:: GROUP_6 semantic presentation

REAL is set
NAT is non empty V41() V42() V43() Element of bool REAL
bool REAL is set
COMPLEX is set
NAT is non empty V41() V42() V43() set
bool NAT is set
bool NAT is set
2 is non empty ext-real V41() V42() V43() V47() V48() V49() integer Element of NAT
[:2,2:] is set
[:[:2,2:],2:] is set
bool [:[:2,2:],2:] is set
INT is set
RAT is set
[:COMPLEX,COMPLEX:] is set
bool [:COMPLEX,COMPLEX:] is set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is set
[:REAL,REAL:] is set
bool [:REAL,REAL:] is set
[:[:REAL,REAL:],REAL:] is set
bool [:[:REAL,REAL:],REAL:] is set
[:RAT,RAT:] is set
bool [:RAT,RAT:] is set
[:[:RAT,RAT:],RAT:] is set
bool [:[:RAT,RAT:],RAT:] is set
[:INT,INT:] is set
bool [:INT,INT:] is set
[:[:INT,INT:],INT:] is set
bool [:[:INT,INT:],INT:] is set
[:NAT,NAT:] is set
[:[:NAT,NAT:],NAT:] is set
bool [:[:NAT,NAT:],NAT:] is set
{} is set
the Function-like functional empty ext-real V41() V42() V43() V45() V46() V47() V48() V49() integer finite V56() set is Function-like functional empty ext-real V41() V42() V43() V45() V46() V47() V48() V49() integer finite V56() set
1 is non empty ext-real V41() V42() V43() V47() V48() V49() integer Element of NAT
0 is ext-real V41() V42() V43() V47() V48() V49() integer Element of NAT
G is non empty set
B is non empty set
[:G,B:] is set
bool [:G,B:] is set
N is Relation-like G -defined B -valued Function-like non empty V22(G) quasi_total Element of bool [:G,B:]
f is Element of G
N . f is Element of B
g is Element of G
N . g is Element of B
f is set
g is set
N . f is set
N . g is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative Subgroup of G
N is non empty unital Group-like associative Subgroup of B
G is non empty unital Group-like associative multMagma
(1). G is non empty finite strict unital Group-like associative Subgroup of G
(Omega). G is non empty strict unital Group-like associative 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative Subgroup of G
the carrier of B is non empty set
N is Element of the carrier of G
f is non empty unital Group-like associative (G,B)
N * f is Element of bool the carrier of G
bool the carrier of G is set
carr f is Element of bool the carrier of G
the carrier of f is non empty set
N * (carr f) is Element of bool the carrier of G
K245( the carrier of G,N) is finite Element of bool the carrier of G
K245( the carrier of G,N) * (carr f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,N) & b₂ in carr f ) } is set
f * N is Element of bool the carrier of G
(carr f) * N is Element of bool the carrier of G
(carr f) * K245( the carrier of G,N) 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 K245( the carrier of G,N) ) } is set
g is Element of the carrier of B
g * f is Element of bool the carrier of B
bool the carrier of B is set
carr f is Element of bool the carrier of B
g * (carr f) is Element of bool the carrier of B
K245( the carrier of B,g) is finite Element of bool the carrier of B
K245( the carrier of B,g) * (carr f) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in carr f ) } is set
f * g is Element of bool the carrier of B
(carr f) * g is Element of bool the carrier of B
(carr f) * K245( the carrier of B,g) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr f & b₂ in K245( the carrier of B,g) ) } is set
J is set
g is Element of the carrier of B
g * g is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (g,g) is Element of the carrier of B
[g,g] is set
{g,g} is finite set
{g} is finite set
{{g,g},{g}} is finite V56() set
the multF of B . [g,g] is set
g is Element of the carrier of G
N * g 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (N,g) is Element of the carrier of G
[N,g] is set
{N,g} is finite set
{N} is finite set
{{N,g},{N}} is finite V56() set
the multF of G . [N,g] is set
J is set
g is Element of the carrier of G
N * g 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (N,g) is Element of the carrier of G
[N,g] is set
{N,g} is finite set
{N} is finite set
{{N,g},{N}} is finite V56() set
the multF of G . [N,g] is set
g is Element of the carrier of f
x is Element of the carrier of B
g * x is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (g,x) is Element of the carrier of B
[g,x] is set
{g,x} is finite set
{g} is finite set
{{g,x},{g}} is finite V56() set
the multF of B . [g,x] is set
J is set
g is Element of the carrier of B
g * g is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (g,g) is Element of the carrier of B
[g,g] is set
{g,g} is finite set
{g} is finite set
{{g,g},{g}} is finite V56() set
the multF of B . [g,g] is set
g is Element of the carrier of G
g * N 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (g,N) is Element of the carrier of G
[g,N] is set
{g,N} is finite set
{g} is finite set
{{g,N},{g}} is finite V56() set
the multF of G . [g,N] is set
J is set
g is Element of the carrier of G
g * N 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (g,N) is Element of the carrier of G
[g,N] is set
{g,N} is finite set
{g} is finite set
{{g,N},{g}} is finite V56() set
the multF of G . [g,N] is set
g is Element of the carrier of f
x is Element of the carrier of B
x * g is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (x,g) is Element of the carrier of B
[x,g] is set
{x,g} is finite set
{x} is finite set
{{x,g},{x}} is finite V56() set
the multF of B . [x,g] is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative Subgroup of G
N is non empty unital Group-like associative (G,B)
f is non empty unital Group-like associative (G,B)
N /\ f is non empty strict unital Group-like associative Subgroup of G
N /\ f is non empty strict unital Group-like associative Subgroup of B
the carrier of (N /\ f) is non empty set
the carrier of B is non empty set
carr N is Element of bool the carrier of B
bool the carrier of B is set
the carrier of N is non empty set
carr f is Element of bool the carrier of B
the carrier of f is non empty set
(carr N) /\ (carr f) is Element of bool the carrier of B
g is non empty unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is Element of the carrier of G
B " is Element of the carrier of G
N is Element of the carrier of G
B * N 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (B,N) is Element of the carrier of G
[B,N] is set
{B,N} is finite set
{B} is finite set
{{B,N},{B}} is finite V56() set
the multF of G . [B,N] is set
(B * N) * (B ") is Element of the carrier of G
the multF of G . ((B * N),(B ")) is Element of the carrier of G
[(B * N),(B ")] is set
{(B * N),(B ")} is finite set
{(B * N)} is finite set
{{(B * N),(B ")},{(B * N)}} is finite V56() set
the multF of G . [(B * N),(B ")] is set
N |^ (B ") is Element of the carrier of G
N * (B ") is Element of the carrier of G
the multF of G . (N,(B ")) is Element of the carrier of G
[N,(B ")] is set
{N,(B ")} is finite set
{N} is finite set
{{N,(B ")},{N}} is finite V56() set
the multF of G . [N,(B ")] is set
B * (N * (B ")) is Element of the carrier of G
the multF of G . (B,(N * (B "))) is Element of the carrier of G
[B,(N * (B "))] is set
{B,(N * (B "))} is finite set
{{B,(N * (B "))},{B}} is finite V56() set
the multF of G . [B,(N * (B "))] is set
(B ") " is Element of the carrier of G
((B ") ") * N is Element of the carrier of G
the multF of G . (((B ") "),N) is Element of the carrier of G
[((B ") "),N] is set
{((B ") "),N} is finite set
{((B ") ")} is finite set
{{((B ") "),N},{((B ") ")}} is finite V56() set
the multF of G . [((B ") "),N] is set
(((B ") ") * N) * (B ") is Element of the carrier of G
the multF of G . ((((B ") ") * N),(B ")) is Element of the carrier of G
[(((B ") ") * N),(B ")] is set
{(((B ") ") * N),(B ")} is finite set
{(((B ") ") * N)} is finite set
{{(((B ") ") * N),(B ")},{(((B ") ") * N)}} is finite V56() set
the multF of G . [(((B ") ") * N),(B ")] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative Subgroup of G
B * B is Element of bool the carrier of G
bool the carrier of G is set
carr B is Element of bool the carrier of G
the carrier of B is non empty set
(carr B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in carr B ) } is set
N is Element of the carrier of G
N * B is Element of bool the carrier of G
N * (carr B) is Element of bool the carrier of G
K245( the carrier of G,N) is finite Element of bool the carrier of G
K245( the carrier of G,N) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,N) & b₂ in carr B ) } is set
(N * B) * B is Element of bool the carrier of G
(N * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * B & b₂ in carr B ) } is set
N * (B * B) is Element of bool the carrier of G
K245( the carrier of G,N) * (B * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,N) & b₂ in B * B ) } is set
(B * B) * N is Element of bool the carrier of G
(B * B) * K245( the carrier of G,N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * B & b₂ in K245( the carrier of G,N) ) } is set
B * N is Element of bool the carrier of G
(carr B) * N is Element of bool the carrier of G
(carr B) * K245( the carrier of G,N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,N) ) } is set
B * (B * N) is Element of bool the carrier of G
(carr B) * (B * N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in B * N ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is set
{ [.b₁,b₂.] where b₁, b₂ is Element of the carrier of G : verum } is set
G ` is non empty strict unital Group-like associative normal Subgroup of G
B is Element of bool the carrier of G
gr B is non empty strict unital Group-like associative Subgroup of G
commutators G is Element of bool the carrier of G
N is set
f is Element of the carrier of G
g is Element of the carrier of G
[.f,g.] is Element of the carrier of G
N is set
f is Element of the carrier of G
g is Element of the carrier of G
[.f,g.] is Element of the carrier of G
G is non empty strict unital Group-like associative multMagma
G ` is non empty strict unital Group-like associative normal Subgroup of G
the carrier of G is non empty set
B is non empty strict unital Group-like associative Subgroup of G
N is Element of the carrier of G
f is Element of the carrier of G
[.N,f.] is Element of the carrier of G
bool the carrier of G is set
{ H₁(b₁,b₂) where b₁, b₂ is Element of the carrier of G : S₁[b₁,b₂] } is set
N is Element of bool the carrier of G
the carrier of B is non empty set
f is set
g is Element of the carrier of G
I is Element of the carrier of G
[.g,I.] is Element of the carrier of G
gr N is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative Subgroup of G
N is non empty unital Group-like associative normal Subgroup of G
the carrier of B is non empty set
f is non empty unital Group-like associative (G,B)
g is Element of the carrier of B
g * f is Element of bool the carrier of B
bool the carrier of B is set
carr f is Element of bool the carrier of B
the carrier of f is non empty set
g * (carr f) is Element of bool the carrier of B
K245( the carrier of B,g) is finite Element of bool the carrier of B
K245( the carrier of B,g) * (carr f) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in carr f ) } is set
f * g is Element of bool the carrier of B
(carr f) * g is Element of bool the carrier of B
(carr f) * K245( the carrier of B,g) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr f & b₂ in K245( the carrier of B,g) ) } is set
I is set
J is Element of the carrier of B
g * J is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (g,J) is Element of the carrier of B
[g,J] is set
{g,J} is finite set
{g} is finite set
{{g,J},{g}} is finite V56() set
the multF of B . [g,J] is set
the carrier of G is non empty set
g is Element of the carrier of G
g is Element of the carrier of G
g * g 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (g,g) is Element of the carrier of G
[g,g] is set
{g,g} is finite set
{g} is finite set
{{g,g},{g}} is finite V56() set
the multF of G . [g,g] is set
g * N is Element of bool the carrier of G
bool the carrier of G is set
carr N is Element of bool the carrier of G
the carrier of N is non empty set
g * (carr N) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr N ) } is set
N * g is Element of bool the carrier of G
(carr N) * g is Element of bool the carrier of G
(carr N) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,g) ) } is set
x is Element of the carrier of G
x * g is Element of the carrier of G
the multF of G . (x,g) is Element of the carrier of G
[x,g] is set
{x,g} is finite set
{x} is finite set
{{x,g},{x}} is finite V56() set
the multF of G . [x,g] is set
b is Element of the carrier of B
b * g is Element of the carrier of B
the multF of B . (b,g) is Element of the carrier of B
[b,g] is set
{b,g} is finite set
{b} is finite set
{{b,g},{b}} is finite V56() set
the multF of B . [b,g] is set
G is non empty unital Group-like associative multMagma
N is non empty unital Group-like associative normal Subgroup of G
the carrier of N is non empty set
the multF of N is Relation-like [: the carrier of N, the carrier of N:] -defined the carrier of N -valued Function-like V22([: the carrier of N, the carrier of N:]) quasi_total associative Element of bool [:[: the carrier of N, the carrier of N:], the carrier of N:]
[: the carrier of N, the carrier of N:] is set
[:[: the carrier of N, the carrier of N:], the carrier of N:] is set
bool [:[: the carrier of N, the carrier of N:], the carrier of N:] is set
multMagma(# the carrier of N, the multF of N #) is strict multMagma
B is non empty unital Group-like associative Subgroup of G
the carrier of G is non empty set
f is non empty unital Group-like associative Subgroup of G
g is Element of the carrier of G
g * f is Element of bool the carrier of G
bool the carrier of G is set
carr f is Element of bool the carrier of G
the carrier of f is non empty set
g * (carr f) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,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 K245( the carrier of G,g) & b₂ in carr f ) } is set
g * N is Element of bool the carrier of G
carr N is Element of bool the carrier of G
g * (carr N) is Element of bool the carrier of G
K245( the carrier of G,g) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr N ) } is set
N * g is Element of bool the carrier of G
(carr N) * g is Element of bool the carrier of G
(carr N) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,g) ) } is set
f * g is Element of bool the carrier of G
(carr f) * g is Element of bool the carrier of G
(carr f) * K245( the carrier of G,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 K245( the carrier of G,g) ) } is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative Subgroup of G
N is non empty unital Group-like associative normal Subgroup of G
B /\ N is non empty strict unital Group-like associative Subgroup of G
N /\ B is non empty strict unital Group-like associative Subgroup of G
the carrier of B is non empty set
f is non empty unital Group-like associative (G,B)
g is Element of the carrier of B
g * f is Element of bool the carrier of B
bool the carrier of B is set
carr f is Element of bool the carrier of B
the carrier of f is non empty set
g * (carr f) is Element of bool the carrier of B
K245( the carrier of B,g) is finite Element of bool the carrier of B
K245( the carrier of B,g) * (carr f) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in carr f ) } is set
f * g is Element of bool the carrier of B
(carr f) * g is Element of bool the carrier of B
(carr f) * K245( the carrier of B,g) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr f & b₂ in K245( the carrier of B,g) ) } is set
I is set
J is Element of the carrier of B
g * J is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (g,J) is Element of the carrier of B
[g,J] is set
{g,J} is finite set
{g} is finite set
{{g,J},{g}} is finite V56() set
the multF of B . [g,J] is set
the carrier of G is non empty set
g is Element of the carrier of G
g is Element of the carrier of G
g * g 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (g,g) is Element of the carrier of G
[g,g] is set
{g,g} is finite set
{g} is finite set
{{g,g},{g}} is finite V56() set
the multF of G . [g,g] is set
g * N is Element of bool the carrier of G
bool the carrier of G is set
carr N is Element of bool the carrier of G
the carrier of N is non empty set
g * (carr N) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr N ) } is set
g " is Element of the carrier of G
g " is Element of the carrier of B
N * g is Element of bool the carrier of G
(carr N) * g is Element of bool the carrier of G
(carr N) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,g) ) } is set
b is Element of the carrier of G
b * g is Element of the carrier of G
the multF of G . (b,g) is Element of the carrier of G
[b,g] is set
{b,g} is finite set
{b} is finite set
{{b,g},{b}} is finite V56() set
the multF of G . [b,g] is set
a1 is Element of the carrier of G
a1 * (g ") is Element of the carrier of G
the multF of G . (a1,(g ")) is Element of the carrier of G
[a1,(g ")] is set
{a1,(g ")} is finite set
{a1} is finite set
{{a1,(g ")},{a1}} is finite V56() set
the multF of G . [a1,(g ")] is set
x is Element of the carrier of B
x * (g ") is Element of the carrier of B
the multF of B . (x,(g ")) is Element of the carrier of B
[x,(g ")] is set
{x,(g ")} is finite set
{x} is finite set
{{x,(g ")},{x}} is finite V56() set
the multF of B . [x,(g ")] is set
a2 is Element of the carrier of B
a2 * g is Element of the carrier of B
the multF of B . (a2,g) is Element of the carrier of B
[a2,g] is set
{a2,g} is finite set
{a2} is finite set
{{a2,g},{a2}} is finite V56() set
the multF of B . [a2,g] is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative Subgroup of G
N is non empty unital Group-like associative normal Subgroup of G
B /\ N is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
N is non empty unital Group-like associative Subgroup of G
B /\ N is non empty strict unital Group-like associative Subgroup of G
G is non empty 1-sorted
the carrier of G is non empty set
the Element of the carrier of G is Element of the carrier of G
N is set
f is set
{ the Element of the carrier of G} is finite set
B is set
{B} is finite set
N is Element of the carrier of G
f is Element of the carrier of G
G is non empty unital Group-like associative multMagma
(1). G is non empty finite strict unital Group-like associative normal Subgroup of G
the carrier of ((1). G) is non empty finite set
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
{(1_ G)} is finite set
G is non empty unital Group-like associative multMagma
(1). G is non empty finite strict unital Group-like associative normal Subgroup of G
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 finite 1 -element strict unital Group-like associative normal Subgroup of the non empty unital Group-like associative multMagma
G is non empty trivial finite 1 -element unital Group-like associative multMagma
card G is non empty ext-real V41() V42() V43() V47() V48() V49() integer cardinal Element of NAT
the carrier of G is non empty trivial finite set
card the carrier of G is cardinal set
B is set
{B} is finite set
G is non empty finite unital Group-like associative multMagma
card G is non empty ext-real V41() V42() V43() V47() V48() V49() integer cardinal Element of NAT
the carrier of G is non empty finite set
card the carrier of G is cardinal set
G is non empty trivial finite 1 -element strict unital Group-like associative multMagma
(1). G is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of G
card G is non empty ext-real V41() V42() V43() V47() V48() V49() integer cardinal Element of NAT
the carrier of G is non empty trivial finite set
card the carrier of G is cardinal set
card ((1). G) is non empty ext-real V41() V42() V43() V47() V48() V49() integer cardinal Element of NAT
the carrier of ((1). G) is non empty trivial finite set
card the carrier of ((1). G) is cardinal set
G is non empty unital Group-like associative multMagma
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
Left_Cosets B is Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is set
bool (bool the carrier of G) is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is set
N is non empty unital Group-like associative normal Subgroup of G
Left_Cosets N is non empty Element of bool (bool the carrier of G)
bool the carrier of G is set
bool (bool the carrier of G) is set
f is Element of the carrier of G
f * N is Element of bool the carrier of G
carr N is Element of bool the carrier of G
the carrier of N is non empty set
f * (carr N) is Element of bool the carrier of G
K245( the carrier of G,f) is finite Element of bool the carrier of G
K245( the carrier of G,f) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,f) & b₂ in carr N ) } is set
N * f is Element of bool the carrier of G
(carr N) * f is Element of bool the carrier of G
(carr N) * K245( the carrier of G,f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,f) ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is Element of the carrier of G
N is non empty unital Group-like associative normal Subgroup of G
B * N is Element of bool the carrier of G
bool the carrier of G is set
carr N is Element of bool the carrier of G
the carrier of N is non empty set
B * (carr N) is Element of bool the carrier of G
K245( the carrier of G,B) is finite Element of bool the carrier of G
K245( the carrier of G,B) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,B) & b₂ in carr N ) } is set
Left_Cosets N is non empty Element of bool (bool the carrier of G)
bool (bool the carrier of G) is set
N * B is Element of bool the carrier of G
(carr N) * B is Element of bool the carrier of G
(carr N) * K245( the carrier of G,B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,B) ) } is set
Right_Cosets N is Element of bool (bool the carrier of G)
G is non empty unital Group-like associative multMagma
B is set
N is non empty unital Group-like associative normal Subgroup of G
Left_Cosets N 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 set
bool (bool the carrier of G) is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is set
B is Element of bool the carrier of G
N is Element of bool the carrier of G
B * N is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B & b₂ in N ) } is set
f is non empty unital Group-like associative normal Subgroup of G
Left_Cosets f is non empty Element of bool (bool the carrier of G)
bool (bool the carrier of G) is set
g is Element of the carrier of G
g * f is Element of bool the carrier of G
carr f is Element of bool the carrier of G
the carrier of f is non empty set
g * (carr f) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,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 K245( the carrier of G,g) & b₂ in carr f ) } is set
f * g is Element of bool the carrier of G
(carr f) * g is Element of bool the carrier of G
(carr f) * K245( the carrier of G,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 K245( the carrier of G,g) ) } is set
I is Element of the carrier of G
I * f is Element of bool the carrier of G
I * (carr f) is Element of bool the carrier of G
K245( the carrier of G,I) is finite Element of bool the carrier of G
K245( the carrier of G,I) * (carr f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in carr f ) } is set
f * I is Element of bool the carrier of G
(carr f) * I is Element of bool the carrier of G
(carr f) * K245( the carrier of G,I) 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 K245( the carrier of G,I) ) } is set
f * (I * f) is Element of bool the carrier of G
(carr f) * (I * f) 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 I * f ) } is set
g * (f * (I * f)) is Element of bool the carrier of G
K245( the carrier of G,g) * (f * (I * f)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in f * (I * f) ) } is set
(I * f) * f is Element of bool the carrier of G
(I * f) * (carr f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in I * f & b₂ in carr f ) } is set
g * ((I * f) * f) is Element of bool the carrier of G
K245( the carrier of G,g) * ((I * f) * f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in (I * f) * f ) } is set
f * f is Element of bool the carrier of G
(carr f) * (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 f & b₂ in carr f ) } is set
I * (f * f) is Element of bool the carrier of G
K245( the carrier of G,I) * (f * f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in f * f ) } is set
g * (I * (f * f)) is Element of bool the carrier of G
K245( the carrier of G,g) * (I * (f * f)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in I * (f * f) ) } is set
g * (I * f) is Element of bool the carrier of G
K245( the carrier of G,g) * (I * f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in I * f ) } is set
g * I 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (g,I) is Element of the carrier of G
[g,I] is set
{g,I} is finite set
{g} is finite set
{{g,I},{g}} is finite V56() set
the multF of G . [g,I] is set
(g * I) * f is Element of bool the carrier of G
(g * I) * (carr f) is Element of bool the carrier of G
K245( the carrier of G,(g * I)) is finite Element of bool the carrier of G
K245( the carrier of G,(g * I)) * (carr f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(g * I)) & b₂ in carr f ) } is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
Left_Cosets B 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 set
bool (bool the carrier of G) is set
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
N is Element of Left_Cosets B
f is Element of Left_Cosets B
g is Element of bool the carrier of G
I is Element of bool the carrier of G
g * I 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 I ) } is set
J is Element of Left_Cosets B
g is Element of bool the carrier of G
g is Element of bool the carrier of G
g * g 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 g ) } is set
N is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
f is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
g is set
I is set
J is Element of Left_Cosets B
g is Element of Left_Cosets B
N . (J,g) is Element of Left_Cosets B
[J,g] is set
{J,g} is finite set
{J} is finite set
{{J,g},{J}} is finite V56() set
N . [J,g] is set
g is Element of bool the carrier of G
x is Element of bool the carrier of G
g * x 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 x ) } is set
N . (g,I) is set
[g,I] is set
{g,I} is finite set
{g} is finite set
{{g,I},{g}} is finite V56() set
N . [g,I] is set
f . (g,I) is set
f . [g,I] is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
the carrier of (G,B) is non empty set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
the carrier of (G,B) is non empty set
[: the carrier of (G,B), the carrier of (G,B):] is set
the multF of (G,B) is Relation-like [: the carrier of (G,B), the carrier of (G,B):] -defined the carrier of (G,B) -valued Function-like V22([: the carrier of (G,B), the carrier of (G,B):]) quasi_total Element of bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):]
[:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
the carrier of (G,B) is non empty set
N is Element of the carrier of (G,B)
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
the carrier of (G,B) is non empty set
N is Element of the carrier of (G,B)
(G,B,N) is Element of bool the carrier of G
f is Element of the carrier of (G,B)
(G,B,f) is Element of bool the carrier of G
(G,B,N) * (G,B,f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,B,N) & b₂ in (G,B,f) ) } is set
N * f is Element of the carrier of (G,B)
the multF of (G,B) is Relation-like [: the carrier of (G,B), the carrier of (G,B):] -defined the carrier of (G,B) -valued Function-like V22([: the carrier of (G,B), the carrier of (G,B):]) quasi_total Element of bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):]
[: the carrier of (G,B), the carrier of (G,B):] is set
[:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
the multF of (G,B) . (N,f) is Element of the carrier of (G,B)
[N,f] is set
{N,f} is finite set
{N} is finite set
{{N,f},{N}} is finite V56() set
the multF of (G,B) . [N,f] is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
the carrier of (G,B) is non empty set
N is Element of the carrier of (G,B)
f is Element of the carrier of (G,B)
N * f is Element of the carrier of (G,B)
the multF of (G,B) is Relation-like [: the carrier of (G,B), the carrier of (G,B):] -defined the carrier of (G,B) -valued Function-like V22([: the carrier of (G,B), the carrier of (G,B):]) quasi_total Element of bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):]
[: the carrier of (G,B), the carrier of (G,B):] is set
[:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
the multF of (G,B) . (N,f) is Element of the carrier of (G,B)
[N,f] is set
{N,f} is finite set
{N} is finite set
{{N,f},{N}} is finite V56() set
the multF of (G,B) . [N,f] is set
(G,B,(N * f)) is Element of bool the carrier of G
(G,B,N) is Element of bool the carrier of G
(G,B,f) is Element of bool the carrier of G
(G,B,N) * (G,B,f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,B,N) & b₂ in (G,B,f) ) } is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
the carrier of (G,B) is non empty set
N is Element of the carrier of (G,B)
f is Element of the carrier of (G,B)
g is Element of the carrier of (G,B)
f * g is Element of the carrier of (G,B)
the multF of (G,B) is Relation-like [: the carrier of (G,B), the carrier of (G,B):] -defined the carrier of (G,B) -valued Function-like V22([: the carrier of (G,B), the carrier of (G,B):]) quasi_total Element of bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):]
[: the carrier of (G,B), the carrier of (G,B):] is set
[:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
the multF of (G,B) . (f,g) is Element of the carrier of (G,B)
[f,g] is set
{f,g} is finite set
{f} is finite set
{{f,g},{f}} is finite V56() set
the multF of (G,B) . [f,g] is set
N * (f * g) is Element of the carrier of (G,B)
the multF of (G,B) . (N,(f * g)) is Element of the carrier of (G,B)
[N,(f * g)] is set
{N,(f * g)} is finite set
{N} is finite set
{{N,(f * g)},{N}} is finite V56() set
the multF of (G,B) . [N,(f * g)] is set
N * f is Element of the carrier of (G,B)
the multF of (G,B) . (N,f) is Element of the carrier of (G,B)
[N,f] is set
{N,f} is finite set
{{N,f},{N}} is finite V56() set
the multF of (G,B) . [N,f] is set
(N * f) * g is Element of the carrier of (G,B)
the multF of (G,B) . ((N * f),g) is Element of the carrier of (G,B)
[(N * f),g] is set
{(N * f),g} is finite set
{(N * f)} is finite set
{{(N * f),g},{(N * f)}} is finite V56() set
the multF of (G,B) . [(N * f),g] is set
I is Element of the carrier of G
I * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
the carrier of B is non empty set
I * (carr B) is Element of bool the carrier of G
K245( the carrier of G,I) is finite Element of bool the carrier of G
K245( the carrier of G,I) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in carr B ) } is set
B * I is Element of bool the carrier of G
(carr B) * I is Element of bool the carrier of G
(carr B) * K245( the carrier of G,I) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,I) ) } is set
J is Element of the carrier of G
J * B is Element of bool the carrier of G
J * (carr B) is Element of bool the carrier of G
K245( the carrier of G,J) is finite Element of bool the carrier of G
K245( the carrier of G,J) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,J) & b₂ in carr B ) } is set
B * J is Element of bool the carrier of G
(carr B) * J is Element of bool the carrier of G
(carr B) * K245( the carrier of G,J) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,J) ) } is set
(G,B,N) is Element of bool the carrier of G
(G,B,(f * g)) is Element of bool the carrier of G
(G,B,N) * (G,B,(f * g)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,B,N) & b₂ in (G,B,(f * g)) ) } is set
(G,B,f) is Element of bool the carrier of G
(G,B,g) is Element of bool the carrier of G
(G,B,f) * (G,B,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,B,f) & b₂ in (G,B,g) ) } is set
(I * B) * ((G,B,f) * (G,B,g)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in I * B & b₂ in (G,B,f) * (G,B,g) ) } is set
(G,B,N) * (G,B,f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,B,N) & b₂ in (G,B,f) ) } is set
((G,B,N) * (G,B,f)) * (J * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,B,N) * (G,B,f) & b₂ in J * B ) } is set
(G,B,(N * f)) is Element of bool the carrier of G
(G,B,(N * f)) * (G,B,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,B,(N * f)) & b₂ in (G,B,g) ) } is set
carr B is Element of bool the carrier of G
the carrier of B is non empty set
N is Element of the carrier of (G,B)
f is Element of the carrier of (G,B)
f * N is Element of the carrier of (G,B)
the multF of (G,B) is Relation-like [: the carrier of (G,B), the carrier of (G,B):] -defined the carrier of (G,B) -valued Function-like V22([: the carrier of (G,B), the carrier of (G,B):]) quasi_total Element of bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):]
[: the carrier of (G,B), the carrier of (G,B):] is set
[:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
the multF of (G,B) . (f,N) is Element of the carrier of (G,B)
[f,N] is set
{f,N} is finite set
{f} is finite set
{{f,N},{f}} is finite V56() set
the multF of (G,B) . [f,N] is set
N * f is Element of the carrier of (G,B)
the multF of (G,B) . (N,f) is Element of the carrier of (G,B)
[N,f] is set
{N,f} is finite set
{N} is finite set
{{N,f},{N}} is finite V56() set
the multF of (G,B) . [N,f] is set
g is Element of the carrier of G
g * B is Element of bool the carrier of G
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
B * g is Element of bool the carrier of G
(carr B) * g is Element of bool the carrier of G
(carr B) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,g) ) } is set
(g * B) * B is Element of bool the carrier of G
(g * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * B & b₂ in carr B ) } is set
B * B is Element of bool the carrier of G
(carr B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in carr B ) } is set
g * (B * B) is Element of bool the carrier of G
K245( the carrier of G,g) * (B * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in B * B ) } is set
B * (B * g) is Element of bool the carrier of G
(carr B) * (B * g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in B * g ) } is set
(B * B) * g is Element of bool the carrier of G
(B * B) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * B & b₂ in K245( the carrier of G,g) ) } is set
g " is Element of the carrier of G
(g ") * B is Element of bool the carrier of G
(g ") * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(g ")) is finite Element of bool the carrier of G
K245( the carrier of G,(g ")) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(g ")) & b₂ in carr B ) } is set
I is Element of the carrier of (G,B)
f * I is Element of the carrier of (G,B)
the multF of (G,B) . (f,I) is Element of the carrier of (G,B)
[f,I] is set
{f,I} is finite set
{{f,I},{f}} is finite V56() set
the multF of (G,B) . [f,I] is set
I * f is Element of the carrier of (G,B)
the multF of (G,B) . (I,f) is Element of the carrier of (G,B)
[I,f] is set
{I,f} is finite set
{I} is finite set
{{I,f},{I}} is finite V56() set
the multF of (G,B) . [I,f] is set
(B * g) * ((g ") * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * g & b₂ in (g ") * B ) } is set
(B * g) * (g ") is Element of bool the carrier of G
(B * g) * K245( the carrier of G,(g ")) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * g & b₂ in K245( the carrier of G,(g ")) ) } is set
((B * g) * (g ")) * B is Element of bool the carrier of G
((B * g) * (g ")) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (B * g) * (g ") & b₂ in carr B ) } is set
g * (g ") 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (g,(g ")) is Element of the carrier of G
[g,(g ")] is set
{g,(g ")} is finite set
{g} is finite set
{{g,(g ")},{g}} is finite V56() set
the multF of G . [g,(g ")] is set
B * (g * (g ")) is Element of bool the carrier of G
(carr B) * (g * (g ")) is Element of bool the carrier of G
K245( the carrier of G,(g * (g "))) is finite Element of bool the carrier of G
(carr B) * K245( the carrier of G,(g * (g "))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,(g * (g "))) ) } is set
(B * (g * (g "))) * B is Element of bool the carrier of G
(B * (g * (g "))) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * (g * (g ")) & b₂ in carr B ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
B * (1_ G) is Element of bool the carrier of G
(carr B) * (1_ G) is Element of bool the carrier of G
K245( the carrier of G,(1_ G)) is finite Element of bool the carrier of G
(carr B) * K245( the carrier of G,(1_ G)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,(1_ G)) ) } is set
(B * (1_ G)) * B is Element of bool the carrier of G
(B * (1_ G)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * (1_ G) & b₂ in carr B ) } is set
(G,B,I) is Element of bool the carrier of G
(G,B,f) is Element of bool the carrier of G
(G,B,I) * (G,B,f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,B,I) & b₂ in (G,B,f) ) } is set
((g ") * B) * g is Element of bool the carrier of G
((g ") * B) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (g ") * B & b₂ in K245( the carrier of G,g) ) } is set
(((g ") * B) * g) * B is Element of bool the carrier of G
(((g ") * B) * g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((g ") * B) * g & b₂ in carr B ) } is set
(g ") * (B * g) is Element of bool the carrier of G
K245( the carrier of G,(g ")) * (B * g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(g ")) & b₂ in B * g ) } is set
((g ") * (B * g)) * B is Element of bool the carrier of G
((g ") * (B * g)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (g ") * (B * g) & b₂ in carr B ) } is set
(g ") * (g * B) is Element of bool the carrier of G
K245( the carrier of G,(g ")) * (g * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(g ")) & b₂ in g * B ) } is set
((g ") * (g * B)) * B is Element of bool the carrier of G
((g ") * (g * B)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (g ") * (g * B) & b₂ in carr B ) } is set
(g ") * g is Element of the carrier of G
the multF of G . ((g "),g) is Element of the carrier of G
[(g "),g] is set
{(g "),g} is finite set
{(g ")} is finite set
{{(g "),g},{(g ")}} is finite V56() set
the multF of G . [(g "),g] is set
((g ") * g) * B is Element of bool the carrier of G
((g ") * g) * (carr B) is Element of bool the carrier of G
K245( the carrier of G,((g ") * g)) is finite Element of bool the carrier of G
K245( the carrier of G,((g ") * g)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((g ") * g)) & b₂ in carr B ) } is set
(((g ") * g) * B) * B is Element of bool the carrier of G
(((g ") * g) * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((g ") * g) * B & b₂ in carr B ) } is set
(1_ G) * B is Element of bool the carrier of G
(1_ G) * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(1_ G)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(1_ G)) & b₂ in carr B ) } is set
((1_ G) * B) * B is Element of bool the carrier of G
((1_ G) * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (1_ G) * B & b₂ in carr B ) } is set
(1_ G) * (B * B) is Element of bool the carrier of G
K245( the carrier of G,(1_ G)) * (B * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(1_ G)) & b₂ in B * B ) } is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict unital Group-like associative multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
the carrier of (G,B) is non empty set
N is Element of the carrier of (G,B)
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
f is non empty unital Group-like associative multMagma
the carrier of f is non empty set
J is non empty unital Group-like associative multMagma
N is non empty unital Group-like associative normal Subgroup of G
B is Element of the carrier of G
N * B is Element of bool the carrier of G
bool the carrier of G is set
carr N is Element of bool the carrier of G
the carrier of N is non empty set
(carr N) * B is Element of bool the carrier of G
K245( the carrier of G,B) is finite Element of bool the carrier of G
(carr N) * K245( the carrier of G,B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,B) ) } is set
(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 set
(G,N) is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V22([:(Left_Cosets N),(Left_Cosets N):]) quasi_total Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
multMagma(# (Left_Cosets N),(G,N) #) is strict multMagma
the carrier of (G,N) is non empty set
I is non empty unital Group-like associative normal Subgroup of f
g is Element of the carrier of f
g * I is Element of bool the carrier of f
bool the carrier of f is set
carr I is Element of bool the carrier of f
the carrier of I is non empty set
g * (carr I) is Element of bool the carrier of f
K245( the carrier of f,g) is finite Element of bool the carrier of f
K245( the carrier of f,g) * (carr I) is Element of bool the carrier of f
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of f : ( b₁ in K245( the carrier of f,g) & b₂ in carr I ) } is set
(f,I) is non empty strict unital Group-like associative multMagma
Left_Cosets I is non empty Element of bool (bool the carrier of f)
bool (bool the carrier of f) is set
(f,I) is Relation-like [:(Left_Cosets I),(Left_Cosets I):] -defined Left_Cosets I -valued Function-like V22([:(Left_Cosets I),(Left_Cosets I):]) quasi_total Element of bool [:[:(Left_Cosets I),(Left_Cosets I):],(Left_Cosets I):]
[:(Left_Cosets I),(Left_Cosets I):] is set
[:[:(Left_Cosets I),(Left_Cosets I):],(Left_Cosets I):] is set
bool [:[:(Left_Cosets I),(Left_Cosets I):],(Left_Cosets I):] is set
multMagma(# (Left_Cosets I),(f,I) #) is strict multMagma
the carrier of (f,I) is non empty set
g is non empty unital Group-like associative normal Subgroup of J
carr g is Element of bool the carrier of J
the carrier of J is non empty set
bool the carrier of J is set
the carrier of g is non empty set
(J,g) is non empty strict unital Group-like associative multMagma
Left_Cosets g is non empty Element of bool (bool the carrier of J)
bool (bool the carrier of J) is set
(J,g) is Relation-like [:(Left_Cosets g),(Left_Cosets g):] -defined Left_Cosets g -valued Function-like V22([:(Left_Cosets g),(Left_Cosets g):]) quasi_total Element of bool [:[:(Left_Cosets g),(Left_Cosets g):],(Left_Cosets g):]
[:(Left_Cosets g),(Left_Cosets g):] is set
[:[:(Left_Cosets g),(Left_Cosets g):],(Left_Cosets g):] is set
bool [:[:(Left_Cosets g),(Left_Cosets g):],(Left_Cosets g):] is set
multMagma(# (Left_Cosets g),(J,g) #) is strict multMagma
the carrier of (J,g) is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is set
N is non empty unital Group-like associative normal Subgroup of G
(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 the carrier of G is set
bool (bool the carrier of G) is set
(G,N) is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V22([:(Left_Cosets N),(Left_Cosets N):]) quasi_total Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
multMagma(# (Left_Cosets N),(G,N) #) is strict multMagma
the carrier of (G,N) is non empty set
f is Element of the carrier of G
f * N is Element of bool the carrier of G
carr N is Element of bool the carrier of G
the carrier of N is non empty set
f * (carr N) is Element of bool the carrier of G
K245( the carrier of G,f) is finite Element of bool the carrier of G
K245( the carrier of G,f) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,f) & b₂ in carr N ) } is set
N * f is Element of bool the carrier of G
(carr N) * f is Element of bool the carrier of G
(carr N) * K245( the carrier of G,f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,f) ) } is set
the carrier of (G,N) is non empty set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict unital Group-like associative multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
1_ (G,B) is non being_of_order_0 Element of the carrier of (G,B)
the carrier of (G,B) is non empty set
carr B is Element of bool the carrier of G
the carrier of B is non empty set
f is Element of the carrier of (G,B)
g is Element of the carrier of G
g * B is Element of bool the carrier of G
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
B * g is Element of bool the carrier of G
(carr B) * g is Element of bool the carrier of G
(carr B) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,g) ) } is set
N is Element of the carrier of (G,B)
f * N is Element of the carrier of (G,B)
the multF of (G,B) is Relation-like [: the carrier of (G,B), the carrier of (G,B):] -defined the carrier of (G,B) -valued Function-like V22([: the carrier of (G,B), the carrier of (G,B):]) quasi_total associative Element of bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):]
[: the carrier of (G,B), the carrier of (G,B):] is set
[:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
the multF of (G,B) . (f,N) is Element of the carrier of (G,B)
[f,N] is set
{f,N} is finite set
{f} is finite set
{{f,N},{f}} is finite V56() set
the multF of (G,B) . [f,N] is set
(g * B) * B is Element of bool the carrier of G
(g * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * B & b₂ in carr B ) } is set
B * B is Element of bool the carrier of G
(carr B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in carr B ) } is set
g * (B * B) is Element of bool the carrier of G
K245( the carrier of G,g) * (B * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in B * B ) } is set
N * f is Element of the carrier of (G,B)
the multF of (G,B) . (N,f) is Element of the carrier of (G,B)
[N,f] is set
{N,f} is finite set
{N} is finite set
{{N,f},{N}} is finite V56() set
the multF of (G,B) . [N,f] is set
B * (B * g) is Element of bool the carrier of G
(carr B) * (B * g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in B * g ) } is set
(B * B) * g is Element of bool the carrier of G
(B * B) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * B & b₂ in K245( the carrier of G,g) ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is Element of the carrier of G
B " is Element of the carrier of G
N is non empty unital Group-like associative normal Subgroup of G
(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 the carrier of G is set
bool (bool the carrier of G) is set
(G,N) is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V22([:(Left_Cosets N),(Left_Cosets N):]) quasi_total Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
multMagma(# (Left_Cosets N),(G,N) #) is strict multMagma
the carrier of (G,N) is non empty set
B * N is Element of bool the carrier of G
carr N is Element of bool the carrier of G
the carrier of N is non empty set
B * (carr N) is Element of bool the carrier of G
K245( the carrier of G,B) is finite Element of bool the carrier of G
K245( the carrier of G,B) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,B) & b₂ in carr N ) } is set
(B ") * N is Element of bool the carrier of G
(B ") * (carr N) is Element of bool the carrier of G
K245( the carrier of G,(B ")) is finite Element of bool the carrier of G
K245( the carrier of G,(B ")) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(B ")) & b₂ in carr N ) } is set
f is Element of the carrier of (G,N)
f " is Element of the carrier of (G,N)
g is Element of the carrier of (G,N)
g * f is Element of the carrier of (G,N)
the multF of (G,N) is Relation-like [: the carrier of (G,N), the carrier of (G,N):] -defined the carrier of (G,N) -valued Function-like V22([: the carrier of (G,N), the carrier of (G,N):]) quasi_total associative Element of bool [:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):]
[: the carrier of (G,N), the carrier of (G,N):] is set
[:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):] is set
bool [:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):] is set
the multF of (G,N) . (g,f) is Element of the carrier of (G,N)
[g,f] is set
{g,f} is finite set
{g} is finite set
{{g,f},{g}} is finite V56() set
the multF of (G,N) . [g,f] is set
(G,N,g) is Element of bool the carrier of G
(G,N,f) is Element of bool the carrier of G
(G,N,g) * (G,N,f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,N,g) & b₂ in (G,N,f) ) } is set
((B ") * N) * B is Element of bool the carrier of G
((B ") * N) * K245( the carrier of G,B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (B ") * N & b₂ in K245( the carrier of G,B) ) } is set
(((B ") * N) * B) * N is Element of bool the carrier of G
(((B ") * N) * B) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((B ") * N) * B & b₂ in carr N ) } is set
N * B is Element of bool the carrier of G
(carr N) * B is Element of bool the carrier of G
(carr N) * K245( the carrier of G,B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,B) ) } is set
(B ") * (N * B) is Element of bool the carrier of G
K245( the carrier of G,(B ")) * (N * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(B ")) & b₂ in N * B ) } is set
((B ") * (N * B)) * N is Element of bool the carrier of G
((B ") * (N * B)) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (B ") * (N * B) & b₂ in carr N ) } is set
(B ") * (B * N) is Element of bool the carrier of G
K245( the carrier of G,(B ")) * (B * N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(B ")) & b₂ in B * N ) } is set
((B ") * (B * N)) * N is Element of bool the carrier of G
((B ") * (B * N)) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (B ") * (B * N) & b₂ in carr N ) } is set
(B ") * B 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((B "),B) is Element of the carrier of G
[(B "),B] is set
{(B "),B} is finite set
{(B ")} is finite set
{{(B "),B},{(B ")}} is finite V56() set
the multF of G . [(B "),B] is set
((B ") * B) * N is Element of bool the carrier of G
((B ") * B) * (carr N) is Element of bool the carrier of G
K245( the carrier of G,((B ") * B)) is finite Element of bool the carrier of G
K245( the carrier of G,((B ") * B)) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((B ") * B)) & b₂ in carr N ) } is set
(((B ") * B) * N) * N is Element of bool the carrier of G
(((B ") * B) * N) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((B ") * B) * N & b₂ in carr N ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * N is Element of bool the carrier of G
(1_ G) * (carr N) is Element of bool the carrier of G
K245( the carrier of G,(1_ G)) is finite Element of bool the carrier of G
K245( the carrier of G,(1_ G)) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(1_ G)) & b₂ in carr N ) } is set
((1_ G) * N) * N is Element of bool the carrier of G
((1_ G) * N) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (1_ G) * N & b₂ in carr N ) } is set
N * N is Element of bool the carrier of G
(carr N) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in carr N ) } is set
(1_ G) * (N * N) is Element of bool the carrier of G
K245( the carrier of G,(1_ G)) * (N * N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(1_ G)) & b₂ in N * N ) } is set
1_ (G,N) is non being_of_order_0 Element of the carrier of (G,N)
f * g is Element of the carrier of (G,N)
the multF of (G,N) . (f,g) is Element of the carrier of (G,N)
[f,g] is set
{f,g} is finite set
{f} is finite set
{{f,g},{f}} is finite V56() set
the multF of (G,N) . [f,g] is set
(G,N,f) * (G,N,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,N,f) & b₂ in (G,N,g) ) } is set
(N * B) * ((B ") * N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * B & b₂ in (B ") * N ) } is set
(N * B) * (B ") is Element of bool the carrier of G
(N * B) * K245( the carrier of G,(B ")) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * B & b₂ in K245( the carrier of G,(B ")) ) } is set
((N * B) * (B ")) * N is Element of bool the carrier of G
((N * B) * (B ")) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (N * B) * (B ") & b₂ in carr N ) } is set
B * (B ") is Element of the carrier of G
the multF of G . (B,(B ")) is Element of the carrier of G
[B,(B ")] is set
{B,(B ")} is finite set
{B} is finite set
{{B,(B ")},{B}} is finite V56() set
the multF of G . [B,(B ")] is set
N * (B * (B ")) is Element of bool the carrier of G
(carr N) * (B * (B ")) is Element of bool the carrier of G
K245( the carrier of G,(B * (B "))) is finite Element of bool the carrier of G
(carr N) * K245( the carrier of G,(B * (B "))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,(B * (B "))) ) } is set
(N * (B * (B "))) * N is Element of bool the carrier of G
(N * (B * (B "))) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * (B * (B ")) & b₂ in carr N ) } is set
N * (1_ G) is Element of bool the carrier of G
(carr N) * (1_ G) is Element of bool the carrier of G
(carr N) * K245( the carrier of G,(1_ G)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,(1_ G)) ) } is set
(N * (1_ G)) * N is Element of bool the carrier of G
(N * (1_ G)) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * (1_ G) & b₂ in carr N ) } is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict unital Group-like associative multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
card (G,B) is cardinal set
the carrier of (G,B) is non empty set
card the carrier of (G,B) is cardinal set
Index B is cardinal set
card (Left_Cosets B) is cardinal set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is non empty strict unital Group-like associative multMagma
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
card (G,B) is cardinal set
the carrier of (G,B) is non empty set
card the carrier of (G,B) is cardinal set
index B is ext-real V41() V42() V43() V47() V48() V49() integer Element of NAT
N is finite set
card N is ext-real V41() V42() V43() V47() V48() V49() integer Element of NAT
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative Subgroup of G
N is non empty strict unital Group-like associative normal Subgroup of G
(G,B,N) is non empty strict unital Group-like associative normal (G,B)
(B,(G,B,N)) is non empty strict unital Group-like associative multMagma
Left_Cosets (G,B,N) is non empty Element of bool (bool the carrier of B)
the carrier of B is non empty set
bool the carrier of B is set
bool (bool the carrier of B) is set
(B,(G,B,N)) is Relation-like [:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):] -defined Left_Cosets (G,B,N) -valued Function-like V22([:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):]) quasi_total Element of bool [:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):]
[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):] is set
[:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):] is set
bool [:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):] is set
multMagma(# (Left_Cosets (G,B,N)),(B,(G,B,N)) #) is 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)
the carrier of G is non empty set
bool the carrier of G is set
bool (bool the carrier of G) is set
(G,N) is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V22([:(Left_Cosets N),(Left_Cosets N):]) quasi_total Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
multMagma(# (Left_Cosets N),(G,N) #) is strict multMagma
the multF of (B,(G,B,N)) is Relation-like [: the carrier of (B,(G,B,N)), the carrier of (B,(G,B,N)):] -defined the carrier of (B,(G,B,N)) -valued Function-like V22([: the carrier of (B,(G,B,N)), the carrier of (B,(G,B,N)):]) quasi_total associative Element of bool [:[: the carrier of (B,(G,B,N)), the carrier of (B,(G,B,N)):], the carrier of (B,(G,B,N)):]
the carrier of (B,(G,B,N)) is non empty set
[: the carrier of (B,(G,B,N)), the carrier of (B,(G,B,N)):] is set
[:[: the carrier of (B,(G,B,N)), the carrier of (B,(G,B,N)):], the carrier of (B,(G,B,N)):] is set
bool [:[: the carrier of (B,(G,B,N)), the carrier of (B,(G,B,N)):], the carrier of (B,(G,B,N)):] is set
the multF of (G,N) is Relation-like [: the carrier of (G,N), the carrier of (G,N):] -defined the carrier of (G,N) -valued Function-like V22([: the carrier of (G,N), the carrier of (G,N):]) quasi_total associative Element of bool [:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):]
the carrier of (G,N) is non empty set
[: the carrier of (G,N), the carrier of (G,N):] is set
[:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):] is set
bool [:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):] is set
g is set
x is Element of the carrier of B
x * (G,B,N) is Element of bool the carrier of B
carr (G,B,N) is Element of bool the carrier of B
the carrier of (G,B,N) is non empty set
x * (carr (G,B,N)) is Element of bool the carrier of B
K245( the carrier of B,x) is finite Element of bool the carrier of B
K245( the carrier of B,x) * (carr (G,B,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,x) & b₂ in carr (G,B,N) ) } is set
(G,B,N) * x is Element of bool the carrier of B
(carr (G,B,N)) * x is Element of bool the carrier of B
(carr (G,B,N)) * K245( the carrier of B,x) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr (G,B,N) & b₂ in K245( the carrier of B,x) ) } is set
b is Element of the carrier of G
b * N is Element of bool the carrier of G
carr N is Element of bool the carrier of G
the carrier of N is non empty set
b * (carr N) is Element of bool the carrier of G
K245( the carrier of G,b) is finite Element of bool the carrier of G
K245( the carrier of G,b) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,b) & b₂ in carr N ) } is set
g is set
dom the multF of (B,(G,B,N)) is Relation-like the carrier of (B,(G,B,N)) -defined the carrier of (B,(G,B,N)) -valued Element of bool [: the carrier of (B,(G,B,N)), the carrier of (B,(G,B,N)):]
bool [: the carrier of (B,(G,B,N)), the carrier of (B,(G,B,N)):] is set
x is set
b is set
[x,b] is set
{x,b} is finite set
{x} is finite set
{{x,b},{x}} is finite V56() set
a1 is Element of the carrier of B
a1 * (G,B,N) is Element of bool the carrier of B
carr (G,B,N) is Element of bool the carrier of B
the carrier of (G,B,N) is non empty set
a1 * (carr (G,B,N)) is Element of bool the carrier of B
K245( the carrier of B,a1) is finite Element of bool the carrier of B
K245( the carrier of B,a1) * (carr (G,B,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,a1) & b₂ in carr (G,B,N) ) } is set
(G,B,N) * a1 is Element of bool the carrier of B
(carr (G,B,N)) * a1 is Element of bool the carrier of B
(carr (G,B,N)) * K245( the carrier of B,a1) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr (G,B,N) & b₂ in K245( the carrier of B,a1) ) } is set
a2 is Element of the carrier of B
a2 * (G,B,N) is Element of bool the carrier of B
a2 * (carr (G,B,N)) is Element of bool the carrier of B
K245( the carrier of B,a2) is finite Element of bool the carrier of B
K245( the carrier of B,a2) * (carr (G,B,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,a2) & b₂ in carr (G,B,N) ) } is set
(G,B,N) * a2 is Element of bool the carrier of B
(carr (G,B,N)) * a2 is Element of bool the carrier of B
(carr (G,B,N)) * K245( the carrier of B,a2) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr (G,B,N) & b₂ in K245( the carrier of B,a2) ) } is set
the multF of (B,(G,B,N)) . g is set
V is Element of Left_Cosets (G,B,N)
V is Element of Left_Cosets (G,B,N)
the multF of (B,(G,B,N)) . (V,V) is set
[V,V] is set
{V,V} is finite set
{V} is finite set
{{V,V},{V}} is finite V56() set
the multF of (B,(G,B,N)) . [V,V] is set
(a2 * (G,B,N)) * ((G,B,N) * a1) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in a2 * (G,B,N) & b₂ in (G,B,N) * a1 ) } is set
(a2 * (G,B,N)) * (G,B,N) is Element of bool the carrier of B
(a2 * (G,B,N)) * (carr (G,B,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in a2 * (G,B,N) & b₂ in carr (G,B,N) ) } is set
((a2 * (G,B,N)) * (G,B,N)) * a1 is Element of bool the carrier of B
((a2 * (G,B,N)) * (G,B,N)) * K245( the carrier of B,a1) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in (a2 * (G,B,N)) * (G,B,N) & b₂ in K245( the carrier of B,a1) ) } is set
(G,B,N) * (G,B,N) is Element of bool the carrier of B
(carr (G,B,N)) * (carr (G,B,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr (G,B,N) & b₂ in carr (G,B,N) ) } is set
a2 * ((G,B,N) * (G,B,N)) is Element of bool the carrier of B
K245( the carrier of B,a2) * ((G,B,N) * (G,B,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,a2) & b₂ in (G,B,N) * (G,B,N) ) } is set
(a2 * ((G,B,N) * (G,B,N))) * a1 is Element of bool the carrier of B
(a2 * ((G,B,N) * (G,B,N))) * K245( the carrier of B,a1) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in a2 * ((G,B,N) * (G,B,N)) & b₂ in K245( the carrier of B,a1) ) } is set
(a2 * (G,B,N)) * a1 is Element of bool the carrier of B
(a2 * (G,B,N)) * K245( the carrier of B,a1) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in a2 * (G,B,N) & b₂ in K245( the carrier of B,a1) ) } is set
a2 * ((G,B,N) * a1) is Element of bool the carrier of B
K245( the carrier of B,a2) * ((G,B,N) * a1) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,a2) & b₂ in (G,B,N) * a1 ) } is set
a2 * (a1 * (G,B,N)) is Element of bool the carrier of B
K245( the carrier of B,a2) * (a1 * (G,B,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,a2) & b₂ in a1 * (G,B,N) ) } is set
a2 * a1 is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (a2,a1) is Element of the carrier of B
[a2,a1] is set
{a2,a1} is finite set
{a2} is finite set
{{a2,a1},{a2}} is finite V56() set
the multF of B . [a2,a1] is set
(a2 * a1) * (G,B,N) is Element of bool the carrier of B
(a2 * a1) * (carr (G,B,N)) is Element of bool the carrier of B
K245( the carrier of B,(a2 * a1)) is finite Element of bool the carrier of B
K245( the carrier of B,(a2 * a1)) * (carr (G,B,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,(a2 * a1)) & b₂ in carr (G,B,N) ) } is set
a9 is Element of the carrier of G
a9 * N is Element of bool the carrier of G
carr N is Element of bool the carrier of G
the carrier of N is non empty set
a9 * (carr N) is Element of bool the carrier of G
K245( the carrier of G,a9) is finite Element of bool the carrier of G
K245( the carrier of G,a9) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,a9) & b₂ in carr N ) } is set
b9 is Element of the carrier of G
b9 * N is Element of bool the carrier of G
b9 * (carr N) is Element of bool the carrier of G
K245( the carrier of G,b9) is finite Element of bool the carrier of G
K245( the carrier of G,b9) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,b9) & b₂ in carr N ) } is set
a9 * b9 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (a9,b9) is Element of the carrier of G
[a9,b9] is set
{a9,b9} is finite set
{a9} is finite set
{{a9,b9},{a9}} is finite V56() set
the multF of G . [a9,b9] is set
N * b9 is Element of bool the carrier of G
(carr N) * b9 is Element of bool the carrier of G
(carr N) * K245( the carrier of G,b9) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,b9) ) } is set
the multF of (G,N) . g is set
V1 is Element of Left_Cosets N
V2 is Element of Left_Cosets N
the multF of (G,N) . (V1,V2) is set
[V1,V2] is set
{V1,V2} is finite set
{V1} is finite set
{{V1,V2},{V1}} is finite V56() set
the multF of (G,N) . [V1,V2] is set
(a9 * N) * (N * b9) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a9 * N & b₂ in N * b9 ) } is set
(a9 * N) * N is Element of bool the carrier of G
(a9 * N) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a9 * N & b₂ in carr N ) } is set
((a9 * N) * N) * b9 is Element of bool the carrier of G
((a9 * N) * N) * K245( the carrier of G,b9) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (a9 * N) * N & b₂ in K245( the carrier of G,b9) ) } is set
N * N is Element of bool the carrier of G
(carr N) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in carr N ) } is set
a9 * (N * N) is Element of bool the carrier of G
K245( the carrier of G,a9) * (N * N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,a9) & b₂ in N * N ) } is set
(a9 * (N * N)) * b9 is Element of bool the carrier of G
(a9 * (N * N)) * K245( the carrier of G,b9) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a9 * (N * N) & b₂ in K245( the carrier of G,b9) ) } is set
(a9 * N) * b9 is Element of bool the carrier of G
(a9 * N) * K245( the carrier of G,b9) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a9 * N & b₂ in K245( the carrier of G,b9) ) } is set
a9 * (N * b9) is Element of bool the carrier of G
K245( the carrier of G,a9) * (N * b9) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,a9) & b₂ in N * b9 ) } is set
a9 * (b9 * N) is Element of bool the carrier of G
K245( the carrier of G,a9) * (b9 * N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,a9) & b₂ in b9 * N ) } is set
(a9 * b9) * N is Element of bool the carrier of G
(a9 * b9) * (carr N) is Element of bool the carrier of G
K245( the carrier of G,(a9 * b9)) is finite Element of bool the carrier of G
K245( the carrier of G,(a9 * b9)) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(a9 * b9)) & b₂ in carr N ) } is set
dom the multF of (G,N) is Relation-like the carrier of (G,N) -defined the carrier of (G,N) -valued Element of bool [: the carrier of (G,N), the carrier of (G,N):]
bool [: the carrier of (G,N), the carrier of (G,N):] is set
(dom the multF of (G,N)) /\ [: the carrier of (B,(G,B,N)), the carrier of (B,(G,B,N)):] is Relation-like the carrier of (G,N) -defined the carrier of (G,N) -valued Element of bool [: the carrier of (G,N), the carrier of (G,N):]
the multF of (G,N) || the carrier of (B,(G,B,N)) is set
the multF of (G,N) | [: the carrier of (B,(G,B,N)), the carrier of (B,(G,B,N)):] is Relation-like Function-like set
G is non empty unital Group-like associative multMagma
N is non empty strict unital Group-like associative normal Subgroup of G
B is non empty strict unital Group-like associative normal Subgroup of G
(G,B,N) is non empty strict unital Group-like associative normal (G,B)
(B,(G,B,N)) is non empty strict unital Group-like associative multMagma
Left_Cosets (G,B,N) is non empty Element of bool (bool the carrier of B)
the carrier of B is non empty set
bool the carrier of B is set
bool (bool the carrier of B) is set
(B,(G,B,N)) is Relation-like [:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):] -defined Left_Cosets (G,B,N) -valued Function-like V22([:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):]) quasi_total Element of bool [:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):]
[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):] is set
[:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):] is set
bool [:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):] is set
multMagma(# (Left_Cosets (G,B,N)),(B,(G,B,N)) #) is 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)
the carrier of G is non empty set
bool the carrier of G is set
bool (bool the carrier of G) is set
(G,N) is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V22([:(Left_Cosets N),(Left_Cosets N):]) quasi_total Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
multMagma(# (Left_Cosets N),(G,N) #) is strict multMagma
the carrier of (G,N) is non empty set
g is non empty unital Group-like associative Subgroup of (G,N)
I is Element of the carrier of (G,N)
I * g is Element of bool the carrier of (G,N)
bool the carrier of (G,N) is set
carr g is Element of bool the carrier of (G,N)
the carrier of g is non empty set
I * (carr g) is Element of bool the carrier of (G,N)
K245( the carrier of (G,N),I) is finite Element of bool the carrier of (G,N)
K245( the carrier of (G,N),I) * (carr g) is Element of bool the carrier of (G,N)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (G,N) : ( b₁ in K245( the carrier of (G,N),I) & b₂ in carr g ) } is set
g * I is Element of bool the carrier of (G,N)
(carr g) * I is Element of bool the carrier of (G,N)
(carr g) * K245( the carrier of (G,N),I) is Element of bool the carrier of (G,N)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (G,N) : ( b₁ in carr g & b₂ in K245( the carrier of (G,N),I) ) } is set
J is set
g is Element of the carrier of (G,N)
I * g is Element of the carrier of (G,N)
the multF of (G,N) is Relation-like [: the carrier of (G,N), the carrier of (G,N):] -defined the carrier of (G,N) -valued Function-like V22([: the carrier of (G,N), the carrier of (G,N):]) quasi_total associative Element of bool [:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):]
[: the carrier of (G,N), the carrier of (G,N):] is set
[:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):] is set
bool [:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):] is set
the multF of (G,N) . (I,g) is Element of the carrier of (G,N)
[I,g] is set
{I,g} is finite set
{I} is finite set
{{I,g},{I}} is finite V56() set
the multF of (G,N) . [I,g] is set
f is non empty unital Group-like associative normal (G,B)
g is Element of the carrier of B
g * f is Element of bool the carrier of B
carr f is Element of bool the carrier of B
the carrier of f is non empty set
g * (carr f) is Element of bool the carrier of B
K245( the carrier of B,g) is finite Element of bool the carrier of B
K245( the carrier of B,g) * (carr f) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in carr f ) } is set
f * g is Element of bool the carrier of B
(carr f) * g is Element of bool the carrier of B
(carr f) * K245( the carrier of B,g) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr f & b₂ in K245( the carrier of B,g) ) } is set
b is Element of the carrier of G
b * N is Element of bool the carrier of G
carr N is Element of bool the carrier of G
the carrier of N is non empty set
b * (carr N) is Element of bool the carrier of G
K245( the carrier of G,b) is finite Element of bool the carrier of G
K245( the carrier of G,b) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,b) & b₂ in carr N ) } is set
N * b is Element of bool the carrier of G
(carr N) * b is Element of bool the carrier of G
(carr N) * K245( the carrier of G,b) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,b) ) } is set
x is Element of the carrier of G
b " is Element of the carrier of G
x * (b ") 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (x,(b ")) is Element of the carrier of G
[x,(b ")] is set
{x,(b ")} is finite set
{x} is finite set
{{x,(b ")},{x}} is finite V56() set
the multF of G . [x,(b ")] is set
b * (x * (b ")) is Element of the carrier of G
the multF of G . (b,(x * (b "))) is Element of the carrier of G
[b,(x * (b "))] is set
{b,(x * (b "))} is finite set
{b} is finite set
{{b,(x * (b "))},{b}} is finite V56() set
the multF of G . [b,(x * (b "))] is set
x |^ (b ") is Element of the carrier of G
N * (b * (x * (b "))) is Element of bool the carrier of G
(carr N) * (b * (x * (b "))) is Element of bool the carrier of G
K245( the carrier of G,(b * (x * (b ")))) is finite Element of bool the carrier of G
(carr N) * K245( the carrier of G,(b * (x * (b ")))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,(b * (x * (b ")))) ) } is set
a2 is Element of the carrier of B
f * a2 is Element of bool the carrier of B
(carr f) * a2 is Element of bool the carrier of B
K245( the carrier of B,a2) is finite Element of bool the carrier of B
(carr f) * K245( the carrier of B,a2) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr f & b₂ in K245( the carrier of B,a2) ) } is set
V is Element of the carrier of g
N * x is Element of bool the carrier of G
(carr N) * x is Element of bool the carrier of G
K245( the carrier of G,x) is finite Element of bool the carrier of G
(carr N) * K245( the carrier of G,x) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,x) ) } is set
(N * b) * (N * x) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * b & b₂ in N * x ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(N * x) * (1_ G) is Element of bool the carrier of G
K245( the carrier of G,(1_ G)) is finite Element of bool the carrier of G
(N * x) * K245( the carrier of G,(1_ G)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * x & b₂ in K245( the carrier of G,(1_ G)) ) } is set
(N * b) * ((N * x) * (1_ G)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * b & b₂ in (N * x) * (1_ G) ) } is set
(b ") * b is Element of the carrier of G
the multF of G . ((b "),b) is Element of the carrier of G
[(b "),b] is set
{(b "),b} is finite set
{(b ")} is finite set
{{(b "),b},{(b ")}} is finite V56() set
the multF of G . [(b "),b] is set
(N * x) * ((b ") * b) is Element of bool the carrier of G
K245( the carrier of G,((b ") * b)) is finite Element of bool the carrier of G
(N * x) * K245( the carrier of G,((b ") * b)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * x & b₂ in K245( the carrier of G,((b ") * b)) ) } is set
(N * b) * ((N * x) * ((b ") * b)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * b & b₂ in (N * x) * ((b ") * b) ) } is set
x * ((b ") * b) is Element of the carrier of G
the multF of G . (x,((b ") * b)) is Element of the carrier of G
[x,((b ") * b)] is set
{x,((b ") * b)} is finite set
{{x,((b ") * b)},{x}} is finite V56() set
the multF of G . [x,((b ") * b)] is set
N * (x * ((b ") * b)) is Element of bool the carrier of G
(carr N) * (x * ((b ") * b)) is Element of bool the carrier of G
K245( the carrier of G,(x * ((b ") * b))) is finite Element of bool the carrier of G
(carr N) * K245( the carrier of G,(x * ((b ") * b))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,(x * ((b ") * b))) ) } is set
(N * b) * (N * (x * ((b ") * b))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * b & b₂ in N * (x * ((b ") * b)) ) } is set
(N * b) * N is Element of bool the carrier of G
(N * b) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * b & b₂ in carr N ) } is set
((N * b) * N) * (x * ((b ") * b)) is Element of bool the carrier of G
((N * b) * N) * K245( the carrier of G,(x * ((b ") * b))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (N * b) * N & b₂ in K245( the carrier of G,(x * ((b ") * b))) ) } is set
N * (b * N) is Element of bool the carrier of G
(carr N) * (b * N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in b * N ) } is set
(N * (b * N)) * (x * ((b ") * b)) is Element of bool the carrier of G
(N * (b * N)) * K245( the carrier of G,(x * ((b ") * b))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * (b * N) & b₂ in K245( the carrier of G,(x * ((b ") * b))) ) } is set
N * (N * b) is Element of bool the carrier of G
(carr N) * (N * b) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in N * b ) } is set
(N * (N * b)) * (x * ((b ") * b)) is Element of bool the carrier of G
(N * (N * b)) * K245( the carrier of G,(x * ((b ") * b))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * (N * b) & b₂ in K245( the carrier of G,(x * ((b ") * b))) ) } is set
(N * b) * (x * ((b ") * b)) is Element of bool the carrier of G
(N * b) * K245( the carrier of G,(x * ((b ") * b))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * b & b₂ in K245( the carrier of G,(x * ((b ") * b))) ) } is set
N * ((N * b) * (x * ((b ") * b))) is Element of bool the carrier of G
(carr N) * ((N * b) * (x * ((b ") * b))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in (N * b) * (x * ((b ") * b)) ) } is set
b * (x * ((b ") * b)) is Element of the carrier of G
the multF of G . (b,(x * ((b ") * b))) is Element of the carrier of G
[b,(x * ((b ") * b))] is set
{b,(x * ((b ") * b))} is finite set
{{b,(x * ((b ") * b))},{b}} is finite V56() set
the multF of G . [b,(x * ((b ") * b))] is set
N * (b * (x * ((b ") * b))) is Element of bool the carrier of G
(carr N) * (b * (x * ((b ") * b))) is Element of bool the carrier of G
K245( the carrier of G,(b * (x * ((b ") * b)))) is finite Element of bool the carrier of G
(carr N) * K245( the carrier of G,(b * (x * ((b ") * b)))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,(b * (x * ((b ") * b)))) ) } is set
N * (N * (b * (x * ((b ") * b)))) is Element of bool the carrier of G
(carr N) * (N * (b * (x * ((b ") * b)))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in N * (b * (x * ((b ") * b))) ) } is set
(x * (b ")) * b is Element of the carrier of G
the multF of G . ((x * (b ")),b) is Element of the carrier of G
[(x * (b ")),b] is set
{(x * (b ")),b} is finite set
{(x * (b "))} is finite set
{{(x * (b ")),b},{(x * (b "))}} is finite V56() set
the multF of G . [(x * (b ")),b] is set
b * ((x * (b ")) * b) is Element of the carrier of G
the multF of G . (b,((x * (b ")) * b)) is Element of the carrier of G
[b,((x * (b ")) * b)] is set
{b,((x * (b ")) * b)} is finite set
{{b,((x * (b ")) * b)},{b}} is finite V56() set
the multF of G . [b,((x * (b ")) * b)] is set
N * (b * ((x * (b ")) * b)) is Element of bool the carrier of G
(carr N) * (b * ((x * (b ")) * b)) is Element of bool the carrier of G
K245( the carrier of G,(b * ((x * (b ")) * b))) is finite Element of bool the carrier of G
(carr N) * K245( the carrier of G,(b * ((x * (b ")) * b))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,(b * ((x * (b ")) * b))) ) } is set
N * (N * (b * ((x * (b ")) * b))) is Element of bool the carrier of G
(carr N) * (N * (b * ((x * (b ")) * b))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in N * (b * ((x * (b ")) * b)) ) } is set
(b * (x * (b "))) * b is Element of the carrier of G
the multF of G . ((b * (x * (b "))),b) is Element of the carrier of G
[(b * (x * (b "))),b] is set
{(b * (x * (b "))),b} is finite set
{(b * (x * (b ")))} is finite set
{{(b * (x * (b "))),b},{(b * (x * (b ")))}} is finite V56() set
the multF of G . [(b * (x * (b "))),b] is set
N * ((b * (x * (b "))) * b) is Element of bool the carrier of G
(carr N) * ((b * (x * (b "))) * b) is Element of bool the carrier of G
K245( the carrier of G,((b * (x * (b "))) * b)) is finite Element of bool the carrier of G
(carr N) * K245( the carrier of G,((b * (x * (b "))) * b)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,((b * (x * (b "))) * b)) ) } is set
N * (N * ((b * (x * (b "))) * b)) is Element of bool the carrier of G
(carr N) * (N * ((b * (x * (b "))) * b)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in N * ((b * (x * (b "))) * b) ) } is set
(N * (b * (x * (b ")))) * b is Element of bool the carrier of G
(N * (b * (x * (b ")))) * K245( the carrier of G,b) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * (b * (x * (b "))) & b₂ in K245( the carrier of G,b) ) } is set
N * ((N * (b * (x * (b ")))) * b) is Element of bool the carrier of G
(carr N) * ((N * (b * (x * (b ")))) * b) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in (N * (b * (x * (b ")))) * b ) } is set
(b * (x * (b "))) * N is Element of bool the carrier of G
(b * (x * (b "))) * (carr N) is Element of bool the carrier of G
K245( the carrier of G,(b * (x * (b ")))) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(b * (x * (b ")))) & b₂ in carr N ) } is set
((b * (x * (b "))) * N) * b is Element of bool the carrier of G
((b * (x * (b "))) * N) * K245( the carrier of G,b) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (b * (x * (b "))) * N & b₂ in K245( the carrier of G,b) ) } is set
N * (((b * (x * (b "))) * N) * b) is Element of bool the carrier of G
(carr N) * (((b * (x * (b "))) * N) * b) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in ((b * (x * (b "))) * N) * b ) } is set
(b * (x * (b "))) * (N * b) is Element of bool the carrier of G
K245( the carrier of G,(b * (x * (b ")))) * (N * b) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(b * (x * (b ")))) & b₂ in N * b ) } is set
N * ((b * (x * (b "))) * (N * b)) is Element of bool the carrier of G
(carr N) * ((b * (x * (b "))) * (N * b)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in (b * (x * (b "))) * (N * b) ) } is set
(N * (b * (x * (b ")))) * (N * b) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * (b * (x * (b "))) & b₂ in N * b ) } is set
V is Element of the carrier of (G,N)
V * I is Element of the carrier of (G,N)
the multF of (G,N) . (V,I) is Element of the carrier of (G,N)
[V,I] is set
{V,I} is finite set
{V} is finite set
{{V,I},{V}} is finite V56() set
the multF of (G,N) . [V,I] is set
G is non empty strict unital Group-like associative multMagma
G ` is non empty strict unital Group-like associative normal Subgroup of G
B is non empty strict unital Group-like associative normal Subgroup of G
(G,B) is non empty strict unital Group-like associative multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
the carrier of (G,B) is non empty set
N is Element of the carrier of G
N * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
the carrier of B is non empty set
N * (carr B) is Element of bool the carrier of G
K245( the carrier of G,N) is finite Element of bool the carrier of G
K245( the carrier of G,N) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,N) & b₂ in carr B ) } is set
f is Element of the carrier of G
f * B is Element of bool the carrier of G
f * (carr B) is Element of bool the carrier of G
K245( the carrier of G,f) is finite Element of bool the carrier of G
K245( the carrier of G,f) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,f) & b₂ in carr B ) } is set
g is Element of the carrier of (G,B)
I is Element of the carrier of (G,B)
[.g,I.] is Element of the carrier of (G,B)
1_ (G,B) is non being_of_order_0 Element of the carrier of (G,B)
g " is Element of the carrier of (G,B)
N " is Element of the carrier of G
(N ") * B is Element of bool the carrier of G
(N ") * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(N ")) is finite Element of bool the carrier of G
K245( the carrier of G,(N ")) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(N ")) & b₂ in carr B ) } is set
I " is Element of the carrier of (G,B)
f " is Element of the carrier of G
(f ") * B is Element of bool the carrier of G
(f ") * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(f ")) is finite Element of bool the carrier of G
K245( the carrier of G,(f ")) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(f ")) & b₂ in carr B ) } is set
(g ") * (I ") is Element of the carrier of (G,B)
the multF of (G,B) is Relation-like [: the carrier of (G,B), the carrier of (G,B):] -defined the carrier of (G,B) -valued Function-like V22([: the carrier of (G,B), the carrier of (G,B):]) quasi_total associative Element of bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):]
[: the carrier of (G,B), the carrier of (G,B):] is set
[:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
the multF of (G,B) . ((g "),(I ")) is Element of the carrier of (G,B)
[(g "),(I ")] is set
{(g "),(I ")} is finite set
{(g ")} is finite set
{{(g "),(I ")},{(g ")}} is finite V56() set
the multF of (G,B) . [(g "),(I ")] is set
((N ") * B) * ((f ") * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (N ") * B & b₂ in (f ") * B ) } is set
g * I is Element of the carrier of (G,B)
the multF of (G,B) . (g,I) is Element of the carrier of (G,B)
[g,I] is set
{g,I} is finite set
{g} is finite set
{{g,I},{g}} is finite V56() set
the multF of (G,B) . [g,I] is set
(N * B) * (f * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in N * B & b₂ in f * B ) } is set
((g ") * (I ")) * (g * I) is Element of the carrier of (G,B)
the multF of (G,B) . (((g ") * (I ")),(g * I)) is Element of the carrier of (G,B)
[((g ") * (I ")),(g * I)] is set
{((g ") * (I ")),(g * I)} is finite set
{((g ") * (I "))} is finite set
{{((g ") * (I ")),(g * I)},{((g ") * (I "))}} is finite V56() set
the multF of (G,B) . [((g ") * (I ")),(g * I)] is set
(((N ") * B) * ((f ") * B)) * ((N * B) * (f * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((N ") * B) * ((f ") * B) & b₂ in (N * B) * (f * B) ) } is set
B * (f * B) is Element of bool the carrier of G
(carr B) * (f * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in f * B ) } is set
N * (B * (f * B)) is Element of bool the carrier of G
K245( the carrier of G,N) * (B * (f * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,N) & b₂ in B * (f * B) ) } is set
(((N ") * B) * ((f ") * B)) * (N * (B * (f * B))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((N ") * B) * ((f ") * B) & b₂ in N * (B * (f * B)) ) } is set
B * f is Element of bool the carrier of G
(carr B) * f is Element of bool the carrier of G
(carr B) * K245( the carrier of G,f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,f) ) } is set
(B * f) * B is Element of bool the carrier of G
(B * f) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * f & b₂ in carr B ) } is set
N * ((B * f) * B) is Element of bool the carrier of G
K245( the carrier of G,N) * ((B * f) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,N) & b₂ in (B * f) * B ) } is set
(((N ") * B) * ((f ") * B)) * (N * ((B * f) * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((N ") * B) * ((f ") * B) & b₂ in N * ((B * f) * B) ) } is set
(f * B) * B is Element of bool the carrier of G
(f * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in f * B & b₂ in carr B ) } is set
N * ((f * B) * B) is Element of bool the carrier of G
K245( the carrier of G,N) * ((f * B) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,N) & b₂ in (f * B) * B ) } is set
(((N ") * B) * ((f ") * B)) * (N * ((f * B) * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((N ") * B) * ((f ") * B) & b₂ in N * ((f * B) * B) ) } is set
N * (f * B) is Element of bool the carrier of G
K245( the carrier of G,N) * (f * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,N) & b₂ in f * B ) } is set
(((N ") * B) * ((f ") * B)) * (N * (f * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((N ") * B) * ((f ") * B) & b₂ in N * (f * B) ) } is set
N * 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (N,f) is Element of the carrier of G
[N,f] is set
{N,f} is finite set
{N} is finite set
{{N,f},{N}} is finite V56() set
the multF of G . [N,f] is set
(N * f) * B is Element of bool the carrier of G
(N * f) * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(N * f)) is finite Element of bool the carrier of G
K245( the carrier of G,(N * f)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(N * f)) & b₂ in carr B ) } is set
(((N ") * B) * ((f ") * B)) * ((N * f) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((N ") * B) * ((f ") * B) & b₂ in (N * f) * B ) } is set
B * ((f ") * B) is Element of bool the carrier of G
(carr B) * ((f ") * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in (f ") * B ) } is set
(N ") * (B * ((f ") * B)) is Element of bool the carrier of G
K245( the carrier of G,(N ")) * (B * ((f ") * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(N ")) & b₂ in B * ((f ") * B) ) } is set
((N ") * (B * ((f ") * B))) * ((N * f) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (N ") * (B * ((f ") * B)) & b₂ in (N * f) * B ) } is set
B * (f ") is Element of bool the carrier of G
(carr B) * (f ") is Element of bool the carrier of G
(carr B) * K245( the carrier of G,(f ")) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,(f ")) ) } is set
(B * (f ")) * B is Element of bool the carrier of G
(B * (f ")) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * (f ") & b₂ in carr B ) } is set
(N ") * ((B * (f ")) * B) is Element of bool the carrier of G
K245( the carrier of G,(N ")) * ((B * (f ")) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(N ")) & b₂ in (B * (f ")) * B ) } is set
((N ") * ((B * (f ")) * B)) * ((N * f) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (N ") * ((B * (f ")) * B) & b₂ in (N * f) * B ) } is set
((f ") * B) * B is Element of bool the carrier of G
((f ") * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (f ") * B & b₂ in carr B ) } is set
(N ") * (((f ") * B) * B) is Element of bool the carrier of G
K245( the carrier of G,(N ")) * (((f ") * B) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(N ")) & b₂ in ((f ") * B) * B ) } is set
((N ") * (((f ") * B) * B)) * ((N * f) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (N ") * (((f ") * B) * B) & b₂ in (N * f) * B ) } is set
(N ") * ((f ") * B) is Element of bool the carrier of G
K245( the carrier of G,(N ")) * ((f ") * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(N ")) & b₂ in (f ") * B ) } is set
((N ") * ((f ") * B)) * ((N * f) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (N ") * ((f ") * B) & b₂ in (N * f) * B ) } is set
(N ") * (f ") is Element of the carrier of G
the multF of G . ((N "),(f ")) is Element of the carrier of G
[(N "),(f ")] is set
{(N "),(f ")} is finite set
{(N ")} is finite set
{{(N "),(f ")},{(N ")}} is finite V56() set
the multF of G . [(N "),(f ")] is set
((N ") * (f ")) * B is Element of bool the carrier of G
((N ") * (f ")) * (carr B) is Element of bool the carrier of G
K245( the carrier of G,((N ") * (f "))) is finite Element of bool the carrier of G
K245( the carrier of G,((N ") * (f "))) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((N ") * (f "))) & b₂ in carr B ) } is set
(((N ") * (f ")) * B) * ((N * f) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((N ") * (f ")) * B & b₂ in (N * f) * B ) } is set
B * ((N * f) * B) is Element of bool the carrier of G
(carr B) * ((N * f) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in (N * f) * B ) } is set
((N ") * (f ")) * (B * ((N * f) * B)) is Element of bool the carrier of G
K245( the carrier of G,((N ") * (f "))) * (B * ((N * f) * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((N ") * (f "))) & b₂ in B * ((N * f) * B) ) } is set
B * (N * f) is Element of bool the carrier of G
(carr B) * (N * f) is Element of bool the carrier of G
(carr B) * K245( the carrier of G,(N * f)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,(N * f)) ) } is set
(B * (N * f)) * B is Element of bool the carrier of G
(B * (N * f)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * (N * f) & b₂ in carr B ) } is set
((N ") * (f ")) * ((B * (N * f)) * B) is Element of bool the carrier of G
K245( the carrier of G,((N ") * (f "))) * ((B * (N * f)) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((N ") * (f "))) & b₂ in (B * (N * f)) * B ) } is set
((N * f) * B) * B is Element of bool the carrier of G
((N * f) * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (N * f) * B & b₂ in carr B ) } is set
((N ") * (f ")) * (((N * f) * B) * B) is Element of bool the carrier of G
K245( the carrier of G,((N ") * (f "))) * (((N * f) * B) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((N ") * (f "))) & b₂ in ((N * f) * B) * B ) } is set
((N ") * (f ")) * ((N * f) * B) is Element of bool the carrier of G
K245( the carrier of G,((N ") * (f "))) * ((N * f) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((N ") * (f "))) & b₂ in (N * f) * B ) } is set
((N ") * (f ")) * (N * f) is Element of the carrier of G
the multF of G . (((N ") * (f ")),(N * f)) is Element of the carrier of G
[((N ") * (f ")),(N * f)] is set
{((N ") * (f ")),(N * f)} is finite set
{((N ") * (f "))} is finite set
{{((N ") * (f ")),(N * f)},{((N ") * (f "))}} is finite V56() set
the multF of G . [((N ") * (f ")),(N * f)] is set
(((N ") * (f ")) * (N * f)) * B is Element of bool the carrier of G
(((N ") * (f ")) * (N * f)) * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(((N ") * (f ")) * (N * f))) is finite Element of bool the carrier of G
K245( the carrier of G,(((N ") * (f ")) * (N * f))) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(((N ") * (f ")) * (N * f))) & b₂ in carr B ) } is set
[.N,f.] is Element of the carrier of G
[.N,f.] * B is Element of bool the carrier of G
[.N,f.] * (carr B) is Element of bool the carrier of G
K245( the carrier of G,[.N,f.]) is finite Element of bool the carrier of G
K245( the carrier of G,[.N,f.]) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,[.N,f.]) & b₂ in carr B ) } is set
the carrier of (G,B) is non empty set
N is Element of the carrier of (G,B)
f is Element of the carrier of (G,B)
[.N,f.] is Element of the carrier of (G,B)
N " is Element of the carrier of (G,B)
f " is Element of the carrier of (G,B)
(N ") * (f ") is Element of the carrier of (G,B)
the multF of (G,B) is Relation-like [: the carrier of (G,B), the carrier of (G,B):] -defined the carrier of (G,B) -valued Function-like V22([: the carrier of (G,B), the carrier of (G,B):]) quasi_total associative Element of bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):]
[: the carrier of (G,B), the carrier of (G,B):] is set
[:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
the multF of (G,B) . ((N "),(f ")) is Element of the carrier of (G,B)
[(N "),(f ")] is set
{(N "),(f ")} is finite set
{(N ")} is finite set
{{(N "),(f ")},{(N ")}} is finite V56() set
the multF of (G,B) . [(N "),(f ")] is set
N * f is Element of the carrier of (G,B)
the multF of (G,B) . (N,f) is Element of the carrier of (G,B)
[N,f] is set
{N,f} is finite set
{N} is finite set
{{N,f},{N}} is finite V56() set
the multF of (G,B) . [N,f] is set
((N ") * (f ")) * (N * f) is Element of the carrier of (G,B)
the multF of (G,B) . (((N ") * (f ")),(N * f)) is Element of the carrier of (G,B)
[((N ") * (f ")),(N * f)] is set
{((N ") * (f ")),(N * f)} is finite set
{((N ") * (f "))} is finite set
{{((N ") * (f ")),(N * f)},{((N ") * (f "))}} is finite V56() set
the multF of (G,B) . [((N ") * (f ")),(N * f)] is set
g is Element of the carrier of G
g * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
the carrier of B is non empty set
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
B * g is Element of bool the carrier of G
(carr B) * g is Element of bool the carrier of G
(carr B) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,g) ) } is set
g " is Element of the carrier of G
(g ") * B is Element of bool the carrier of G
(g ") * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(g ")) is finite Element of bool the carrier of G
K245( the carrier of G,(g ")) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(g ")) & b₂ in carr B ) } is set
I is Element of the carrier of G
I * B is Element of bool the carrier of G
I * (carr B) is Element of bool the carrier of G
K245( the carrier of G,I) is finite Element of bool the carrier of G
K245( the carrier of G,I) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in carr B ) } is set
B * I is Element of bool the carrier of G
(carr B) * I is Element of bool the carrier of G
(carr B) * K245( the carrier of G,I) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,I) ) } is set
I " is Element of the carrier of G
(I ") * B is Element of bool the carrier of G
(I ") * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(I ")) is finite Element of bool the carrier of G
K245( the carrier of G,(I ")) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(I ")) & b₂ in carr B ) } is set
((I ") * B) * ((g ") * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (I ") * B & b₂ in (g ") * B ) } is set
[.I,g.] is Element of the carrier of G
[.I,g.] * B is Element of bool the carrier of G
[.I,g.] * (carr B) is Element of bool the carrier of G
K245( the carrier of G,[.I,g.]) is finite Element of bool the carrier of G
K245( the carrier of G,[.I,g.]) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,[.I,g.]) & b₂ in carr B ) } is set
(I ") * (g ") 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((I "),(g ")) is Element of the carrier of G
[(I "),(g ")] is set
{(I "),(g ")} is finite set
{(I ")} is finite set
{{(I "),(g ")},{(I ")}} is finite V56() set
the multF of G . [(I "),(g ")] is set
I * g is Element of the carrier of G
the multF of G . (I,g) is Element of the carrier of G
[I,g] is set
{I,g} is finite set
{I} is finite set
{{I,g},{I}} is finite V56() set
the multF of G . [I,g] is set
((I ") * (g ")) * (I * g) is Element of the carrier of G
the multF of G . (((I ") * (g ")),(I * g)) is Element of the carrier of G
[((I ") * (g ")),(I * g)] is set
{((I ") * (g ")),(I * g)} is finite set
{((I ") * (g "))} is finite set
{{((I ") * (g ")),(I * g)},{((I ") * (g "))}} is finite V56() set
the multF of G . [((I ") * (g ")),(I * g)] is set
(((I ") * (g ")) * (I * g)) * B is Element of bool the carrier of G
(((I ") * (g ")) * (I * g)) * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(((I ") * (g ")) * (I * g))) is finite Element of bool the carrier of G
K245( the carrier of G,(((I ") * (g ")) * (I * g))) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(((I ") * (g ")) * (I * g))) & b₂ in carr B ) } is set
(I * g) * B is Element of bool the carrier of G
(I * g) * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(I * g)) is finite Element of bool the carrier of G
K245( the carrier of G,(I * g)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(I * g)) & b₂ in carr B ) } is set
((I ") * (g ")) * ((I * g) * B) is Element of bool the carrier of G
K245( the carrier of G,((I ") * (g "))) is finite Element of bool the carrier of G
K245( the carrier of G,((I ") * (g "))) * ((I * g) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((I ") * (g "))) & b₂ in (I * g) * B ) } is set
((I * g) * B) * B is Element of bool the carrier of G
((I * g) * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (I * g) * B & b₂ in carr B ) } is set
((I ") * (g ")) * (((I * g) * B) * B) is Element of bool the carrier of G
K245( the carrier of G,((I ") * (g "))) * (((I * g) * B) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((I ") * (g "))) & b₂ in ((I * g) * B) * B ) } is set
B * (I * g) is Element of bool the carrier of G
(carr B) * (I * g) is Element of bool the carrier of G
(carr B) * K245( the carrier of G,(I * g)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,(I * g)) ) } is set
(B * (I * g)) * B is Element of bool the carrier of G
(B * (I * g)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * (I * g) & b₂ in carr B ) } is set
((I ") * (g ")) * ((B * (I * g)) * B) is Element of bool the carrier of G
K245( the carrier of G,((I ") * (g "))) * ((B * (I * g)) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((I ") * (g "))) & b₂ in (B * (I * g)) * B ) } is set
B * ((I * g) * B) is Element of bool the carrier of G
(carr B) * ((I * g) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in (I * g) * B ) } is set
((I ") * (g ")) * (B * ((I * g) * B)) is Element of bool the carrier of G
K245( the carrier of G,((I ") * (g "))) * (B * ((I * g) * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((I ") * (g "))) & b₂ in B * ((I * g) * B) ) } is set
((I ") * (g ")) * B is Element of bool the carrier of G
((I ") * (g ")) * (carr B) is Element of bool the carrier of G
K245( the carrier of G,((I ") * (g "))) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,((I ") * (g "))) & b₂ in carr B ) } is set
(((I ") * (g ")) * B) * ((I * g) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((I ") * (g ")) * B & b₂ in (I * g) * B ) } is set
(I ") * ((g ") * B) is Element of bool the carrier of G
K245( the carrier of G,(I ")) * ((g ") * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(I ")) & b₂ in (g ") * B ) } is set
((I ") * ((g ") * B)) * ((I * g) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (I ") * ((g ") * B) & b₂ in (I * g) * B ) } is set
((g ") * B) * B is Element of bool the carrier of G
((g ") * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (g ") * B & b₂ in carr B ) } is set
(I ") * (((g ") * B) * B) is Element of bool the carrier of G
K245( the carrier of G,(I ")) * (((g ") * B) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(I ")) & b₂ in ((g ") * B) * B ) } is set
((I ") * (((g ") * B) * B)) * ((I * g) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (I ") * (((g ") * B) * B) & b₂ in (I * g) * B ) } is set
B * (g ") is Element of bool the carrier of G
(carr B) * (g ") is Element of bool the carrier of G
(carr B) * K245( the carrier of G,(g ")) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,(g ")) ) } is set
(B * (g ")) * B is Element of bool the carrier of G
(B * (g ")) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * (g ") & b₂ in carr B ) } is set
(I ") * ((B * (g ")) * B) is Element of bool the carrier of G
K245( the carrier of G,(I ")) * ((B * (g ")) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(I ")) & b₂ in (B * (g ")) * B ) } is set
((I ") * ((B * (g ")) * B)) * ((I * g) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (I ") * ((B * (g ")) * B) & b₂ in (I * g) * B ) } is set
B * ((g ") * B) is Element of bool the carrier of G
(carr B) * ((g ") * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in (g ") * B ) } is set
(I ") * (B * ((g ") * B)) is Element of bool the carrier of G
K245( the carrier of G,(I ")) * (B * ((g ") * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(I ")) & b₂ in B * ((g ") * B) ) } is set
((I ") * (B * ((g ") * B))) * ((I * g) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (I ") * (B * ((g ") * B)) & b₂ in (I * g) * B ) } is set
(((I ") * B) * ((g ") * B)) * ((I * g) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((I ") * B) * ((g ") * B) & b₂ in (I * g) * B ) } is set
I * (g * B) is Element of bool the carrier of G
K245( the carrier of G,I) * (g * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in g * B ) } is set
(((I ") * B) * ((g ") * B)) * (I * (g * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((I ") * B) * ((g ") * B) & b₂ in I * (g * B) ) } is set
(g * B) * B is Element of bool the carrier of G
(g * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * B & b₂ in carr B ) } is set
I * ((g * B) * B) is Element of bool the carrier of G
K245( the carrier of G,I) * ((g * B) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in (g * B) * B ) } is set
(((I ") * B) * ((g ") * B)) * (I * ((g * B) * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((I ") * B) * ((g ") * B) & b₂ in I * ((g * B) * B) ) } is set
(B * g) * B is Element of bool the carrier of G
(B * g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in B * g & b₂ in carr B ) } is set
I * ((B * g) * B) is Element of bool the carrier of G
K245( the carrier of G,I) * ((B * g) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in (B * g) * B ) } is set
(((I ") * B) * ((g ") * B)) * (I * ((B * g) * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((I ") * B) * ((g ") * B) & b₂ in I * ((B * g) * B) ) } is set
B * (g * B) is Element of bool the carrier of G
(carr B) * (g * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in g * B ) } is set
I * (B * (g * B)) is Element of bool the carrier of G
K245( the carrier of G,I) * (B * (g * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in B * (g * B) ) } is set
(((I ") * B) * ((g ") * B)) * (I * (B * (g * B))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((I ") * B) * ((g ") * B) & b₂ in I * (B * (g * B)) ) } is set
(I * B) * (g * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in I * B & b₂ in g * B ) } is set
(((I ") * B) * ((g ") * B)) * ((I * B) * (g * B)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((I ") * B) * ((g ") * B) & b₂ in (I * B) * (g * B) ) } is set
1_ (G,B) is non being_of_order_0 Element of the carrier of (G,B)
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
1_ B is non being_of_order_0 Element of the carrier of B
N is Element of the carrier of G
f is Element of the carrier of G
N * 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (N,f) is Element of the carrier of G
[N,f] is set
{N,f} is finite set
{N} is finite set
{{N,f},{N}} is finite V56() set
the multF of G . [N,f] is set
g is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of B:]
g . (N * f) is Element of the carrier of B
g . N is Element of the carrier of B
g . f is Element of the carrier of B
(g . N) * (g . f) is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . ((g . N),(g . f)) is Element of the carrier of B
[(g . N),(g . f)] is set
{(g . N),(g . f)} is finite set
{(g . N)} is finite set
{{(g . N),(g . f)},{(g . N)}} is finite V56() set
the multF of B . [(g . N),(g . f)] is set
(1_ B) * (1_ B) is Element of the carrier of B
the multF of B . ((1_ B),(1_ B)) is Element of the carrier of B
[(1_ B),(1_ B)] is set
{(1_ B),(1_ B)} is finite set
{(1_ B)} is finite set
{{(1_ B),(1_ B)},{(1_ B)}} is finite V56() set
the multF of B . [(1_ B),(1_ B)] is set
(g . N) * (1_ B) is Element of the carrier of B
the multF of B . ((g . N),(1_ B)) is Element of the carrier of B
[(g . N),(1_ B)] is set
{(g . N),(1_ B)} is finite set
{{(g . N),(1_ B)},{(g . N)}} is finite V56() set
the multF of B . [(g . N),(1_ B)] is set
G is non empty multMagma
the carrier of G is non empty set
B is non empty multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
1_ B is non being_of_order_0 Element of the carrier of B
the carrier of G --> (1_ B) is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of B:]
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of B:]
f is Element of the carrier of G
N . f is Element of the carrier of B
f is Element of the carrier of G
g is Element of the carrier of G
f * g 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (f,g) is Element of the carrier of G
[f,g] is set
{f,g} is finite set
{f} is finite set
{{f,g},{f}} is finite V56() set
the multF of G . [f,g] is set
N . (f * g) is Element of the carrier of B
N . f is Element of the carrier of B
N . g is Element of the carrier of B
(N . f) * (N . g) is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . ((N . f),(N . g)) is Element of the carrier of B
[(N . f),(N . g)] is set
{(N . f),(N . g)} is finite set
{(N . f)} is finite set
{{(N . f),(N . g)},{(N . f)}} is finite V56() set
the multF of B . [(N . f),(N . g)] is set
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
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
1_ B is non being_of_order_0 Element of the carrier of B
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total (G,B) Element of bool [: the carrier of G, the carrier of B:]
N . (1_ G) is Element of the carrier of B
(1_ G) * (1_ G) 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((1_ G),(1_ G)) is Element of the carrier of G
[(1_ G),(1_ G)] is set
{(1_ G),(1_ G)} is finite set
{(1_ G)} is finite set
{{(1_ G),(1_ G)},{(1_ G)}} is finite V56() set
the multF of G . [(1_ G),(1_ G)] is set
N . ((1_ G) * (1_ G)) is Element of the carrier of B
(N . (1_ G)) * (N . (1_ G)) is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . ((N . (1_ G)),(N . (1_ G))) is Element of the carrier of B
[(N . (1_ G)),(N . (1_ G))] is set
{(N . (1_ G)),(N . (1_ G))} is finite set
{(N . (1_ G))} is finite set
{{(N . (1_ G)),(N . (1_ G))},{(N . (1_ G))}} is finite V56() set
the multF of B . [(N . (1_ G)),(N . (1_ G))] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total (G,B) Element of bool [: the carrier of G, the carrier of B:]
1_ G is non being_of_order_0 Element of the carrier of G
N . (1_ G) is Element of the carrier of B
1_ B is non being_of_order_0 Element of the carrier of B
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Element of the carrier of G
N " is Element of the carrier of G
f is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
f . (N ") is Element of the carrier of B
f . N is Element of the carrier of B
(f . N) " is Element of the carrier of B
(f . (N ")) * (f . N) is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . ((f . (N ")),(f . N)) is Element of the carrier of B
[(f . (N ")),(f . N)] is set
{(f . (N ")),(f . N)} is finite set
{(f . (N "))} is finite set
{{(f . (N ")),(f . N)},{(f . (N "))}} is finite V56() set
the multF of B . [(f . (N ")),(f . N)] is set
(N ") * N 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((N "),N) is Element of the carrier of G
[(N "),N] is set
{(N "),N} is finite set
{(N ")} is finite set
{{(N "),N},{(N ")}} is finite V56() set
the multF of G . [(N "),N] is set
f . ((N ") * N) is Element of the carrier of B
1_ G is non being_of_order_0 Element of the carrier of G
f . (1_ G) is Element of the carrier of B
1_ B is non being_of_order_0 Element of the carrier of B
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Element of the carrier of G
f is Element of the carrier of G
N |^ f is Element of the carrier of G
g is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
g . (N |^ f) is Element of the carrier of B
g . N is Element of the carrier of B
g . f is Element of the carrier of B
(g . N) |^ (g . f) is Element of the carrier of B
f " is Element of the carrier of G
(f ") * N 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((f "),N) is Element of the carrier of G
[(f "),N] is set
{(f "),N} is finite set
{(f ")} is finite set
{{(f "),N},{(f ")}} is finite V56() set
the multF of G . [(f "),N] is set
((f ") * N) * f is Element of the carrier of G
the multF of G . (((f ") * N),f) is Element of the carrier of G
[((f ") * N),f] is set
{((f ") * N),f} is finite set
{((f ") * N)} is finite set
{{((f ") * N),f},{((f ") * N)}} is finite V56() set
the multF of G . [((f ") * N),f] is set
g . (((f ") * N) * f) is Element of the carrier of B
g . ((f ") * N) is Element of the carrier of B
(g . ((f ") * N)) * (g . f) is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . ((g . ((f ") * N)),(g . f)) is Element of the carrier of B
[(g . ((f ") * N)),(g . f)] is set
{(g . ((f ") * N)),(g . f)} is finite set
{(g . ((f ") * N))} is finite set
{{(g . ((f ") * N)),(g . f)},{(g . ((f ") * N))}} is finite V56() set
the multF of B . [(g . ((f ") * N)),(g . f)] is set
g . (f ") is Element of the carrier of B
(g . (f ")) * (g . N) is Element of the carrier of B
the multF of B . ((g . (f ")),(g . N)) is Element of the carrier of B
[(g . (f ")),(g . N)] is set
{(g . (f ")),(g . N)} is finite set
{(g . (f "))} is finite set
{{(g . (f ")),(g . N)},{(g . (f "))}} is finite V56() set
the multF of B . [(g . (f ")),(g . N)] is set
((g . (f ")) * (g . N)) * (g . f) is Element of the carrier of B
the multF of B . (((g . (f ")) * (g . N)),(g . f)) is Element of the carrier of B
[((g . (f ")) * (g . N)),(g . f)] is set
{((g . (f ")) * (g . N)),(g . f)} is finite set
{((g . (f ")) * (g . N))} is finite set
{{((g . (f ")) * (g . N)),(g . f)},{((g . (f ")) * (g . N))}} is finite V56() set
the multF of B . [((g . (f ")) * (g . N)),(g . f)] is set
(g . f) " is Element of the carrier of B
((g . f) ") * (g . N) is Element of the carrier of B
the multF of B . (((g . f) "),(g . N)) is Element of the carrier of B
[((g . f) "),(g . N)] is set
{((g . f) "),(g . N)} is finite set
{((g . f) ")} is finite set
{{((g . f) "),(g . N)},{((g . f) ")}} is finite V56() set
the multF of B . [((g . f) "),(g . N)] is set
(((g . f) ") * (g . N)) * (g . f) is Element of the carrier of B
the multF of B . ((((g . f) ") * (g . N)),(g . f)) is Element of the carrier of B
[(((g . f) ") * (g . N)),(g . f)] is set
{(((g . f) ") * (g . N)),(g . f)} is finite set
{(((g . f) ") * (g . N))} is finite set
{{(((g . f) ") * (g . N)),(g . f)},{(((g . f) ") * (g . N))}} is finite V56() set
the multF of B . [(((g . f) ") * (g . N)),(g . f)] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Element of the carrier of G
f is Element of the carrier of G
[.N,f.] is Element of the carrier of G
g is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
g . [.N,f.] is Element of the carrier of B
g . N is Element of the carrier of B
g . f is Element of the carrier of B
[.(g . N),(g . f).] is Element of the carrier of B
N " is Element of the carrier of G
f " is Element of the carrier of G
(N ") * (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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((N "),(f ")) is Element of the carrier of G
[(N "),(f ")] is set
{(N "),(f ")} is finite set
{(N ")} is finite set
{{(N "),(f ")},{(N ")}} is finite V56() set
the multF of G . [(N "),(f ")] is set
((N ") * (f ")) * N is Element of the carrier of G
the multF of G . (((N ") * (f ")),N) is Element of the carrier of G
[((N ") * (f ")),N] is set
{((N ") * (f ")),N} is finite set
{((N ") * (f "))} is finite set
{{((N ") * (f ")),N},{((N ") * (f "))}} is finite V56() set
the multF of G . [((N ") * (f ")),N] is set
(((N ") * (f ")) * N) * f is Element of the carrier of G
the multF of G . ((((N ") * (f ")) * N),f) is Element of the carrier of G
[(((N ") * (f ")) * N),f] is set
{(((N ") * (f ")) * N),f} is finite set
{(((N ") * (f ")) * N)} is finite set
{{(((N ") * (f ")) * N),f},{(((N ") * (f ")) * N)}} is finite V56() set
the multF of G . [(((N ") * (f ")) * N),f] is set
g . ((((N ") * (f ")) * N) * f) is Element of the carrier of B
g . (((N ") * (f ")) * N) is Element of the carrier of B
(g . (((N ") * (f ")) * N)) * (g . f) is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . ((g . (((N ") * (f ")) * N)),(g . f)) is Element of the carrier of B
[(g . (((N ") * (f ")) * N)),(g . f)] is set
{(g . (((N ") * (f ")) * N)),(g . f)} is finite set
{(g . (((N ") * (f ")) * N))} is finite set
{{(g . (((N ") * (f ")) * N)),(g . f)},{(g . (((N ") * (f ")) * N))}} is finite V56() set
the multF of B . [(g . (((N ") * (f ")) * N)),(g . f)] is set
g . ((N ") * (f ")) is Element of the carrier of B
(g . ((N ") * (f "))) * (g . N) is Element of the carrier of B
the multF of B . ((g . ((N ") * (f "))),(g . N)) is Element of the carrier of B
[(g . ((N ") * (f "))),(g . N)] is set
{(g . ((N ") * (f "))),(g . N)} is finite set
{(g . ((N ") * (f ")))} is finite set
{{(g . ((N ") * (f "))),(g . N)},{(g . ((N ") * (f ")))}} is finite V56() set
the multF of B . [(g . ((N ") * (f "))),(g . N)] is set
((g . ((N ") * (f "))) * (g . N)) * (g . f) is Element of the carrier of B
the multF of B . (((g . ((N ") * (f "))) * (g . N)),(g . f)) is Element of the carrier of B
[((g . ((N ") * (f "))) * (g . N)),(g . f)] is set
{((g . ((N ") * (f "))) * (g . N)),(g . f)} is finite set
{((g . ((N ") * (f "))) * (g . N))} is finite set
{{((g . ((N ") * (f "))) * (g . N)),(g . f)},{((g . ((N ") * (f "))) * (g . N))}} is finite V56() set
the multF of B . [((g . ((N ") * (f "))) * (g . N)),(g . f)] is set
g . (N ") is Element of the carrier of B
g . (f ") is Element of the carrier of B
(g . (N ")) * (g . (f ")) is Element of the carrier of B
the multF of B . ((g . (N ")),(g . (f "))) is Element of the carrier of B
[(g . (N ")),(g . (f "))] is set
{(g . (N ")),(g . (f "))} is finite set
{(g . (N "))} is finite set
{{(g . (N ")),(g . (f "))},{(g . (N "))}} is finite V56() set
the multF of B . [(g . (N ")),(g . (f "))] is set
((g . (N ")) * (g . (f "))) * (g . N) is Element of the carrier of B
the multF of B . (((g . (N ")) * (g . (f "))),(g . N)) is Element of the carrier of B
[((g . (N ")) * (g . (f "))),(g . N)] is set
{((g . (N ")) * (g . (f "))),(g . N)} is finite set
{((g . (N ")) * (g . (f ")))} is finite set
{{((g . (N ")) * (g . (f "))),(g . N)},{((g . (N ")) * (g . (f ")))}} is finite V56() set
the multF of B . [((g . (N ")) * (g . (f "))),(g . N)] is set
(((g . (N ")) * (g . (f "))) * (g . N)) * (g . f) is Element of the carrier of B
the multF of B . ((((g . (N ")) * (g . (f "))) * (g . N)),(g . f)) is Element of the carrier of B
[(((g . (N ")) * (g . (f "))) * (g . N)),(g . f)] is set
{(((g . (N ")) * (g . (f "))) * (g . N)),(g . f)} is finite set
{(((g . (N ")) * (g . (f "))) * (g . N))} is finite set
{{(((g . (N ")) * (g . (f "))) * (g . N)),(g . f)},{(((g . (N ")) * (g . (f "))) * (g . N))}} is finite V56() set
the multF of B . [(((g . (N ")) * (g . (f "))) * (g . N)),(g . f)] is set
(g . N) " is Element of the carrier of B
((g . N) ") * (g . (f ")) is Element of the carrier of B
the multF of B . (((g . N) "),(g . (f "))) is Element of the carrier of B
[((g . N) "),(g . (f "))] is set
{((g . N) "),(g . (f "))} is finite set
{((g . N) ")} is finite set
{{((g . N) "),(g . (f "))},{((g . N) ")}} is finite V56() set
the multF of B . [((g . N) "),(g . (f "))] is set
(((g . N) ") * (g . (f "))) * (g . N) is Element of the carrier of B
the multF of B . ((((g . N) ") * (g . (f "))),(g . N)) is Element of the carrier of B
[(((g . N) ") * (g . (f "))),(g . N)] is set
{(((g . N) ") * (g . (f "))),(g . N)} is finite set
{(((g . N) ") * (g . (f ")))} is finite set
{{(((g . N) ") * (g . (f "))),(g . N)},{(((g . N) ") * (g . (f ")))}} is finite V56() set
the multF of B . [(((g . N) ") * (g . (f "))),(g . N)] is set
((((g . N) ") * (g . (f "))) * (g . N)) * (g . f) is Element of the carrier of B
the multF of B . (((((g . N) ") * (g . (f "))) * (g . N)),(g . f)) is Element of the carrier of B
[((((g . N) ") * (g . (f "))) * (g . N)),(g . f)] is set
{((((g . N) ") * (g . (f "))) * (g . N)),(g . f)} is finite set
{((((g . N) ") * (g . (f "))) * (g . N))} is finite set
{{((((g . N) ") * (g . (f "))) * (g . N)),(g . f)},{((((g . N) ") * (g . (f "))) * (g . N))}} is finite V56() set
the multF of B . [((((g . N) ") * (g . (f "))) * (g . N)),(g . f)] is set
(g . f) " is Element of the carrier of B
((g . N) ") * ((g . f) ") is Element of the carrier of B
the multF of B . (((g . N) "),((g . f) ")) is Element of the carrier of B
[((g . N) "),((g . f) ")] is set
{((g . N) "),((g . f) ")} is finite set
{{((g . N) "),((g . f) ")},{((g . N) ")}} is finite V56() set
the multF of B . [((g . N) "),((g . f) ")] is set
(((g . N) ") * ((g . f) ")) * (g . N) is Element of the carrier of B
the multF of B . ((((g . N) ") * ((g . f) ")),(g . N)) is Element of the carrier of B
[(((g . N) ") * ((g . f) ")),(g . N)] is set
{(((g . N) ") * ((g . f) ")),(g . N)} is finite set
{(((g . N) ") * ((g . f) "))} is finite set
{{(((g . N) ") * ((g . f) ")),(g . N)},{(((g . N) ") * ((g . f) "))}} is finite V56() set
the multF of B . [(((g . N) ") * ((g . f) ")),(g . N)] is set
((((g . N) ") * ((g . f) ")) * (g . N)) * (g . f) is Element of the carrier of B
the multF of B . (((((g . N) ") * ((g . f) ")) * (g . N)),(g . f)) is Element of the carrier of B
[((((g . N) ") * ((g . f) ")) * (g . N)),(g . f)] is set
{((((g . N) ") * ((g . f) ")) * (g . N)),(g . f)} is finite set
{((((g . N) ") * ((g . f) ")) * (g . N))} is finite set
{{((((g . N) ") * ((g . f) ")) * (g . N)),(g . f)},{((((g . N) ") * ((g . f) ")) * (g . N))}} is finite V56() set
the multF of B . [((((g . N) ") * ((g . f) ")) * (g . N)),(g . f)] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Element of the carrier of G
f is Element of the carrier of G
g is Element of the carrier of G
[.N,f,g.] is Element of the carrier of G
I is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
I . [.N,f,g.] is Element of the carrier of B
I . N is Element of the carrier of B
I . f is Element of the carrier of B
I . g is Element of the carrier of B
[.(I . N),(I . f),(I . g).] is Element of the carrier of B
[.N,f.] is Element of the carrier of G
[.[.N,f.],g.] is Element of the carrier of G
I . [.[.N,f.],g.] is Element of the carrier of B
I . [.N,f.] is Element of the carrier of B
[.(I . [.N,f.]),(I . g).] is Element of the carrier of B
[.(I . N),(I . f).] is Element of the carrier of B
[.[.(I . N),(I . f).],(I . g).] is Element of the carrier of B
G is ext-real V41() V42() V43() V47() V48() V49() integer Element of NAT
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
N is non empty unital Group-like associative multMagma
the carrier of N is non empty set
[: the carrier of B, the carrier of N:] is set
bool [: the carrier of B, the carrier of N:] is set
f is Element of the carrier of B
f |^ G is Element of the carrier of B
power B is Relation-like [: the carrier of B,NAT:] -defined the carrier of B -valued Function-like V22([: the carrier of B,NAT:]) quasi_total Element of bool [:[: the carrier of B,NAT:], the carrier of B:]
[: the carrier of B,NAT:] is set
[:[: the carrier of B,NAT:], the carrier of B:] is set
bool [:[: the carrier of B,NAT:], the carrier of B:] is set
(power B) . (f,G) is set
[f,G] is set
{f,G} is finite set
{f} is finite set
{{f,G},{f}} is finite V56() set
(power B) . [f,G] is set
g is Relation-like the carrier of B -defined the carrier of N -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,N) Element of bool [: the carrier of B, the carrier of N:]
g . (f |^ G) is Element of the carrier of N
g . f is Element of the carrier of N
(g . f) |^ G is Element of the carrier of N
power N is Relation-like [: the carrier of N,NAT:] -defined the carrier of N -valued Function-like V22([: the carrier of N,NAT:]) quasi_total Element of bool [:[: the carrier of N,NAT:], the carrier of N:]
[: the carrier of N,NAT:] is set
[:[: the carrier of N,NAT:], the carrier of N:] is set
bool [:[: the carrier of N,NAT:], the carrier of N:] is set
(power N) . ((g . f),G) is set
[(g . f),G] is set
{(g . f),G} is finite set
{(g . f)} is finite set
{{(g . f),G},{(g . f)}} is finite V56() set
(power N) . [(g . f),G] is set
I is ext-real V41() V42() V43() V47() V48() V49() integer Element of NAT
f |^ I is Element of the carrier of B
(power B) . (f,I) is set
[f,I] is set
{f,I} is finite set
{{f,I},{f}} is finite V56() set
(power B) . [f,I] is set
g . (f |^ I) is Element of the carrier of N
(g . f) |^ I is Element of the carrier of N
(power N) . ((g . f),I) is set
[(g . f),I] is set
{(g . f),I} is finite set
{{(g . f),I},{(g . f)}} is finite V56() set
(power N) . [(g . f),I] is set
I + 1 is ext-real V41() V42() V43() V47() V48() V49() integer Element of NAT
f |^ (I + 1) is Element of the carrier of B
(power B) . (f,(I + 1)) is set
[f,(I + 1)] is set
{f,(I + 1)} is finite set
{{f,(I + 1)},{f}} is finite V56() set
(power B) . [f,(I + 1)] is set
g . (f |^ (I + 1)) is Element of the carrier of N
(g . f) |^ (I + 1) is Element of the carrier of N
(power N) . ((g . f),(I + 1)) is set
[(g . f),(I + 1)] is set
{(g . f),(I + 1)} is finite set
{{(g . f),(I + 1)},{(g . f)}} is finite V56() set
(power N) . [(g . f),(I + 1)] is set
(f |^ I) * f is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . ((f |^ I),f) is Element of the carrier of B
[(f |^ I),f] is set
{(f |^ I),f} is finite set
{(f |^ I)} is finite set
{{(f |^ I),f},{(f |^ I)}} is finite V56() set
the multF of B . [(f |^ I),f] is set
g . ((f |^ I) * f) is Element of the carrier of N
((g . f) |^ I) * (g . f) is Element of the carrier of N
the multF of N is Relation-like [: the carrier of N, the carrier of N:] -defined the carrier of N -valued Function-like V22([: the carrier of N, the carrier of N:]) quasi_total associative Element of bool [:[: the carrier of N, the carrier of N:], the carrier of N:]
[: the carrier of N, the carrier of N:] is set
[:[: the carrier of N, the carrier of N:], the carrier of N:] is set
bool [:[: the carrier of N, the carrier of N:], the carrier of N:] is set
the multF of N . (((g . f) |^ I),(g . f)) is Element of the carrier of N
[((g . f) |^ I),(g . f)] is set
{((g . f) |^ I),(g . f)} is finite set
{((g . f) |^ I)} is finite set
{{((g . f) |^ I),(g . f)},{((g . f) |^ I)}} is finite V56() set
the multF of N . [((g . f) |^ I),(g . f)] is set
f |^ 0 is Element of the carrier of B
(power B) . (f,0) is set
[f,0] is set
{f,0} is finite set
{{f,0},{f}} is finite V56() set
(power B) . [f,0] is set
g . (f |^ 0) is Element of the carrier of N
1_ B is non being_of_order_0 Element of the carrier of B
g . (1_ B) is Element of the carrier of N
1_ N is non being_of_order_0 Element of the carrier of N
(g . f) |^ 0 is Element of the carrier of N
(power N) . ((g . f),0) is set
[(g . f),0] is set
{(g . f),0} is finite set
{{(g . f),0},{(g . f)}} is finite V56() set
(power N) . [(g . f),0] is set
G is ext-real V48() V49() integer set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
N is non empty unital Group-like associative multMagma
the carrier of N is non empty set
[: the carrier of B, the carrier of N:] is set
bool [: the carrier of B, the carrier of N:] is set
f is Element of the carrier of B
f |^ G is Element of the carrier of B
g is Relation-like the carrier of B -defined the carrier of N -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,N) Element of bool [: the carrier of B, the carrier of N:]
g . (f |^ G) is Element of the carrier of N
g . f is Element of the carrier of N
(g . f) |^ G is Element of the carrier of N
abs G is ext-real V41() V42() V43() V47() V48() V49() integer Element of NAT
f |^ (abs G) is Element of the carrier of B
power B is Relation-like [: the carrier of B,NAT:] -defined the carrier of B -valued Function-like V22([: the carrier of B,NAT:]) quasi_total Element of bool [:[: the carrier of B,NAT:], the carrier of B:]
[: the carrier of B,NAT:] is set
[:[: the carrier of B,NAT:], the carrier of B:] is set
bool [:[: the carrier of B,NAT:], the carrier of B:] is set
(power B) . (f,(abs G)) is set
[f,(abs G)] is set
{f,(abs G)} is finite set
{f} is finite set
{{f,(abs G)},{f}} is finite V56() set
(power B) . [f,(abs G)] is set
g . (f |^ (abs G)) is Element of the carrier of N
(g . f) |^ (abs G) is Element of the carrier of N
power N is Relation-like [: the carrier of N,NAT:] -defined the carrier of N -valued Function-like V22([: the carrier of N,NAT:]) quasi_total Element of bool [:[: the carrier of N,NAT:], the carrier of N:]
[: the carrier of N,NAT:] is set
[:[: the carrier of N,NAT:], the carrier of N:] is set
bool [:[: the carrier of N,NAT:], the carrier of N:] is set
(power N) . ((g . f),(abs G)) is set
[(g . f),(abs G)] is set
{(g . f),(abs G)} is finite set
{(g . f)} is finite set
{{(g . f),(abs G)},{(g . f)}} is finite V56() set
(power N) . [(g . f),(abs G)] is set
abs G is ext-real V41() V42() V43() V47() V48() V49() integer Element of NAT
f |^ (abs G) is Element of the carrier of B
power B is Relation-like [: the carrier of B,NAT:] -defined the carrier of B -valued Function-like V22([: the carrier of B,NAT:]) quasi_total Element of bool [:[: the carrier of B,NAT:], the carrier of B:]
[: the carrier of B,NAT:] is set
[:[: the carrier of B,NAT:], the carrier of B:] is set
bool [:[: the carrier of B,NAT:], the carrier of B:] is set
(power B) . (f,(abs G)) is set
[f,(abs G)] is set
{f,(abs G)} is finite set
{f} is finite set
{{f,(abs G)},{f}} is finite V56() set
(power B) . [f,(abs G)] is set
(f |^ (abs G)) " is Element of the carrier of B
g . ((f |^ (abs G)) ") is Element of the carrier of N
g . (f |^ (abs G)) is Element of the carrier of N
(g . (f |^ (abs G))) " is Element of the carrier of N
(g . f) |^ (abs G) is Element of the carrier of N
power N is Relation-like [: the carrier of N,NAT:] -defined the carrier of N -valued Function-like V22([: the carrier of N,NAT:]) quasi_total Element of bool [:[: the carrier of N,NAT:], the carrier of N:]
[: the carrier of N,NAT:] is set
[:[: the carrier of N,NAT:], the carrier of N:] is set
bool [:[: the carrier of N,NAT:], the carrier of N:] is set
(power N) . ((g . f),(abs G)) is set
[(g . f),(abs G)] is set
{(g . f),(abs G)} is finite set
{(g . f)} is finite set
{{(g . f),(abs G)},{(g . f)}} is finite V56() set
(power N) . [(g . f),(abs G)] is set
((g . f) |^ (abs G)) " is Element of the carrier of N
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
id the carrier of G is Relation-like the carrier of G -defined the carrier of G -valued V6() V8() V9() V13() Function-like one-to-one non empty V22( the carrier of G) quasi_total onto bijective Element of bool [: the carrier of G, the carrier of G:]
[: the carrier of G, the carrier of G:] is set
bool [: the carrier of G, the carrier of G:] is set
N is Element of the carrier of G
(id the carrier of G) . N is Element of the carrier of G
f is Element of the carrier of G
N * 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (N,f) is Element of the carrier of G
[N,f] is set
{N,f} is finite set
{N} is finite set
{{N,f},{N}} is finite V56() set
the multF of G . [N,f] is set
(id the carrier of G) . (N * f) is Element of the carrier of G
(id the carrier of G) . f is Element of the carrier of G
((id the carrier of G) . N) * ((id the carrier of G) . f) is Element of the carrier of G
the multF of G . (((id the carrier of G) . N),((id the carrier of G) . f)) is Element of the carrier of G
[((id the carrier of G) . N),((id the carrier of G) . f)] is set
{((id the carrier of G) . N),((id the carrier of G) . f)} is finite set
{((id the carrier of G) . N)} is finite set
{{((id the carrier of G) . N),((id the carrier of G) . f)},{((id the carrier of G) . N)}} is finite V56() set
the multF of G . [((id the carrier of G) . N),((id the carrier of G) . f)] is set
((id the carrier of G) . N) * f is Element of the carrier of G
the multF of G . (((id the carrier of G) . N),f) is Element of the carrier of G
[((id the carrier of G) . N),f] is set
{((id the carrier of G) . N),f} is finite set
{{((id the carrier of G) . N),f},{((id the carrier of G) . N)}} is finite V56() set
the multF of G . [((id the carrier of G) . N),f] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is non empty unital Group-like associative multMagma
the carrier of N is non empty set
[: the carrier of B, the carrier of N:] is set
bool [: the carrier of B, the carrier of N:] is set
f is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
g is Relation-like the carrier of B -defined the carrier of N -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,N) Element of bool [: the carrier of B, the carrier of N:]
g * f is Relation-like the carrier of G -defined the carrier of N -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of N:]
[: the carrier of G, the carrier of N:] is set
bool [: the carrier of G, the carrier of N:] is set
J is Element of the carrier of G
(g * f) . J is Element of the carrier of N
g is Element of the carrier of G
J * g 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (J,g) is Element of the carrier of G
[J,g] is set
{J,g} is finite set
{J} is finite set
{{J,g},{J}} is finite V56() set
the multF of G . [J,g] is set
(g * f) . (J * g) is Element of the carrier of N
(g * f) . g is Element of the carrier of N
((g * f) . J) * ((g * f) . g) is Element of the carrier of N
the multF of N is Relation-like [: the carrier of N, the carrier of N:] -defined the carrier of N -valued Function-like V22([: the carrier of N, the carrier of N:]) quasi_total associative Element of bool [:[: the carrier of N, the carrier of N:], the carrier of N:]
[: the carrier of N, the carrier of N:] is set
[:[: the carrier of N, the carrier of N:], the carrier of N:] is set
bool [:[: the carrier of N, the carrier of N:], the carrier of N:] is set
the multF of N . (((g * f) . J),((g * f) . g)) is Element of the carrier of N
[((g * f) . J),((g * f) . g)] is set
{((g * f) . J),((g * f) . g)} is finite set
{((g * f) . J)} is finite set
{{((g * f) . J),((g * f) . g)},{((g * f) . J)}} is finite V56() set
the multF of N . [((g * f) . J),((g * f) . g)] is set
f . (J * g) is Element of the carrier of B
g . (f . (J * g)) is Element of the carrier of N
f . J is Element of the carrier of B
f . g is Element of the carrier of B
(f . J) * (f . g) is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . ((f . J),(f . g)) is Element of the carrier of B
[(f . J),(f . g)] is set
{(f . J),(f . g)} is finite set
{(f . J)} is finite set
{{(f . J),(f . g)},{(f . J)}} is finite V56() set
the multF of B . [(f . J),(f . g)] is set
g . ((f . J) * (f . g)) is Element of the carrier of N
g . (f . J) is Element of the carrier of N
g . (f . g) is Element of the carrier of N
(g . (f . J)) * (g . (f . g)) is Element of the carrier of N
the multF of N . ((g . (f . J)),(g . (f . g))) is Element of the carrier of N
[(g . (f . J)),(g . (f . g))] is set
{(g . (f . J)),(g . (f . g))} is finite set
{(g . (f . J))} is finite set
{{(g . (f . J)),(g . (f . g))},{(g . (f . J))}} is finite V56() set
the multF of N . [(g . (f . J)),(g . (f . g))] is set
((g * f) . J) * (g . (f . g)) is Element of the carrier of N
the multF of N . (((g * f) . J),(g . (f . g))) is Element of the carrier of N
[((g * f) . J),(g . (f . g))] is set
{((g * f) . J),(g . (f . g))} is finite set
{{((g * f) . J),(g . (f . g))},{((g * f) . J)}} is finite V56() set
the multF of N . [((g * f) . J),(g . (f . g))] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is non empty unital Group-like associative multMagma
the carrier of N is non empty set
[: the carrier of B, the carrier of N:] is set
bool [: the carrier of B, the carrier of N:] is set
[: the carrier of G, the carrier of N:] is set
bool [: the carrier of G, the carrier of N:] is set
f is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
g is Relation-like the carrier of B -defined the carrier of N -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,N) Element of bool [: the carrier of B, the carrier of N:]
g * f is Relation-like the carrier of G -defined the carrier of N -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of N:]
I is Relation-like the carrier of G -defined the carrier of N -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of N:]
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
1_ B is non being_of_order_0 Element of the carrier of B
the carrier of G --> (1_ B) is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of B:]
N is Element of the carrier of G
( the carrier of G --> (1_ B)) . N is Element of the carrier of B
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of B:]
f is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of B:]
g is Element of the carrier of G
N . g is Element of the carrier of B
f . g is Element of the carrier of B
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative multMagma
(G,B) is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of B:]
the carrier of G is non empty set
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
f is Element of the carrier of G
(G,B) . f is Element of the carrier of B
g is Element of the carrier of G
f * g 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (f,g) is Element of the carrier of G
[f,g] is set
{f,g} is finite set
{f} is finite set
{{f,g},{f}} is finite V56() set
the multF of G . [f,g] is set
(G,B) . (f * g) is Element of the carrier of B
(G,B) . g is Element of the carrier of B
((G,B) . f) * ((G,B) . g) is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (((G,B) . f),((G,B) . g)) is Element of the carrier of B
[((G,B) . f),((G,B) . g)] is set
{((G,B) . f),((G,B) . g)} is finite set
{((G,B) . f)} is finite set
{{((G,B) . f),((G,B) . g)},{((G,B) . f)}} is finite V56() set
the multF of B . [((G,B) . f),((G,B) . g)] is set
1_ B is non being_of_order_0 Element of the carrier of B
I is Element of the carrier of G
(G,B) . I is Element of the carrier of B
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
(G,B) is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
N is non empty unital Group-like associative multMagma
the carrier of N is non empty set
[: the carrier of B, the carrier of N:] is set
bool [: the carrier of B, the carrier of N:] is set
(G,N) is Relation-like the carrier of G -defined the carrier of N -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,N) Element of bool [: the carrier of G, the carrier of N:]
[: the carrier of G, the carrier of N:] is set
bool [: the carrier of G, the carrier of N:] is set
(B,N) is Relation-like the carrier of B -defined the carrier of N -valued Function-like non empty V22( the carrier of B) quasi_total unity-preserving (B,N) Element of bool [: the carrier of B, the carrier of N:]
f is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
(B,N) * f is Relation-like the carrier of G -defined the carrier of G -defined the carrier of N -valued the carrier of N -valued Function-like non empty V22( the carrier of G) V22( the carrier of G) quasi_total quasi_total unity-preserving (G,N) Element of bool [: the carrier of G, the carrier of N:]
g is Relation-like the carrier of B -defined the carrier of N -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,N) Element of bool [: the carrier of B, the carrier of N:]
g * (G,B) is Relation-like the carrier of G -defined the carrier of G -defined the carrier of N -valued the carrier of N -valued Function-like non empty V22( the carrier of G) V22( the carrier of G) quasi_total quasi_total unity-preserving (G,N) Element of bool [: the carrier of G, the carrier of N:]
I is Element of the carrier of G
((B,N) * f) . I is Element of the carrier of N
f . I is Element of the carrier of B
(B,N) . (f . I) is Element of the carrier of N
1_ N is non being_of_order_0 Element of the carrier of N
I is Element of the carrier of G
(g * (G,B)) . I is Element of the carrier of N
(G,B) . I is Element of the carrier of B
g . ((G,B) . I) is Element of the carrier of N
1_ B is non being_of_order_0 Element of the carrier of B
g . (1_ B) is Element of the carrier of N
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict unital Group-like associative multMagma
Left_Cosets B is non empty Element of bool (bool the carrier of G)
bool the carrier of G is set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
the carrier of (G,B) is non empty set
[: the carrier of G, the carrier of (G,B):] is set
bool [: the carrier of G, the carrier of (G,B):] is set
N is set
f is Element of the carrier of G
f * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
the carrier of B is non empty set
f * (carr B) is Element of bool the carrier of G
K245( the carrier of G,f) is finite Element of bool the carrier of G
K245( the carrier of G,f) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,f) & b₂ in carr B ) } is set
g is set
N is Relation-like Function-like set
dom N is set
rng N is set
f is set
g is set
N . g is set
I is Element of the carrier of G
I * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
the carrier of B is non empty set
I * (carr B) is Element of bool the carrier of G
K245( the carrier of G,I) is finite Element of bool the carrier of G
K245( the carrier of G,I) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in carr B ) } is set
B * I is Element of bool the carrier of G
(carr B) * I is Element of bool the carrier of G
(carr B) * K245( the carrier of G,I) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,I) ) } is set
f is Relation-like the carrier of G -defined the carrier of (G,B) -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of (G,B):]
g is Element of the carrier of G
f . g is Element of the carrier of (G,B)
g * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
the carrier of B is non empty set
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
I is Element of the carrier of G
I * B is Element of bool the carrier of G
I * (carr B) is Element of bool the carrier of G
K245( the carrier of G,I) is finite Element of bool the carrier of G
K245( the carrier of G,I) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in carr B ) } is set
N is Relation-like the carrier of G -defined the carrier of (G,B) -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of (G,B):]
f is Relation-like the carrier of G -defined the carrier of (G,B) -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of (G,B):]
g is Element of the carrier of G
N . g is Element of the carrier of (G,B)
g * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
the carrier of B is non empty set
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
f . g is Element of the carrier of (G,B)
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict unital Group-like associative multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
(G,B) is Relation-like the carrier of G -defined the carrier of (G,B) -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of (G,B):]
the carrier of (G,B) is non empty set
[: the carrier of G, the carrier of (G,B):] is set
bool [: the carrier of G, the carrier of (G,B):] is set
f is Element of the carrier of G
(G,B) . f is Element of the carrier of (G,B)
g is Element of the carrier of G
f * g 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (f,g) is Element of the carrier of G
[f,g] is set
{f,g} is finite set
{f} is finite set
{{f,g},{f}} is finite V56() set
the multF of G . [f,g] is set
(G,B) . (f * g) is Element of the carrier of (G,B)
(G,B) . g is Element of the carrier of (G,B)
((G,B) . f) * ((G,B) . g) is Element of the carrier of (G,B)
the multF of (G,B) is Relation-like [: the carrier of (G,B), the carrier of (G,B):] -defined the carrier of (G,B) -valued Function-like V22([: the carrier of (G,B), the carrier of (G,B):]) quasi_total associative Element of bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):]
[: the carrier of (G,B), the carrier of (G,B):] is set
[:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
the multF of (G,B) . (((G,B) . f),((G,B) . g)) is Element of the carrier of (G,B)
[((G,B) . f),((G,B) . g)] is set
{((G,B) . f),((G,B) . g)} is finite set
{((G,B) . f)} is finite set
{{((G,B) . f),((G,B) . g)},{((G,B) . f)}} is finite V56() set
the multF of (G,B) . [((G,B) . f),((G,B) . g)] is set
f * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
the carrier of B is non empty set
f * (carr B) is Element of bool the carrier of G
K245( the carrier of G,f) is finite Element of bool the carrier of G
K245( the carrier of G,f) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,f) & b₂ in carr B ) } is set
g * B is Element of bool the carrier of G
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
(f * g) * B is Element of bool the carrier of G
(f * g) * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(f * g)) is finite Element of bool the carrier of G
K245( the carrier of G,(f * g)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(f * g)) & b₂ in carr B ) } is set
(G,B,((G,B) . f)) is Element of bool the carrier of G
(G,B,((G,B) . g)) is Element of bool the carrier of G
(G,B,((G,B) . f)) * (G,B,((G,B) . g)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,B,((G,B) . f)) & b₂ in (G,B,((G,B) . g)) ) } is set
(f * B) * g is Element of bool the carrier of G
(f * B) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in f * B & b₂ in K245( the carrier of G,g) ) } is set
((f * B) * g) * B is Element of bool the carrier of G
((f * B) * g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (f * B) * g & b₂ in carr B ) } is set
B * g is Element of bool the carrier of G
(carr B) * g is Element of bool the carrier of G
(carr B) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,g) ) } is set
f * (B * g) is Element of bool the carrier of G
K245( the carrier of G,f) * (B * g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,f) & b₂ in B * g ) } is set
(f * (B * g)) * B is Element of bool the carrier of G
(f * (B * g)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in f * (B * g) & b₂ in carr B ) } is set
f * (g * B) is Element of bool the carrier of G
K245( the carrier of G,f) * (g * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,f) & b₂ in g * B ) } is set
(f * (g * B)) * B is Element of bool the carrier of G
(f * (g * B)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in f * (g * B) & b₂ in carr B ) } is set
(g * B) * B is Element of bool the carrier of G
(g * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * B & b₂ in carr B ) } is set
f * ((g * B) * B) is Element of bool the carrier of G
K245( the carrier of G,f) * ((g * B) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,f) & b₂ in (g * B) * B ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
1_ B is non being_of_order_0 Element of the carrier of B
{ b₁ where b₁ is Element of the carrier of G : N . b₁ = 1_ B } is set
bool the carrier of G is set
{ b₁ where b₁ is Element of the carrier of G : S₁[b₁] } is set
g is Element of the carrier of G
f is Element of bool the carrier of G
I is Element of the carrier of G
g * I 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (g,I) is Element of the carrier of G
[g,I] is set
{g,I} is finite set
{g} is finite set
{{g,I},{g}} is finite V56() set
the multF of G . [g,I] is set
N . (g * I) is Element of the carrier of B
N . g is Element of the carrier of B
N . I is Element of the carrier of B
(N . g) * (N . I) is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . ((N . g),(N . I)) is Element of the carrier of B
[(N . g),(N . I)] is set
{(N . g),(N . I)} is finite set
{(N . g)} is finite set
{{(N . g),(N . I)},{(N . g)}} is finite V56() set
the multF of B . [(N . g),(N . I)] is set
J is Element of the carrier of G
N . J is Element of the carrier of B
g is Element of the carrier of G
N . g is Element of the carrier of B
g is Element of the carrier of G
g " is Element of the carrier of G
N . (g ") is Element of the carrier of B
(1_ B) " is Element of the carrier of B
I is Element of the carrier of G
N . I is Element of the carrier of B
1_ G is non being_of_order_0 Element of the carrier of G
N . (1_ G) is Element of the carrier of B
g is non empty strict unital Group-like associative Subgroup of G
the carrier of g is non empty set
I is non empty strict unital Group-like associative Subgroup of G
the carrier of I is non empty set
f is non empty strict unital Group-like associative Subgroup of G
the carrier of f is non empty set
g is non empty strict unital Group-like associative Subgroup of G
the carrier of g is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
(G,B,N) is non empty strict unital Group-like associative Subgroup of G
1_ B is non being_of_order_0 Element of the carrier of B
bool the carrier of G is set
{ b₁ where b₁ is Element of the carrier of G : S₁[b₁] } is set
g is Element of the carrier of G
f is Element of bool the carrier of G
I is Element of the carrier of G
g * I 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (g,I) is Element of the carrier of G
[g,I] is set
{g,I} is finite set
{g} is finite set
{{g,I},{g}} is finite V56() set
the multF of G . [g,I] is set
N . (g * I) is Element of the carrier of B
N . g is Element of the carrier of B
N . I is Element of the carrier of B
(N . g) * (N . I) is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . ((N . g),(N . I)) is Element of the carrier of B
[(N . g),(N . I)] is set
{(N . g),(N . I)} is finite set
{(N . g)} is finite set
{{(N . g),(N . I)},{(N . g)}} is finite V56() set
the multF of B . [(N . g),(N . I)] is set
J is Element of the carrier of G
N . J is Element of the carrier of B
g is Element of the carrier of G
N . g is Element of the carrier of B
g is Element of the carrier of G
g " is Element of the carrier of G
N . (g ") is Element of the carrier of B
(1_ B) " is Element of the carrier of B
I is Element of the carrier of G
N . I is Element of the carrier of B
1_ G is non being_of_order_0 Element of the carrier of G
N . (1_ G) is Element of the carrier of B
g is non empty strict unital Group-like associative Subgroup of G
the carrier of g is non empty set
I is Element of the carrier of G
g |^ I is non empty strict unital Group-like associative Subgroup of G
J is Element of the carrier of G
g is Element of the carrier of G
g |^ I is Element of the carrier of G
N . J is Element of the carrier of B
N . I is Element of the carrier of B
(1_ B) |^ (N . I) is Element of the carrier of B
g is Element of the carrier of G
N . g is Element of the carrier of B
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
1_ B is non being_of_order_0 Element of the carrier of B
N is Element of the carrier of G
f is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
(G,B,f) is non empty strict unital Group-like associative normal Subgroup of G
f . N is Element of the carrier of B
the carrier of (G,B,f) is non empty set
{ b₁ where b₁ is Element of the carrier of G : f . b₁ = 1_ B } is set
g is Element of the carrier of G
f . g is Element of the carrier of B
{ b₁ where b₁ is Element of the carrier of G : f . b₁ = 1_ B } is set
the carrier of (G,B,f) is non empty set
G is non empty strict unital Group-like associative multMagma
B is non empty strict unital Group-like associative multMagma
(G,B) is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
the carrier of G is non empty set
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
(G,B,(G,B)) is non empty strict unital Group-like associative normal Subgroup of G
N is Element of the carrier of G
(G,B) . N is Element of the carrier of B
1_ B is non being_of_order_0 Element of the carrier of B
G is non empty unital Group-like associative multMagma
B is non empty strict unital Group-like associative normal Subgroup of G
(G,B) is non empty strict unital Group-like associative multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
(G,B) is Relation-like the carrier of G -defined the carrier of (G,B) -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,(G,B)) Element of bool [: the carrier of G, the carrier of (G,B):]
the carrier of (G,B) is non empty set
[: the carrier of G, the carrier of (G,B):] is set
bool [: the carrier of G, the carrier of (G,B):] is set
(G,(G,B),(G,B)) is non empty strict unital Group-like associative normal Subgroup of G
N is Element of the carrier of G
(G,B) . N is Element of the carrier of (G,B)
1_ (G,B) is non being_of_order_0 Element of the carrier of (G,B)
N * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
the carrier of B is non empty set
N * (carr B) is Element of bool the carrier of G
K245( the carrier of G,N) is finite Element of bool the carrier of G
K245( the carrier of G,N) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,N) & b₂ in carr B ) } is set
N * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
the carrier of B is non empty set
N * (carr B) is Element of bool the carrier of G
K245( the carrier of G,N) is finite Element of bool the carrier of G
K245( the carrier of G,N) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,N) & b₂ in carr B ) } is set
(G,B) . N is Element of the carrier of (G,B)
1_ (G,B) is non being_of_order_0 Element of the carrier of (G,B)
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
N .: the carrier of G is Element of bool the carrier of B
bool the carrier of B is set
bool the carrier of G is set
f is Element of bool the carrier of G
N .: f is Element of bool the carrier of B
I is Element of the carrier of B
J is Element of the carrier of B
g is Element of the carrier of G
N . g is Element of the carrier of B
g is Element of the carrier of G
N . g is Element of the carrier of B
I * J is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (I,J) is Element of the carrier of B
[I,J] is set
{I,J} is finite set
{I} is finite set
{{I,J},{I}} is finite V56() set
the multF of B . [I,J] is set
g * g 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (g,g) is Element of the carrier of G
[g,g] is set
{g,g} is finite set
{g} is finite set
{{g,g},{g}} is finite V56() set
the multF of G . [g,g] is set
N . (g * g) is Element of the carrier of B
I is Element of the carrier of B
J is Element of the carrier of G
N . J is Element of the carrier of B
J " is Element of the carrier of G
N . (J ") is Element of the carrier of B
I " is Element of the carrier of B
dom N is Element of bool the carrier of G
I is non empty strict unital Group-like associative Subgroup of B
the carrier of I is non empty set
f is non empty strict unital Group-like associative Subgroup of B
the carrier of f is non empty set
g is non empty strict unital Group-like associative Subgroup of B
the carrier of g is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
rng N is Element of bool the carrier of B
bool the carrier of B is set
(G,B,N) is non empty strict unital Group-like associative Subgroup of B
the carrier of (G,B,N) is non empty set
N .: the carrier of G is Element of bool the carrier of B
dom N is Element of bool the carrier of G
bool the carrier of G is set
N .: (dom N) is Element of bool the carrier of B
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
N is set
f is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
(B,G,f) is non empty strict unital Group-like associative Subgroup of G
the carrier of (B,G,f) is non empty set
f .: the carrier of B is Element of bool the carrier of G
bool the carrier of G is set
dom f is Element of bool the carrier of B
bool the carrier of B is set
g is set
f . g is set
I is Element of the carrier of B
f . I is Element of the carrier of G
g is Element of the carrier of B
f . g is Element of the carrier of G
dom f is Element of bool the carrier of B
bool the carrier of B is set
f .: the carrier of B is Element of bool the carrier of G
bool the carrier of G is set
the carrier of (B,G,f) is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
(G,B,N) is non empty strict unital Group-like associative Subgroup of B
rng N is Element of bool the carrier of B
bool the carrier of B is set
gr (rng N) is non empty strict unital Group-like associative Subgroup of B
the carrier of (G,B,N) is non empty set
carr (G,B,N) is Element of bool the carrier of B
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative multMagma
(G,B) is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
the carrier of G is non empty set
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
(G,B,(G,B)) is non empty strict unital Group-like associative Subgroup of B
(1). B is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of B
the carrier of (G,B,(G,B)) is non empty set
1_ B is non being_of_order_0 Element of the carrier of B
{(1_ B)} is finite set
f is set
(G,B) .: the carrier of G is Element of bool the carrier of B
bool the carrier of B is set
dom (G,B) is Element of bool the carrier of G
bool the carrier of G is set
g is set
(G,B) . g is set
I is Element of the carrier of G
(G,B) . I is Element of the carrier of B
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict unital Group-like associative multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
(G,B) is Relation-like the carrier of G -defined the carrier of (G,B) -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,(G,B)) Element of bool [: the carrier of G, the carrier of (G,B):]
the carrier of (G,B) is non empty set
[: the carrier of G, the carrier of (G,B):] is set
bool [: the carrier of G, the carrier of (G,B):] is set
(G,(G,B),(G,B)) is non empty strict unital Group-like associative Subgroup of (G,B)
N is Element of the carrier of (G,B)
f is Element of the carrier of G
f * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
the carrier of B is non empty set
f * (carr B) is Element of bool the carrier of G
K245( the carrier of G,f) is finite Element of bool the carrier of G
K245( the carrier of G,f) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,f) & b₂ in carr B ) } is set
B * f is Element of bool the carrier of G
(carr B) * f is Element of bool the carrier of G
(carr B) * K245( the carrier of G,f) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,f) ) } is set
(G,B) . f is Element of the carrier of (G,B)
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
N is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
(B,G,N) is non empty strict unital Group-like associative Subgroup of G
the carrier of (B,G,N) is non empty set
[: the carrier of B, the carrier of (B,G,N):] is set
bool [: the carrier of B, the carrier of (B,G,N):] is set
rng N is Element of bool the carrier of G
bool the carrier of G is set
dom N is Element of bool the carrier of B
bool the carrier of B is set
f is Relation-like the carrier of B -defined the carrier of (B,G,N) -valued Function-like non empty V22( the carrier of B) quasi_total Element of bool [: the carrier of B, the carrier of (B,G,N):]
g is Element of the carrier of B
f . g is Element of the carrier of (B,G,N)
I is Element of the carrier of B
f . I is Element of the carrier of (B,G,N)
(f . g) * (f . I) is Element of the carrier of (B,G,N)
the multF of (B,G,N) is Relation-like [: the carrier of (B,G,N), the carrier of (B,G,N):] -defined the carrier of (B,G,N) -valued Function-like V22([: the carrier of (B,G,N), the carrier of (B,G,N):]) quasi_total associative Element of bool [:[: the carrier of (B,G,N), the carrier of (B,G,N):], the carrier of (B,G,N):]
[: the carrier of (B,G,N), the carrier of (B,G,N):] is set
[:[: the carrier of (B,G,N), the carrier of (B,G,N):], the carrier of (B,G,N):] is set
bool [:[: the carrier of (B,G,N), the carrier of (B,G,N):], the carrier of (B,G,N):] is set
the multF of (B,G,N) . ((f . g),(f . I)) is Element of the carrier of (B,G,N)
[(f . g),(f . I)] is set
{(f . g),(f . I)} is finite set
{(f . g)} is finite set
{{(f . g),(f . I)},{(f . g)}} is finite V56() set
the multF of (B,G,N) . [(f . g),(f . I)] is set
N . g is Element of the carrier of G
N . I is Element of the carrier of G
(N . g) * (N . I) 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((N . g),(N . I)) is Element of the carrier of G
[(N . g),(N . I)] is set
{(N . g),(N . I)} is finite set
{(N . g)} is finite set
{{(N . g),(N . I)},{(N . g)}} is finite V56() set
the multF of G . [(N . g),(N . I)] is set
g * I is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (g,I) is Element of the carrier of B
[g,I] is set
{g,I} is finite set
{g} is finite set
{{g,I},{g}} is finite V56() set
the multF of B . [g,I] is set
f . (g * I) is Element of the carrier of (B,G,N)
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
N is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
(B,G,N) is non empty strict unital Group-like associative Subgroup of G
N .: the carrier of B is Element of bool the carrier of G
bool the carrier of G is set
G is non empty finite unital Group-like associative multMagma
the carrier of G is non empty finite set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total finite unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
(G,B,N) is non empty strict unital Group-like associative Subgroup of B
G is non empty unital Group-like associative commutative multMagma
the carrier of G is non empty set
B is Element of the carrier of G
N is Element of the carrier of G
B * N 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (B,N) is Element of the carrier of G
[B,N] is set
{B,N} is finite set
{B} is finite set
{{B,N},{B}} is finite V56() set
the multF of G . [B,N] is set
N * B is Element of the carrier of G
the multF of G . (N,B) is Element of the carrier of G
[N,B] is set
{N,B} is finite set
{N} is finite set
{{N,B},{N}} is finite V56() set
the multF of G . [N,B] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
N is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
(B,G,N) is non empty strict unital Group-like associative Subgroup of G
the carrier of (B,G,N) is non empty set
f is Element of the carrier of (B,G,N)
g is Element of the carrier of (B,G,N)
f * g is Element of the carrier of (B,G,N)
the multF of (B,G,N) is Relation-like [: the carrier of (B,G,N), the carrier of (B,G,N):] -defined the carrier of (B,G,N) -valued Function-like V22([: the carrier of (B,G,N), the carrier of (B,G,N):]) quasi_total associative Element of bool [:[: the carrier of (B,G,N), the carrier of (B,G,N):], the carrier of (B,G,N):]
[: the carrier of (B,G,N), the carrier of (B,G,N):] is set
[:[: the carrier of (B,G,N), the carrier of (B,G,N):], the carrier of (B,G,N):] is set
bool [:[: the carrier of (B,G,N), the carrier of (B,G,N):], the carrier of (B,G,N):] is set
the multF of (B,G,N) . (f,g) is Element of the carrier of (B,G,N)
[f,g] is set
{f,g} is finite set
{f} is finite set
{{f,g},{f}} is finite V56() set
the multF of (B,G,N) . [f,g] is set
g * f is Element of the carrier of (B,G,N)
the multF of (B,G,N) . (g,f) is Element of the carrier of (B,G,N)
[g,f] is set
{g,f} is finite set
{g} is finite set
{{g,f},{g}} is finite V56() set
the multF of (B,G,N) . [g,f] is set
g is Element of the carrier of B
N . g is Element of the carrier of G
g is Element of the carrier of B
N . g is Element of the carrier of G
I is Element of the carrier of G
J is Element of the carrier of G
I * J 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (I,J) is Element of the carrier of G
[I,J] is set
{I,J} is finite set
{I} is finite set
{{I,J},{I}} is finite V56() set
the multF of G . [I,J] is set
g * g is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (g,g) is Element of the carrier of B
[g,g] is set
{g,g} is finite set
{g} is finite set
{{g,g},{g}} is finite V56() set
the multF of B . [g,g] is set
N . (g * g) is Element of the carrier of G
g * g is Element of the carrier of B
the multF of B . (g,g) is Element of the carrier of B
[g,g] is set
{g,g} is finite set
{g} is finite set
{{g,g},{g}} is finite V56() set
the multF of B . [g,g] is set
N . (g * g) is Element of the carrier of G
J * I is Element of the carrier of G
the multF of G . (J,I) is Element of the carrier of G
[J,I] is set
{J,I} is finite set
{J} is finite set
{{J,I},{J}} is finite V56() set
the multF of G . [J,I] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
card B is cardinal set
card the carrier of B is cardinal set
N is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
(B,G,N) is non empty strict unital Group-like associative Subgroup of G
card (B,G,N) is cardinal set
the carrier of (B,G,N) is non empty set
card the carrier of (B,G,N) is cardinal set
N .: the carrier of B is Element of bool the carrier of G
bool the carrier of G is set
card (N .: the carrier of B) is cardinal set
G is non empty finite unital Group-like associative multMagma
the carrier of G is non empty finite set
card G is non empty ext-real V41() V42() V43() V47() V48() V49() integer cardinal Element of NAT
card the carrier of G is cardinal set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total finite unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
(G,B,N) is non empty finite strict unital Group-like associative Subgroup of B
card (G,B,N) is non empty ext-real V41() V42() V43() V47() V48() V49() integer cardinal Element of NAT
the carrier of (G,B,N) is non empty finite set
card the carrier of (G,B,N) is cardinal set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Element of the carrier of B
f is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
(G,B,f) is non empty strict unital Group-like associative Subgroup of B
f " is Relation-like Function-like set
(f ") . N is set
f . ((f ") . N) is set
the carrier of (G,B,f) is non empty set
[: the carrier of G, the carrier of (G,B,f):] is set
bool [: the carrier of G, the carrier of (G,B,f):] is set
g is Relation-like the carrier of G -defined the carrier of (G,B,f) -valued Function-like non empty V22( the carrier of G) quasi_total Element of bool [: the carrier of G, the carrier of (G,B,f):]
rng g is Element of bool the carrier of (G,B,f)
bool the carrier of (G,B,f) is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
N is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
N " is Relation-like Function-like set
(B,G,N) is non empty strict unital Group-like associative Subgroup of G
the carrier of (B,G,N) is non empty set
[: the carrier of (B,G,N), the carrier of B:] is set
bool [: the carrier of (B,G,N), the carrier of B:] is set
f is non empty unital Group-like associative multMagma
the carrier of f is non empty set
[: the carrier of B, the carrier of f:] is set
bool [: the carrier of B, the carrier of f:] is set
rng N is Element of bool the carrier of G
bool the carrier of G is set
[: the carrier of f, the carrier of B:] is set
bool [: the carrier of f, the carrier of B:] is set
I is Element of the carrier of f
J is Element of the carrier of f
g is Element of the carrier of G
x is Element of the carrier of B
N . x is Element of the carrier of G
g is Element of the carrier of G
b is Element of the carrier of B
N . b is Element of the carrier of G
g is Relation-like the carrier of f -defined the carrier of B -valued Function-like non empty V22( the carrier of f) quasi_total Element of bool [: the carrier of f, the carrier of B:]
I * J 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 V22([: the carrier of f, the carrier of f:]) quasi_total associative Element of bool [:[: the carrier of f, the carrier of f:], the carrier of f:]
[: the carrier of f, the carrier of f:] is set
[:[: the carrier of f, the carrier of f:], the carrier of f:] is set
bool [:[: the carrier of f, the carrier of f:], the carrier of f:] is set
the multF of f . (I,J) is Element of the carrier of f
[I,J] is set
{I,J} is finite set
{I} is finite set
{{I,J},{I}} is finite V56() set
the multF of f . [I,J] is set
g . (I * J) is Element of the carrier of B
(N . x) * (N . b) 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((N . x),(N . b)) is Element of the carrier of G
[(N . x),(N . b)] is set
{(N . x),(N . b)} is finite set
{(N . x)} is finite set
{{(N . x),(N . b)},{(N . x)}} is finite V56() set
the multF of G . [(N . x),(N . b)] is set
g . ((N . x) * (N . b)) is set
x * b is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (x,b) is Element of the carrier of B
[x,b] is set
{x,b} is finite set
{x} is finite set
{{x,b},{x}} is finite V56() set
the multF of B . [x,b] is set
N . (x * b) is Element of the carrier of G
g . (N . (x * b)) is set
g . I is Element of the carrier of B
(g . I) * b is Element of the carrier of B
the multF of B . ((g . I),b) is Element of the carrier of B
[(g . I),b] is set
{(g . I),b} is finite set
{(g . I)} is finite set
{{(g . I),b},{(g . I)}} is finite V56() set
the multF of B . [(g . I),b] is set
g . J is Element of the carrier of B
(g . I) * (g . J) is Element of the carrier of B
the multF of B . ((g . I),(g . J)) is Element of the carrier of B
[(g . I),(g . J)] is set
{(g . I),(g . J)} is finite set
{{(g . I),(g . J)},{(g . I)}} is finite V56() set
the multF of B . [(g . I),(g . J)] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
(1). B is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of B
N is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
(B,G,N) is non empty strict unital Group-like associative normal Subgroup of B
f is set
the carrier of (B,G,N) is non empty set
1_ B is non being_of_order_0 Element of the carrier of B
g is Element of the carrier of B
N . g is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
N . (1_ B) is Element of the carrier of G
g is Element of the carrier of B
N . g is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ B)} is finite set
the carrier of (B,G,N) is non empty set
1_ B is non being_of_order_0 Element of the carrier of B
{(1_ B)} is finite set
f is Element of the carrier of B
g is Element of the carrier of B
N . f is Element of the carrier of G
N . g is Element of the carrier of G
f " is Element of the carrier of B
N . (f ") is Element of the carrier of G
(N . f) * (N . (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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((N . f),(N . (f "))) is Element of the carrier of G
[(N . f),(N . (f "))] is set
{(N . f),(N . (f "))} is finite set
{(N . f)} is finite set
{{(N . f),(N . (f "))},{(N . f)}} is finite V56() set
the multF of G . [(N . f),(N . (f "))] is set
f * (f ") is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (f,(f ")) is Element of the carrier of B
[f,(f ")] is set
{f,(f ")} is finite set
{f} is finite set
{{f,(f ")},{f}} is finite V56() set
the multF of B . [f,(f ")] is set
N . (f * (f ")) is Element of the carrier of G
N . (1_ B) is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
g * (f ") is Element of the carrier of B
the multF of B . (g,(f ")) is Element of the carrier of B
[g,(f ")] is set
{g,(f ")} is finite set
{g} is finite set
{{g,(f ")},{g}} is finite V56() set
the multF of B . [g,(f ")] is set
N . (g * (f ")) is Element of the carrier of G
(1_ B) * f is Element of the carrier of B
the multF of B . ((1_ B),f) is Element of the carrier of B
[(1_ B),f] is set
{(1_ B),f} is finite set
{{(1_ B),f},{(1_ B)}} is finite V56() set
the multF of B . [(1_ B),f] is set
(g * (f ")) * f is Element of the carrier of B
the multF of B . ((g * (f ")),f) is Element of the carrier of B
[(g * (f ")),f] is set
{(g * (f ")),f} is finite set
{(g * (f "))} is finite set
{{(g * (f ")),f},{(g * (f "))}} is finite V56() set
the multF of B . [(g * (f ")),f] is set
(f ") * f is Element of the carrier of B
the multF of B . ((f "),f) is Element of the carrier of B
[(f "),f] is set
{(f "),f} is finite set
{(f ")} is finite set
{{(f "),f},{(f ")}} is finite V56() set
the multF of B . [(f "),f] is set
g * ((f ") * f) is Element of the carrier of B
the multF of B . (g,((f ") * f)) is Element of the carrier of B
[g,((f ") * f)] is set
{g,((f ") * f)} is finite set
{{g,((f ") * f)},{g}} is finite V56() set
the multF of B . [g,((f ") * f)] is set
g * (1_ B) is Element of the carrier of B
the multF of B . (g,(1_ B)) is Element of the carrier of B
[g,(1_ B)] is set
{g,(1_ B)} is finite set
{{g,(1_ B)},{g}} is finite V56() set
the multF of B . [g,(1_ B)] is set
f is Element of the carrier of B
N . f is Element of the carrier of G
g is Element of the carrier of B
N . 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
B is non empty strict unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
(G,B,N) is non empty strict unital Group-like associative Subgroup of B
rng N is Element of bool the carrier of B
bool the carrier of B is set
the carrier of (G,B,N) is non empty set
rng N is Element of bool the carrier of B
bool the carrier of B is set
f is set
N .: the carrier of G is Element of bool the carrier of B
dom N is Element of bool the carrier of G
bool the carrier of G is set
g is set
N . g is set
G is non empty set
B is non empty set
[:G,B:] is set
bool [:G,B:] is set
N is Relation-like G -defined B -valued Function-like non empty V22(G) quasi_total Element of bool [:G,B:]
rng N is Element of bool B
bool B is set
f is Element of B
g is set
N . g is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict unital Group-like associative multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
the carrier of (G,B) is non empty set
(G,B) is Relation-like the carrier of G -defined the carrier of (G,B) -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,(G,B)) Element of bool [: the carrier of G, the carrier of (G,B):]
[: the carrier of G, the carrier of (G,B):] is set
bool [: the carrier of G, the carrier of (G,B):] is set
(G,(G,B),(G,B)) is non empty strict unital Group-like associative Subgroup of (G,B)
G is set
B is set
[:G,B:] is set
bool [:G,B:] is set
N is Relation-like G -defined B -valued Function-like quasi_total Element of bool [:G,B:]
rng N is Element of bool B
bool B is set
G is set
B is non empty set
[:G,B:] is set
bool [:G,B:] is set
N is Relation-like G -defined B -valued Function-like V22(G) quasi_total Element of bool [:G,B:]
dom N is Element of bool G
bool G is set
rng N is Element of bool B
bool B is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty strict unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
N " is Relation-like Function-like set
(G,B,N) is non empty strict unital Group-like associative Subgroup of B
G is set
B is non empty set
[:G,B:] is set
bool [:G,B:] is set
[:B,G:] is set
bool [:B,G:] is set
N is Relation-like G -defined B -valued Function-like V22(G) quasi_total Element of bool [:G,B:]
N " is Relation-like Function-like set
f is Relation-like B -defined G -valued Function-like quasi_total Element of bool [:B,G:]
dom N is Element of bool G
bool G is set
rng f is Element of bool G
G is set
B is non empty set
[:G,B:] is set
bool [:G,B:] is set
N is non empty set
[:B,N:] is set
bool [:B,N:] is set
f is Relation-like G -defined B -valued Function-like V22(G) quasi_total Element of bool [:G,B:]
g is Relation-like B -defined N -valued Function-like non empty V22(B) quasi_total Element of bool [:B,N:]
g * f is Relation-like G -defined N -valued Function-like V22(G) quasi_total Element of bool [:G,N:]
[:G,N:] is set
bool [:G,N:] is set
dom g is Element of bool B
bool B is set
rng f is Element of bool B
rng (g * f) is Element of bool N
bool N is set
rng g is Element of bool N
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
(1). G is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of G
(G,((1). G)) is non empty strict unital Group-like associative multMagma
Left_Cosets ((1). G) is non empty Element of bool (bool the carrier of G)
bool the carrier of G is set
bool (bool the carrier of G) is set
(G,((1). G)) is Relation-like [:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):] -defined Left_Cosets ((1). G) -valued Function-like V22([:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):]) quasi_total Element of bool [:[:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):],(Left_Cosets ((1). G)):]
[:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):] is set
[:[:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):],(Left_Cosets ((1). G)):] is set
bool [:[:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):],(Left_Cosets ((1). G)):] is set
multMagma(# (Left_Cosets ((1). G)),(G,((1). G)) #) is strict multMagma
the carrier of (G,((1). G)) is non empty set
(G,((1). G)) is Relation-like the carrier of G -defined the carrier of (G,((1). G)) -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,(G,((1). G))) Element of bool [: the carrier of G, the carrier of (G,((1). G)):]
[: the carrier of G, the carrier of (G,((1). G)):] is set
bool [: the carrier of G, the carrier of (G,((1). G)):] is set
(G,(G,((1). G)),(G,((1). G))) is non empty strict unital Group-like associative normal Subgroup of G
N is non empty unital Group-like associative multMagma
the carrier of N is non empty set
[: the carrier of N, the carrier of N:] is set
bool [: the carrier of N, the carrier of N:] is set
id the carrier of N is Relation-like the carrier of N -defined the carrier of N -valued V6() V8() V9() V13() Function-like one-to-one non empty V22( the carrier of N) quasi_total onto bijective Element of bool [: the carrier of N, the carrier of N:]
f is Relation-like the carrier of N -defined the carrier of N -valued Function-like non empty V22( the carrier of N) quasi_total unity-preserving (N,N) Element of bool [: the carrier of N, the carrier of N:]
G is non empty strict unital Group-like associative multMagma
B is non empty strict unital Group-like associative multMagma
the carrier of G is non empty set
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
N " is Relation-like Function-like set
f is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
N is non empty strict unital Group-like associative multMagma
f is non empty strict unital Group-like associative multMagma
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative multMagma
N is non empty unital Group-like associative multMagma
the carrier of G is non empty set
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
f is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
the carrier of N is non empty set
[: the carrier of B, the carrier of N:] is set
bool [: the carrier of B, the carrier of N:] is set
g is Relation-like the carrier of B -defined the carrier of N -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,N) Element of bool [: the carrier of B, the carrier of N:]
g * f is Relation-like the carrier of G -defined the carrier of G -defined the carrier of N -valued the carrier of N -valued Function-like non empty V22( the carrier of G) V22( the carrier of G) quasi_total quasi_total unity-preserving (G,N) Element of bool [: the carrier of G, the carrier of N:]
[: the carrier of G, the carrier of N:] is set
bool [: the carrier of G, the carrier of N:] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
N is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
(B,G,N) is non empty strict unital Group-like associative Subgroup of G
the carrier of (B,G,N) is non empty set
[: the carrier of B, the carrier of (B,G,N):] is set
bool [: the carrier of B, the carrier of (B,G,N):] is set
f is Relation-like the carrier of B -defined the carrier of (B,G,N) -valued Function-like non empty V22( the carrier of B) quasi_total unity-preserving (B,(B,G,N)) Element of bool [: the carrier of B, the carrier of (B,G,N):]
rng f is Element of bool the carrier of (B,G,N)
bool the carrier of (B,G,N) is set
G is non empty strict unital Group-like associative multMagma
B is non empty strict unital Group-like associative multMagma
(G,B) is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
the carrier of G is non empty set
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
(1). B is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of B
(1). G is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of G
f is Element of the carrier of G
(G,B) . f is Element of the carrier of B
g is Element of the carrier of G
(G,B) . g is Element of the carrier of B
the carrier of ((1). G) is non empty trivial finite set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is finite set
rng (G,B) is Element of bool the carrier of B
bool the carrier of B is set
(G,B) . (1_ G) is Element of the carrier of B
{((G,B) . (1_ G))} is finite set
1_ B is non being_of_order_0 Element of the carrier of B
{(1_ B)} is finite set
the carrier of ((1). B) is non empty trivial finite set
G is non empty unital Group-like associative multMagma
(1). G is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of G
B is non empty unital Group-like associative multMagma
(1). B is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of B
G is non empty strict unital Group-like associative multMagma
(1). G is non empty trivial finite 1 -element strict unital Group-like associative normal Subgroup of G
(G,((1). G)) is non empty strict unital Group-like associative multMagma
Left_Cosets ((1). G) 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 set
bool (bool the carrier of G) is set
(G,((1). G)) is Relation-like [:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):] -defined Left_Cosets ((1). G) -valued Function-like V22([:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):]) quasi_total Element of bool [:[:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):],(Left_Cosets ((1). G)):]
[:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):] is set
[:[:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):],(Left_Cosets ((1). G)):] is set
bool [:[:(Left_Cosets ((1). G)),(Left_Cosets ((1). G)):],(Left_Cosets ((1). G)):] is set
multMagma(# (Left_Cosets ((1). G)),(G,((1). G)) #) is strict multMagma
the carrier of (G,((1). G)) is non empty set
(G,((1). G)) is Relation-like the carrier of G -defined the carrier of (G,((1). G)) -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,(G,((1). G))) Element of bool [: the carrier of G, the carrier of (G,((1). G)):]
[: the carrier of G, the carrier of (G,((1). G)):] is set
bool [: the carrier of G, the carrier of (G,((1). G)):] 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
(G,((Omega). G)) is non empty strict unital Group-like associative multMagma
Left_Cosets ((Omega). G) is non empty Element of bool (bool the carrier of G)
bool the carrier of G is set
bool (bool the carrier of G) is set
(G,((Omega). G)) is Relation-like [:(Left_Cosets ((Omega). G)),(Left_Cosets ((Omega). G)):] -defined Left_Cosets ((Omega). G) -valued Function-like V22([:(Left_Cosets ((Omega). G)),(Left_Cosets ((Omega). G)):]) quasi_total Element of bool [:[:(Left_Cosets ((Omega). G)),(Left_Cosets ((Omega). G)):],(Left_Cosets ((Omega). G)):]
[:(Left_Cosets ((Omega). G)),(Left_Cosets ((Omega). G)):] is set
[:[:(Left_Cosets ((Omega). G)),(Left_Cosets ((Omega). G)):],(Left_Cosets ((Omega). G)):] is set
bool [:[:(Left_Cosets ((Omega). G)),(Left_Cosets ((Omega). G)):],(Left_Cosets ((Omega). G)):] is set
multMagma(# (Left_Cosets ((Omega). G)),(G,((Omega). G)) #) is strict multMagma
the carrier of (G,((Omega). G)) is non empty set
{ the carrier of G} is finite set
G is non empty strict unital Group-like associative multMagma
B is non empty strict unital Group-like associative multMagma
card G is cardinal set
the carrier of G is non empty set
card the carrier of G is cardinal set
card B is cardinal set
the carrier of B is non empty set
card the carrier of B is cardinal set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
f is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
(B,G,f) is non empty strict unital Group-like associative Subgroup of G
(G,B,N) is non empty strict unital Group-like associative Subgroup of B
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative multMagma
the carrier of G is non empty set
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
rng N is Element of bool the carrier of B
bool the carrier of B is set
G is non empty strict unital Group-like associative multMagma
B is non empty strict unital Group-like associative multMagma
card G is cardinal set
the carrier of G is non empty set
card the carrier of G is cardinal set
card B is cardinal set
the carrier of B is non empty set
card the carrier of B is cardinal set
G is non empty trivial finite 1 -element strict unital Group-like associative multMagma
B is non empty strict unital Group-like associative multMagma
card G is non empty ext-real V41() V42() V43() V47() V48() V49() integer cardinal Element of NAT
the carrier of G is non empty trivial finite set
card the carrier of G is cardinal set
N is non empty finite unital Group-like associative multMagma
card N is non empty ext-real V41() V42() V43() V47() V48() V49() integer cardinal Element of NAT
the carrier of N is non empty finite set
card the carrier of N is cardinal set
G is non empty unital Group-like associative multMagma
B is non empty strict unital Group-like associative multMagma
the carrier of G is non empty set
the carrier of B is non empty set
[: the carrier of G, the carrier of B:] is set
bool [: the carrier of G, the carrier of B:] is set
N is Relation-like the carrier of G -defined the carrier of B -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,B) Element of bool [: the carrier of G, the carrier of B:]
f is Element of the carrier of B
I is Element of the carrier of G
N . I is Element of the carrier of B
g is Element of the carrier of B
J is Element of the carrier of G
N . J is Element of the carrier of B
f * g is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (f,g) is Element of the carrier of B
[f,g] is set
{f,g} is finite set
{f} is finite set
{{f,g},{f}} is finite V56() set
the multF of B . [f,g] is set
I * J 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (I,J) is Element of the carrier of G
[I,J] is set
{I,J} is finite set
{I} is finite set
{{I,J},{I}} is finite V56() set
the multF of G . [I,J] is set
N . (I * J) is Element of the carrier of B
J * I is Element of the carrier of G
the multF of G . (J,I) is Element of the carrier of G
[J,I] is set
{J,I} is finite set
{J} is finite set
{{J,I},{J}} is finite V56() set
the multF of G . [J,I] is set
N . (J * I) is Element of the carrier of B
g * f is Element of the carrier of B
the multF of B . (g,f) is Element of the carrier of B
[g,f] is set
{g,f} is finite set
{g} is finite set
{{g,f},{g}} is finite V56() set
the multF of B . [g,f] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
N is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
(B,G,N) is non empty strict unital Group-like associative normal Subgroup of B
(B,(B,G,N)) is non empty strict unital Group-like associative multMagma
Left_Cosets (B,G,N) is non empty Element of bool (bool the carrier of B)
bool the carrier of B is set
bool (bool the carrier of B) is set
(B,(B,G,N)) is Relation-like [:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):] -defined Left_Cosets (B,G,N) -valued Function-like V22([:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):]) quasi_total Element of bool [:[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):],(Left_Cosets (B,G,N)):]
[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):] is set
[:[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):],(Left_Cosets (B,G,N)):] is set
bool [:[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):],(Left_Cosets (B,G,N)):] is set
multMagma(# (Left_Cosets (B,G,N)),(B,(B,G,N)) #) is strict multMagma
(B,G,N) is non empty strict unital Group-like associative Subgroup of G
the carrier of (B,(B,G,N)) is non empty set
the carrier of (B,G,N) is non empty set
[: the carrier of (B,(B,G,N)), the carrier of (B,G,N):] is set
bool [: the carrier of (B,(B,G,N)), the carrier of (B,G,N):] is set
(B,(B,G,N)) is Relation-like the carrier of B -defined the carrier of (B,(B,G,N)) -valued Function-like non empty V22( the carrier of B) quasi_total unity-preserving (B,(B,(B,G,N))) Element of bool [: the carrier of B, the carrier of (B,(B,G,N)):]
[: the carrier of B, the carrier of (B,(B,G,N)):] is set
bool [: the carrier of B, the carrier of (B,(B,G,N)):] is set
dom (B,(B,G,N)) is Element of bool the carrier of B
I is Element of the carrier of (B,(B,G,N))
J is Element of the carrier of B
J * (B,G,N) is Element of bool the carrier of B
carr (B,G,N) is Element of bool the carrier of B
the carrier of (B,G,N) is non empty set
J * (carr (B,G,N)) is Element of bool the carrier of B
K245( the carrier of B,J) is finite Element of bool the carrier of B
K245( the carrier of B,J) * (carr (B,G,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,J) & b₂ in carr (B,G,N) ) } is set
(B,G,N) * J is Element of bool the carrier of B
(carr (B,G,N)) * J is Element of bool the carrier of B
(carr (B,G,N)) * K245( the carrier of B,J) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr (B,G,N) & b₂ in K245( the carrier of B,J) ) } is set
N . J is Element of the carrier of G
g is Element of the carrier of (B,G,N)
g is Element of the carrier of B
g * (B,G,N) is Element of bool the carrier of B
g * (carr (B,G,N)) is Element of bool the carrier of B
K245( the carrier of B,g) is finite Element of bool the carrier of B
K245( the carrier of B,g) * (carr (B,G,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in carr (B,G,N) ) } is set
N . g is Element of the carrier of G
J " is Element of the carrier of B
(J ") * g is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . ((J "),g) is Element of the carrier of B
[(J "),g] is set
{(J "),g} is finite set
{(J ")} is finite set
{{(J "),g},{(J ")}} is finite V56() set
the multF of B . [(J "),g] is set
1_ G is non being_of_order_0 Element of the carrier of G
N . ((J ") * g) is Element of the carrier of G
N . (J ") is Element of the carrier of G
(N . (J ")) * (N . g) 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((N . (J ")),(N . g)) is Element of the carrier of G
[(N . (J ")),(N . g)] is set
{(N . (J ")),(N . g)} is finite set
{(N . (J "))} is finite set
{{(N . (J ")),(N . g)},{(N . (J "))}} is finite V56() set
the multF of G . [(N . (J ")),(N . g)] is set
(N . J) " is Element of the carrier of G
((N . J) ") * (N . g) is Element of the carrier of G
the multF of G . (((N . J) "),(N . g)) is Element of the carrier of G
[((N . J) "),(N . g)] is set
{((N . J) "),(N . g)} is finite set
{((N . J) ")} is finite set
{{((N . J) "),(N . g)},{((N . J) ")}} is finite V56() set
the multF of G . [((N . J) "),(N . g)] is set
((N . J) ") " is Element of the carrier of G
I is Relation-like the carrier of (B,(B,G,N)) -defined the carrier of (B,G,N) -valued Function-like non empty V22( the carrier of (B,(B,G,N))) quasi_total Element of bool [: the carrier of (B,(B,G,N)), the carrier of (B,G,N):]
J is Element of the carrier of (B,(B,G,N))
g is Element of the carrier of B
g * (B,G,N) is Element of bool the carrier of B
carr (B,G,N) is Element of bool the carrier of B
the carrier of (B,G,N) is non empty set
g * (carr (B,G,N)) is Element of bool the carrier of B
K245( the carrier of B,g) is finite Element of bool the carrier of B
K245( the carrier of B,g) * (carr (B,G,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in carr (B,G,N) ) } is set
(B,G,N) * g is Element of bool the carrier of B
(carr (B,G,N)) * g is Element of bool the carrier of B
(carr (B,G,N)) * K245( the carrier of B,g) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr (B,G,N) & b₂ in K245( the carrier of B,g) ) } is set
g is Element of the carrier of (B,(B,G,N))
x is Element of the carrier of B
x * (B,G,N) is Element of bool the carrier of B
x * (carr (B,G,N)) is Element of bool the carrier of B
K245( the carrier of B,x) is finite Element of bool the carrier of B
K245( the carrier of B,x) * (carr (B,G,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,x) & b₂ in carr (B,G,N) ) } is set
(B,G,N) * x is Element of bool the carrier of B
(carr (B,G,N)) * x is Element of bool the carrier of B
(carr (B,G,N)) * K245( the carrier of B,x) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr (B,G,N) & b₂ in K245( the carrier of B,x) ) } is set
I . g is Element of the carrier of (B,G,N)
N . x is Element of the carrier of G
I . J is Element of the carrier of (B,G,N)
N . g is Element of the carrier of G
(I . J) * (I . g) is Element of the carrier of (B,G,N)
the multF of (B,G,N) is Relation-like [: the carrier of (B,G,N), the carrier of (B,G,N):] -defined the carrier of (B,G,N) -valued Function-like V22([: the carrier of (B,G,N), the carrier of (B,G,N):]) quasi_total associative Element of bool [:[: the carrier of (B,G,N), the carrier of (B,G,N):], the carrier of (B,G,N):]
[: the carrier of (B,G,N), the carrier of (B,G,N):] is set
[:[: the carrier of (B,G,N), the carrier of (B,G,N):], the carrier of (B,G,N):] is set
bool [:[: the carrier of (B,G,N), the carrier of (B,G,N):], the carrier of (B,G,N):] is set
the multF of (B,G,N) . ((I . J),(I . g)) is Element of the carrier of (B,G,N)
[(I . J),(I . g)] is set
{(I . J),(I . g)} is finite set
{(I . J)} is finite set
{{(I . J),(I . g)},{(I . J)}} is finite V56() set
the multF of (B,G,N) . [(I . J),(I . g)] is set
(N . g) * (N . x) 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((N . g),(N . x)) is Element of the carrier of G
[(N . g),(N . x)] is set
{(N . g),(N . x)} is finite set
{(N . g)} is finite set
{{(N . g),(N . x)},{(N . g)}} is finite V56() set
the multF of G . [(N . g),(N . x)] is set
g * x is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (g,x) is Element of the carrier of B
[g,x] is set
{g,x} is finite set
{g} is finite set
{{g,x},{g}} is finite V56() set
the multF of B . [g,x] is set
N . (g * x) is Element of the carrier of G
J * g is Element of the carrier of (B,(B,G,N))
the multF of (B,(B,G,N)) is Relation-like [: the carrier of (B,(B,G,N)), the carrier of (B,(B,G,N)):] -defined the carrier of (B,(B,G,N)) -valued Function-like V22([: the carrier of (B,(B,G,N)), the carrier of (B,(B,G,N)):]) quasi_total associative Element of bool [:[: the carrier of (B,(B,G,N)), the carrier of (B,(B,G,N)):], the carrier of (B,(B,G,N)):]
[: the carrier of (B,(B,G,N)), the carrier of (B,(B,G,N)):] is set
[:[: the carrier of (B,(B,G,N)), the carrier of (B,(B,G,N)):], the carrier of (B,(B,G,N)):] is set
bool [:[: the carrier of (B,(B,G,N)), the carrier of (B,(B,G,N)):], the carrier of (B,(B,G,N)):] is set
the multF of (B,(B,G,N)) . (J,g) is Element of the carrier of (B,(B,G,N))
[J,g] is set
{J,g} is finite set
{J} is finite set
{{J,g},{J}} is finite V56() set
the multF of (B,(B,G,N)) . [J,g] is set
(g * (B,G,N)) * ((B,G,N) * x) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in g * (B,G,N) & b₂ in (B,G,N) * x ) } is set
(g * (B,G,N)) * (B,G,N) is Element of bool the carrier of B
(g * (B,G,N)) * (carr (B,G,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in g * (B,G,N) & b₂ in carr (B,G,N) ) } is set
((g * (B,G,N)) * (B,G,N)) * x is Element of bool the carrier of B
((g * (B,G,N)) * (B,G,N)) * K245( the carrier of B,x) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in (g * (B,G,N)) * (B,G,N) & b₂ in K245( the carrier of B,x) ) } is set
(g * (B,G,N)) * x is Element of bool the carrier of B
(g * (B,G,N)) * K245( the carrier of B,x) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in g * (B,G,N) & b₂ in K245( the carrier of B,x) ) } is set
g * ((B,G,N) * x) is Element of bool the carrier of B
K245( the carrier of B,g) * ((B,G,N) * x) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in (B,G,N) * x ) } is set
g * (x * (B,G,N)) is Element of bool the carrier of B
K245( the carrier of B,g) * (x * (B,G,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in x * (B,G,N) ) } is set
(g * x) * (B,G,N) is Element of bool the carrier of B
(g * x) * (carr (B,G,N)) is Element of bool the carrier of B
K245( the carrier of B,(g * x)) is finite Element of bool the carrier of B
K245( the carrier of B,(g * x)) * (carr (B,G,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,(g * x)) & b₂ in carr (B,G,N) ) } is set
I . (J * g) is Element of the carrier of (B,G,N)
J is Relation-like the carrier of (B,(B,G,N)) -defined the carrier of (B,G,N) -valued Function-like non empty V22( the carrier of (B,(B,G,N))) quasi_total unity-preserving ((B,(B,G,N)),(B,G,N)) Element of bool [: the carrier of (B,(B,G,N)), the carrier of (B,G,N):]
g is set
dom J is set
J . g is set
g is set
J . g is set
dom J is Element of bool the carrier of (B,(B,G,N))
bool the carrier of (B,(B,G,N)) is set
x is Element of the carrier of (B,(B,G,N))
a1 is Element of the carrier of B
a1 * (B,G,N) is Element of bool the carrier of B
a1 * (carr (B,G,N)) is Element of bool the carrier of B
K245( the carrier of B,a1) is finite Element of bool the carrier of B
K245( the carrier of B,a1) * (carr (B,G,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,a1) & b₂ in carr (B,G,N) ) } is set
(B,G,N) * a1 is Element of bool the carrier of B
(carr (B,G,N)) * a1 is Element of bool the carrier of B
(carr (B,G,N)) * K245( the carrier of B,a1) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr (B,G,N) & b₂ in K245( the carrier of B,a1) ) } is set
b is Element of the carrier of (B,(B,G,N))
a2 is Element of the carrier of B
a2 * (B,G,N) is Element of bool the carrier of B
a2 * (carr (B,G,N)) is Element of bool the carrier of B
K245( the carrier of B,a2) is finite Element of bool the carrier of B
K245( the carrier of B,a2) * (carr (B,G,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,a2) & b₂ in carr (B,G,N) ) } is set
(B,G,N) * a2 is Element of bool the carrier of B
(carr (B,G,N)) * a2 is Element of bool the carrier of B
(carr (B,G,N)) * K245( the carrier of B,a2) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr (B,G,N) & b₂ in K245( the carrier of B,a2) ) } is set
J . b is Element of the carrier of (B,G,N)
N . a2 is Element of the carrier of G
J . x is Element of the carrier of (B,G,N)
N . a1 is Element of the carrier of G
(N . a2) " is Element of the carrier of G
((N . a2) ") * (N . a1) is Element of the carrier of G
the multF of G . (((N . a2) "),(N . a1)) is Element of the carrier of G
[((N . a2) "),(N . a1)] is set
{((N . a2) "),(N . a1)} is finite set
{((N . a2) ")} is finite set
{{((N . a2) "),(N . a1)},{((N . a2) ")}} is finite V56() set
the multF of G . [((N . a2) "),(N . a1)] is set
1_ G is non being_of_order_0 Element of the carrier of G
a2 " is Element of the carrier of B
N . (a2 ") is Element of the carrier of G
(N . (a2 ")) * (N . a1) is Element of the carrier of G
the multF of G . ((N . (a2 ")),(N . a1)) is Element of the carrier of G
[(N . (a2 ")),(N . a1)] is set
{(N . (a2 ")),(N . a1)} is finite set
{(N . (a2 "))} is finite set
{{(N . (a2 ")),(N . a1)},{(N . (a2 "))}} is finite V56() set
the multF of G . [(N . (a2 ")),(N . a1)] is set
(a2 ") * a1 is Element of the carrier of B
the multF of B . ((a2 "),a1) is Element of the carrier of B
[(a2 "),a1] is set
{(a2 "),a1} is finite set
{(a2 ")} is finite set
{{(a2 "),a1},{(a2 ")}} is finite V56() set
the multF of B . [(a2 "),a1] is set
N . ((a2 ") * a1) is Element of the carrier of G
dom N is Element of bool the carrier of B
g is set
(B,(B,G,N)) . g is set
dom J is Element of bool the carrier of (B,(B,G,N))
bool the carrier of (B,(B,G,N)) is set
rng (B,(B,G,N)) is Element of bool the carrier of (B,(B,G,N))
rng J is Element of bool the carrier of (B,G,N)
bool the carrier of (B,G,N) is set
g is set
g is Element of the carrier of B
N . g is Element of the carrier of G
g * (B,G,N) is Element of bool the carrier of B
g * (carr (B,G,N)) is Element of bool the carrier of B
K245( the carrier of B,g) is finite Element of bool the carrier of B
K245( the carrier of B,g) * (carr (B,G,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in carr (B,G,N) ) } is set
x is Element of the carrier of (B,(B,G,N))
J . x is Element of the carrier of (B,G,N)
J * (B,(B,G,N)) is Relation-like the carrier of B -defined the carrier of B -defined the carrier of (B,G,N) -valued the carrier of (B,G,N) -valued Function-like non empty V22( the carrier of B) V22( the carrier of B) quasi_total quasi_total unity-preserving (B,(B,G,N)) Element of bool [: the carrier of B, the carrier of (B,G,N):]
[: the carrier of B, the carrier of (B,G,N):] is set
bool [: the carrier of B, the carrier of (B,G,N):] is set
g is set
g is Element of the carrier of B
(B,(B,G,N)) . g is Element of the carrier of (B,(B,G,N))
g * (B,G,N) is Element of bool the carrier of B
g * (carr (B,G,N)) is Element of bool the carrier of B
K245( the carrier of B,g) is finite Element of bool the carrier of B
K245( the carrier of B,g) * (carr (B,G,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in carr (B,G,N) ) } is set
N . g is set
(B,(B,G,N)) . g is set
J . ((B,(B,G,N)) . g) is set
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
N is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
(B,G,N) is non empty strict unital Group-like associative normal Subgroup of B
(B,(B,G,N)) is non empty strict unital Group-like associative multMagma
Left_Cosets (B,G,N) is non empty Element of bool (bool the carrier of B)
bool the carrier of B is set
bool (bool the carrier of B) is set
(B,(B,G,N)) is Relation-like [:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):] -defined Left_Cosets (B,G,N) -valued Function-like V22([:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):]) quasi_total Element of bool [:[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):],(Left_Cosets (B,G,N)):]
[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):] is set
[:[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):],(Left_Cosets (B,G,N)):] is set
bool [:[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):],(Left_Cosets (B,G,N)):] is set
multMagma(# (Left_Cosets (B,G,N)),(B,(B,G,N)) #) is strict multMagma
(B,G,N) is non empty strict unital Group-like associative Subgroup of G
B is non empty unital Group-like associative multMagma
the carrier of B is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
[: the carrier of B, the carrier of G:] is set
bool [: the carrier of B, the carrier of G:] is set
N is Relation-like the carrier of B -defined the carrier of G -valued Function-like non empty V22( the carrier of B) quasi_total quasi_total unity-preserving (B,G) Element of bool [: the carrier of B, the carrier of G:]
(B,G,N) is non empty strict unital Group-like associative normal Subgroup of B
(B,(B,G,N)) is non empty strict unital Group-like associative multMagma
Left_Cosets (B,G,N) is non empty Element of bool (bool the carrier of B)
bool the carrier of B is set
bool (bool the carrier of B) is set
(B,(B,G,N)) is Relation-like [:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):] -defined Left_Cosets (B,G,N) -valued Function-like V22([:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):]) quasi_total Element of bool [:[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):],(Left_Cosets (B,G,N)):]
[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):] is set
[:[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):],(Left_Cosets (B,G,N)):] is set
bool [:[:(Left_Cosets (B,G,N)),(Left_Cosets (B,G,N)):],(Left_Cosets (B,G,N)):] is set
multMagma(# (Left_Cosets (B,G,N)),(B,(B,G,N)) #) is strict multMagma
the carrier of (B,(B,G,N)) is non empty set
(B,G,N) is non empty strict unital Group-like associative Subgroup of G
the carrier of (B,G,N) is non empty set
[: the carrier of (B,(B,G,N)), the carrier of (B,G,N):] is set
bool [: the carrier of (B,(B,G,N)), the carrier of (B,G,N):] is set
(B,(B,G,N)) is Relation-like the carrier of B -defined the carrier of (B,(B,G,N)) -valued Function-like non empty V22( the carrier of B) quasi_total unity-preserving (B,(B,(B,G,N))) Element of bool [: the carrier of B, the carrier of (B,(B,G,N)):]
[: the carrier of B, the carrier of (B,(B,G,N)):] is set
bool [: the carrier of B, the carrier of (B,(B,G,N)):] is set
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative normal Subgroup of G
(G,B) is non empty strict unital Group-like associative multMagma
Left_Cosets B 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 set
bool (bool the carrier of G) is set
(G,B) is Relation-like [:(Left_Cosets B),(Left_Cosets B):] -defined Left_Cosets B -valued Function-like V22([:(Left_Cosets B),(Left_Cosets B):]) quasi_total Element of bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):]
[:(Left_Cosets B),(Left_Cosets B):] is set
[:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
bool [:[:(Left_Cosets B),(Left_Cosets B):],(Left_Cosets B):] is set
multMagma(# (Left_Cosets B),(G,B) #) is strict multMagma
N is non empty strict unital Group-like associative normal Subgroup of G
(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)
(G,N) is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V22([:(Left_Cosets N),(Left_Cosets N):]) quasi_total Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
multMagma(# (Left_Cosets N),(G,N) #) is strict multMagma
(G,B,N) is non empty strict unital Group-like associative normal (G,B)
(B,(G,B,N)) is non empty strict unital Group-like associative multMagma
Left_Cosets (G,B,N) is non empty Element of bool (bool the carrier of B)
the carrier of B is non empty set
bool the carrier of B is set
bool (bool the carrier of B) is set
(B,(G,B,N)) is Relation-like [:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):] -defined Left_Cosets (G,B,N) -valued Function-like V22([:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):]) quasi_total Element of bool [:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):]
[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):] is set
[:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):] is set
bool [:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):] is set
multMagma(# (Left_Cosets (G,B,N)),(B,(G,B,N)) #) is strict multMagma
f is non empty strict unital Group-like associative normal Subgroup of (G,N)
((G,N),f) is non empty strict unital Group-like associative multMagma
Left_Cosets f is non empty Element of bool (bool the carrier of (G,N))
the carrier of (G,N) is non empty set
bool the carrier of (G,N) is set
bool (bool the carrier of (G,N)) is set
((G,N),f) is Relation-like [:(Left_Cosets f),(Left_Cosets f):] -defined Left_Cosets f -valued Function-like V22([:(Left_Cosets f),(Left_Cosets f):]) quasi_total Element of bool [:[:(Left_Cosets f),(Left_Cosets f):],(Left_Cosets f):]
[:(Left_Cosets f),(Left_Cosets f):] is set
[:[:(Left_Cosets f),(Left_Cosets f):],(Left_Cosets f):] is set
bool [:[:(Left_Cosets f),(Left_Cosets f):],(Left_Cosets f):] is set
multMagma(# (Left_Cosets f),((G,N),f) #) is strict multMagma
the carrier of (G,B) is non empty set
g is Element of the carrier of (G,N)
I is Element of the carrier of G
I * N is Element of bool the carrier of G
carr N is Element of bool the carrier of G
the carrier of N is non empty set
I * (carr N) is Element of bool the carrier of G
K245( the carrier of G,I) is finite Element of bool the carrier of G
K245( the carrier of G,I) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in carr N ) } is set
N * I is Element of bool the carrier of G
(carr N) * I is Element of bool the carrier of G
(carr N) * K245( the carrier of G,I) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,I) ) } is set
I * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
I * (carr B) is Element of bool the carrier of G
K245( the carrier of G,I) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,I) & b₂ in carr B ) } is set
J is Element of the carrier of (G,B)
g is Element of the carrier of G
g * N is Element of bool the carrier of G
g * (carr N) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr N ) } is set
g * B is Element of bool the carrier of G
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
I " is Element of the carrier of G
(I ") * g 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . ((I "),g) is Element of the carrier of G
[(I "),g] is set
{(I "),g} is finite set
{(I ")} is finite set
{{(I "),g},{(I ")}} is finite V56() set
the multF of G . [(I "),g] is set
[: the carrier of (G,N), the carrier of (G,B):] is set
bool [: the carrier of (G,N), the carrier of (G,B):] is set
g is Relation-like the carrier of (G,N) -defined the carrier of (G,B) -valued Function-like non empty V22( the carrier of (G,N)) quasi_total Element of bool [: the carrier of (G,N), the carrier of (G,B):]
I is Element of the carrier of (G,N)
g is Element of the carrier of G
g * N is Element of bool the carrier of G
carr N is Element of bool the carrier of G
the carrier of N is non empty set
g * (carr N) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr N ) } is set
N * g is Element of bool the carrier of G
(carr N) * g is Element of bool the carrier of G
(carr N) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,g) ) } is set
J is Element of the carrier of (G,N)
g is Element of the carrier of G
g * N is Element of bool the carrier of G
g * (carr N) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr N ) } is set
N * g is Element of bool the carrier of G
(carr N) * g is Element of bool the carrier of G
(carr N) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,g) ) } is set
I * J is Element of the carrier of (G,N)
the multF of (G,N) is Relation-like [: the carrier of (G,N), the carrier of (G,N):] -defined the carrier of (G,N) -valued Function-like V22([: the carrier of (G,N), the carrier of (G,N):]) quasi_total associative Element of bool [:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):]
[: the carrier of (G,N), the carrier of (G,N):] is set
[:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):] is set
bool [:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):] is set
the multF of (G,N) . (I,J) is Element of the carrier of (G,N)
[I,J] is set
{I,J} is finite set
{I} is finite set
{{I,J},{I}} is finite V56() set
the multF of (G,N) . [I,J] is set
(G,N,I) is Element of bool the carrier of G
(G,N,J) is Element of bool the carrier of G
(G,N,I) * (G,N,J) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,N,I) & b₂ in (G,N,J) ) } is set
(g * N) * g is Element of bool the carrier of G
(g * N) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * N & b₂ in K245( the carrier of G,g) ) } is set
((g * N) * g) * N is Element of bool the carrier of G
((g * N) * g) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (g * N) * g & b₂ in carr N ) } is set
g * (N * g) is Element of bool the carrier of G
K245( the carrier of G,g) * (N * g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in N * g ) } is set
(g * (N * g)) * N is Element of bool the carrier of G
(g * (N * g)) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * (N * g) & b₂ in carr N ) } is set
g * (g * N) is Element of bool the carrier of G
K245( the carrier of G,g) * (g * N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in g * N ) } is set
(g * (g * N)) * N is Element of bool the carrier of G
(g * (g * N)) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * (g * N) & b₂ in carr N ) } is set
(g * N) * N is Element of bool the carrier of G
(g * N) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * N & b₂ in carr N ) } is set
g * ((g * N) * N) is Element of bool the carrier of G
K245( the carrier of G,g) * ((g * N) * N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in (g * N) * N ) } is set
g * g 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (g,g) is Element of the carrier of G
[g,g] is set
{g,g} is finite set
{g} is finite set
{{g,g},{g}} is finite V56() set
the multF of G . [g,g] is set
(g * g) * N is Element of bool the carrier of G
(g * g) * (carr N) is Element of bool the carrier of G
K245( the carrier of G,(g * g)) is finite Element of bool the carrier of G
K245( the carrier of G,(g * g)) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(g * g)) & b₂ in carr N ) } is set
g . J is Element of the carrier of (G,B)
g * B is Element of bool the carrier of G
carr B is Element of bool the carrier of G
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
g . I is Element of the carrier of (G,B)
g * B is Element of bool the carrier of G
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
(g . I) * (g . J) is Element of the carrier of (G,B)
the multF of (G,B) is Relation-like [: the carrier of (G,B), the carrier of (G,B):] -defined the carrier of (G,B) -valued Function-like V22([: the carrier of (G,B), the carrier of (G,B):]) quasi_total associative Element of bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):]
[: the carrier of (G,B), the carrier of (G,B):] is set
[:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
bool [:[: the carrier of (G,B), the carrier of (G,B):], the carrier of (G,B):] is set
the multF of (G,B) . ((g . I),(g . J)) is Element of the carrier of (G,B)
[(g . I),(g . J)] is set
{(g . I),(g . J)} is finite set
{(g . I)} is finite set
{{(g . I),(g . J)},{(g . I)}} is finite V56() set
the multF of (G,B) . [(g . I),(g . J)] is set
(G,B,(g . I)) is Element of bool the carrier of G
(G,B,(g . J)) is Element of bool the carrier of G
(G,B,(g . I)) * (G,B,(g . J)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,B,(g . I)) & b₂ in (G,B,(g . J)) ) } is set
(g * B) * g is Element of bool the carrier of G
(g * B) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * B & b₂ in K245( the carrier of G,g) ) } is set
((g * B) * g) * B is Element of bool the carrier of G
((g * B) * g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (g * B) * g & b₂ in carr B ) } is set
B * g is Element of bool the carrier of G
(carr B) * g is Element of bool the carrier of G
(carr B) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,g) ) } is set
g * (B * g) is Element of bool the carrier of G
K245( the carrier of G,g) * (B * g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in B * g ) } is set
(g * (B * g)) * B is Element of bool the carrier of G
(g * (B * g)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * (B * g) & b₂ in carr B ) } is set
g * (g * B) is Element of bool the carrier of G
K245( the carrier of G,g) * (g * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in g * B ) } is set
(g * (g * B)) * B is Element of bool the carrier of G
(g * (g * B)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * (g * B) & b₂ in carr B ) } is set
(g * B) * B is Element of bool the carrier of G
(g * B) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in g * B & b₂ in carr B ) } is set
g * ((g * B) * B) is Element of bool the carrier of G
K245( the carrier of G,g) * ((g * B) * B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in (g * B) * B ) } is set
(g * g) * B is Element of bool the carrier of G
(g * g) * (carr B) is Element of bool the carrier of G
K245( the carrier of G,(g * g)) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(g * g)) & b₂ in carr B ) } is set
g . (I * J) is Element of the carrier of (G,B)
I is Relation-like the carrier of (G,N) -defined the carrier of (G,B) -valued Function-like non empty V22( the carrier of (G,N)) quasi_total unity-preserving ((G,N),(G,B)) Element of bool [: the carrier of (G,N), the carrier of (G,B):]
((G,N),(G,B),I) is non empty strict unital Group-like associative normal Subgroup of (G,N)
J is Element of the carrier of (G,N)
I . J is Element of the carrier of (G,B)
1_ (G,B) is non being_of_order_0 Element of the carrier of (G,B)
g is Element of the carrier of G
g * N is Element of bool the carrier of G
g * (carr N) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr N ) } is set
N * g is Element of bool the carrier of G
(carr N) * g is Element of bool the carrier of G
(carr N) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,g) ) } is set
g * B is Element of bool the carrier of G
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
g is Element of the carrier of B
g * (G,B,N) is Element of bool the carrier of B
carr (G,B,N) is Element of bool the carrier of B
the carrier of (G,B,N) is non empty set
g * (carr (G,B,N)) is Element of bool the carrier of B
K245( the carrier of B,g) is finite Element of bool the carrier of B
K245( the carrier of B,g) * (carr (G,B,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in carr (G,B,N) ) } is set
(G,B,N) * g is Element of bool the carrier of B
(carr (G,B,N)) * g is Element of bool the carrier of B
(carr (G,B,N)) * K245( the carrier of B,g) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr (G,B,N) & b₂ in K245( the carrier of B,g) ) } is set
g is Element of the carrier of B
g * (G,B,N) is Element of bool the carrier of B
carr (G,B,N) is Element of bool the carrier of B
the carrier of (G,B,N) is non empty set
g * (carr (G,B,N)) is Element of bool the carrier of B
K245( the carrier of B,g) is finite Element of bool the carrier of B
K245( the carrier of B,g) * (carr (G,B,N)) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in K245( the carrier of B,g) & b₂ in carr (G,B,N) ) } is set
(G,B,N) * g is Element of bool the carrier of B
(carr (G,B,N)) * g is Element of bool the carrier of B
(carr (G,B,N)) * K245( the carrier of B,g) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in carr (G,B,N) & b₂ in K245( the carrier of B,g) ) } is set
g is Element of the carrier of G
g * N is Element of bool the carrier of G
g * (carr N) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr N ) } is set
I . J is Element of the carrier of (G,B)
g * B is Element of bool the carrier of G
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
1_ (G,B) is non being_of_order_0 Element of the carrier of (G,B)
rng I is Element of bool the carrier of (G,B)
bool the carrier of (G,B) is set
J is set
g is Element of the carrier of G
g * B is Element of bool the carrier of G
g * (carr B) is Element of bool the carrier of G
K245( the carrier of G,g) is finite Element of bool the carrier of G
K245( the carrier of G,g) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr B ) } is set
B * g is Element of bool the carrier of G
(carr B) * g is Element of bool the carrier of G
(carr B) * K245( the carrier of G,g) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in K245( the carrier of G,g) ) } is set
g * N is Element of bool the carrier of G
g * (carr N) is Element of bool the carrier of G
K245( the carrier of G,g) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,g) & b₂ in carr N ) } is set
g is Element of the carrier of (G,N)
I . g is Element of the carrier of (G,B)
dom I is Element of bool the carrier of (G,N)
((G,N),(G,B),I) is non empty strict unital Group-like associative Subgroup of (G,B)
G is non empty unital Group-like associative multMagma
B is non empty unital Group-like associative Subgroup of G
N is non empty strict unital Group-like associative normal Subgroup of G
B "\/" N is non empty strict unital Group-like associative Subgroup of G
the carrier of G is non empty set
carr B is Element of bool the carrier of G
bool the carrier of G is set
the carrier of B is non empty set
carr N is Element of bool the carrier of G
the carrier of N is non empty set
(carr B) \/ (carr N) is Element of bool the carrier of G
gr ((carr B) \/ (carr N)) is non empty strict unital Group-like associative Subgroup of G
(G,(B "\/" N),N) is non empty strict unital Group-like associative normal (G,B "\/" N)
((B "\/" N),(G,(B "\/" N),N)) is non empty strict unital Group-like associative multMagma
Left_Cosets (G,(B "\/" N),N) is non empty Element of bool (bool the carrier of (B "\/" N))
the carrier of (B "\/" N) is non empty set
bool the carrier of (B "\/" N) is set
bool (bool the carrier of (B "\/" N)) is set
((B "\/" N),(G,(B "\/" N),N)) is Relation-like [:(Left_Cosets (G,(B "\/" N),N)),(Left_Cosets (G,(B "\/" N),N)):] -defined Left_Cosets (G,(B "\/" N),N) -valued Function-like V22([:(Left_Cosets (G,(B "\/" N),N)),(Left_Cosets (G,(B "\/" N),N)):]) quasi_total Element of bool [:[:(Left_Cosets (G,(B "\/" N),N)),(Left_Cosets (G,(B "\/" N),N)):],(Left_Cosets (G,(B "\/" N),N)):]
[:(Left_Cosets (G,(B "\/" N),N)),(Left_Cosets (G,(B "\/" N),N)):] is set
[:[:(Left_Cosets (G,(B "\/" N),N)),(Left_Cosets (G,(B "\/" N),N)):],(Left_Cosets (G,(B "\/" N),N)):] is set
bool [:[:(Left_Cosets (G,(B "\/" N),N)),(Left_Cosets (G,(B "\/" N),N)):],(Left_Cosets (G,(B "\/" N),N)):] is set
multMagma(# (Left_Cosets (G,(B "\/" N),N)),((B "\/" N),(G,(B "\/" N),N)) #) is strict multMagma
(G,B,N) is non empty strict unital Group-like associative normal (G,B)
(B,(G,B,N)) is non empty strict unital Group-like associative multMagma
Left_Cosets (G,B,N) is non empty Element of bool (bool the carrier of B)
bool the carrier of B is set
bool (bool the carrier of B) is set
(B,(G,B,N)) is Relation-like [:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):] -defined Left_Cosets (G,B,N) -valued Function-like V22([:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):]) quasi_total Element of bool [:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):]
[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):] is set
[:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):] is set
bool [:[:(Left_Cosets (G,B,N)),(Left_Cosets (G,B,N)):],(Left_Cosets (G,B,N)):] is set
multMagma(# (Left_Cosets (G,B,N)),(B,(G,B,N)) #) is strict multMagma
(G,N) is Relation-like the carrier of G -defined the carrier of (G,N) -valued Function-like non empty V22( the carrier of G) quasi_total unity-preserving (G,(G,N)) Element of bool [: the carrier of G, the carrier of (G,N):]
(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 set
(G,N) is Relation-like [:(Left_Cosets N),(Left_Cosets N):] -defined Left_Cosets N -valued Function-like V22([:(Left_Cosets N),(Left_Cosets N):]) quasi_total Element of bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):]
[:(Left_Cosets N),(Left_Cosets N):] is set
[:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
bool [:[:(Left_Cosets N),(Left_Cosets N):],(Left_Cosets N):] is set
multMagma(# (Left_Cosets N),(G,N) #) is strict multMagma
the carrier of (G,N) is non empty set
[: the carrier of G, the carrier of (G,N):] is set
bool [: the carrier of G, the carrier of (G,N):] is set
(G,N) | the carrier of B is Relation-like the carrier of G -defined the carrier of (G,N) -valued Function-like Element of bool [: the carrier of G, the carrier of (G,N):]
dom ((G,N) | the carrier of B) is Element of bool the carrier of G
dom (G,N) is Element of bool the carrier of G
(dom (G,N)) /\ the carrier of B is Element of bool the carrier of G
rng ((G,N) | the carrier of B) is Element of bool the carrier of (G,N)
bool the carrier of (G,N) is set
the carrier of ((B "\/" N),(G,(B "\/" N),N)) is non empty set
g is set
g is set
((G,N) | the carrier of B) . g is set
x is Element of the carrier of B
a1 is Element of the carrier of (B "\/" N)
a1 * (G,(B "\/" N),N) is Element of bool the carrier of (B "\/" N)
carr (G,(B "\/" N),N) is Element of bool the carrier of (B "\/" N)
the carrier of (G,(B "\/" N),N) is non empty set
a1 * (carr (G,(B "\/" N),N)) is Element of bool the carrier of (B "\/" N)
K245( the carrier of (B "\/" N),a1) is finite Element of bool the carrier of (B "\/" N)
K245( the carrier of (B "\/" N),a1) * (carr (G,(B "\/" N),N)) is Element of bool the carrier of (B "\/" N)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (B "\/" N) : ( b₁ in K245( the carrier of (B "\/" N),a1) & b₂ in carr (G,(B "\/" N),N) ) } is set
b is Element of the carrier of G
b * N is Element of bool the carrier of G
b * (carr N) is Element of bool the carrier of G
K245( the carrier of G,b) is finite Element of bool the carrier of G
K245( the carrier of G,b) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,b) & b₂ in carr N ) } is set
N * b is Element of bool the carrier of G
(carr N) * b is Element of bool the carrier of G
(carr N) * K245( the carrier of G,b) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in K245( the carrier of G,b) ) } is set
(G,(B "\/" N),N) * a1 is Element of bool the carrier of (B "\/" N)
(carr (G,(B "\/" N),N)) * a1 is Element of bool the carrier of (B "\/" N)
(carr (G,(B "\/" N),N)) * K245( the carrier of (B "\/" N),a1) is Element of bool the carrier of (B "\/" N)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (B "\/" N) : ( b₁ in carr (G,(B "\/" N),N) & b₂ in K245( the carrier of (B "\/" N),a1) ) } is set
((G,N) | the carrier of B) . x is set
(G,N) . b is Element of the carrier of (G,N)
[: the carrier of B, the carrier of ((B "\/" N),(G,(B "\/" N),N)):] is set
bool [: the carrier of B, the carrier of ((B "\/" N),(G,(B "\/" N),N)):] is set
g is Element of the carrier of B
x is Element of the carrier of B
b is Element of the carrier of G
(G,N) . b is Element of the carrier of (G,N)
g is Relation-like the carrier of B -defined the carrier of ((B "\/" N),(G,(B "\/" N),N)) -valued Function-like non empty V22( the carrier of B) quasi_total Element of bool [: the carrier of B, the carrier of ((B "\/" N),(G,(B "\/" N),N)):]
g . g is Element of the carrier of ((B "\/" N),(G,(B "\/" N),N))
a1 is Element of the carrier of G
(G,N) . a1 is Element of the carrier of (G,N)
g . x is Element of the carrier of ((B "\/" N),(G,(B "\/" N),N))
g * x is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V22([: the carrier of B, the carrier of B:]) quasi_total associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is set
the multF of B . (g,x) is Element of the carrier of B
[g,x] is set
{g,x} is finite set
{g} is finite set
{{g,x},{g}} is finite V56() set
the multF of B . [g,x] is set
b * a1 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 V22([: the carrier of G, the carrier of G:]) quasi_total associative Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is set
the multF of G . (b,a1) is Element of the carrier of G
[b,a1] is set
{b,a1} is finite set
{b} is finite set
{{b,a1},{b}} is finite V56() set
the multF of G . [b,a1] is set
g . (g * x) is Element of the carrier of ((B "\/" N),(G,(B "\/" N),N))
(G,N) . (b * a1) is Element of the carrier of (G,N)
((G,N) . b) * ((G,N) . a1) is Element of the carrier of (G,N)
the multF of (G,N) is Relation-like [: the carrier of (G,N), the carrier of (G,N):] -defined the carrier of (G,N) -valued Function-like V22([: the carrier of (G,N), the carrier of (G,N):]) quasi_total associative Element of bool [:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):]
[: the carrier of (G,N), the carrier of (G,N):] is set
[:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):] is set
bool [:[: the carrier of (G,N), the carrier of (G,N):], the carrier of (G,N):] is set
the multF of (G,N) . (((G,N) . b),((G,N) . a1)) is Element of the carrier of (G,N)
[((G,N) . b),((G,N) . a1)] is set
{((G,N) . b),((G,N) . a1)} is finite set
{((G,N) . b)} is finite set
{{((G,N) . b),((G,N) . a1)},{((G,N) . b)}} is finite V56() set
the multF of (G,N) . [((G,N) . b),((G,N) . a1)] is set
(g . g) * (g . x) is Element of the carrier of ((B "\/" N),(G,(B "\/" N),N))
the multF of ((B "\/" N),(G,(B "\/" N),N)) is Relation-like [: the carrier of ((B "\/" N),(G,(B "\/" N),N)), the carrier of ((B "\/" N),(G,(B "\/" N),N)):] -defined the carrier of ((B "\/" N),(G,(B "\/" N),N)) -valued Function-like V22([: the carrier of ((B "\/" N),(G,(B "\/" N),N)), the carrier of ((B "\/" N),(G,(B "\/" N),N)):]) quasi_total associative Element of bool [:[: the carrier of ((B "\/" N),(G,(B "\/" N),N)), the carrier of ((B "\/" N),(G,(B "\/" N),N)):], the carrier of ((B "\/" N),(G,(B "\/" N),N)):]
[: the carrier of ((B "\/" N),(G,(B "\/" N),N)), the carrier of ((B "\/" N),(G,(B "\/" N),N)):] is set
[:[: the carrier of ((B "\/" N),(G,(B "\/" N),N)), the carrier of ((B "\/" N),(G,(B "\/" N),N)):], the carrier of ((B "\/" N),(G,(B "\/" N),N)):] is set
bool [:[: the carrier of ((B "\/" N),(G,(B "\/" N),N)), the carrier of ((B "\/" N),(G,(B "\/" N),N)):], the carrier of ((B "\/" N),(G,(B "\/" N),N)):] is set
the multF of ((B "\/" N),(G,(B "\/" N),N)) . ((g . g),(g . x)) is Element of the carrier of ((B "\/" N),(G,(B "\/" N),N))
[(g . g),(g . x)] is set
{(g . g),(g . x)} is finite set
{(g . g)} is finite set
{{(g . g),(g . x)},{(g . g)}} is finite V56() set
the multF of ((B "\/" N),(G,(B "\/" N),N)) . [(g . g),(g . x)] is set
g is Relation-like the carrier of B -defined the carrier of ((B "\/" N),(G,(B "\/" N),N)) -valued Function-like non empty V22( the carrier of B) quasi_total unity-preserving (B,((B "\/" N),(G,(B "\/" N),N))) Element of bool [: the carrier of B, the carrier of ((B "\/" N),(G,(B "\/" N),N)):]
(B,((B "\/" N),(G,(B "\/" N),N)),g) is non empty strict unital Group-like associative normal Subgroup of B
x is Element of the carrier of B
g . x is Element of the carrier of ((B "\/" N),(G,(B "\/" N),N))
b is Element of the carrier of G
(G,N) . b is Element of the carrier of (G,N)
b * N is Element of bool the carrier of G
b * (carr N) is Element of bool the carrier of G
K245( the carrier of G,b) is finite Element of bool the carrier of G
K245( the carrier of G,b) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,b) & b₂ in carr N ) } is set
1_ ((B "\/" N),(G,(B "\/" N),N)) is non being_of_order_0 Element of the carrier of ((B "\/" N),(G,(B "\/" N),N))
carr (G,(B "\/" N),N) is Element of bool the carrier of (B "\/" N)
the carrier of (G,(B "\/" N),N) is non empty set
carr (G,(B "\/" N),N) is Element of bool the carrier of (B "\/" N)
the carrier of (G,(B "\/" N),N) is non empty set
1_ ((B "\/" N),(G,(B "\/" N),N)) is non being_of_order_0 Element of the carrier of ((B "\/" N),(G,(B "\/" N),N))
rng g is Element of bool the carrier of ((B "\/" N),(G,(B "\/" N),N))
bool the carrier of ((B "\/" N),(G,(B "\/" N),N)) is set
x is set
b is Element of the carrier of (B "\/" N)
b * (G,(B "\/" N),N) is Element of bool the carrier of (B "\/" N)
carr (G,(B "\/" N),N) is Element of bool the carrier of (B "\/" N)
the carrier of (G,(B "\/" N),N) is non empty set
b * (carr (G,(B "\/" N),N)) is Element of bool the carrier of (B "\/" N)
K245( the carrier of (B "\/" N),b) is finite Element of bool the carrier of (B "\/" N)
K245( the carrier of (B "\/" N),b) * (carr (G,(B "\/" N),N)) is Element of bool the carrier of (B "\/" N)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (B "\/" N) : ( b₁ in K245( the carrier of (B "\/" N),b) & b₂ in carr (G,(B "\/" N),N) ) } is set
(G,(B "\/" N),N) * b is Element of bool the carrier of (B "\/" N)
(carr (G,(B "\/" N),N)) * b is Element of bool the carrier of (B "\/" N)
(carr (G,(B "\/" N),N)) * K245( the carrier of (B "\/" N),b) is Element of bool the carrier of (B "\/" N)
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of (B "\/" N) : ( b₁ in carr (G,(B "\/" N),N) & b₂ in K245( the carrier of (B "\/" N),b) ) } is set
B * N is Element of bool the carrier of G
(carr B) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr B & b₂ in carr N ) } is set
N * B is Element of bool the carrier of G
(carr N) * (carr B) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr N & b₂ in carr B ) } is set
a1 is Element of the carrier of G
a2 is Element of the carrier of G
a1 * a2 is Element of the carrier of G
the multF of G . (a1,a2) is Element of the carrier of G
[a1,a2] is set
{a1,a2} is finite set
{a1} is finite set
{{a1,a2},{a1}} is finite V56() set
the multF of G . [a1,a2] is set
(a1 * a2) * N is Element of bool the carrier of G
(a1 * a2) * (carr N) is Element of bool the carrier of G
K245( the carrier of G,(a1 * a2)) is finite Element of bool the carrier of G
K245( the carrier of G,(a1 * a2)) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,(a1 * a2)) & b₂ in carr N ) } is set
a2 * N is Element of bool the carrier of G
a2 * (carr N) is Element of bool the carrier of G
K245( the carrier of G,a2) is finite Element of bool the carrier of G
K245( the carrier of G,a2) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,a2) & b₂ in carr N ) } is set
a1 * (a2 * N) is Element of bool the carrier of G
K245( the carrier of G,a1) is finite Element of bool the carrier of G
K245( the carrier of G,a1) * (a2 * N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,a1) & b₂ in a2 * N ) } is set
a1 * N is Element of bool the carrier of G
a1 * (carr N) is Element of bool the carrier of G
K245( the carrier of G,a1) * (carr N) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in K245( the carrier of G,a1) & b₂ in carr N ) } is set
(G,N) . a1 is Element of the carrier of (G,N)
g . a1 is set
(B,((B "\/" N),(G,(B "\/" N),N)),g) is non empty strict unital Group-like associative Subgroup of ((B "\/" N),(G,(B "\/" N),N))