of (G ./. R):], the carrier of (G ./. R):] is non empty set
K84( the carrier of (G ./. R), the multF of (G ./. R),H2,N) is Element of the carrier of (G ./. R)
N * H2 is Element of the carrier of (G ./. R)
K84( the carrier of (G ./. R), the multF of (G ./. R),N,H2) is Element of the carrier of (G ./. R)
N9 is Element of the carrier of G
N9 * R is Element of bool the carrier of G
carr R is Element of bool the carrier of G
the carrier of R is non empty set
N9 * (carr R) is Element of bool the carrier of G
{N9} is non empty Element of bool the carrier of G
{N9} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {N9} & b2 in carr R ) } is set
R * N9 is Element of bool the carrier of G
(carr R) * N9 is Element of bool the carrier of G
(carr R) * {N9} is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in {N9} ) } is set
H1 is non empty unital Group-like associative normal Subgroup of F
G1 is Element of the carrier of F
G1 * H1 is Element of bool the carrier of F
carr H1 is Element of bool the carrier of F
the carrier of H1 is non empty set
G1 * (carr H1) is Element of bool the carrier of F
{G1} is non empty Element of bool the carrier of F
{G1} * (carr H1) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {G1} & b2 in carr H1 ) } is set
H1 * G1 is Element of bool the carrier of F
(carr H1) * G1 is Element of bool the carrier of F
(carr H1) * {G1} is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in carr H1 & b2 in {G1} ) } is set
G2 is Element of the carrier of G
[.G2,N9.] is Element of the carrier of G
G2 " is Element of the carrier of G
N9 " is Element of the carrier of G
(G2 ") * (N9 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(G2 "),(N9 ")) is Element of the carrier of G
((G2 ") * (N9 ")) * G2 is Element of the carrier of G
K84( the carrier of G, the multF of G,((G2 ") * (N9 ")),G2) is Element of the carrier of G
(((G2 ") * (N9 ")) * G2) * N9 is Element of the carrier of G
K84( the carrier of G, the multF of G,(((G2 ") * (N9 ")) * G2),N9) is Element of the carrier of G
@ N is Element of bool the carrier of G
@ H2 is Element of bool the carrier of G
G2 * R is Element of bool the carrier of G
G2 * (carr R) is Element of bool the carrier of G
{G2} is non empty Element of bool the carrier of G
{G2} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {G2} & b2 in carr R ) } is set
R * G2 is Element of bool the carrier of G
(carr R) * G2 is Element of bool the carrier of G
(carr R) * {G2} is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in {G2} ) } is set
(N9 * R) * (G2 * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in N9 * R & b2 in G2 * R ) } is set
N9 * G2 is Element of the carrier of G
K84( the carrier of G, the multF of G,N9,G2) is Element of the carrier of G
(N9 * G2) * R is Element of bool the carrier of G
(N9 * G2) * (carr R) is Element of bool the carrier of G
{(N9 * G2)} is non empty Element of bool the carrier of G
{(N9 * G2)} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(N9 * G2)} & b2 in carr R ) } is set
(G2 * R) * (N9 * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in G2 * R & b2 in N9 * R ) } is set
G2 * N9 is Element of the carrier of G
K84( the carrier of G, the multF of G,G2,N9) is Element of the carrier of G
(G2 * N9) * R is Element of bool the carrier of G
(G2 * N9) * (carr R) is Element of bool the carrier of G
{(G2 * N9)} is non empty Element of bool the carrier of G
{(G2 * N9)} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(G2 * N9)} & b2 in carr R ) } is set
(N9 * G2) * [.G2,N9.] is Element of the carrier of G
K84( the carrier of G, the multF of G,(N9 * G2),[.G2,N9.]) is Element of the carrier of G
((N9 * G2) * [.G2,N9.]) * R is Element of bool the carrier of G
((N9 * G2) * [.G2,N9.]) * (carr R) is Element of bool the carrier of G
{((N9 * G2) * [.G2,N9.])} is non empty Element of bool the carrier of G
{((N9 * G2) * [.G2,N9.])} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((N9 * G2) * [.G2,N9.])} & b2 in carr R ) } is set
[.G2,N9.] * R is Element of bool the carrier of G
[.G2,N9.] * (carr R) is Element of bool the carrier of G
{[.G2,N9.]} is non empty Element of bool the carrier of G
{[.G2,N9.]} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {[.G2,N9.]} & b2 in carr R ) } is set
(N9 * G2) * ([.G2,N9.] * R) is Element of bool the carrier of G
{(N9 * G2)} * ([.G2,N9.] * R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(N9 * G2)} & b2 in [.G2,N9.] * R ) } is set
((G2 ") * (N9 ")) * (G2 * N9) is Element of the carrier of G
K84( the carrier of G, the multF of G,((G2 ") * (N9 ")),(G2 * N9)) is Element of the carrier of G
(N9 * G2) * (((G2 ") * (N9 ")) * (G2 * N9)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(N9 * G2),(((G2 ") * (N9 ")) * (G2 * N9))) is Element of the carrier of G
((N9 * G2) * (((G2 ") * (N9 ")) * (G2 * N9))) * R is Element of bool the carrier of G
((N9 * G2) * (((G2 ") * (N9 ")) * (G2 * N9))) * (carr R) is Element of bool the carrier of G
{((N9 * G2) * (((G2 ") * (N9 ")) * (G2 * N9)))} is non empty Element of bool the carrier of G
{((N9 * G2) * (((G2 ") * (N9 ")) * (G2 * N9)))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((N9 * G2) * (((G2 ") * (N9 ")) * (G2 * N9)))} & b2 in carr R ) } is set
(N9 * G2) * ((G2 ") * (N9 ")) is Element of the carrier of G
K84( the carrier of G, the multF of G,(N9 * G2),((G2 ") * (N9 "))) is Element of the carrier of G
((N9 * G2) * ((G2 ") * (N9 "))) * (G2 * N9) is Element of the carrier of G
K84( the carrier of G, the multF of G,((N9 * G2) * ((G2 ") * (N9 "))),(G2 * N9)) is Element of the carrier of G
(((N9 * G2) * ((G2 ") * (N9 "))) * (G2 * N9)) * R is Element of bool the carrier of G
(((N9 * G2) * ((G2 ") * (N9 "))) * (G2 * N9)) * (carr R) is Element of bool the carrier of G
{(((N9 * G2) * ((G2 ") * (N9 "))) * (G2 * N9))} is non empty Element of bool the carrier of G
{(((N9 * G2) * ((G2 ") * (N9 "))) * (G2 * N9))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(((N9 * G2) * ((G2 ") * (N9 "))) * (G2 * N9))} & b2 in carr R ) } is set
G2 * ((G2 ") * (N9 ")) is Element of the carrier of G
K84( the carrier of G, the multF of G,G2,((G2 ") * (N9 "))) is Element of the carrier of G
N9 * (G2 * ((G2 ") * (N9 "))) is Element of the carrier of G
K84( the carrier of G, the multF of G,N9,(G2 * ((G2 ") * (N9 ")))) is Element of the carrier of G
(N9 * (G2 * ((G2 ") * (N9 ")))) * (G2 * N9) is Element of the carrier of G
K84( the carrier of G, the multF of G,(N9 * (G2 * ((G2 ") * (N9 ")))),(G2 * N9)) is Element of the carrier of G
((N9 * (G2 * ((G2 ") * (N9 ")))) * (G2 * N9)) * R is Element of bool the carrier of G
((N9 * (G2 * ((G2 ") * (N9 ")))) * (G2 * N9)) * (carr R) is Element of bool the carrier of G
{((N9 * (G2 * ((G2 ") * (N9 ")))) * (G2 * N9))} is non empty Element of bool the carrier of G
{((N9 * (G2 * ((G2 ") * (N9 ")))) * (G2 * N9))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((N9 * (G2 * ((G2 ") * (N9 ")))) * (G2 * N9))} & b2 in carr R ) } is set
G2 * (G2 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,G2,(G2 ")) is Element of the carrier of G
(G2 * (G2 ")) * (N9 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(G2 * (G2 ")),(N9 ")) is Element of the carrier of G
N9 * ((G2 * (G2 ")) * (N9 ")) is Element of the carrier of G
K84( the carrier of G, the multF of G,N9,((G2 * (G2 ")) * (N9 "))) is Element of the carrier of G
(N9 * ((G2 * (G2 ")) * (N9 "))) * (G2 * N9) is Element of the carrier of G
K84( the carrier of G, the multF of G,(N9 * ((G2 * (G2 ")) * (N9 "))),(G2 * N9)) is Element of the carrier of G
((N9 * ((G2 * (G2 ")) * (N9 "))) * (G2 * N9)) * R is Element of bool the carrier of G
((N9 * ((G2 * (G2 ")) * (N9 "))) * (G2 * N9)) * (carr R) is Element of bool the carrier of G
{((N9 * ((G2 * (G2 ")) * (N9 "))) * (G2 * N9))} is non empty Element of bool the carrier of G
{((N9 * ((G2 * (G2 ")) * (N9 "))) * (G2 * N9))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((N9 * ((G2 * (G2 ")) * (N9 "))) * (G2 * N9))} & b2 in carr R ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * (N9 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(1_ G),(N9 ")) is Element of the carrier of G
N9 * ((1_ G) * (N9 ")) is Element of the carrier of G
K84( the carrier of G, the multF of G,N9,((1_ G) * (N9 "))) is Element of the carrier of G
(N9 * ((1_ G) * (N9 "))) * (G2 * N9) is Element of the carrier of G
K84( the carrier of G, the multF of G,(N9 * ((1_ G) * (N9 "))),(G2 * N9)) is Element of the carrier of G
((N9 * ((1_ G) * (N9 "))) * (G2 * N9)) * R is Element of bool the carrier of G
((N9 * ((1_ G) * (N9 "))) * (G2 * N9)) * (carr R) is Element of bool the carrier of G
{((N9 * ((1_ G) * (N9 "))) * (G2 * N9))} is non empty Element of bool the carrier of G
{((N9 * ((1_ G) * (N9 "))) * (G2 * N9))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((N9 * ((1_ G) * (N9 "))) * (G2 * N9))} & b2 in carr R ) } is set
N9 * (N9 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,N9,(N9 ")) is Element of the carrier of G
(N9 * (N9 ")) * (G2 * N9) is Element of the carrier of G
K84( the carrier of G, the multF of G,(N9 * (N9 ")),(G2 * N9)) is Element of the carrier of G
((N9 * (N9 ")) * (G2 * N9)) * R is Element of bool the carrier of G
((N9 * (N9 ")) * (G2 * N9)) * (carr R) is Element of bool the carrier of G
{((N9 * (N9 ")) * (G2 * N9))} is non empty Element of bool the carrier of G
{((N9 * (N9 ")) * (G2 * N9))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((N9 * (N9 ")) * (G2 * N9))} & b2 in carr R ) } is set
(1_ G) * (G2 * N9) is Element of the carrier of G
K84( the carrier of G, the multF of G,(1_ G),(G2 * N9)) is Element of the carrier of G
((1_ G) * (G2 * N9)) * R is Element of bool the carrier of G
((1_ G) * (G2 * N9)) * (carr R) is Element of bool the carrier of G
{((1_ G) * (G2 * N9))} is non empty Element of bool the carrier of G
{((1_ G) * (G2 * N9))} * (carr R) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((1_ G) * (G2 * N9))} & b2 in carr R ) } is set
G is non empty unital Group-like associative multMagma
(G) is non empty set
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
(1). G is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic Subgroup of G
F is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len F is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
F . 1 is set
F . (len F) is set
dom F is Element of bool NAT
R is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
dom R is Element of bool NAT
c4 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
c4 + 1 is V31() V32() V33() ext-real set
R . c4 is set
R . (c4 + 1) is set
H1 is non empty strict unital Group-like associative normal Subgroup of G
H2 is non empty strict unital Group-like associative normal Subgroup of G
(G,H1,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr H1 is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of H1 is non empty set
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
[.(carr H1),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr H1),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr H1 & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr H1),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
N is non empty strict unital Group-like associative normal Subgroup of G
(H1,N) `*` is non empty strict unital Group-like associative normal Subgroup of H1
H1 ./. ((H1,N) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((H1,N) `*`) is non empty Element of bool (bool the carrier of H1)
bool the carrier of H1 is non empty set
bool (bool the carrier of H1) is non empty set
CosOp ((H1,N) `*`) is Relation-like [:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):] -defined Left_Cosets ((H1,N) `*`) -valued Function-like V18([:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):], Left_Cosets ((H1,N) `*`)) Element of bool [:[:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):],(Left_Cosets ((H1,N) `*`)):]
[:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):] is non empty set
[:[:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):],(Left_Cosets ((H1,N) `*`)):] is non empty set
bool [:[:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):],(Left_Cosets ((H1,N) `*`)):] is non empty set
multMagma(# (Left_Cosets ((H1,N) `*`)),(CosOp ((H1,N) `*`)) #) is non empty strict multMagma
G ./. N is non empty strict unital Group-like associative multMagma
Left_Cosets N is non empty Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
CosOp N is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V18([:(Left_Cosets N),(Left_Cosets N):], Left_Cosets N) Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is non empty set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
multMagma(# (Left_Cosets N),(CosOp N) #) is non empty strict multMagma
center (G ./. N) is non empty strict unital Group-like associative Subgroup of G ./. N
N is non empty strict unital Group-like associative normal Subgroup of G
(H1,N) `*` is non empty strict unital Group-like associative normal Subgroup of H1
H1 ./. ((H1,N) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((H1,N) `*`) is non empty Element of bool (bool the carrier of H1)
bool the carrier of H1 is non empty set
bool (bool the carrier of H1) is non empty set
CosOp ((H1,N) `*`) is Relation-like [:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):] -defined Left_Cosets ((H1,N) `*`) -valued Function-like V18([:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):], Left_Cosets ((H1,N) `*`)) Element of bool [:[:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):],(Left_Cosets ((H1,N) `*`)):]
[:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):] is non empty set
[:[:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):],(Left_Cosets ((H1,N) `*`)):] is non empty set
bool [:[:(Left_Cosets ((H1,N) `*`)),(Left_Cosets ((H1,N) `*`)):],(Left_Cosets ((H1,N) `*`)):] is non empty set
multMagma(# (Left_Cosets ((H1,N) `*`)),(CosOp ((H1,N) `*`)) #) is non empty strict multMagma
G ./. N is non empty strict unital Group-like associative multMagma
Left_Cosets N is non empty Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
CosOp N is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V18([:(Left_Cosets N),(Left_Cosets N):], Left_Cosets N) Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is non empty set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
multMagma(# (Left_Cosets N),(CosOp N) #) is non empty strict multMagma
center (G ./. N) is non empty strict unital Group-like associative Subgroup of G ./. N
c4 is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len c4 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
c4 . 1 is set
c4 . (len c4) is set
dom c4 is Element of bool NAT
H1 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
H1 + 1 is V31() V32() V33() ext-real set
H2 is non empty strict unital Group-like associative normal Subgroup of G
c4 . H1 is set
N is non empty strict unital Group-like associative normal Subgroup of G
c4 . (H1 + 1) is set
(G,H2,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr H2 is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of H2 is non empty set
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
[.(carr H2),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr H2),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr H2 & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr H2),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
F is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len F is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
F . 1 is set
F . (len F) is set
dom F is Element of bool NAT
R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R + 1 is V31() V32() V33() ext-real set
F . R is set
F . (R + 1) is set
c4 is non empty strict unital Group-like associative normal Subgroup of G
H1 is non empty strict unital Group-like associative normal Subgroup of G
(c4,H1) `*` is non empty strict unital Group-like associative normal Subgroup of c4
c4 ./. ((c4,H1) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((c4,H1) `*`) is non empty Element of bool (bool the carrier of c4)
the carrier of c4 is non empty set
bool the carrier of c4 is non empty set
bool (bool the carrier of c4) is non empty set
CosOp ((c4,H1) `*`) is Relation-like [:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] -defined Left_Cosets ((c4,H1) `*`) -valued Function-like V18([:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):], Left_Cosets ((c4,H1) `*`)) Element of bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):]
[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] is non empty set
[:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
multMagma(# (Left_Cosets ((c4,H1) `*`)),(CosOp ((c4,H1) `*`)) #) is non empty strict multMagma
G ./. H1 is non empty strict unital Group-like associative multMagma
Left_Cosets H1 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp H1 is Relation-like [:(Left_Cosets H1),(Left_Cosets H1):] -defined Left_Cosets H1 -valued Function-like V18([:(Left_Cosets H1),(Left_Cosets H1):], Left_Cosets H1) Element of bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):]
[:(Left_Cosets H1),(Left_Cosets H1):] is non empty set
[:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
multMagma(# (Left_Cosets H1),(CosOp H1) #) is non empty strict multMagma
center (G ./. H1) is non empty strict unital Group-like associative Subgroup of G ./. H1
(G,c4,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr c4 is Element of bool the carrier of G
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
[.(carr c4),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr c4),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr c4 & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr c4),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
R is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R . 1 is set
R . (len R) is set
dom R is Element of bool NAT
c4 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
c4 + 1 is V31() V32() V33() ext-real set
H1 is non empty strict unital Group-like associative normal Subgroup of G
R . c4 is set
H2 is non empty strict unital Group-like associative normal Subgroup of G
R . (c4 + 1) is set
(H1,H2) `*` is non empty strict unital Group-like associative normal Subgroup of H1
H1 ./. ((H1,H2) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((H1,H2) `*`) is non empty Element of bool (bool the carrier of H1)
the carrier of H1 is non empty set
bool the carrier of H1 is non empty set
bool (bool the carrier of H1) is non empty set
CosOp ((H1,H2) `*`) is Relation-like [:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):] -defined Left_Cosets ((H1,H2) `*`) -valued Function-like V18([:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):], Left_Cosets ((H1,H2) `*`)) Element of bool [:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):]
[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):] is non empty set
[:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):] is non empty set
bool [:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):] is non empty set
multMagma(# (Left_Cosets ((H1,H2) `*`)),(CosOp ((H1,H2) `*`)) #) is non empty strict multMagma
G ./. H2 is non empty strict unital Group-like associative multMagma
Left_Cosets H2 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp H2 is Relation-like [:(Left_Cosets H2),(Left_Cosets H2):] -defined Left_Cosets H2 -valued Function-like V18([:(Left_Cosets H2),(Left_Cosets H2):], Left_Cosets H2) Element of bool [:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):]
[:(Left_Cosets H2),(Left_Cosets H2):] is non empty set
[:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):] is non empty set
bool [:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):] is non empty set
multMagma(# (Left_Cosets H2),(CosOp H2) #) is non empty strict multMagma
center (G ./. H2) is non empty strict unital Group-like associative Subgroup of G ./. H2
F is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len F is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
F . 1 is set
F . (len F) is set
dom F is Element of bool NAT
c4 is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len c4 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
c4 . 1 is set
c4 . (len c4) is set
dom c4 is Element of bool NAT
G is non empty unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
R is non empty unital Group-like associative Subgroup of G
R /\ F is non empty strict unital Group-like associative Subgroup of G
c4 is non empty strict unital Group-like associative normal Subgroup of G
(R,c4) `*` is non empty strict unital Group-like associative normal Subgroup of R
R ./. ((R,c4) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((R,c4) `*`) is non empty Element of bool (bool the carrier of R)
the carrier of R is non empty set
bool the carrier of R is non empty set
bool (bool the carrier of R) is non empty set
CosOp ((R,c4) `*`) is Relation-like [:(Left_Cosets ((R,c4) `*`)),(Left_Cosets ((R,c4) `*`)):] -defined Left_Cosets ((R,c4) `*`) -valued Function-like V18([:(Left_Cosets ((R,c4) `*`)),(Left_Cosets ((R,c4) `*`)):], Left_Cosets ((R,c4) `*`)) Element of bool [:[:(Left_Cosets ((R,c4) `*`)),(Left_Cosets ((R,c4) `*`)):],(Left_Cosets ((R,c4) `*`)):]
[:(Left_Cosets ((R,c4) `*`)),(Left_Cosets ((R,c4) `*`)):] is non empty set
[:[:(Left_Cosets ((R,c4) `*`)),(Left_Cosets ((R,c4) `*`)):],(Left_Cosets ((R,c4) `*`)):] is non empty set
bool [:[:(Left_Cosets ((R,c4) `*`)),(Left_Cosets ((R,c4) `*`)):],(Left_Cosets ((R,c4) `*`)):] is non empty set
multMagma(# (Left_Cosets ((R,c4) `*`)),(CosOp ((R,c4) `*`)) #) is non empty strict multMagma
G ./. c4 is non empty strict unital Group-like associative multMagma
Left_Cosets c4 is non empty Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp c4 is Relation-like [:(Left_Cosets c4),(Left_Cosets c4):] -defined Left_Cosets c4 -valued Function-like V18([:(Left_Cosets c4),(Left_Cosets c4):], Left_Cosets c4) Element of bool [:[:(Left_Cosets c4),(Left_Cosets c4):],(Left_Cosets c4):]
[:(Left_Cosets c4),(Left_Cosets c4):] is non empty set
[:[:(Left_Cosets c4),(Left_Cosets c4):],(Left_Cosets c4):] is non empty set
bool [:[:(Left_Cosets c4),(Left_Cosets c4):],(Left_Cosets c4):] is non empty set
multMagma(# (Left_Cosets c4),(CosOp c4) #) is non empty strict multMagma
center (G ./. c4) is non empty strict unital Group-like associative Subgroup of G ./. c4
c4 /\ F is non empty strict unital Group-like associative normal Subgroup of F
H1 is non empty unital Group-like associative Subgroup of F
H2 is non empty unital Group-like associative normal Subgroup of F
(H1,H2) `*` is non empty strict unital Group-like associative normal Subgroup of H1
H1 ./. ((H1,H2) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((H1,H2) `*`) is non empty Element of bool (bool the carrier of H1)
the carrier of H1 is non empty set
bool the carrier of H1 is non empty set
bool (bool the carrier of H1) is non empty set
CosOp ((H1,H2) `*`) is Relation-like [:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):] -defined Left_Cosets ((H1,H2) `*`) -valued Function-like V18([:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):], Left_Cosets ((H1,H2) `*`)) Element of bool [:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):]
[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):] is non empty set
[:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):] is non empty set
bool [:[:(Left_Cosets ((H1,H2) `*`)),(Left_Cosets ((H1,H2) `*`)):],(Left_Cosets ((H1,H2) `*`)):] is non empty set
multMagma(# (Left_Cosets ((H1,H2) `*`)),(CosOp ((H1,H2) `*`)) #) is non empty strict multMagma
F ./. H2 is non empty strict unital Group-like associative multMagma
Left_Cosets H2 is non empty Element of bool (bool the carrier of F)
the carrier of F is non empty set
bool the carrier of F is non empty set
bool (bool the carrier of F) is non empty set
CosOp H2 is Relation-like [:(Left_Cosets H2),(Left_Cosets H2):] -defined Left_Cosets H2 -valued Function-like V18([:(Left_Cosets H2),(Left_Cosets H2):], Left_Cosets H2) Element of bool [:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):]
[:(Left_Cosets H2),(Left_Cosets H2):] is non empty set
[:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):] is non empty set
bool [:[:(Left_Cosets H2),(Left_Cosets H2):],(Left_Cosets H2):] is non empty set
multMagma(# (Left_Cosets H2),(CosOp H2) #) is non empty strict multMagma
center (F ./. H2) is non empty strict unital Group-like associative Subgroup of F ./. H2
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
(G,R,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr R is Element of bool the carrier of G
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
[.(carr R),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr R),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr R & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr R),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
the carrier of (F ./. H2) is non empty set
N9 is non empty unital Group-like associative Subgroup of F ./. H2
G1 is Element of the carrier of (F ./. H2)
G2 is Element of the carrier of (F ./. H2)
G2 * G1 is Element of the carrier of (F ./. H2)
the multF of (F ./. H2) is Relation-like [: the carrier of (F ./. H2), the carrier of (F ./. H2):] -defined the carrier of (F ./. H2) -valued Function-like V18([: the carrier of (F ./. H2), the carrier of (F ./. H2):], the carrier of (F ./. H2)) V22( the carrier of (F ./. H2)) Element of bool [:[: the carrier of (F ./. H2), the carrier of (F ./. H2):], the carrier of (F ./. H2):]
[: the carrier of (F ./. H2), the carrier of (F ./. H2):] is non empty set
[:[: the carrier of (F ./. H2), the carrier of (F ./. H2):], the carrier of (F ./. H2):] is non empty set
bool [:[: the carrier of (F ./. H2), the carrier of (F ./. H2):], the carrier of (F ./. H2):] is non empty set
K84( the carrier of (F ./. H2), the multF of (F ./. H2),G2,G1) is Element of the carrier of (F ./. H2)
G1 * G2 is Element of the carrier of (F ./. H2)
K84( the carrier of (F ./. H2), the multF of (F ./. H2),G1,G2) is Element of the carrier of (F ./. H2)
g1 is Element of the carrier of F
g1 * H2 is Element of bool the carrier of F
carr H2 is Element of bool the carrier of F
the carrier of H2 is non empty set
g1 * (carr H2) is Element of bool the carrier of F
{g1} is non empty Element of bool the carrier of F
{g1} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {g1} & b2 in carr H2 ) } is set
H2 * g1 is Element of bool the carrier of F
(carr H2) * g1 is Element of bool the carrier of F
(carr H2) * {g1} is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in carr H2 & b2 in {g1} ) } is set
N is non empty unital Group-like associative normal Subgroup of H1
g is Element of the carrier of H1
g * N is Element of bool the carrier of H1
carr N is Element of bool the carrier of H1
the carrier of N is non empty set
g * (carr N) is Element of bool the carrier of H1
{g} is non empty Element of bool the carrier of H1
{g} * (carr N) is Element of bool the carrier of H1
{ (b1 * b2) where b1, b2 is Element of the carrier of H1 : ( b1 in {g} & b2 in carr N ) } is set
N * g is Element of bool the carrier of H1
(carr N) * g is Element of bool the carrier of H1
(carr N) * {g} is Element of bool the carrier of H1
{ (b1 * b2) where b1, b2 is Element of the carrier of H1 : ( b1 in carr N & b2 in {g} ) } is set
@ G2 is Element of bool the carrier of F
@ G1 is Element of bool the carrier of F
h1 is Element of the carrier of F
h1 * H2 is Element of bool the carrier of F
h1 * (carr H2) is Element of bool the carrier of F
{h1} is non empty Element of bool the carrier of F
{h1} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {h1} & b2 in carr H2 ) } is set
(g1 * H2) * (h1 * H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in g1 * H2 & b2 in h1 * H2 ) } is set
g1 * h1 is Element of the carrier of F
the multF of F is Relation-like [: the carrier of F, the carrier of F:] -defined the carrier of F -valued Function-like V18([: the carrier of F, the carrier of F:], the carrier of F) V22( 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
K84( the carrier of F, the multF of F,g1,h1) is Element of the carrier of F
(g1 * h1) * H2 is Element of bool the carrier of F
(g1 * h1) * (carr H2) is Element of bool the carrier of F
{(g1 * h1)} is non empty Element of bool the carrier of F
{(g1 * h1)} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {(g1 * h1)} & b2 in carr H2 ) } is set
(h1 * H2) * (g1 * H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in h1 * H2 & b2 in g1 * H2 ) } is set
h1 * g1 is Element of the carrier of F
K84( the carrier of F, the multF of F,h1,g1) is Element of the carrier of F
(h1 * g1) * H2 is Element of bool the carrier of F
(h1 * g1) * (carr H2) is Element of bool the carrier of F
{(h1 * g1)} is non empty Element of bool the carrier of F
{(h1 * g1)} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {(h1 * g1)} & b2 in carr H2 ) } is set
[.h1,g1.] is Element of the carrier of F
h1 " is Element of the carrier of F
g1 " is Element of the carrier of F
(h1 ") * (g1 ") is Element of the carrier of F
K84( the carrier of F, the multF of F,(h1 "),(g1 ")) is Element of the carrier of F
((h1 ") * (g1 ")) * h1 is Element of the carrier of F
K84( the carrier of F, the multF of F,((h1 ") * (g1 ")),h1) is Element of the carrier of F
(((h1 ") * (g1 ")) * h1) * g1 is Element of the carrier of F
K84( the carrier of F, the multF of F,(((h1 ") * (g1 ")) * h1),g1) is Element of the carrier of F
h2 is Element of the carrier of G
a is Element of the carrier of R
a1 is Element of the carrier of G
[.a1,h2.] is Element of the carrier of G
a1 " is Element of the carrier of G
h2 " is Element of the carrier of G
(a1 ") * (h2 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(a1 "),(h2 ")) is Element of the carrier of G
((a1 ") * (h2 ")) * a1 is Element of the carrier of G
K84( the carrier of G, the multF of G,((a1 ") * (h2 ")),a1) is Element of the carrier of G
(((a1 ") * (h2 ")) * a1) * h2 is Element of the carrier of G
K84( the carrier of G, the multF of G,(((a1 ") * (h2 ")) * a1),h2) is Element of the carrier of G
a1 * h2 is Element of the carrier of G
K84( the carrier of G, the multF of G,a1,h2) is Element of the carrier of G
((h1 ") * (g1 ")) * (h1 * g1) is Element of the carrier of F
K84( the carrier of F, the multF of F,((h1 ") * (g1 ")),(h1 * g1)) is Element of the carrier of F
((a1 ") * (h2 ")) * (a1 * h2) is Element of the carrier of G
K84( the carrier of G, the multF of G,((a1 ") * (h2 ")),(a1 * h2)) is Element of the carrier of G
[.h1,g1.] * H2 is Element of bool the carrier of F
[.h1,g1.] * (carr H2) is Element of bool the carrier of F
{[.h1,g1.]} is non empty Element of bool the carrier of F
{[.h1,g1.]} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {[.h1,g1.]} & b2 in carr H2 ) } is set
(g1 * h1) * ([.h1,g1.] * H2) is Element of bool the carrier of F
{(g1 * h1)} * ([.h1,g1.] * H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {(g1 * h1)} & b2 in [.h1,g1.] * H2 ) } is set
(((h1 ") * (g1 ")) * (h1 * g1)) * H2 is Element of bool the carrier of F
(((h1 ") * (g1 ")) * (h1 * g1)) * (carr H2) is Element of bool the carrier of F
{(((h1 ") * (g1 ")) * (h1 * g1))} is non empty Element of bool the carrier of F
{(((h1 ") * (g1 ")) * (h1 * g1))} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {(((h1 ") * (g1 ")) * (h1 * g1))} & b2 in carr H2 ) } is set
(g1 * h1) * ((((h1 ") * (g1 ")) * (h1 * g1)) * H2) is Element of bool the carrier of F
{(g1 * h1)} * ((((h1 ") * (g1 ")) * (h1 * g1)) * H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {(g1 * h1)} & b2 in (((h1 ") * (g1 ")) * (h1 * g1)) * H2 ) } is set
(g1 * h1) * (((h1 ") * (g1 ")) * (h1 * g1)) is Element of the carrier of F
K84( the carrier of F, the multF of F,(g1 * h1),(((h1 ") * (g1 ")) * (h1 * g1))) is Element of the carrier of F
((g1 * h1) * (((h1 ") * (g1 ")) * (h1 * g1))) * H2 is Element of bool the carrier of F
((g1 * h1) * (((h1 ") * (g1 ")) * (h1 * g1))) * (carr H2) is Element of bool the carrier of F
{((g1 * h1) * (((h1 ") * (g1 ")) * (h1 * g1)))} is non empty Element of bool the carrier of F
{((g1 * h1) * (((h1 ") * (g1 ")) * (h1 * g1)))} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {((g1 * h1) * (((h1 ") * (g1 ")) * (h1 * g1)))} & b2 in carr H2 ) } is set
(g1 * h1) * ((h1 ") * (g1 ")) is Element of the carrier of F
K84( the carrier of F, the multF of F,(g1 * h1),((h1 ") * (g1 "))) is Element of the carrier of F
((g1 * h1) * ((h1 ") * (g1 "))) * (h1 * g1) is Element of the carrier of F
K84( the carrier of F, the multF of F,((g1 * h1) * ((h1 ") * (g1 "))),(h1 * g1)) is Element of the carrier of F
(((g1 * h1) * ((h1 ") * (g1 "))) * (h1 * g1)) * H2 is Element of bool the carrier of F
(((g1 * h1) * ((h1 ") * (g1 "))) * (h1 * g1)) * (carr H2) is Element of bool the carrier of F
{(((g1 * h1) * ((h1 ") * (g1 "))) * (h1 * g1))} is non empty Element of bool the carrier of F
{(((g1 * h1) * ((h1 ") * (g1 "))) * (h1 * g1))} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {(((g1 * h1) * ((h1 ") * (g1 "))) * (h1 * g1))} & b2 in carr H2 ) } is set
h1 * ((h1 ") * (g1 ")) is Element of the carrier of F
K84( the carrier of F, the multF of F,h1,((h1 ") * (g1 "))) is Element of the carrier of F
g1 * (h1 * ((h1 ") * (g1 "))) is Element of the carrier of F
K84( the carrier of F, the multF of F,g1,(h1 * ((h1 ") * (g1 ")))) is Element of the carrier of F
(g1 * (h1 * ((h1 ") * (g1 ")))) * (h1 * g1) is Element of the carrier of F
K84( the carrier of F, the multF of F,(g1 * (h1 * ((h1 ") * (g1 ")))),(h1 * g1)) is Element of the carrier of F
((g1 * (h1 * ((h1 ") * (g1 ")))) * (h1 * g1)) * H2 is Element of bool the carrier of F
((g1 * (h1 * ((h1 ") * (g1 ")))) * (h1 * g1)) * (carr H2) is Element of bool the carrier of F
{((g1 * (h1 * ((h1 ") * (g1 ")))) * (h1 * g1))} is non empty Element of bool the carrier of F
{((g1 * (h1 * ((h1 ") * (g1 ")))) * (h1 * g1))} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {((g1 * (h1 * ((h1 ") * (g1 ")))) * (h1 * g1))} & b2 in carr H2 ) } is set
h1 * (h1 ") is Element of the carrier of F
K84( the carrier of F, the multF of F,h1,(h1 ")) is Element of the carrier of F
(h1 * (h1 ")) * (g1 ") is Element of the carrier of F
K84( the carrier of F, the multF of F,(h1 * (h1 ")),(g1 ")) is Element of the carrier of F
g1 * ((h1 * (h1 ")) * (g1 ")) is Element of the carrier of F
K84( the carrier of F, the multF of F,g1,((h1 * (h1 ")) * (g1 "))) is Element of the carrier of F
(g1 * ((h1 * (h1 ")) * (g1 "))) * (h1 * g1) is Element of the carrier of F
K84( the carrier of F, the multF of F,(g1 * ((h1 * (h1 ")) * (g1 "))),(h1 * g1)) is Element of the carrier of F
((g1 * ((h1 * (h1 ")) * (g1 "))) * (h1 * g1)) * H2 is Element of bool the carrier of F
((g1 * ((h1 * (h1 ")) * (g1 "))) * (h1 * g1)) * (carr H2) is Element of bool the carrier of F
{((g1 * ((h1 * (h1 ")) * (g1 "))) * (h1 * g1))} is non empty Element of bool the carrier of F
{((g1 * ((h1 * (h1 ")) * (g1 "))) * (h1 * g1))} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {((g1 * ((h1 * (h1 ")) * (g1 "))) * (h1 * g1))} & b2 in carr H2 ) } is set
1_ F is non being_of_order_0 Element of the carrier of F
(1_ F) * (g1 ") is Element of the carrier of F
K84( the carrier of F, the multF of F,(1_ F),(g1 ")) is Element of the carrier of F
g1 * ((1_ F) * (g1 ")) is Element of the carrier of F
K84( the carrier of F, the multF of F,g1,((1_ F) * (g1 "))) is Element of the carrier of F
(g1 * ((1_ F) * (g1 "))) * (h1 * g1) is Element of the carrier of F
K84( the carrier of F, the multF of F,(g1 * ((1_ F) * (g1 "))),(h1 * g1)) is Element of the carrier of F
((g1 * ((1_ F) * (g1 "))) * (h1 * g1)) * H2 is Element of bool the carrier of F
((g1 * ((1_ F) * (g1 "))) * (h1 * g1)) * (carr H2) is Element of bool the carrier of F
{((g1 * ((1_ F) * (g1 "))) * (h1 * g1))} is non empty Element of bool the carrier of F
{((g1 * ((1_ F) * (g1 "))) * (h1 * g1))} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {((g1 * ((1_ F) * (g1 "))) * (h1 * g1))} & b2 in carr H2 ) } is set
g1 * (g1 ") is Element of the carrier of F
K84( the carrier of F, the multF of F,g1,(g1 ")) is Element of the carrier of F
(g1 * (g1 ")) * (h1 * g1) is Element of the carrier of F
K84( the carrier of F, the multF of F,(g1 * (g1 ")),(h1 * g1)) is Element of the carrier of F
((g1 * (g1 ")) * (h1 * g1)) * H2 is Element of bool the carrier of F
((g1 * (g1 ")) * (h1 * g1)) * (carr H2) is Element of bool the carrier of F
{((g1 * (g1 ")) * (h1 * g1))} is non empty Element of bool the carrier of F
{((g1 * (g1 ")) * (h1 * g1))} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {((g1 * (g1 ")) * (h1 * g1))} & b2 in carr H2 ) } is set
(1_ F) * (h1 * g1) is Element of the carrier of F
K84( the carrier of F, the multF of F,(1_ F),(h1 * g1)) is Element of the carrier of F
((1_ F) * (h1 * g1)) * H2 is Element of bool the carrier of F
((1_ F) * (h1 * g1)) * (carr H2) is Element of bool the carrier of F
{((1_ F) * (h1 * g1))} is non empty Element of bool the carrier of F
{((1_ F) * (h1 * g1))} * (carr H2) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {((1_ F) * (h1 * g1))} & b2 in carr H2 ) } is set
G is non empty unital Group-like associative multMagma
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
F is non empty unital Group-like associative Subgroup of G
((Omega). G) /\ F is non empty strict unital Group-like associative normal Subgroup of F
(Omega). F is non empty strict unital Group-like associative normal Subgroup of F
the carrier of F is non empty set
the multF of F is Relation-like [: the carrier of F, the carrier of F:] -defined the carrier of F -valued Function-like V18([: the carrier of F, the carrier of F:], the carrier of F) V22( 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
multMagma(# the carrier of F, the multF of F #) is non empty strict unital Group-like associative multMagma
the multF of G || the carrier of F is set
the multF of G | [: the carrier of F, the carrier of F:] is Relation-like set
R is non empty strict unital Group-like associative multMagma
((Omega). G) /\ F is non empty strict unital Group-like associative Subgroup of G
the carrier of (((Omega). G) /\ F) is non empty set
the multF of (((Omega). G) /\ F) is Relation-like [: the carrier of (((Omega). G) /\ F), the carrier of (((Omega). G) /\ F):] -defined the carrier of (((Omega). G) /\ F) -valued Function-like V18([: the carrier of (((Omega). G) /\ F), the carrier of (((Omega). G) /\ F):], the carrier of (((Omega). G) /\ F)) V22( the carrier of (((Omega). G) /\ F)) Element of bool [:[: the carrier of (((Omega). G) /\ F), the carrier of (((Omega). G) /\ F):], the carrier of (((Omega). G) /\ F):]
[: the carrier of (((Omega). G) /\ F), the carrier of (((Omega). G) /\ F):] is non empty set
[:[: the carrier of (((Omega). G) /\ F), the carrier of (((Omega). G) /\ F):], the carrier of (((Omega). G) /\ F):] is non empty set
bool [:[: the carrier of (((Omega). G) /\ F), the carrier of (((Omega). G) /\ F):], the carrier of (((Omega). G) /\ F):] is non empty set
multMagma(# the carrier of (((Omega). G) /\ F), the multF of (((Omega). G) /\ F) #) is non empty strict unital Group-like associative multMagma
F /\ ((Omega). G) is non empty strict unital Group-like associative Subgroup of G
the carrier of (F /\ ((Omega). G)) is non empty set
the multF of (F /\ ((Omega). G)) is Relation-like [: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):] -defined the carrier of (F /\ ((Omega). G)) -valued Function-like V18([: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):], the carrier of (F /\ ((Omega). G))) V22( the carrier of (F /\ ((Omega). G))) Element of bool [:[: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):], the carrier of (F /\ ((Omega). G)):]
[: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):] is non empty set
[:[: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):], the carrier of (F /\ ((Omega). G)):] is non empty set
bool [:[: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):], the carrier of (F /\ ((Omega). G)):] is non empty set
multMagma(# the carrier of (F /\ ((Omega). G)), the multF of (F /\ ((Omega). G)) #) is non empty strict unital Group-like associative multMagma
F /\ ((Omega). G) is non empty strict unital Group-like associative normal Subgroup of F
the carrier of (F /\ ((Omega). G)) is non empty set
the multF of (F /\ ((Omega). G)) is Relation-like [: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):] -defined the carrier of (F /\ ((Omega). G)) -valued Function-like V18([: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):], the carrier of (F /\ ((Omega). G))) V22( the carrier of (F /\ ((Omega). G))) Element of bool [:[: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):], the carrier of (F /\ ((Omega). G)):]
[: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):] is non empty set
[:[: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):], the carrier of (F /\ ((Omega). G)):] is non empty set
bool [:[: the carrier of (F /\ ((Omega). G)), the carrier of (F /\ ((Omega). G)):], the carrier of (F /\ ((Omega). G)):] is non empty set
multMagma(# the carrier of (F /\ ((Omega). G)), the multF of (F /\ ((Omega). G)) #) is non empty strict unital Group-like associative multMagma
c4 is non empty strict unital Group-like associative Subgroup of R
(Omega). R is non empty strict unital Group-like associative normal Subgroup of R
the carrier of R is non empty set
the multF of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like V18([: the carrier of R, the carrier of R:], the carrier of R) V22( the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
multMagma(# the carrier of R, the multF of R #) is non empty strict unital Group-like associative multMagma
c4 /\ ((Omega). R) is non empty strict unital Group-like associative normal Subgroup of c4
the carrier of (c4 /\ ((Omega). R)) is non empty set
the multF of (c4 /\ ((Omega). R)) is Relation-like [: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):] -defined the carrier of (c4 /\ ((Omega). R)) -valued Function-like V18([: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):], the carrier of (c4 /\ ((Omega). R))) V22( the carrier of (c4 /\ ((Omega). R))) Element of bool [:[: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):], the carrier of (c4 /\ ((Omega). R)):]
[: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):] is non empty set
[:[: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):], the carrier of (c4 /\ ((Omega). R)):] is non empty set
bool [:[: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):], the carrier of (c4 /\ ((Omega). R)):] is non empty set
multMagma(# the carrier of (c4 /\ ((Omega). R)), the multF of (c4 /\ ((Omega). R)) #) is non empty strict unital Group-like associative multMagma
c4 /\ ((Omega). R) is non empty strict unital Group-like associative Subgroup of R
the carrier of (c4 /\ ((Omega). R)) is non empty set
the multF of (c4 /\ ((Omega). R)) is Relation-like [: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):] -defined the carrier of (c4 /\ ((Omega). R)) -valued Function-like V18([: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):], the carrier of (c4 /\ ((Omega). R))) V22( the carrier of (c4 /\ ((Omega). R))) Element of bool [:[: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):], the carrier of (c4 /\ ((Omega). R)):]
[: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):] is non empty set
[:[: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):], the carrier of (c4 /\ ((Omega). R)):] is non empty set
bool [:[: the carrier of (c4 /\ ((Omega). R)), the carrier of (c4 /\ ((Omega). R)):], the carrier of (c4 /\ ((Omega). R)):] is non empty set
multMagma(# the carrier of (c4 /\ ((Omega). R)), the multF of (c4 /\ ((Omega). R)) #) is non empty strict unital Group-like associative multMagma
((Omega). R) /\ c4 is non empty strict unital Group-like associative Subgroup of R
the carrier of (((Omega). R) /\ c4) is non empty set
the multF of (((Omega). R) /\ c4) is Relation-like [: the carrier of (((Omega). R) /\ c4), the carrier of (((Omega). R) /\ c4):] -defined the carrier of (((Omega). R) /\ c4) -valued Function-like V18([: the carrier of (((Omega). R) /\ c4), the carrier of (((Omega). R) /\ c4):], the carrier of (((Omega). R) /\ c4)) V22( the carrier of (((Omega). R) /\ c4)) Element of bool [:[: the carrier of (((Omega). R) /\ c4), the carrier of (((Omega). R) /\ c4):], the carrier of (((Omega). R) /\ c4):]
[: the carrier of (((Omega). R) /\ c4), the carrier of (((Omega). R) /\ c4):] is non empty set
[:[: the carrier of (((Omega). R) /\ c4), the carrier of (((Omega). R) /\ c4):], the carrier of (((Omega). R) /\ c4):] is non empty set
bool [:[: the carrier of (((Omega). R) /\ c4), the carrier of (((Omega). R) /\ c4):], the carrier of (((Omega). R) /\ c4):] is non empty set
multMagma(# the carrier of (((Omega). R) /\ c4), the multF of (((Omega). R) /\ c4) #) is non empty strict unital Group-like associative multMagma
G is non empty unital Group-like associative multMagma
F is non empty strict unital Group-like associative Subgroup of G
c4 is non empty unital Group-like associative Subgroup of G
R is non empty unital Group-like associative Subgroup of G
F /\ R is non empty strict unital Group-like associative Subgroup of G
c4 /\ R is non empty strict unital Group-like associative Subgroup of G
F /\ c4 is non empty strict unital Group-like associative Subgroup of G
(F /\ c4) /\ R is non empty strict unital Group-like associative Subgroup of G
F /\ (c4 /\ R) is non empty strict unital Group-like associative Subgroup of G
H2 is non empty unital Group-like associative normal Subgroup of c4
H1 is non empty unital Group-like associative Subgroup of c4
H2 /\ H1 is non empty strict unital Group-like associative normal Subgroup of H1
G is non empty unital Group-like associative () multMagma
F is non empty unital Group-like associative Subgroup of G
(G) is non empty set
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
(1). G is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic Subgroup of G
R is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R . 1 is set
R . (len R) is set
dom R is Element of bool NAT
Seg (len R) is V37() V44( len R) Element of bool NAT
{ b1 where b1 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT : ( 1 <= b1 & b1 <= len R ) } is set
(F) is non empty set
c4 is V24() V25() V26() V30() V31() V32() V33() ext-real set
R . c4 is set
H1 is non empty strict unital Group-like associative normal Subgroup of G
H1 /\ F is non empty strict unital Group-like associative normal Subgroup of F
H2 is non empty strict unital Group-like associative Subgroup of F
N is Element of (F)
c4 is Relation-like NAT -defined (F) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (F)
dom c4 is Element of bool NAT
len c4 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
c4 . 1 is set
(Omega). F is non empty strict unital Group-like associative normal Subgroup of F
the carrier of F is non empty set
the multF of F is Relation-like [: the carrier of F, the carrier of F:] -defined the carrier of F -valued Function-like V18([: the carrier of F, the carrier of F:], the carrier of F) V22( 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
multMagma(# the carrier of F, the multF of F #) is non empty strict unital Group-like associative multMagma
H1 is non empty strict unital Group-like associative normal Subgroup of G
H1 /\ F is non empty strict unital Group-like associative normal Subgroup of F
c4 . (len c4) is set
(1). F is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic Subgroup of F
R . (len c4) is set
H1 is non empty strict unital Group-like associative normal Subgroup of G
H1 /\ F is non empty strict unital Group-like associative normal Subgroup of F
H1 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
H1 + 1 is V31() V32() V33() ext-real set
c4 . H1 is set
c4 . (H1 + 1) is set
H2 is non empty strict unital Group-like associative normal Subgroup of F
N is non empty strict unital Group-like associative normal Subgroup of F
(H2,N) `*` is non empty strict unital Group-like associative normal Subgroup of H2
H2 ./. ((H2,N) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((H2,N) `*`) is non empty Element of bool (bool the carrier of H2)
the carrier of H2 is non empty set
bool the carrier of H2 is non empty set
bool (bool the carrier of H2) is non empty set
CosOp ((H2,N) `*`) is Relation-like [:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):] -defined Left_Cosets ((H2,N) `*`) -valued Function-like V18([:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):], Left_Cosets ((H2,N) `*`)) Element of bool [:[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):],(Left_Cosets ((H2,N) `*`)):]
[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):] is non empty set
[:[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):],(Left_Cosets ((H2,N) `*`)):] is non empty set
bool [:[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):],(Left_Cosets ((H2,N) `*`)):] is non empty set
multMagma(# (Left_Cosets ((H2,N) `*`)),(CosOp ((H2,N) `*`)) #) is non empty strict multMagma
F ./. N is non empty strict unital Group-like associative multMagma
Left_Cosets N is non empty Element of bool (bool the carrier of F)
bool the carrier of F is non empty set
bool (bool the carrier of F) is non empty set
CosOp N is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V18([:(Left_Cosets N),(Left_Cosets N):], Left_Cosets N) Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is non empty set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
multMagma(# (Left_Cosets N),(CosOp N) #) is non empty strict multMagma
center (F ./. N) is non empty strict unital Group-like associative Subgroup of F ./. N
R . H1 is set
N9 is non empty strict unital Group-like associative normal Subgroup of G
N9 /\ F is non empty strict unital Group-like associative normal Subgroup of F
R . (H1 + 1) is set
G1 is non empty strict unital Group-like associative normal Subgroup of G
G1 /\ F is non empty strict unital Group-like associative normal Subgroup of F
(N9,G1) `*` is non empty strict unital Group-like associative normal Subgroup of N9
N9 ./. ((N9,G1) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((N9,G1) `*`) is non empty Element of bool (bool the carrier of N9)
the carrier of N9 is non empty set
bool the carrier of N9 is non empty set
bool (bool the carrier of N9) is non empty set
CosOp ((N9,G1) `*`) is Relation-like [:(Left_Cosets ((N9,G1) `*`)),(Left_Cosets ((N9,G1) `*`)):] -defined Left_Cosets ((N9,G1) `*`) -valued Function-like V18([:(Left_Cosets ((N9,G1) `*`)),(Left_Cosets ((N9,G1) `*`)):], Left_Cosets ((N9,G1) `*`)) Element of bool [:[:(Left_Cosets ((N9,G1) `*`)),(Left_Cosets ((N9,G1) `*`)):],(Left_Cosets ((N9,G1) `*`)):]
[:(Left_Cosets ((N9,G1) `*`)),(Left_Cosets ((N9,G1) `*`)):] is non empty set
[:[:(Left_Cosets ((N9,G1) `*`)),(Left_Cosets ((N9,G1) `*`)):],(Left_Cosets ((N9,G1) `*`)):] is non empty set
bool [:[:(Left_Cosets ((N9,G1) `*`)),(Left_Cosets ((N9,G1) `*`)):],(Left_Cosets ((N9,G1) `*`)):] is non empty set
multMagma(# (Left_Cosets ((N9,G1) `*`)),(CosOp ((N9,G1) `*`)) #) is non empty strict multMagma
G ./. G1 is non empty strict unital Group-like associative multMagma
Left_Cosets G1 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp G1 is Relation-like [:(Left_Cosets G1),(Left_Cosets G1):] -defined Left_Cosets G1 -valued Function-like V18([:(Left_Cosets G1),(Left_Cosets G1):], Left_Cosets G1) Element of bool [:[:(Left_Cosets G1),(Left_Cosets G1):],(Left_Cosets G1):]
[:(Left_Cosets G1),(Left_Cosets G1):] is non empty set
[:[:(Left_Cosets G1),(Left_Cosets G1):],(Left_Cosets G1):] is non empty set
bool [:[:(Left_Cosets G1),(Left_Cosets G1):],(Left_Cosets G1):] is non empty set
multMagma(# (Left_Cosets G1),(CosOp G1) #) is non empty strict multMagma
center (G ./. G1) is non empty strict unital Group-like associative Subgroup of G ./. G1
H1 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
H1 + 1 is V31() V32() V33() ext-real set
H2 is non empty strict unital Group-like associative normal Subgroup of F
c4 . H1 is set
N is non empty strict unital Group-like associative normal Subgroup of F
c4 . (H1 + 1) is set
(H2,N) `*` is non empty strict unital Group-like associative normal Subgroup of H2
H2 ./. ((H2,N) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((H2,N) `*`) is non empty Element of bool (bool the carrier of H2)
the carrier of H2 is non empty set
bool the carrier of H2 is non empty set
bool (bool the carrier of H2) is non empty set
CosOp ((H2,N) `*`) is Relation-like [:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):] -defined Left_Cosets ((H2,N) `*`) -valued Function-like V18([:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):], Left_Cosets ((H2,N) `*`)) Element of bool [:[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):],(Left_Cosets ((H2,N) `*`)):]
[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):] is non empty set
[:[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):],(Left_Cosets ((H2,N) `*`)):] is non empty set
bool [:[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):],(Left_Cosets ((H2,N) `*`)):] is non empty set
multMagma(# (Left_Cosets ((H2,N) `*`)),(CosOp ((H2,N) `*`)) #) is non empty strict multMagma
F ./. N is non empty strict unital Group-like associative multMagma
Left_Cosets N is non empty Element of bool (bool the carrier of F)
bool the carrier of F is non empty set
bool (bool the carrier of F) is non empty set
CosOp N is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V18([:(Left_Cosets N),(Left_Cosets N):], Left_Cosets N) Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is non empty set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
multMagma(# (Left_Cosets N),(CosOp N) #) is non empty strict multMagma
center (F ./. N) is non empty strict unital Group-like associative Subgroup of F ./. N
G is non empty unital Group-like associative multMagma
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
(G) is non empty set
(1). G is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic Subgroup of G
<*((Omega). G),((1). G)*> is Relation-like NAT -defined Function-like non empty V37() V44(2) FinSequence-like FinSubsequence-like set
<*((Omega). G)*> is Relation-like NAT -defined Function-like non empty V37() V44(1) FinSequence-like FinSubsequence-like set
[1,((Omega). G)] is set
{1,((Omega). G)} is non empty set
{{1,((Omega). G)},{1}} is non empty set
{[1,((Omega). G)]} is non empty set
<*((1). G)*> is Relation-like NAT -defined Function-like non empty V37() V44(1) FinSequence-like FinSubsequence-like set
[1,((1). G)] is set
{1,((1). G)} is non empty set
{{1,((1). G)},{1}} is non empty set
{[1,((1). G)]} is non empty set
K195(<*((Omega). G)*>,<*((1). G)*>) is Relation-like NAT -defined Function-like non empty V37() V44(K402(1,1)) FinSequence-like FinSubsequence-like set
K402(1,1) is non empty V24() V25() V26() V30() V31() V32() V33() ext-real positive non negative Element of NAT
F is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len F is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
F . 1 is set
F . 2 is set
dom F is Element of bool NAT
R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R + 1 is V31() V32() V33() ext-real set
F . R is set
F . (R + 1) is set
c4 is non empty unital Group-like associative Subgroup of G
H1 is non empty unital Group-like associative Subgroup of G
{1,2} is non empty Element of bool NAT
commutators (c4,((Omega). G)) is Element of bool the carrier of G
bool the carrier of G is non empty set
carr c4 is Element of bool the carrier of G
the carrier of c4 is non empty set
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
commutators ((carr c4),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr c4 & b2 in carr ((Omega). G) ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty Element of bool the carrier of G
(G,c4,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
[.(carr c4),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
gr (commutators ((carr c4),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
{(1_ G)} \ {(1_ G)} is Element of bool the carrier of G
gr ({(1_ G)} \ {(1_ G)}) is non empty strict unital Group-like associative Subgroup of G
{} the carrier of G is functional empty proper V24() V25() V26() V28() V29() V30() V31() V32() V33() FinSequence-membered ext-real non positive non negative Element of bool the carrier of G
gr ({} the carrier of G) is non empty strict unital Group-like associative Subgroup of G
H2 is non empty strict unital Group-like associative normal Subgroup of G
N is non empty strict unital Group-like associative normal Subgroup of G
(G,H2,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr H2 is Element of bool the carrier of G
the carrier of H2 is non empty set
[.(carr H2),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr H2),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr H2 & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr H2),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
c4 is non empty strict unital Group-like associative normal Subgroup of G
H1 is non empty strict unital Group-like associative normal Subgroup of G
(G,c4,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr c4 is Element of bool the carrier of G
the carrier of c4 is non empty set
[.(carr c4),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr c4),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr c4 & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr c4),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
G is non empty strict unital Group-like associative multMagma
the carrier of G is non empty set
F is non empty strict unital Group-like associative multMagma
the carrier of F is non empty set
[: the carrier of G, the carrier of F:] is non empty set
bool [: the carrier of G, the carrier of F:] is non empty set
R is Relation-like the carrier of G -defined the carrier of F -valued Function-like V18( the carrier of G, the carrier of F) unity-preserving V152(G,F) Element of bool [: the carrier of G, the carrier of F:]
c4 is non empty strict unital Group-like associative Subgroup of G
R .: c4 is non empty strict unital Group-like associative Subgroup of F
R | c4 is Relation-like the carrier of c4 -defined the carrier of F -valued Function-like V18( the carrier of c4, the carrier of F) unity-preserving V152(c4,F) Element of bool [: the carrier of c4, the carrier of F:]
the carrier of c4 is non empty set
[: the carrier of c4, the carrier of F:] is non empty set
bool [: the carrier of c4, the carrier of F:] is non empty set
R | the carrier of c4 is Relation-like the carrier of G -defined the carrier of F -valued Element of bool [: the carrier of G, the carrier of F:]
Image (R | c4) is non empty strict unital Group-like associative Subgroup of F
H1 is Element of the carrier of G
R . H1 is Element of the carrier of F
H2 is Element of the carrier of G
R . H2 is Element of the carrier of F
(R . H1) * (R . H2) is Element of the carrier of F
the multF of F is Relation-like [: the carrier of F, the carrier of F:] -defined the carrier of F -valued Function-like V18([: the carrier of F, the carrier of F:], the carrier of F) V22( 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
K84( the carrier of F, the multF of F,(R . H1),(R . H2)) is Element of the carrier of F
((R . H1) * (R . H2)) * (R .: c4) is Element of bool the carrier of F
bool the carrier of F is non empty set
carr (R .: c4) is Element of bool the carrier of F
the carrier of (R .: c4) is non empty set
((R . H1) * (R . H2)) * (carr (R .: c4)) is Element of bool the carrier of F
{((R . H1) * (R . H2))} is non empty Element of bool the carrier of F
{((R . H1) * (R . H2))} * (carr (R .: c4)) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {((R . H1) * (R . H2))} & b2 in carr (R .: c4) ) } is set
H1 * H2 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,H1,H2) is Element of the carrier of G
(H1 * H2) * c4 is Element of bool the carrier of G
bool the carrier of G is non empty set
carr c4 is Element of bool the carrier of G
(H1 * H2) * (carr c4) is Element of bool the carrier of G
{(H1 * H2)} is non empty Element of bool the carrier of G
{(H1 * H2)} * (carr c4) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(H1 * H2)} & b2 in carr c4 ) } is set
R .: ((H1 * H2) * c4) is Element of bool the carrier of F
(R .: c4) * (R . H1) is Element of bool the carrier of F
(carr (R .: c4)) * (R . H1) is Element of bool the carrier of F
{(R . H1)} is non empty Element of bool the carrier of F
(carr (R .: c4)) * {(R . H1)} is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in carr (R .: c4) & b2 in {(R . H1)} ) } is set
((R .: c4) * (R . H1)) * (R . H2) is Element of bool the carrier of F
{(R . H2)} is non empty Element of bool the carrier of F
((R .: c4) * (R . H1)) * {(R . H2)} is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in (R .: c4) * (R . H1) & b2 in {(R . H2)} ) } is set
c4 * H1 is Element of bool the carrier of G
(carr c4) * H1 is Element of bool the carrier of G
{H1} is non empty Element of bool the carrier of G
(carr c4) * {H1} is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in carr c4 & b2 in {H1} ) } is set
(c4 * H1) * H2 is Element of bool the carrier of G
{H2} is non empty Element of bool the carrier of G
(c4 * H1) * {H2} is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in c4 * H1 & b2 in {H2} ) } is set
R .: ((c4 * H1) * H2) is Element of bool the carrier of F
R . (H1 * H2) is Element of the carrier of F
(R . (H1 * H2)) * (R .: c4) is Element of bool the carrier of F
(R . (H1 * H2)) * (carr (R .: c4)) is Element of bool the carrier of F
{(R . (H1 * H2))} is non empty Element of bool the carrier of F
{(R . (H1 * H2))} * (carr (R .: c4)) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {(R . (H1 * H2))} & b2 in carr (R .: c4) ) } is set
(R .: c4) * ((R . H1) * (R . H2)) is Element of bool the carrier of F
(carr (R .: c4)) * ((R . H1) * (R . H2)) is Element of bool the carrier of F
(carr (R .: c4)) * {((R . H1) * (R . H2))} is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in carr (R .: c4) & b2 in {((R . H1) * (R . H2))} ) } is set
(R .: c4) * (R . (H1 * H2)) is Element of bool the carrier of F
(carr (R .: c4)) * (R . (H1 * H2)) is Element of bool the carrier of F
(carr (R .: c4)) * {(R . (H1 * H2))} is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in carr (R .: c4) & b2 in {(R . (H1 * H2))} ) } is set
c4 * (H1 * H2) is Element of bool the carrier of G
(carr c4) * (H1 * H2) is Element of bool the carrier of G
(carr c4) * {(H1 * H2)} is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in carr c4 & b2 in {(H1 * H2)} ) } is set
R .: (c4 * (H1 * H2)) is Element of bool the carrier of F
G is non empty strict unital Group-like associative multMagma
the carrier of G is non empty set
F is non empty strict unital Group-like associative multMagma
the carrier of F is non empty set
[: the carrier of G, the carrier of F:] is non empty set
bool [: the carrier of G, the carrier of F:] is non empty set
R is Relation-like the carrier of G -defined the carrier of F -valued Function-like V18( the carrier of G, the carrier of F) unity-preserving V152(G,F) Element of bool [: the carrier of G, the carrier of F:]
Image R is non empty strict unital Group-like associative Subgroup of F
the carrier of (Image R) is non empty set
c4 is non empty strict unital Group-like associative Subgroup of G
R .: c4 is non empty strict unital Group-like associative Subgroup of F
R | c4 is Relation-like the carrier of c4 -defined the carrier of F -valued Function-like V18( the carrier of c4, the carrier of F) unity-preserving V152(c4,F) Element of bool [: the carrier of c4, the carrier of F:]
the carrier of c4 is non empty set
[: the carrier of c4, the carrier of F:] is non empty set
bool [: the carrier of c4, the carrier of F:] is non empty set
R | the carrier of c4 is Relation-like the carrier of G -defined the carrier of F -valued Element of bool [: the carrier of G, the carrier of F:]
Image (R | c4) is non empty strict unital Group-like associative Subgroup of F
H1 is Element of the carrier of G
R . H1 is Element of the carrier of F
H2 is Element of the carrier of G
R . H2 is Element of the carrier of F
(R . H1) * (R . H2) is Element of the carrier of F
the multF of F is Relation-like [: the carrier of F, the carrier of F:] -defined the carrier of F -valued Function-like V18([: the carrier of F, the carrier of F:], the carrier of F) V22( 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
K84( the carrier of F, the multF of F,(R . H1),(R . H2)) is Element of the carrier of F
((R . H1) * (R . H2)) * (R .: c4) is Element of bool the carrier of F
bool the carrier of F is non empty set
carr (R .: c4) is Element of bool the carrier of F
the carrier of (R .: c4) is non empty set
((R . H1) * (R . H2)) * (carr (R .: c4)) is Element of bool the carrier of F
{((R . H1) * (R . H2))} is non empty Element of bool the carrier of F
{((R . H1) * (R . H2))} * (carr (R .: c4)) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {((R . H1) * (R . H2))} & b2 in carr (R .: c4) ) } is set
N is non empty unital Group-like associative Subgroup of Image R
N9 is Element of the carrier of (Image R)
G1 is Element of the carrier of (Image R)
N9 * G1 is Element of the carrier of (Image R)
the multF of (Image R) is Relation-like [: the carrier of (Image R), the carrier of (Image R):] -defined the carrier of (Image R) -valued Function-like V18([: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R)) V22( the carrier of (Image R)) Element of bool [:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):]
[: the carrier of (Image R), the carrier of (Image R):] is non empty set
[:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):] is non empty set
bool [:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):] is non empty set
K84( the carrier of (Image R), the multF of (Image R),N9,G1) is Element of the carrier of (Image R)
(N9 * G1) * N is Element of bool the carrier of (Image R)
bool the carrier of (Image R) is non empty set
carr N is Element of bool the carrier of (Image R)
the carrier of N is non empty set
(N9 * G1) * (carr N) is Element of bool the carrier of (Image R)
{(N9 * G1)} is non empty Element of bool the carrier of (Image R)
{(N9 * G1)} * (carr N) is Element of bool the carrier of (Image R)
{ (b1 * b2) where b1, b2 is Element of the carrier of (Image R) : ( b1 in {(N9 * G1)} & b2 in carr N ) } is set
H1 * H2 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,H1,H2) is Element of the carrier of G
R . (H1 * H2) is Element of the carrier of F
(H1 * H2) * c4 is Element of bool the carrier of G
bool the carrier of G is non empty set
carr c4 is Element of bool the carrier of G
(H1 * H2) * (carr c4) is Element of bool the carrier of G
{(H1 * H2)} is non empty Element of bool the carrier of G
{(H1 * H2)} * (carr c4) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(H1 * H2)} & b2 in carr c4 ) } is set
R .: ((H1 * H2) * c4) is Element of bool the carrier of F
(R . (H1 * H2)) * (R .: c4) is Element of bool the carrier of F
(R . (H1 * H2)) * (carr (R .: c4)) is Element of bool the carrier of F
{(R . (H1 * H2))} is non empty Element of bool the carrier of F
{(R . (H1 * H2))} * (carr (R .: c4)) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {(R . (H1 * H2))} & b2 in carr (R .: c4) ) } is set
g is set
h1 is Element of the carrier of (Image R)
(N9 * G1) * h1 is Element of the carrier of (Image R)
K84( the carrier of (Image R), the multF of (Image R),(N9 * G1),h1) is Element of the carrier of (Image R)
h2 is Element of the carrier of c4
(R | c4) . h2 is Element of the carrier of F
a is Element of the carrier of G
R . a is Element of the carrier of F
(R . (H1 * H2)) * (R . a) is Element of the carrier of F
K84( the carrier of F, the multF of F,(R . (H1 * H2)),(R . a)) is Element of the carrier of F
(H1 * H2) * a is Element of the carrier of G
K84( the carrier of G, the multF of G,(H1 * H2),a) is Element of the carrier of G
R . ((H1 * H2) * a) is Element of the carrier of F
g is set
h1 is set
R . h1 is set
h2 is Element of the carrier of G
a is Element of the carrier of G
(H1 * H2) * a is Element of the carrier of G
K84( the carrier of G, the multF of G,(H1 * H2),a) is Element of the carrier of G
R . a is Element of the carrier of F
R .: the carrier of c4 is Element of bool the carrier of F
(R . (H1 * H2)) * (R . a) is Element of the carrier of F
K84( the carrier of F, the multF of F,(R . (H1 * H2)),(R . a)) is Element of the carrier of F
a1 is Element of the carrier of (Image R)
(N9 * G1) * a1 is Element of the carrier of (Image R)
K84( the carrier of (Image R), the multF of (Image R),(N9 * G1),a1) is Element of the carrier of (Image R)
G is non empty strict unital Group-like associative multMagma
the carrier of G is non empty set
F is non empty strict unital Group-like associative multMagma
the carrier of F is non empty set
[: the carrier of G, the carrier of F:] is non empty set
bool [: the carrier of G, the carrier of F:] is non empty set
R is Relation-like the carrier of G -defined the carrier of F -valued Function-like V18( the carrier of G, the carrier of F) unity-preserving V152(G,F) Element of bool [: the carrier of G, the carrier of F:]
Image R is non empty strict unital Group-like associative Subgroup of F
c4 is non empty strict unital Group-like associative Subgroup of G
R .: c4 is non empty strict unital Group-like associative Subgroup of F
R | c4 is Relation-like the carrier of c4 -defined the carrier of F -valued Function-like V18( the carrier of c4, the carrier of F) unity-preserving V152(c4,F) Element of bool [: the carrier of c4, the carrier of F:]
the carrier of c4 is non empty set
[: the carrier of c4, the carrier of F:] is non empty set
bool [: the carrier of c4, the carrier of F:] is non empty set
R | the carrier of c4 is Relation-like the carrier of G -defined the carrier of F -valued Element of bool [: the carrier of G, the carrier of F:]
Image (R | c4) is non empty strict unital Group-like associative Subgroup of F
H1 is non empty strict unital Group-like associative normal Subgroup of G
(c4,H1) `*` is non empty strict unital Group-like associative normal Subgroup of c4
c4 ./. ((c4,H1) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((c4,H1) `*`) is non empty Element of bool (bool the carrier of c4)
bool the carrier of c4 is non empty set
bool (bool the carrier of c4) is non empty set
CosOp ((c4,H1) `*`) is Relation-like [:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] -defined Left_Cosets ((c4,H1) `*`) -valued Function-like V18([:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):], Left_Cosets ((c4,H1) `*`)) Element of bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):]
[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] is non empty set
[:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
multMagma(# (Left_Cosets ((c4,H1) `*`)),(CosOp ((c4,H1) `*`)) #) is non empty strict multMagma
G ./. H1 is non empty strict unital Group-like associative multMagma
Left_Cosets H1 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp H1 is Relation-like [:(Left_Cosets H1),(Left_Cosets H1):] -defined Left_Cosets H1 -valued Function-like V18([:(Left_Cosets H1),(Left_Cosets H1):], Left_Cosets H1) Element of bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):]
[:(Left_Cosets H1),(Left_Cosets H1):] is non empty set
[:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
multMagma(# (Left_Cosets H1),(CosOp H1) #) is non empty strict multMagma
center (G ./. H1) is non empty strict unital Group-like associative Subgroup of G ./. H1
R .: H1 is non empty strict unital Group-like associative Subgroup of F
R | H1 is Relation-like the carrier of H1 -defined the carrier of F -valued Function-like V18( the carrier of H1, the carrier of F) unity-preserving V152(H1,F) Element of bool [: the carrier of H1, the carrier of F:]
the carrier of H1 is non empty set
[: the carrier of H1, the carrier of F:] is non empty set
bool [: the carrier of H1, the carrier of F:] is non empty set
R | the carrier of H1 is Relation-like the carrier of G -defined the carrier of F -valued Element of bool [: the carrier of G, the carrier of F:]
Image (R | H1) is non empty strict unital Group-like associative Subgroup of F
H2 is non empty strict unital Group-like associative Subgroup of Image R
N is non empty strict unital Group-like associative normal Subgroup of Image R
(H2,N) `*` is non empty strict unital Group-like associative normal Subgroup of H2
H2 ./. ((H2,N) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((H2,N) `*`) is non empty Element of bool (bool the carrier of H2)
the carrier of H2 is non empty set
bool the carrier of H2 is non empty set
bool (bool the carrier of H2) is non empty set
CosOp ((H2,N) `*`) is Relation-like [:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):] -defined Left_Cosets ((H2,N) `*`) -valued Function-like V18([:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):], Left_Cosets ((H2,N) `*`)) Element of bool [:[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):],(Left_Cosets ((H2,N) `*`)):]
[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):] is non empty set
[:[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):],(Left_Cosets ((H2,N) `*`)):] is non empty set
bool [:[:(Left_Cosets ((H2,N) `*`)),(Left_Cosets ((H2,N) `*`)):],(Left_Cosets ((H2,N) `*`)):] is non empty set
multMagma(# (Left_Cosets ((H2,N) `*`)),(CosOp ((H2,N) `*`)) #) is non empty strict multMagma
(Image R) ./. N is non empty strict unital Group-like associative multMagma
Left_Cosets N is non empty Element of bool (bool the carrier of (Image R))
the carrier of (Image R) is non empty set
bool the carrier of (Image R) is non empty set
bool (bool the carrier of (Image R)) is non empty set
CosOp N is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V18([:(Left_Cosets N),(Left_Cosets N):], Left_Cosets N) Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is non empty set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
multMagma(# (Left_Cosets N),(CosOp N) #) is non empty strict multMagma
center ((Image R) ./. N) is non empty strict unital Group-like associative Subgroup of (Image R) ./. N
the carrier of ((Image R) ./. N) is non empty set
G1 is non empty unital Group-like associative Subgroup of (Image R) ./. N
G2 is Element of the carrier of ((Image R) ./. N)
g1 is Element of the carrier of ((Image R) ./. N)
g1 * G2 is Element of the carrier of ((Image R) ./. N)
the multF of ((Image R) ./. N) is Relation-like [: the carrier of ((Image R) ./. N), the carrier of ((Image R) ./. N):] -defined the carrier of ((Image R) ./. N) -valued Function-like V18([: the carrier of ((Image R) ./. N), the carrier of ((Image R) ./. N):], the carrier of ((Image R) ./. N)) V22( the carrier of ((Image R) ./. N)) Element of bool [:[: the carrier of ((Image R) ./. N), the carrier of ((Image R) ./. N):], the carrier of ((Image R) ./. N):]
[: the carrier of ((Image R) ./. N), the carrier of ((Image R) ./. N):] is non empty set
[:[: the carrier of ((Image R) ./. N), the carrier of ((Image R) ./. N):], the carrier of ((Image R) ./. N):] is non empty set
bool [:[: the carrier of ((Image R) ./. N), the carrier of ((Image R) ./. N):], the carrier of ((Image R) ./. N):] is non empty set
K84( the carrier of ((Image R) ./. N), the multF of ((Image R) ./. N),g1,G2) is Element of the carrier of ((Image R) ./. N)
G2 * g1 is Element of the carrier of ((Image R) ./. N)
K84( the carrier of ((Image R) ./. N), the multF of ((Image R) ./. N),G2,g1) is Element of the carrier of ((Image R) ./. N)
g is Element of the carrier of (Image R)
g * N is Element of bool the carrier of (Image R)
carr N is Element of bool the carrier of (Image R)
the carrier of N is non empty set
g * (carr N) is Element of bool the carrier of (Image R)
{g} is non empty Element of bool the carrier of (Image R)
{g} * (carr N) is Element of bool the carrier of (Image R)
{ (b1 * b2) where b1, b2 is Element of the carrier of (Image R) : ( b1 in {g} & b2 in carr N ) } is set
N * g is Element of bool the carrier of (Image R)
(carr N) * g is Element of bool the carrier of (Image R)
(carr N) * {g} is Element of bool the carrier of (Image R)
{ (b1 * b2) where b1, b2 is Element of the carrier of (Image R) : ( b1 in carr N & b2 in {g} ) } is set
N9 is non empty unital Group-like associative normal Subgroup of H2
h1 is Element of the carrier of H2
h1 * N9 is Element of bool the carrier of H2
carr N9 is Element of bool the carrier of H2
the carrier of N9 is non empty set
h1 * (carr N9) is Element of bool the carrier of H2
{h1} is non empty Element of bool the carrier of H2
{h1} * (carr N9) is Element of bool the carrier of H2
{ (b1 * b2) where b1, b2 is Element of the carrier of H2 : ( b1 in {h1} & b2 in carr N9 ) } is set
N9 * h1 is Element of bool the carrier of H2
(carr N9) * h1 is Element of bool the carrier of H2
(carr N9) * {h1} is Element of bool the carrier of H2
{ (b1 * b2) where b1, b2 is Element of the carrier of H2 : ( b1 in carr N9 & b2 in {h1} ) } is set
@ g1 is Element of bool the carrier of (Image R)
@ G2 is Element of bool the carrier of (Image R)
h2 is Element of the carrier of (Image R)
h2 * N is Element of bool the carrier of (Image R)
h2 * (carr N) is Element of bool the carrier of (Image R)
{h2} is non empty Element of bool the carrier of (Image R)
{h2} * (carr N) is Element of bool the carrier of (Image R)
{ (b1 * b2) where b1, b2 is Element of the carrier of (Image R) : ( b1 in {h2} & b2 in carr N ) } is set
(g * N) * (h2 * N) is Element of bool the carrier of (Image R)
{ (b1 * b2) where b1, b2 is Element of the carrier of (Image R) : ( b1 in g * N & b2 in h2 * N ) } is set
g * h2 is Element of the carrier of (Image R)
the multF of (Image R) is Relation-like [: the carrier of (Image R), the carrier of (Image R):] -defined the carrier of (Image R) -valued Function-like V18([: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R)) V22( the carrier of (Image R)) Element of bool [:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):]
[: the carrier of (Image R), the carrier of (Image R):] is non empty set
[:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):] is non empty set
bool [:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):] is non empty set
K84( the carrier of (Image R), the multF of (Image R),g,h2) is Element of the carrier of (Image R)
(g * h2) * N is Element of bool the carrier of (Image R)
(g * h2) * (carr N) is Element of bool the carrier of (Image R)
{(g * h2)} is non empty Element of bool the carrier of (Image R)
{(g * h2)} * (carr N) is Element of bool the carrier of (Image R)
{ (b1 * b2) where b1, b2 is Element of the carrier of (Image R) : ( b1 in {(g * h2)} & b2 in carr N ) } is set
(h2 * N) * (g * N) is Element of bool the carrier of (Image R)
{ (b1 * b2) where b1, b2 is Element of the carrier of (Image R) : ( b1 in h2 * N & b2 in g * N ) } is set
h2 * g is Element of the carrier of (Image R)
K84( the carrier of (Image R), the multF of (Image R),h2,g) is Element of the carrier of (Image R)
(h2 * g) * N is Element of bool the carrier of (Image R)
(h2 * g) * (carr N) is Element of bool the carrier of (Image R)
{(h2 * g)} is non empty Element of bool the carrier of (Image R)
{(h2 * g)} * (carr N) is Element of bool the carrier of (Image R)
{ (b1 * b2) where b1, b2 is Element of the carrier of (Image R) : ( b1 in {(h2 * g)} & b2 in carr N ) } is set
a is Element of the carrier of G
R . a is Element of the carrier of F
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
a1 is Element of the carrier of c4
(R | c4) . a1 is Element of the carrier of F
a2 is Element of the carrier of G
R . a2 is Element of the carrier of F
(R . a) * (R . a2) is Element of the carrier of F
the multF of F is Relation-like [: the carrier of F, the carrier of F:] -defined the carrier of F -valued Function-like V18([: the carrier of F, the carrier of F:], the carrier of F) V22( 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
K84( the carrier of F, the multF of F,(R . a),(R . a2)) is Element of the carrier of F
((R . a) * (R . a2)) * (R .: H1) is Element of bool the carrier of F
bool the carrier of F is non empty set
carr (R .: H1) is Element of bool the carrier of F
the carrier of (R .: H1) is non empty set
((R . a) * (R . a2)) * (carr (R .: H1)) is Element of bool the carrier of F
{((R . a) * (R . a2))} is non empty Element of bool the carrier of F
{((R . a) * (R . a2))} * (carr (R .: H1)) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {((R . a) * (R . a2))} & b2 in carr (R .: H1) ) } is set
a * a2 is Element of the carrier of G
K84( the carrier of G, the multF of G,a,a2) is Element of the carrier of G
(a * a2) * H1 is Element of bool the carrier of G
carr H1 is Element of bool the carrier of G
(a * a2) * (carr H1) is Element of bool the carrier of G
{(a * a2)} is non empty Element of bool the carrier of G
{(a * a2)} * (carr H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(a * a2)} & b2 in carr H1 ) } is set
R .: ((a * a2) * H1) is Element of bool the carrier of F
(R . a2) * (R . a) is Element of the carrier of F
K84( the carrier of F, the multF of F,(R . a2),(R . a)) is Element of the carrier of F
((R . a2) * (R . a)) * (R .: H1) is Element of bool the carrier of F
((R . a2) * (R . a)) * (carr (R .: H1)) is Element of bool the carrier of F
{((R . a2) * (R . a))} is non empty Element of bool the carrier of F
{((R . a2) * (R . a))} * (carr (R .: H1)) is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in {((R . a2) * (R . a))} & b2 in carr (R .: H1) ) } is set
a2 * a is Element of the carrier of G
K84( the carrier of G, the multF of G,a2,a) is Element of the carrier of G
(a2 * a) * H1 is Element of bool the carrier of G
(a2 * a) * (carr H1) is Element of bool the carrier of G
{(a2 * a)} is non empty Element of bool the carrier of G
{(a2 * a)} * (carr H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(a2 * a)} & b2 in carr H1 ) } is set
R .: ((a2 * a) * H1) is Element of bool the carrier of F
(G,c4,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr c4 is Element of bool the carrier of G
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
[.(carr c4),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr c4),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr c4 & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr c4),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
[.a2,a.] is Element of the carrier of G
a2 " is Element of the carrier of G
a " is Element of the carrier of G
(a2 ") * (a ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(a2 "),(a ")) is Element of the carrier of G
((a2 ") * (a ")) * a2 is Element of the carrier of G
K84( the carrier of G, the multF of G,((a2 ") * (a ")),a2) is Element of the carrier of G
(((a2 ") * (a ")) * a2) * a is Element of the carrier of G
K84( the carrier of G, the multF of G,(((a2 ") * (a ")) * a2),a) is Element of the carrier of G
[.a2,a.] * H1 is Element of bool the carrier of G
[.a2,a.] * (carr H1) is Element of bool the carrier of G
{[.a2,a.]} is non empty Element of bool the carrier of G
{[.a2,a.]} * (carr H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {[.a2,a.]} & b2 in carr H1 ) } is set
(a * a2) * ([.a2,a.] * H1) is Element of bool the carrier of G
{(a * a2)} * ([.a2,a.] * H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(a * a2)} & b2 in [.a2,a.] * H1 ) } is set
((a2 ") * (a ")) * (a2 * a) is Element of the carrier of G
K84( the carrier of G, the multF of G,((a2 ") * (a ")),(a2 * a)) is Element of the carrier of G
(((a2 ") * (a ")) * (a2 * a)) * H1 is Element of bool the carrier of G
(((a2 ") * (a ")) * (a2 * a)) * (carr H1) is Element of bool the carrier of G
{(((a2 ") * (a ")) * (a2 * a))} is non empty Element of bool the carrier of G
{(((a2 ") * (a ")) * (a2 * a))} * (carr H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(((a2 ") * (a ")) * (a2 * a))} & b2 in carr H1 ) } is set
(a * a2) * ((((a2 ") * (a ")) * (a2 * a)) * H1) is Element of bool the carrier of G
{(a * a2)} * ((((a2 ") * (a ")) * (a2 * a)) * H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(a * a2)} & b2 in (((a2 ") * (a ")) * (a2 * a)) * H1 ) } is set
(a * a2) * (((a2 ") * (a ")) * (a2 * a)) is Element of the carrier of G
K84( the carrier of G, the multF of G,(a * a2),(((a2 ") * (a ")) * (a2 * a))) is Element of the carrier of G
((a * a2) * (((a2 ") * (a ")) * (a2 * a))) * H1 is Element of bool the carrier of G
((a * a2) * (((a2 ") * (a ")) * (a2 * a))) * (carr H1) is Element of bool the carrier of G
{((a * a2) * (((a2 ") * (a ")) * (a2 * a)))} is non empty Element of bool the carrier of G
{((a * a2) * (((a2 ") * (a ")) * (a2 * a)))} * (carr H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((a * a2) * (((a2 ") * (a ")) * (a2 * a)))} & b2 in carr H1 ) } is set
(a * a2) * ((a2 ") * (a ")) is Element of the carrier of G
K84( the carrier of G, the multF of G,(a * a2),((a2 ") * (a "))) is Element of the carrier of G
((a * a2) * ((a2 ") * (a "))) * (a2 * a) is Element of the carrier of G
K84( the carrier of G, the multF of G,((a * a2) * ((a2 ") * (a "))),(a2 * a)) is Element of the carrier of G
(((a * a2) * ((a2 ") * (a "))) * (a2 * a)) * H1 is Element of bool the carrier of G
(((a * a2) * ((a2 ") * (a "))) * (a2 * a)) * (carr H1) is Element of bool the carrier of G
{(((a * a2) * ((a2 ") * (a "))) * (a2 * a))} is non empty Element of bool the carrier of G
{(((a * a2) * ((a2 ") * (a "))) * (a2 * a))} * (carr H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {(((a * a2) * ((a2 ") * (a "))) * (a2 * a))} & b2 in carr H1 ) } is set
a2 * ((a2 ") * (a ")) is Element of the carrier of G
K84( the carrier of G, the multF of G,a2,((a2 ") * (a "))) is Element of the carrier of G
a * (a2 * ((a2 ") * (a "))) is Element of the carrier of G
K84( the carrier of G, the multF of G,a,(a2 * ((a2 ") * (a ")))) is Element of the carrier of G
(a * (a2 * ((a2 ") * (a ")))) * (a2 * a) is Element of the carrier of G
K84( the carrier of G, the multF of G,(a * (a2 * ((a2 ") * (a ")))),(a2 * a)) is Element of the carrier of G
((a * (a2 * ((a2 ") * (a ")))) * (a2 * a)) * H1 is Element of bool the carrier of G
((a * (a2 * ((a2 ") * (a ")))) * (a2 * a)) * (carr H1) is Element of bool the carrier of G
{((a * (a2 * ((a2 ") * (a ")))) * (a2 * a))} is non empty Element of bool the carrier of G
{((a * (a2 * ((a2 ") * (a ")))) * (a2 * a))} * (carr H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((a * (a2 * ((a2 ") * (a ")))) * (a2 * a))} & b2 in carr H1 ) } is set
a2 * (a2 ") is Element of the carrier of G
K84( the carrier of G, the multF of G,a2,(a2 ")) is Element of the carrier of G
(a2 * (a2 ")) * (a ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(a2 * (a2 ")),(a ")) is Element of the carrier of G
a * ((a2 * (a2 ")) * (a ")) is Element of the carrier of G
K84( the carrier of G, the multF of G,a,((a2 * (a2 ")) * (a "))) is Element of the carrier of G
(a * ((a2 * (a2 ")) * (a "))) * (a2 * a) is Element of the carrier of G
K84( the carrier of G, the multF of G,(a * ((a2 * (a2 ")) * (a "))),(a2 * a)) is Element of the carrier of G
((a * ((a2 * (a2 ")) * (a "))) * (a2 * a)) * H1 is Element of bool the carrier of G
((a * ((a2 * (a2 ")) * (a "))) * (a2 * a)) * (carr H1) is Element of bool the carrier of G
{((a * ((a2 * (a2 ")) * (a "))) * (a2 * a))} is non empty Element of bool the carrier of G
{((a * ((a2 * (a2 ")) * (a "))) * (a2 * a))} * (carr H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((a * ((a2 * (a2 ")) * (a "))) * (a2 * a))} & b2 in carr H1 ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * (a ") is Element of the carrier of G
K84( the carrier of G, the multF of G,(1_ G),(a ")) is Element of the carrier of G
a * ((1_ G) * (a ")) is Element of the carrier of G
K84( the carrier of G, the multF of G,a,((1_ G) * (a "))) is Element of the carrier of G
(a * ((1_ G) * (a "))) * (a2 * a) is Element of the carrier of G
K84( the carrier of G, the multF of G,(a * ((1_ G) * (a "))),(a2 * a)) is Element of the carrier of G
((a * ((1_ G) * (a "))) * (a2 * a)) * H1 is Element of bool the carrier of G
((a * ((1_ G) * (a "))) * (a2 * a)) * (carr H1) is Element of bool the carrier of G
{((a * ((1_ G) * (a "))) * (a2 * a))} is non empty Element of bool the carrier of G
{((a * ((1_ G) * (a "))) * (a2 * a))} * (carr H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((a * ((1_ G) * (a "))) * (a2 * a))} & b2 in carr H1 ) } is set
a * (a ") is Element of the carrier of G
K84( the carrier of G, the multF of G,a,(a ")) is Element of the carrier of G
(a * (a ")) * (a2 * a) is Element of the carrier of G
K84( the carrier of G, the multF of G,(a * (a ")),(a2 * a)) is Element of the carrier of G
((a * (a ")) * (a2 * a)) * H1 is Element of bool the carrier of G
((a * (a ")) * (a2 * a)) * (carr H1) is Element of bool the carrier of G
{((a * (a ")) * (a2 * a))} is non empty Element of bool the carrier of G
{((a * (a ")) * (a2 * a))} * (carr H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((a * (a ")) * (a2 * a))} & b2 in carr H1 ) } is set
(1_ G) * (a2 * a) is Element of the carrier of G
K84( the carrier of G, the multF of G,(1_ G),(a2 * a)) is Element of the carrier of G
((1_ G) * (a2 * a)) * H1 is Element of bool the carrier of G
((1_ G) * (a2 * a)) * (carr H1) is Element of bool the carrier of G
{((1_ G) * (a2 * a))} is non empty Element of bool the carrier of G
{((1_ G) * (a2 * a))} * (carr H1) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {((1_ G) * (a2 * a))} & b2 in carr H1 ) } is set
G is non empty strict unital Group-like associative multMagma
the carrier of G is non empty set
F is non empty strict unital Group-like associative multMagma
the carrier of F is non empty set
[: the carrier of G, the carrier of F:] is non empty set
bool [: the carrier of G, the carrier of F:] is non empty set
R is Relation-like the carrier of G -defined the carrier of F -valued Function-like V18( the carrier of G, the carrier of F) unity-preserving V152(G,F) Element of bool [: the carrier of G, the carrier of F:]
Image R is non empty strict unital Group-like associative Subgroup of F
c4 is non empty strict unital Group-like associative normal Subgroup of G
R .: c4 is non empty strict unital Group-like associative Subgroup of F
R | c4 is Relation-like the carrier of c4 -defined the carrier of F -valued Function-like V18( the carrier of c4, the carrier of F) unity-preserving V152(c4,F) Element of bool [: the carrier of c4, the carrier of F:]
the carrier of c4 is non empty set
[: the carrier of c4, the carrier of F:] is non empty set
bool [: the carrier of c4, the carrier of F:] is non empty set
R | the carrier of c4 is Relation-like the carrier of G -defined the carrier of F -valued Element of bool [: the carrier of G, the carrier of F:]
Image (R | c4) is non empty strict unital Group-like associative Subgroup of F
the carrier of (Image R) is non empty set
H1 is non empty strict unital Group-like associative Subgroup of Image R
H2 is Element of the carrier of (Image R)
H2 * H1 is Element of bool the carrier of (Image R)
bool the carrier of (Image R) is non empty set
carr H1 is Element of bool the carrier of (Image R)
the carrier of H1 is non empty set
H2 * (carr H1) is Element of bool the carrier of (Image R)
{H2} is non empty Element of bool the carrier of (Image R)
{H2} * (carr H1) is Element of bool the carrier of (Image R)
{ (b1 * b2) where b1, b2 is Element of the carrier of (Image R) : ( b1 in {H2} & b2 in carr H1 ) } is set
H1 * H2 is Element of bool the carrier of (Image R)
(carr H1) * H2 is Element of bool the carrier of (Image R)
(carr H1) * {H2} is Element of bool the carrier of (Image R)
{ (b1 * b2) where b1, b2 is Element of the carrier of (Image R) : ( b1 in carr H1 & b2 in {H2} ) } is set
N is Element of the carrier of G
R . N is Element of the carrier of F
N9 is set
G1 is Element of the carrier of (Image R)
H2 * G1 is Element of the carrier of (Image R)
the multF of (Image R) is Relation-like [: the carrier of (Image R), the carrier of (Image R):] -defined the carrier of (Image R) -valued Function-like V18([: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R)) V22( the carrier of (Image R)) Element of bool [:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):]
[: the carrier of (Image R), the carrier of (Image R):] is non empty set
[:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):] is non empty set
bool [:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):] is non empty set
K84( the carrier of (Image R), the multF of (Image R),H2,G1) is Element of the carrier of (Image R)
G2 is Element of the carrier of c4
(R | c4) . G2 is Element of the carrier of F
g1 is Element of the carrier of G
R . g1 is Element of the carrier of F
(R . N) * (R . g1) is Element of the carrier of F
the multF of F is Relation-like [: the carrier of F, the carrier of F:] -defined the carrier of F -valued Function-like V18([: the carrier of F, the carrier of F:], the carrier of F) V22( 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
K84( the carrier of F, the multF of F,(R . N),(R . g1)) is Element of the carrier of F
N * g1 is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
K84( the carrier of G, the multF of G,N,g1) is Element of the carrier of G
R . (N * g1) is Element of the carrier of F
N * c4 is Element of bool the carrier of G
bool the carrier of G is non empty set
carr c4 is Element of bool the carrier of G
N * (carr c4) is Element of bool the carrier of G
{N} is non empty Element of bool the carrier of G
{N} * (carr c4) is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in {N} & b2 in carr c4 ) } is set
c4 * N is Element of bool the carrier of G
(carr c4) * N is Element of bool the carrier of G
(carr c4) * {N} is Element of bool the carrier of G
{ (b1 * b2) where b1, b2 is Element of the carrier of G : ( b1 in carr c4 & b2 in {N} ) } is set
R .: (c4 * N) is Element of bool the carrier of F
bool the carrier of F is non empty set
(R .: c4) * (R . N) is Element of bool the carrier of F
carr (R .: c4) is Element of bool the carrier of F
the carrier of (R .: c4) is non empty set
(carr (R .: c4)) * (R . N) is Element of bool the carrier of F
{(R . N)} is non empty Element of bool the carrier of F
(carr (R .: c4)) * {(R . N)} is Element of bool the carrier of F
{ (b1 * b2) where b1, b2 is Element of the carrier of F : ( b1 in carr (R .: c4) & b2 in {(R . N)} ) } is set
G is non empty strict unital Group-like associative () multMagma
the carrier of G is non empty set
F is non empty strict unital Group-like associative multMagma
the carrier of F is non empty set
[: the carrier of G, the carrier of F:] is non empty set
bool [: the carrier of G, the carrier of F:] is non empty set
R is Relation-like the carrier of G -defined the carrier of F -valued Function-like V18( the carrier of G, the carrier of F) unity-preserving V152(G,F) Element of bool [: the carrier of G, the carrier of F:]
Image R is non empty strict unital Group-like associative Subgroup of F
(G) is non empty set
(Omega). G is non empty strict unital Group-like associative normal () Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
(1). G is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic () Subgroup of G
c4 is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len c4 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
c4 . 1 is set
c4 . (len c4) is set
dom c4 is Element of bool NAT
Seg (len c4) is V37() V44( len c4) Element of bool NAT
{ b1 where b1 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT : ( 1 <= b1 & b1 <= len c4 ) } is set
((Image R)) is non empty set
H1 is V24() V25() V26() V30() V31() V32() V33() ext-real set
c4 . H1 is set
H2 is non empty strict unital Group-like associative normal () Subgroup of G
R .: H2 is non empty strict unital Group-like associative Subgroup of F
R | H2 is Relation-like the carrier of H2 -defined the carrier of F -valued Function-like V18( the carrier of H2, the carrier of F) unity-preserving V152(H2,F) Element of bool [: the carrier of H2, the carrier of F:]
the carrier of H2 is non empty set
[: the carrier of H2, the carrier of F:] is non empty set
bool [: the carrier of H2, the carrier of F:] is non empty set
R | the carrier of H2 is Relation-like the carrier of G -defined the carrier of F -valued Element of bool [: the carrier of G, the carrier of F:]
Image (R | H2) is non empty strict unital Group-like associative Subgroup of F
H1 is Relation-like NAT -defined ((Image R)) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of ((Image R))
dom H1 is Element of bool NAT
len H1 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
Seg (len H1) is V37() V44( len H1) Element of bool NAT
{ b1 where b1 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT : ( 1 <= b1 & b1 <= len H1 ) } is set
H2 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
H2 + 1 is V31() V32() V33() ext-real set
H1 . H2 is set
H1 . (H2 + 1) is set
N is non empty strict unital Group-like associative normal Subgroup of Image R
N9 is non empty strict unital Group-like associative normal Subgroup of Image R
(N,N9) `*` is non empty strict unital Group-like associative normal Subgroup of N
N ./. ((N,N9) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((N,N9) `*`) is non empty Element of bool (bool the carrier of N)
the carrier of N is non empty set
bool the carrier of N is non empty set
bool (bool the carrier of N) is non empty set
CosOp ((N,N9) `*`) is Relation-like [:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):] -defined Left_Cosets ((N,N9) `*`) -valued Function-like V18([:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):], Left_Cosets ((N,N9) `*`)) Element of bool [:[:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):],(Left_Cosets ((N,N9) `*`)):]
[:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):] is non empty set
[:[:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):],(Left_Cosets ((N,N9) `*`)):] is non empty set
bool [:[:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):],(Left_Cosets ((N,N9) `*`)):] is non empty set
multMagma(# (Left_Cosets ((N,N9) `*`)),(CosOp ((N,N9) `*`)) #) is non empty strict multMagma
(Image R) ./. N9 is non empty strict unital Group-like associative multMagma
Left_Cosets N9 is non empty Element of bool (bool the carrier of (Image R))
the carrier of (Image R) is non empty set
bool the carrier of (Image R) is non empty set
bool (bool the carrier of (Image R)) is non empty set
CosOp N9 is Relation-like [:(Left_Cosets N9),(Left_Cosets N9):] -defined Left_Cosets N9 -valued Function-like V18([:(Left_Cosets N9),(Left_Cosets N9):], Left_Cosets N9) Element of bool [:[:(Left_Cosets N9),(Left_Cosets N9):],(Left_Cosets N9):]
[:(Left_Cosets N9),(Left_Cosets N9):] is non empty set
[:[:(Left_Cosets N9),(Left_Cosets N9):],(Left_Cosets N9):] is non empty set
bool [:[:(Left_Cosets N9),(Left_Cosets N9):],(Left_Cosets N9):] is non empty set
multMagma(# (Left_Cosets N9),(CosOp N9) #) is non empty strict multMagma
center ((Image R) ./. N9) is non empty strict unital Group-like associative Subgroup of (Image R) ./. N9
c4 . H2 is set
G1 is non empty strict unital Group-like associative normal () Subgroup of G
R .: G1 is non empty strict unital Group-like associative Subgroup of F
R | G1 is Relation-like the carrier of G1 -defined the carrier of F -valued Function-like V18( the carrier of G1, the carrier of F) unity-preserving V152(G1,F) Element of bool [: the carrier of G1, the carrier of F:]
the carrier of G1 is non empty set
[: the carrier of G1, the carrier of F:] is non empty set
bool [: the carrier of G1, the carrier of F:] is non empty set
R | the carrier of G1 is Relation-like the carrier of G -defined the carrier of F -valued Element of bool [: the carrier of G, the carrier of F:]
Image (R | G1) is non empty strict unital Group-like associative Subgroup of F
c4 . (H2 + 1) is set
G2 is non empty strict unital Group-like associative normal () Subgroup of G
R .: G2 is non empty strict unital Group-like associative Subgroup of F
R | G2 is Relation-like the carrier of G2 -defined the carrier of F -valued Function-like V18( the carrier of G2, the carrier of F) unity-preserving V152(G2,F) Element of bool [: the carrier of G2, the carrier of F:]
the carrier of G2 is non empty set
[: the carrier of G2, the carrier of F:] is non empty set
bool [: the carrier of G2, the carrier of F:] is non empty set
R | the carrier of G2 is Relation-like the carrier of G -defined the carrier of F -valued Element of bool [: the carrier of G, the carrier of F:]
Image (R | G2) is non empty strict unital Group-like associative Subgroup of F
(G1,G2) `*` is non empty strict unital Group-like associative normal () Subgroup of G1
G1 ./. ((G1,G2) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((G1,G2) `*`) is non empty Element of bool (bool the carrier of G1)
bool the carrier of G1 is non empty set
bool (bool the carrier of G1) is non empty set
CosOp ((G1,G2) `*`) is Relation-like [:(Left_Cosets ((G1,G2) `*`)),(Left_Cosets ((G1,G2) `*`)):] -defined Left_Cosets ((G1,G2) `*`) -valued Function-like V18([:(Left_Cosets ((G1,G2) `*`)),(Left_Cosets ((G1,G2) `*`)):], Left_Cosets ((G1,G2) `*`)) Element of bool [:[:(Left_Cosets ((G1,G2) `*`)),(Left_Cosets ((G1,G2) `*`)):],(Left_Cosets ((G1,G2) `*`)):]
[:(Left_Cosets ((G1,G2) `*`)),(Left_Cosets ((G1,G2) `*`)):] is non empty set
[:[:(Left_Cosets ((G1,G2) `*`)),(Left_Cosets ((G1,G2) `*`)):],(Left_Cosets ((G1,G2) `*`)):] is non empty set
bool [:[:(Left_Cosets ((G1,G2) `*`)),(Left_Cosets ((G1,G2) `*`)):],(Left_Cosets ((G1,G2) `*`)):] is non empty set
multMagma(# (Left_Cosets ((G1,G2) `*`)),(CosOp ((G1,G2) `*`)) #) is non empty strict multMagma
G ./. G2 is non empty strict unital Group-like associative multMagma
Left_Cosets G2 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp G2 is Relation-like [:(Left_Cosets G2),(Left_Cosets G2):] -defined Left_Cosets G2 -valued Function-like V18([:(Left_Cosets G2),(Left_Cosets G2):], Left_Cosets G2) Element of bool [:[:(Left_Cosets G2),(Left_Cosets G2):],(Left_Cosets G2):]
[:(Left_Cosets G2),(Left_Cosets G2):] is non empty set
[:[:(Left_Cosets G2),(Left_Cosets G2):],(Left_Cosets G2):] is non empty set
bool [:[:(Left_Cosets G2),(Left_Cosets G2):],(Left_Cosets G2):] is non empty set
multMagma(# (Left_Cosets G2),(CosOp G2) #) is non empty strict multMagma
center (G ./. G2) is non empty strict unital Group-like associative Subgroup of G ./. G2
H1 . 1 is set
(Omega). (Image R) is non empty strict unital Group-like associative normal Subgroup of Image R
the carrier of (Image R) is non empty set
the multF of (Image R) is Relation-like [: the carrier of (Image R), the carrier of (Image R):] -defined the carrier of (Image R) -valued Function-like V18([: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R)) V22( the carrier of (Image R)) Element of bool [:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):]
[: the carrier of (Image R), the carrier of (Image R):] is non empty set
[:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):] is non empty set
bool [:[: the carrier of (Image R), the carrier of (Image R):], the carrier of (Image R):] is non empty set
multMagma(# the carrier of (Image R), the multF of (Image R) #) is non empty strict unital Group-like associative multMagma
H1 . (len H1) is set
(1). (Image R) is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic () Subgroup of Image R
H2 is non empty strict unital Group-like associative normal () Subgroup of G
R .: H2 is non empty strict unital Group-like associative Subgroup of F
R | H2 is Relation-like the carrier of H2 -defined the carrier of F -valued Function-like V18( the carrier of H2, the carrier of F) unity-preserving V152(H2,F) Element of bool [: the carrier of H2, the carrier of F:]
the carrier of H2 is non empty set
[: the carrier of H2, the carrier of F:] is non empty set
bool [: the carrier of H2, the carrier of F:] is non empty set
R | the carrier of H2 is Relation-like the carrier of G -defined the carrier of F -valued Element of bool [: the carrier of G, the carrier of F:]
Image (R | H2) is non empty strict unital Group-like associative Subgroup of F
H1 . (len c4) is set
(1). F is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic () Subgroup of F
H2 is non empty strict unital Group-like associative normal () Subgroup of G
R .: H2 is non empty strict unital Group-like associative Subgroup of F
R | H2 is Relation-like the carrier of H2 -defined the carrier of F -valued Function-like V18( the carrier of H2, the carrier of F) unity-preserving V152(H2,F) Element of bool [: the carrier of H2, the carrier of F:]
the carrier of H2 is non empty set
[: the carrier of H2, the carrier of F:] is non empty set
bool [: the carrier of H2, the carrier of F:] is non empty set
R | the carrier of H2 is Relation-like the carrier of G -defined the carrier of F -valued Element of bool [: the carrier of G, the carrier of F:]
Image (R | H2) is non empty strict unital Group-like associative Subgroup of F
H2 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
H2 + 1 is V31() V32() V33() ext-real set
N is non empty strict unital Group-like associative normal Subgroup of Image R
H1 . H2 is set
N9 is non empty strict unital Group-like associative normal Subgroup of Image R
H1 . (H2 + 1) is set
(N,N9) `*` is non empty strict unital Group-like associative normal Subgroup of N
N ./. ((N,N9) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((N,N9) `*`) is non empty Element of bool (bool the carrier of N)
the carrier of N is non empty set
bool the carrier of N is non empty set
bool (bool the carrier of N) is non empty set
CosOp ((N,N9) `*`) is Relation-like [:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):] -defined Left_Cosets ((N,N9) `*`) -valued Function-like V18([:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):], Left_Cosets ((N,N9) `*`)) Element of bool [:[:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):],(Left_Cosets ((N,N9) `*`)):]
[:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):] is non empty set
[:[:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):],(Left_Cosets ((N,N9) `*`)):] is non empty set
bool [:[:(Left_Cosets ((N,N9) `*`)),(Left_Cosets ((N,N9) `*`)):],(Left_Cosets ((N,N9) `*`)):] is non empty set
multMagma(# (Left_Cosets ((N,N9) `*`)),(CosOp ((N,N9) `*`)) #) is non empty strict multMagma
(Image R) ./. N9 is non empty strict unital Group-like associative multMagma
Left_Cosets N9 is non empty Element of bool (bool the carrier of (Image R))
bool the carrier of (Image R) is non empty set
bool (bool the carrier of (Image R)) is non empty set
CosOp N9 is Relation-like [:(Left_Cosets N9),(Left_Cosets N9):] -defined Left_Cosets N9 -valued Function-like V18([:(Left_Cosets N9),(Left_Cosets N9):], Left_Cosets N9) Element of bool [:[:(Left_Cosets N9),(Left_Cosets N9):],(Left_Cosets N9):]
[:(Left_Cosets N9),(Left_Cosets N9):] is non empty set
[:[:(Left_Cosets N9),(Left_Cosets N9):],(Left_Cosets N9):] is non empty set
bool [:[:(Left_Cosets N9),(Left_Cosets N9):],(Left_Cosets N9):] is non empty set
multMagma(# (Left_Cosets N9),(CosOp N9) #) is non empty strict multMagma
center ((Image R) ./. N9) is non empty strict unital Group-like associative Subgroup of (Image R) ./. N9
G is non empty strict unital Group-like associative () multMagma
F is non empty strict unital Group-like associative normal () Subgroup of G
G ./. F is non empty strict unital Group-like associative multMagma
Left_Cosets F is non empty Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp F is Relation-like [:(Left_Cosets F),(Left_Cosets F):] -defined Left_Cosets F -valued Function-like V18([:(Left_Cosets F),(Left_Cosets F):], Left_Cosets F) Element of bool [:[:(Left_Cosets F),(Left_Cosets F):],(Left_Cosets F):]
[:(Left_Cosets F),(Left_Cosets F):] is non empty set
[:[:(Left_Cosets F),(Left_Cosets F):],(Left_Cosets F):] is non empty set
bool [:[:(Left_Cosets F),(Left_Cosets F):],(Left_Cosets F):] is non empty set
multMagma(# (Left_Cosets F),(CosOp F) #) is non empty strict multMagma
nat_hom F is Relation-like the carrier of G -defined the carrier of (G ./. F) -valued Function-like V18( the carrier of G, the carrier of (G ./. F)) unity-preserving V152(G,G ./. F) Element of bool [: the carrier of G, the carrier of (G ./. F):]
the carrier of (G ./. F) is non empty set
[: the carrier of G, the carrier of (G ./. F):] is non empty set
bool [: the carrier of G, the carrier of (G ./. F):] is non empty set
Image (nat_hom F) is non empty strict unital Group-like associative () Subgroup of G ./. F
G is non empty unital Group-like associative multMagma
(G) is non empty set
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
(1). G is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic () Subgroup of G
F is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len F is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
F . 1 is set
F . (len F) is set
dom F is Element of bool NAT
R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R + 1 is V31() V32() V33() ext-real set
F . R is set
F . (R + 1) is set
c4 is non empty strict unital Group-like associative normal Subgroup of G
H1 is non empty strict unital Group-like associative normal Subgroup of G
(G,c4,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr c4 is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of c4 is non empty set
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
[.(carr c4),(carr ((Omega). G)).] is non empty strict unital Group-like associative Subgroup of G
commutators ((carr c4),(carr ((Omega). G))) is Element of bool the carrier of G
{ [.b1,b2.] where b1, b2 is Element of the carrier of G : ( b1 in carr c4 & b2 in carr ((Omega). G) ) } is set
gr (commutators ((carr c4),(carr ((Omega). G)))) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
(G) is non empty set
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
(1). G is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic () Subgroup of G
F is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len F is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
F . 1 is set
F . (len F) is set
dom F is Element of bool NAT
R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R + 1 is V31() V32() V33() ext-real set
F . R is set
F . (R + 1) is set
c4 is non empty strict unital Group-like associative normal Subgroup of G
H1 is non empty strict unital Group-like associative normal Subgroup of G
(c4,H1) `*` is non empty strict unital Group-like associative normal Subgroup of c4
c4 ./. ((c4,H1) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((c4,H1) `*`) is non empty Element of bool (bool the carrier of c4)
the carrier of c4 is non empty set
bool the carrier of c4 is non empty set
bool (bool the carrier of c4) is non empty set
CosOp ((c4,H1) `*`) is Relation-like [:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] -defined Left_Cosets ((c4,H1) `*`) -valued Function-like V18([:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):], Left_Cosets ((c4,H1) `*`)) Element of bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):]
[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] is non empty set
[:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
multMagma(# (Left_Cosets ((c4,H1) `*`)),(CosOp ((c4,H1) `*`)) #) is non empty strict multMagma
G ./. H1 is non empty strict unital Group-like associative multMagma
Left_Cosets H1 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp H1 is Relation-like [:(Left_Cosets H1),(Left_Cosets H1):] -defined Left_Cosets H1 -valued Function-like V18([:(Left_Cosets H1),(Left_Cosets H1):], Left_Cosets H1) Element of bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):]
[:(Left_Cosets H1),(Left_Cosets H1):] is non empty set
[:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
multMagma(# (Left_Cosets H1),(CosOp H1) #) is non empty strict multMagma
center (G ./. H1) is non empty strict unital Group-like associative Subgroup of G ./. H1
R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R + 1 is V31() V32() V33() ext-real set
c4 is non empty strict unital Group-like associative normal Subgroup of G
F . R is set
H1 is non empty strict unital Group-like associative normal Subgroup of G
F . (R + 1) is set
(c4,H1) `*` is non empty strict unital Group-like associative normal Subgroup of c4
c4 ./. ((c4,H1) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((c4,H1) `*`) is non empty Element of bool (bool the carrier of c4)
the carrier of c4 is non empty set
bool the carrier of c4 is non empty set
bool (bool the carrier of c4) is non empty set
CosOp ((c4,H1) `*`) is Relation-like [:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] -defined Left_Cosets ((c4,H1) `*`) -valued Function-like V18([:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):], Left_Cosets ((c4,H1) `*`)) Element of bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):]
[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] is non empty set
[:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
multMagma(# (Left_Cosets ((c4,H1) `*`)),(CosOp ((c4,H1) `*`)) #) is non empty strict multMagma
G ./. H1 is non empty strict unital Group-like associative multMagma
Left_Cosets H1 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp H1 is Relation-like [:(Left_Cosets H1),(Left_Cosets H1):] -defined Left_Cosets H1 -valued Function-like V18([:(Left_Cosets H1),(Left_Cosets H1):], Left_Cosets H1) Element of bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):]
[:(Left_Cosets H1),(Left_Cosets H1):] is non empty set
[:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
multMagma(# (Left_Cosets H1),(CosOp H1) #) is non empty strict multMagma
center (G ./. H1) is non empty strict unital Group-like associative Subgroup of G ./. H1
G is non empty unital Group-like associative multMagma
(G) is non empty set
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
(1). G is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic () Subgroup of G
F is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len F is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
F . 1 is set
F . (len F) is set
dom F is Element of bool NAT
R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R + 1 is V31() V32() V33() ext-real set
F . R is set
F . (R + 1) is set
c4 is non empty strict unital Group-like associative normal Subgroup of G
H1 is non empty strict unital Group-like associative normal Subgroup of G
(c4,H1) `*` is non empty strict unital Group-like associative normal Subgroup of c4
c4 ./. ((c4,H1) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((c4,H1) `*`) is non empty Element of bool (bool the carrier of c4)
the carrier of c4 is non empty set
bool the carrier of c4 is non empty set
bool (bool the carrier of c4) is non empty set
CosOp ((c4,H1) `*`) is Relation-like [:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] -defined Left_Cosets ((c4,H1) `*`) -valued Function-like V18([:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):], Left_Cosets ((c4,H1) `*`)) Element of bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):]
[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] is non empty set
[:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
multMagma(# (Left_Cosets ((c4,H1) `*`)),(CosOp ((c4,H1) `*`)) #) is non empty strict multMagma
G ./. H1 is non empty strict unital Group-like associative multMagma
Left_Cosets H1 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp H1 is Relation-like [:(Left_Cosets H1),(Left_Cosets H1):] -defined Left_Cosets H1 -valued Function-like V18([:(Left_Cosets H1),(Left_Cosets H1):], Left_Cosets H1) Element of bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):]
[:(Left_Cosets H1),(Left_Cosets H1):] is non empty set
[:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
multMagma(# (Left_Cosets H1),(CosOp H1) #) is non empty strict multMagma
center (G ./. H1) is non empty strict unital Group-like associative Subgroup of G ./. H1
R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R + 1 is V31() V32() V33() ext-real set
c4 is non empty strict unital Group-like associative normal Subgroup of G
F . R is set
H1 is non empty strict unital Group-like associative normal Subgroup of G
F . (R + 1) is set
(c4,H1) `*` is non empty strict unital Group-like associative normal Subgroup of c4
c4 ./. ((c4,H1) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((c4,H1) `*`) is non empty Element of bool (bool the carrier of c4)
the carrier of c4 is non empty set
bool the carrier of c4 is non empty set
bool (bool the carrier of c4) is non empty set
CosOp ((c4,H1) `*`) is Relation-like [:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] -defined Left_Cosets ((c4,H1) `*`) -valued Function-like V18([:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):], Left_Cosets ((c4,H1) `*`)) Element of bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):]
[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):] is non empty set
[:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
bool [:[:(Left_Cosets ((c4,H1) `*`)),(Left_Cosets ((c4,H1) `*`)):],(Left_Cosets ((c4,H1) `*`)):] is non empty set
multMagma(# (Left_Cosets ((c4,H1) `*`)),(CosOp ((c4,H1) `*`)) #) is non empty strict multMagma
G ./. H1 is non empty strict unital Group-like associative multMagma
Left_Cosets H1 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp H1 is Relation-like [:(Left_Cosets H1),(Left_Cosets H1):] -defined Left_Cosets H1 -valued Function-like V18([:(Left_Cosets H1),(Left_Cosets H1):], Left_Cosets H1) Element of bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):]
[:(Left_Cosets H1),(Left_Cosets H1):] is non empty set
[:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
bool [:[:(Left_Cosets H1),(Left_Cosets H1):],(Left_Cosets H1):] is non empty set
multMagma(# (Left_Cosets H1),(CosOp H1) #) is non empty strict multMagma
center (G ./. H1) is non empty strict unital Group-like associative Subgroup of G ./. H1
G is non empty unital Group-like associative multMagma
(G) is non empty set
(Omega). G is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V18([: the carrier of G, the carrier of G:], the carrier of G) V22( the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is non empty strict unital Group-like associative multMagma
(1). G is non empty trivial V78() 1 -element strict unital Group-like associative commutative normal cyclic () Subgroup of G
F is Relation-like NAT -defined (G) -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of (G)
len F is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
F . 1 is set
F . (len F) is set
dom F is Element of bool NAT
Subgroups G is non empty set
R is Relation-like NAT -defined Subgroups G -valued Function-like V37() FinSequence-like FinSubsequence-like FinSequence of Subgroups G
dom R is Element of bool NAT
c4 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
c4 + 1 is V31() V32() V33() ext-real set
R . c4 is set
R . (c4 + 1) is set
F . c4 is set
F . (c4 + 1) is set
H1 is non empty unital Group-like associative Subgroup of G
H2 is non empty unital Group-like associative Subgroup of G
N is non empty unital Group-like associative normal Subgroup of H1
H1 ./. N is non empty strict unital Group-like associative multMagma
Left_Cosets N is non empty Element of bool (bool the carrier of H1)
the carrier of H1 is non empty set
bool the carrier of H1 is non empty set
bool (bool the carrier of H1) is non empty set
CosOp N is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V18([:(Left_Cosets N),(Left_Cosets N):], Left_Cosets N) Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is non empty set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
multMagma(# (Left_Cosets N),(CosOp N) #) is non empty strict multMagma
N9 is non empty strict unital Group-like associative normal Subgroup of G
(H1,N9) `*` is non empty strict unital Group-like associative normal Subgroup of H1
H1 ./. ((H1,N9) `*`) is non empty strict unital Group-like associative multMagma
Left_Cosets ((H1,N9) `*`) is non empty Element of bool (bool the carrier of H1)
CosOp ((H1,N9) `*`) is Relation-like [:(Left_Cosets ((H1,N9) `*`)),(Left_Cosets ((H1,N9) `*`)):] -defined Left_Cosets ((H1,N9) `*`) -valued Function-like V18([:(Left_Cosets ((H1,N9) `*`)),(Left_Cosets ((H1,N9) `*`)):], Left_Cosets ((H1,N9) `*`)) Element of bool [:[:(Left_Cosets ((H1,N9) `*`)),(Left_Cosets ((H1,N9) `*`)):],(Left_Cosets ((H1,N9) `*`)):]
[:(Left_Cosets ((H1,N9) `*`)),(Left_Cosets ((H1,N9) `*`)):] is non empty set
[:[:(Left_Cosets ((H1,N9) `*`)),(Left_Cosets ((H1,N9) `*`)):],(Left_Cosets ((H1,N9) `*`)):] is non empty set
bool [:[:(Left_Cosets ((H1,N9) `*`)),(Left_Cosets ((H1,N9) `*`)):],(Left_Cosets ((H1,N9) `*`)):] is non empty set
multMagma(# (Left_Cosets ((H1,N9) `*`)),(CosOp ((H1,N9) `*`)) #) is non empty strict multMagma
G ./. N9 is non empty strict unital Group-like associative multMagma
Left_Cosets N9 is non empty Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
CosOp N9 is Relation-like [:(Left_Cosets N9),(Left_Cosets N9):] -defined Left_Cosets N9 -valued Function-like V18([:(Left_Cosets N9),(Left_Cosets N9):], Left_Cosets N9) Element of bool [:[:(Left_Cosets N9),(Left_Cosets N9):],(Left_Cosets N9):]
[:(Left_Cosets N9),(Left_Cosets N9):] is non empty set
[:[:(Left_Cosets N9),(Left_Cosets N9):],(Left_Cosets N9):] is non empty set
bool [:[:(Left_Cosets N9),(Left_Cosets N9):],(Left_Cosets N9):] is non empty set
multMagma(# (Left_Cosets N9),(CosOp N9) #) is non empty strict multMagma
center (G ./. N9) is non empty strict unital Group-like associative Subgroup of G ./. N9
N is non empty unital Group-like associative normal Subgroup of H1
H1 ./. N is non empty strict unital Group-like associative multMagma
Left_Cosets N is non empty Element of bool (bool the carrier of H1)
the carrier of H1 is non empty set
bool the carrier of H1 is non empty set
bool (bool the carrier of H1) is non empty set
CosOp N is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V18([:(Left_Cosets N),(Left_Cosets N):], Left_Cosets N) Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is non empty set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
multMagma(# (Left_Cosets N),(CosOp N) #) is non empty strict multMagma
len R is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R . 1 is set
R . (len R) is set
H1 is non empty strict unital Group-like associative Subgroup of G
c4 is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
K403(c4,1) is V24() V25() V26() V30() V31() V32() V33() ext-real Element of NAT
R . c4 is set
H2 is non empty strict unital Group-like associative Subgroup of G
R . K403(c4,1) is set
N is non empty unital Group-like associative normal Subgroup of H1
H1 ./. N is non empty strict unital Group-like associative multMagma
Left_Cosets N is non empty Element of bool (bool the carrier of H1)
the carrier of H1 is non empty set
bool the carrier of H1 is non empty set
bool (bool the carrier of H1) is non empty set
CosOp N is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V18([:(Left_Cosets N),(Left_Cosets N):], Left_Cosets N) Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is non empty set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is non empty set
multMagma(# (Left_Cosets N),(CosOp N) #) is non empty strict multMagma