:: GROUP_3 semantic presentation

K109() is set
K113() is non empty V12() V13() V14() Element of bool K109()
bool K109() is non empty set
K110() is set
K46() is non empty V12() V13() V14() set
bool K46() is non empty set
K111() is set
K112() is set
[:K110(),K110():] is set
bool [:K110(),K110():] is non empty set
[:[:K110(),K110():],K110():] is set
bool [:[:K110(),K110():],K110():] is non empty set
[:K109(),K109():] is set
bool [:K109(),K109():] is non empty set
[:[:K109(),K109():],K109():] is set
bool [:[:K109(),K109():],K109():] is non empty set
[:K111(),K111():] is set
bool [:K111(),K111():] is non empty set
[:[:K111(),K111():],K111():] is set
bool [:[:K111(),K111():],K111():] is non empty set
[:K112(),K112():] is set
bool [:K112(),K112():] is non empty set
[:[:K112(),K112():],K112():] is set
bool [:[:K112(),K112():],K112():] is non empty set
[:K113(),K113():] is non empty set
[:[:K113(),K113():],K113():] is non empty set
bool [:[:K113(),K113():],K113():] is non empty set
{} is empty V12() V13() V14() V16() V17() V18() Function-like functional ext-real V37() V38() integer finite V45() set
the empty V12() V13() V14() V16() V17() V18() Function-like functional ext-real V37() V38() integer finite V45() set is empty V12() V13() V14() V16() V17() V18() Function-like functional ext-real V37() V38() integer finite V45() set
1 is non empty V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
K114() is empty V12() V13() V14() V16() V17() V18() Function-like functional ext-real V37() V38() integer finite V45() Element of K113()
2 is non empty V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
H * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H,a) is Element of the carrier of G
[H,a] is set
{H,a} is non empty finite set
{H} is non empty finite set
{{H,a},{H}} is non empty finite V45() set
the multF of G . [H,a] is set
a " is Element of the carrier of G
(H * a) * (a ") is Element of the carrier of G
the multF of G . ((H * a),(a ")) is Element of the carrier of G
[(H * a),(a ")] is set
{(H * a),(a ")} is non empty finite set
{(H * a)} is non empty finite set
{{(H * a),(a ")},{(H * a)}} is non empty finite V45() set
the multF of G . [(H * a),(a ")] is set
H * (a ") is Element of the carrier of G
the multF of G . (H,(a ")) is Element of the carrier of G
[H,(a ")] is set
{H,(a ")} is non empty finite set
{{H,(a ")},{H}} is non empty finite V45() set
the multF of G . [H,(a ")] is set
(H * (a ")) * a is Element of the carrier of G
the multF of G . ((H * (a ")),a) is Element of the carrier of G
[(H * (a ")),a] is set
{(H * (a ")),a} is non empty finite set
{(H * (a "))} is non empty finite set
{{(H * (a ")),a},{(H * (a "))}} is non empty finite V45() set
the multF of G . [(H * (a ")),a] is set
(a ") * a is Element of the carrier of G
the multF of G . ((a "),a) is Element of the carrier of G
[(a "),a] is set
{(a "),a} is non empty finite set
{(a ")} is non empty finite set
{{(a "),a},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),a] is set
((a ") * a) * H is Element of the carrier of G
the multF of G . (((a ") * a),H) is Element of the carrier of G
[((a ") * a),H] is set
{((a ") * a),H} is non empty finite set
{((a ") * a)} is non empty finite set
{{((a ") * a),H},{((a ") * a)}} is non empty finite V45() set
the multF of G . [((a ") * a),H] is set
a * (a ") is Element of the carrier of G
the multF of G . (a,(a ")) is Element of the carrier of G
[a,(a ")] is set
{a,(a ")} is non empty finite set
{a} is non empty finite set
{{a,(a ")},{a}} is non empty finite V45() set
the multF of G . [a,(a ")] is set
(a * (a ")) * H is Element of the carrier of G
the multF of G . ((a * (a ")),H) is Element of the carrier of G
[(a * (a ")),H] is set
{(a * (a ")),H} is non empty finite set
{(a * (a "))} is non empty finite set
{{(a * (a ")),H},{(a * (a "))}} is non empty finite V45() set
the multF of G . [(a * (a ")),H] is set
H * (a * (a ")) is Element of the carrier of G
the multF of G . (H,(a * (a "))) is Element of the carrier of G
[H,(a * (a "))] is set
{H,(a * (a "))} is non empty finite set
{{H,(a * (a "))},{H}} is non empty finite V45() set
the multF of G . [H,(a * (a "))] is set
H * ((a ") * a) is Element of the carrier of G
the multF of G . (H,((a ") * a)) is Element of the carrier of G
[H,((a ") * a)] is set
{H,((a ") * a)} is non empty finite set
{{H,((a ") * a)},{H}} is non empty finite V45() set
the multF of G . [H,((a ") * a)] is set
a * H is Element of the carrier of G
the multF of G . (a,H) is Element of the carrier of G
[a,H] is set
{a,H} is non empty finite set
{{a,H},{a}} is non empty finite V45() set
the multF of G . [a,H] is set
(a ") * (a * H) is Element of the carrier of G
the multF of G . ((a "),(a * H)) is Element of the carrier of G
[(a "),(a * H)] is set
{(a "),(a * H)} is non empty finite set
{{(a "),(a * H)},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(a * H)] is set
(a ") * H is Element of the carrier of G
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
a * ((a ") * H) is Element of the carrier of G
the multF of G . (a,((a ") * H)) is Element of the carrier of G
[a,((a ") * H)] is set
{a,((a ") * H)} is non empty finite set
{{a,((a ") * H)},{a}} is non empty finite V45() set
the multF of G . [a,((a ") * H)] is set
1_ G is non being_of_order_0 Element of the carrier of G
H * (1_ G) is Element of the carrier of G
the multF of G . (H,(1_ G)) is Element of the carrier of G
[H,(1_ G)] is set
{H,(1_ G)} is non empty finite set
{{H,(1_ G)},{H}} is non empty finite V45() set
the multF of G . [H,(1_ G)] is set
(1_ G) * H is Element of the carrier of G
the multF of G . ((1_ G),H) is Element of the carrier of G
[(1_ G),H] is set
{(1_ G),H} is non empty finite set
{(1_ G)} is non empty finite set
{{(1_ G),H},{(1_ G)}} is non empty finite V45() set
the multF of G . [(1_ G),H] is set
G is non empty unital Group-like associative commutative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
H * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H,a) is Element of the carrier of G
[H,a] is set
{H,a} is non empty finite set
{H} is non empty finite set
{{H,a},{H}} is non empty finite V45() set
the multF of G . [H,a] is set
a * H is Element of the carrier of G
the multF of G . (a,H) is Element of the carrier of G
[a,H] is set
{a,H} is non empty finite set
{a} is non empty finite set
{{a,H},{a}} is non empty finite V45() set
the multF of G . [a,H] is set
G is non empty unital Group-like associative multMagma
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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
the multF of G . (H,a) is Element of the carrier of G
[H,a] is set
{H,a} is non empty finite set
{H} is non empty finite set
{{H,a},{H}} is non empty finite V45() set
the multF of G . [H,a] is set
the multF of G . (a,H) is Element of the carrier of G
[a,H] is set
{a,H} is non empty finite set
{a} is non empty finite set
{{a,H},{a}} is non empty finite V45() set
the multF of G . [a,H] is set
the multF of G . (H,a) is Element of the carrier of G
H * a is Element of the carrier of G
a * H is Element of the carrier of G
the multF of G . (a,H) is Element of the carrier of G
H is Element of the carrier of G
a is Element of the carrier of G
H * a is Element of the carrier of G
the multF of G . (H,a) is Element of the carrier of G
[H,a] is set
{H,a} is non empty finite set
{H} is non empty finite set
{{H,a},{H}} is non empty finite V45() set
the multF of G . [H,a] is set
a * H is Element of the carrier of G
the multF of G . (a,H) is Element of the carrier of G
[a,H] is set
{a,H} is non empty finite set
{a} is non empty finite set
{{a,H},{a}} is non empty finite V45() set
the multF of G . [a,H] is set
G is non empty unital Group-like associative multMagma
(1). G is non empty finite strict unital Group-like associative Subgroup of G
the carrier of ((1). G) is non empty finite set
H is Element of the carrier of ((1). G)
a is Element of the carrier of ((1). G)
H * a is Element of the carrier of ((1). G)
the multF of ((1). G) is Relation-like [: the carrier of ((1). G), the carrier of ((1). G):] -defined the carrier of ((1). G) -valued Function-like V28([: the carrier of ((1). G), the carrier of ((1). G):], the carrier of ((1). G)) associative finite Element of bool [:[: the carrier of ((1). G), the carrier of ((1). G):], the carrier of ((1). G):]
[: the carrier of ((1). G), the carrier of ((1). G):] is non empty finite set
[:[: the carrier of ((1). G), the carrier of ((1). G):], the carrier of ((1). G):] is non empty finite set
bool [:[: the carrier of ((1). G), the carrier of ((1). G):], the carrier of ((1). G):] is non empty finite V45() set
the multF of ((1). G) . (H,a) is Element of the carrier of ((1). G)
[H,a] is set
{H,a} is non empty finite set
{H} is non empty finite set
{{H,a},{H}} is non empty finite V45() set
the multF of ((1). G) . [H,a] is set
a * H is Element of the carrier of ((1). G)
the multF of ((1). G) . (a,H) is Element of the carrier of ((1). G)
[a,H] is set
{a,H} is non empty finite set
{a} is non empty finite set
{{a,H},{a}} is non empty finite V45() set
the multF of ((1). G) . [a,H] is set
the carrier of G is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
bool 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
bool the carrier of G is non empty set
H is Element of bool the carrier of G
a is Element of bool the carrier of G
c₄ is Element of bool the carrier of G
H * c₄ is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in c₄ ) } is set
y is Element of bool the carrier of G
a * y is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in y ) } is set
H1 is set
a is Element of the carrier of G
b is Element of the carrier of G
a * 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (a,b) is Element of the carrier of G
[a,b] is set
{a,b} is non empty finite set
{a} is non empty finite set
{{a,b},{a}} is non empty finite V45() set
the multF of G . [a,b] 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 non empty set
a is Element of bool the carrier of G
c₄ is Element of bool the carrier of G
H is Element of the carrier of G
H * a is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
{H} * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in a ) } is set
H * c₄ is Element of bool the carrier of G
{H} * c₄ is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in c₄ ) } is set
a * H is Element of bool the carrier of G
a * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {H} ) } is set
c₄ * H is Element of bool the carrier of G
c₄ * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in c₄ & b₂ in {H} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is non empty unital Group-like associative Subgroup of G
H * a is Element of bool the carrier of G
bool the carrier of G is non empty set
carr a is Element of bool the carrier of G
the carrier of a is non empty set
H * (carr a) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
{H} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr a ) } is set
a * H is Element of bool the carrier of G
(carr a) * H is Element of bool the carrier of G
(carr a) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in {H} ) } is set
c₄ is non empty unital Group-like associative Subgroup of G
H * c₄ is Element of bool the carrier of G
carr c₄ is Element of bool the carrier of G
the carrier of c₄ is non empty set
H * (carr c₄) is Element of bool the carrier of G
{H} * (carr c₄) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr c₄ ) } is set
c₄ * H is Element of bool the carrier of G
(carr c₄) * H is Element of bool the carrier of G
(carr c₄) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr c₄ & b₂ in {H} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
a is non empty unital Group-like associative Subgroup of G
H is Element of the carrier of G
H * a is Element of bool the carrier of G
bool the carrier of G is non empty set
carr a is Element of bool the carrier of G
the carrier of a is non empty set
H * (carr a) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
{H} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr a ) } is set
{H} * a is Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
a is non empty unital Group-like associative Subgroup of G
H is Element of the carrier of G
a * H is Element of bool the carrier of G
bool the carrier of G is non empty set
carr a is Element of bool the carrier of G
the carrier of a is non empty set
(carr a) * H is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(carr a) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in {H} ) } is set
a * {H} is Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
H is Element of the carrier of G
a is Element of bool the carrier of G
a * H is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
a * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {H} ) } is set
c₄ is non empty unital Group-like associative Subgroup of G
(a * H) * c₄ is Element of bool the carrier of G
carr c₄ is Element of bool the carrier of G
the carrier of c₄ is non empty set
(a * H) * (carr c₄) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a * H & b₂ in carr c₄ ) } is set
H * c₄ is Element of bool the carrier of G
H * (carr c₄) is Element of bool the carrier of G
{H} * (carr c₄) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr c₄ ) } is set
a * (H * c₄) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in H * c₄ ) } is set
{H} * c₄ is Element of bool the carrier of G
a * ({H} * c₄) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {H} * c₄ ) } 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 non empty set
H is Element of the carrier of G
a is Element of bool the carrier of G
c₄ is non empty unital Group-like associative Subgroup of G
H * c₄ is Element of bool the carrier of G
carr c₄ is Element of bool the carrier of G
the carrier of c₄ is non empty set
H * (carr c₄) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
{H} * (carr c₄) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr c₄ ) } is set
(H * c₄) * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H * c₄ & b₂ in a ) } is set
c₄ * a is Element of bool the carrier of G
(carr c₄) * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr c₄ & b₂ in a ) } is set
H * (c₄ * a) is Element of bool the carrier of G
{H} * (c₄ * a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in c₄ * a ) } is set
{H} * c₄ is Element of bool the carrier of G
({H} * c₄) * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} * c₄ & b₂ in a ) } 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 non empty set
H is Element of the carrier of G
a is Element of bool the carrier of G
c₄ is non empty unital Group-like associative Subgroup of G
a * c₄ is Element of bool the carrier of G
carr c₄ is Element of bool the carrier of G
the carrier of c₄ is non empty set
a * (carr c₄) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in carr c₄ ) } is set
(a * c₄) * H is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(a * c₄) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a * c₄ & b₂ in {H} ) } is set
c₄ * H is Element of bool the carrier of G
(carr c₄) * H is Element of bool the carrier of G
(carr c₄) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr c₄ & b₂ in {H} ) } is set
a * (c₄ * H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in c₄ * H ) } is set
c₄ * {H} is Element of bool the carrier of G
a * (c₄ * {H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in c₄ * {H} ) } 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 non empty set
H is Element of the carrier of G
a is Element of bool the carrier of G
H * a is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
{H} * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in a ) } is set
c₄ is non empty unital Group-like associative Subgroup of G
c₄ * H is Element of bool the carrier of G
carr c₄ is Element of bool the carrier of G
the carrier of c₄ is non empty set
(carr c₄) * H is Element of bool the carrier of G
(carr c₄) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr c₄ & b₂ in {H} ) } is set
(c₄ * H) * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in c₄ * H & b₂ in a ) } is set
c₄ * (H * a) is Element of bool the carrier of G
(carr c₄) * (H * a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr c₄ & b₂ in H * a ) } is set
c₄ * {H} is Element of bool the carrier of G
(c₄ * {H}) * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in c₄ * {H} & b₂ in a ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
c₄ is non empty unital Group-like associative Subgroup of G
a is non empty unital Group-like associative Subgroup of G
H is Element of the carrier of G
a * H is Element of bool the carrier of G
bool the carrier of G is non empty set
carr a is Element of bool the carrier of G
the carrier of a is non empty set
(carr a) * H is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(carr a) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in {H} ) } is set
(a * H) * c₄ is Element of bool the carrier of G
carr c₄ is Element of bool the carrier of G
the carrier of c₄ is non empty set
(a * H) * (carr c₄) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a * H & b₂ in carr c₄ ) } is set
H * c₄ is Element of bool the carrier of G
H * (carr c₄) is Element of bool the carrier of G
{H} * (carr c₄) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr c₄ ) } is set
a * (H * c₄) is Element of bool the carrier of G
(carr a) * (H * c₄) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in H * c₄ ) } 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 non empty set
H is set
c₄ is set
H1 is non empty strict unital Group-like associative Subgroup of G
a is set
the carrier of H1 is non empty set
y is set
a is non empty strict unital Group-like associative Subgroup of G
the carrier of a is non empty set
a is Relation-like Function-like set
c₄ is set
y is non empty strict unital Group-like associative Subgroup of G
the carrier of y is non empty set
H1 is set
[c₄,H1] is set
{c₄,H1} is non empty finite set
{c₄} is non empty finite set
{{c₄,H1},{c₄}} is non empty finite V45() set
y is set
[c₄,y] is set
{c₄,y} is non empty finite set
{c₄} is non empty finite set
{{c₄,y},{c₄}} is non empty finite V45() set
dom a is set
rng a is set
c₄ is set
y is set
a . y is set
[y,c₄] is set
{y,c₄} is non empty finite set
{y} is non empty finite set
{{y,c₄},{y}} is non empty finite V45() set
H1 is non empty strict unital Group-like associative Subgroup of G
the carrier of H1 is non empty set
y is non empty strict unital Group-like associative Subgroup of G
the carrier of y is non empty set
H1 is set
[H1,c₄] is set
{H1,c₄} is non empty finite set
{H1} is non empty finite set
{{H1,c₄},{H1}} is non empty finite V45() set
a . H1 is set
H is set
a is set
G is non empty unital Group-like associative multMagma
(G) is set
the non empty strict unital Group-like associative Subgroup of G is non empty strict unital Group-like associative Subgroup of G
G is non empty strict unital Group-like associative multMagma
(G) is non empty set
G is non empty unital Group-like associative multMagma
(G) is non empty set
H is set
a is non empty strict unital Group-like associative Subgroup of G
the carrier of a is non empty set
c₄ is set
H is Relation-like Function-like set
dom H is set
rng H is set
the carrier of G is non empty set
bool the carrier of G is non empty set
a is set
c₄ is set
H . c₄ is set
y is non empty strict unital Group-like associative Subgroup of G
the carrier of y is non empty set
a is set
H . a is set
c₄ is set
H . c₄ is set
y is non empty strict unital Group-like associative Subgroup of G
the carrier of y is non empty set
H1 is non empty strict unital Group-like associative Subgroup of G
the carrier of H1 is non empty set
card (G) is cardinal set
card (bool the carrier of G) is cardinal set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
a is Element of the carrier of G
a " is Element of the carrier of G
H is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
c₄ is Element of the carrier of G
(G,c₄,a) is Element of the carrier of G
(a ") * c₄ is Element of the carrier of G
the multF of G . ((a "),c₄) is Element of the carrier of G
[(a "),c₄] is set
{(a "),c₄} is non empty finite set
{{(a "),c₄},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),c₄] is set
((a ") * c₄) * a is Element of the carrier of G
the multF of G . (((a ") * c₄),a) is Element of the carrier of G
[((a ") * c₄),a] is set
{((a ") * c₄),a} is non empty finite set
{((a ") * c₄)} is non empty finite set
{{((a ") * c₄),a},{((a ") * c₄)}} is non empty finite V45() set
the multF of G . [((a ") * c₄),a] 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
H is Element of the carrier of G
(G,(1_ G),H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * (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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),(1_ G)) is Element of the carrier of G
[(H "),(1_ G)] is set
{(H "),(1_ G)} is non empty finite set
{(H ")} is non empty finite set
{{(H "),(1_ G)},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),(1_ G)] is set
((H ") * (1_ G)) * H is Element of the carrier of G
the multF of G . (((H ") * (1_ G)),H) is Element of the carrier of G
[((H ") * (1_ G)),H] is set
{((H ") * (1_ G)),H} is non empty finite set
{((H ") * (1_ G))} is non empty finite set
{{((H ") * (1_ G)),H},{((H ") * (1_ G))}} is non empty finite V45() set
the multF of G . [((H ") * (1_ G)),H] is set
(H ") * H is Element of the carrier of G
the multF of G . ((H "),H) is Element of the carrier of G
[(H "),H] is set
{(H "),H} is non empty finite set
{{(H "),H},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),H] 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
H is Element of the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] 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
H is Element of the carrier of G
(G,H,(1_ G)) is Element of the carrier of G
(1_ G) " is Element of the carrier of G
((1_ G) ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (((1_ G) "),H) is Element of the carrier of G
[((1_ G) "),H] is set
{((1_ G) "),H} is non empty finite set
{((1_ G) ")} is non empty finite set
{{((1_ G) "),H},{((1_ G) ")}} is non empty finite V45() set
the multF of G . [((1_ G) "),H] is set
(((1_ G) ") * H) * (1_ G) is Element of the carrier of G
the multF of G . ((((1_ G) ") * H),(1_ G)) is Element of the carrier of G
[(((1_ G) ") * H),(1_ G)] is set
{(((1_ G) ") * H),(1_ G)} is non empty finite set
{(((1_ G) ") * H)} is non empty finite set
{{(((1_ G) ") * H),(1_ G)},{(((1_ G) ") * H)}} is non empty finite V45() set
the multF of G . [(((1_ G) ") * H),(1_ G)] is set
(1_ G) * H is Element of the carrier of G
the multF of G . ((1_ G),H) is Element of the carrier of G
[(1_ G),H] is set
{(1_ G),H} is non empty finite set
{(1_ G)} is non empty finite set
{{(1_ G),H},{(1_ G)}} is non empty finite V45() set
the multF of G . [(1_ G),H] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
(G,H,H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),H) is Element of the carrier of G
[(H "),H] is set
{(H "),H} is non empty finite set
{(H ")} is non empty finite set
{{(H "),H},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),H] is set
((H ") * H) * H is Element of the carrier of G
the multF of G . (((H ") * H),H) is Element of the carrier of G
[((H ") * H),H] is set
{((H ") * H),H} is non empty finite set
{((H ") * H)} is non empty finite set
{{((H ") * H),H},{((H ") * H)}} is non empty finite V45() set
the multF of G . [((H ") * H),H] is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * H is Element of the carrier of G
the multF of G . ((1_ G),H) is Element of the carrier of G
[(1_ G),H] is set
{(1_ G),H} is non empty finite set
{(1_ G)} is non empty finite set
{{(1_ G),H},{(1_ G)}} is non empty finite V45() set
the multF of G . [(1_ G),H] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
a is non empty unital Group-like associative multMagma
the carrier of a is non empty set
H is Element of the carrier of G
H " is Element of the carrier of G
(G,H,(H ")) is Element of the carrier of G
(H ") " is Element of the carrier of G
((H ") ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (((H ") "),H) is Element of the carrier of G
[((H ") "),H] is set
{((H ") "),H} is non empty finite set
{((H ") ")} is non empty finite set
{{((H ") "),H},{((H ") ")}} is non empty finite V45() set
the multF of G . [((H ") "),H] is set
(((H ") ") * H) * (H ") is Element of the carrier of G
the multF of G . ((((H ") ") * H),(H ")) is Element of the carrier of G
[(((H ") ") * H),(H ")] is set
{(((H ") ") * H),(H ")} is non empty finite set
{(((H ") ") * H)} is non empty finite set
{{(((H ") ") * H),(H ")},{(((H ") ") * H)}} is non empty finite V45() set
the multF of G . [(((H ") ") * H),(H ")] is set
c₄ is Element of the carrier of a
c₄ " is Element of the carrier of a
(a,(c₄ "),c₄) is Element of the carrier of a
(c₄ ") * (c₄ ") is Element of the carrier of a
the multF of a is Relation-like [: the carrier of a, the carrier of a:] -defined the carrier of a -valued Function-like V28([: the carrier of a, the carrier of a:], the carrier of a) associative Element of bool [:[: the carrier of a, the carrier of a:], the carrier of a:]
[: the carrier of a, the carrier of a:] is non empty set
[:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
bool [:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
the multF of a . ((c₄ "),(c₄ ")) is Element of the carrier of a
[(c₄ "),(c₄ ")] is set
{(c₄ "),(c₄ ")} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),(c₄ ")},{(c₄ ")}} is non empty finite V45() set
the multF of a . [(c₄ "),(c₄ ")] is set
((c₄ ") * (c₄ ")) * c₄ is Element of the carrier of a
the multF of a . (((c₄ ") * (c₄ ")),c₄) is Element of the carrier of a
[((c₄ ") * (c₄ ")),c₄] is set
{((c₄ ") * (c₄ ")),c₄} is non empty finite set
{((c₄ ") * (c₄ "))} is non empty finite set
{{((c₄ ") * (c₄ ")),c₄},{((c₄ ") * (c₄ "))}} is non empty finite V45() set
the multF of a . [((c₄ ") * (c₄ ")),c₄] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
H * a is Element of the carrier of G
the multF of G . (H,a) is Element of the carrier of G
[H,a] is set
{H,a} is non empty finite set
{H} is non empty finite set
{{H,a},{H}} is non empty finite V45() set
the multF of G . [H,a] is set
a * H is Element of the carrier of G
the multF of G . (a,H) is Element of the carrier of G
[a,H] is set
{a,H} is non empty finite set
{a} is non empty finite set
{{a,H},{a}} is non empty finite V45() set
the multF of G . [a,H] is set
(a ") * (H * a) is Element of the carrier of G
the multF of G . ((a "),(H * a)) is Element of the carrier of G
[(a "),(H * a)] is set
{(a "),(H * a)} is non empty finite set
{{(a "),(H * a)},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(H * a)] is set
(a ") * (H * a) is Element of the carrier of G
the multF of G . ((a "),(H * a)) is Element of the carrier of G
[(a "),(H * a)] is set
{(a "),(H * a)} is non empty finite set
{{(a "),(H * a)},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(H * a)] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
H * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H,a) is Element of the carrier of G
[H,a] is set
{H,a} is non empty finite set
{H} is non empty finite set
{{H,a},{H}} is non empty finite V45() set
the multF of G . [H,a] is set
c₄ is Element of the carrier of G
(G,(H * a),c₄) is Element of the carrier of G
c₄ " is Element of the carrier of G
(c₄ ") * (H * a) is Element of the carrier of G
the multF of G . ((c₄ "),(H * a)) is Element of the carrier of G
[(c₄ "),(H * a)] is set
{(c₄ "),(H * a)} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),(H * a)},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),(H * a)] is set
((c₄ ") * (H * a)) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * (H * a)),c₄) is Element of the carrier of G
[((c₄ ") * (H * a)),c₄] is set
{((c₄ ") * (H * a)),c₄} is non empty finite set
{((c₄ ") * (H * a))} is non empty finite set
{{((c₄ ") * (H * a)),c₄},{((c₄ ") * (H * a))}} is non empty finite V45() set
the multF of G . [((c₄ ") * (H * a)),c₄] is set
(G,H,c₄) is Element of the carrier of G
(c₄ ") * H is Element of the carrier of G
the multF of G . ((c₄ "),H) is Element of the carrier of G
[(c₄ "),H] is set
{(c₄ "),H} is non empty finite set
{{(c₄ "),H},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),H] is set
((c₄ ") * H) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * H),c₄) is Element of the carrier of G
[((c₄ ") * H),c₄] is set
{((c₄ ") * H),c₄} is non empty finite set
{((c₄ ") * H)} is non empty finite set
{{((c₄ ") * H),c₄},{((c₄ ") * H)}} is non empty finite V45() set
the multF of G . [((c₄ ") * H),c₄] is set
(G,a,c₄) is Element of the carrier of G
(c₄ ") * a is Element of the carrier of G
the multF of G . ((c₄ "),a) is Element of the carrier of G
[(c₄ "),a] is set
{(c₄ "),a} is non empty finite set
{{(c₄ "),a},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),a] is set
((c₄ ") * a) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * a),c₄) is Element of the carrier of G
[((c₄ ") * a),c₄] is set
{((c₄ ") * a),c₄} is non empty finite set
{((c₄ ") * a)} is non empty finite set
{{((c₄ ") * a),c₄},{((c₄ ") * a)}} is non empty finite V45() set
the multF of G . [((c₄ ") * a),c₄] is set
(G,H,c₄) * (G,a,c₄) is Element of the carrier of G
the multF of G . ((G,H,c₄),(G,a,c₄)) is Element of the carrier of G
[(G,H,c₄),(G,a,c₄)] is set
{(G,H,c₄),(G,a,c₄)} is non empty finite set
{(G,H,c₄)} is non empty finite set
{{(G,H,c₄),(G,a,c₄)},{(G,H,c₄)}} is non empty finite V45() set
the multF of G . [(G,H,c₄),(G,a,c₄)] is set
1_ G is non being_of_order_0 Element of the carrier of G
H * (1_ G) is Element of the carrier of G
the multF of G . (H,(1_ G)) is Element of the carrier of G
[H,(1_ G)] is set
{H,(1_ G)} is non empty finite set
{{H,(1_ G)},{H}} is non empty finite V45() set
the multF of G . [H,(1_ G)] is set
(H * (1_ G)) * a is Element of the carrier of G
the multF of G . ((H * (1_ G)),a) is Element of the carrier of G
[(H * (1_ G)),a] is set
{(H * (1_ G)),a} is non empty finite set
{(H * (1_ G))} is non empty finite set
{{(H * (1_ G)),a},{(H * (1_ G))}} is non empty finite V45() set
the multF of G . [(H * (1_ G)),a] is set
(c₄ ") * ((H * (1_ G)) * a) is Element of the carrier of G
the multF of G . ((c₄ "),((H * (1_ G)) * a)) is Element of the carrier of G
[(c₄ "),((H * (1_ G)) * a)] is set
{(c₄ "),((H * (1_ G)) * a)} is non empty finite set
{{(c₄ "),((H * (1_ G)) * a)},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),((H * (1_ G)) * a)] is set
((c₄ ") * ((H * (1_ G)) * a)) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * ((H * (1_ G)) * a)),c₄) is Element of the carrier of G
[((c₄ ") * ((H * (1_ G)) * a)),c₄] is set
{((c₄ ") * ((H * (1_ G)) * a)),c₄} is non empty finite set
{((c₄ ") * ((H * (1_ G)) * a))} is non empty finite set
{{((c₄ ") * ((H * (1_ G)) * a)),c₄},{((c₄ ") * ((H * (1_ G)) * a))}} is non empty finite V45() set
the multF of G . [((c₄ ") * ((H * (1_ G)) * a)),c₄] is set
c₄ * (c₄ ") is Element of the carrier of G
the multF of G . (c₄,(c₄ ")) is Element of the carrier of G
[c₄,(c₄ ")] is set
{c₄,(c₄ ")} is non empty finite set
{c₄} is non empty finite set
{{c₄,(c₄ ")},{c₄}} is non empty finite V45() set
the multF of G . [c₄,(c₄ ")] is set
H * (c₄ * (c₄ ")) is Element of the carrier of G
the multF of G . (H,(c₄ * (c₄ "))) is Element of the carrier of G
[H,(c₄ * (c₄ "))] is set
{H,(c₄ * (c₄ "))} is non empty finite set
{{H,(c₄ * (c₄ "))},{H}} is non empty finite V45() set
the multF of G . [H,(c₄ * (c₄ "))] is set
(H * (c₄ * (c₄ "))) * a is Element of the carrier of G
the multF of G . ((H * (c₄ * (c₄ "))),a) is Element of the carrier of G
[(H * (c₄ * (c₄ "))),a] is set
{(H * (c₄ * (c₄ "))),a} is non empty finite set
{(H * (c₄ * (c₄ ")))} is non empty finite set
{{(H * (c₄ * (c₄ "))),a},{(H * (c₄ * (c₄ ")))}} is non empty finite V45() set
the multF of G . [(H * (c₄ * (c₄ "))),a] is set
(c₄ ") * ((H * (c₄ * (c₄ "))) * a) is Element of the carrier of G
the multF of G . ((c₄ "),((H * (c₄ * (c₄ "))) * a)) is Element of the carrier of G
[(c₄ "),((H * (c₄ * (c₄ "))) * a)] is set
{(c₄ "),((H * (c₄ * (c₄ "))) * a)} is non empty finite set
{{(c₄ "),((H * (c₄ * (c₄ "))) * a)},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),((H * (c₄ * (c₄ "))) * a)] is set
((c₄ ") * ((H * (c₄ * (c₄ "))) * a)) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * ((H * (c₄ * (c₄ "))) * a)),c₄) is Element of the carrier of G
[((c₄ ") * ((H * (c₄ * (c₄ "))) * a)),c₄] is set
{((c₄ ") * ((H * (c₄ * (c₄ "))) * a)),c₄} is non empty finite set
{((c₄ ") * ((H * (c₄ * (c₄ "))) * a))} is non empty finite set
{{((c₄ ") * ((H * (c₄ * (c₄ "))) * a)),c₄},{((c₄ ") * ((H * (c₄ * (c₄ "))) * a))}} is non empty finite V45() set
the multF of G . [((c₄ ") * ((H * (c₄ * (c₄ "))) * a)),c₄] is set
H * c₄ is Element of the carrier of G
the multF of G . (H,c₄) is Element of the carrier of G
[H,c₄] is set
{H,c₄} is non empty finite set
{{H,c₄},{H}} is non empty finite V45() set
the multF of G . [H,c₄] is set
(H * c₄) * (c₄ ") is Element of the carrier of G
the multF of G . ((H * c₄),(c₄ ")) is Element of the carrier of G
[(H * c₄),(c₄ ")] is set
{(H * c₄),(c₄ ")} is non empty finite set
{(H * c₄)} is non empty finite set
{{(H * c₄),(c₄ ")},{(H * c₄)}} is non empty finite V45() set
the multF of G . [(H * c₄),(c₄ ")] is set
((H * c₄) * (c₄ ")) * a is Element of the carrier of G
the multF of G . (((H * c₄) * (c₄ ")),a) is Element of the carrier of G
[((H * c₄) * (c₄ ")),a] is set
{((H * c₄) * (c₄ ")),a} is non empty finite set
{((H * c₄) * (c₄ "))} is non empty finite set
{{((H * c₄) * (c₄ ")),a},{((H * c₄) * (c₄ "))}} is non empty finite V45() set
the multF of G . [((H * c₄) * (c₄ ")),a] is set
(c₄ ") * (((H * c₄) * (c₄ ")) * a) is Element of the carrier of G
the multF of G . ((c₄ "),(((H * c₄) * (c₄ ")) * a)) is Element of the carrier of G
[(c₄ "),(((H * c₄) * (c₄ ")) * a)] is set
{(c₄ "),(((H * c₄) * (c₄ ")) * a)} is non empty finite set
{{(c₄ "),(((H * c₄) * (c₄ ")) * a)},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),(((H * c₄) * (c₄ ")) * a)] is set
((c₄ ") * (((H * c₄) * (c₄ ")) * a)) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * (((H * c₄) * (c₄ ")) * a)),c₄) is Element of the carrier of G
[((c₄ ") * (((H * c₄) * (c₄ ")) * a)),c₄] is set
{((c₄ ") * (((H * c₄) * (c₄ ")) * a)),c₄} is non empty finite set
{((c₄ ") * (((H * c₄) * (c₄ ")) * a))} is non empty finite set
{{((c₄ ") * (((H * c₄) * (c₄ ")) * a)),c₄},{((c₄ ") * (((H * c₄) * (c₄ ")) * a))}} is non empty finite V45() set
the multF of G . [((c₄ ") * (((H * c₄) * (c₄ ")) * a)),c₄] is set
(H * c₄) * ((c₄ ") * a) is Element of the carrier of G
the multF of G . ((H * c₄),((c₄ ") * a)) is Element of the carrier of G
[(H * c₄),((c₄ ") * a)] is set
{(H * c₄),((c₄ ") * a)} is non empty finite set
{{(H * c₄),((c₄ ") * a)},{(H * c₄)}} is non empty finite V45() set
the multF of G . [(H * c₄),((c₄ ") * a)] is set
(c₄ ") * ((H * c₄) * ((c₄ ") * a)) is Element of the carrier of G
the multF of G . ((c₄ "),((H * c₄) * ((c₄ ") * a))) is Element of the carrier of G
[(c₄ "),((H * c₄) * ((c₄ ") * a))] is set
{(c₄ "),((H * c₄) * ((c₄ ") * a))} is non empty finite set
{{(c₄ "),((H * c₄) * ((c₄ ") * a))},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),((H * c₄) * ((c₄ ") * a))] is set
((c₄ ") * ((H * c₄) * ((c₄ ") * a))) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * ((H * c₄) * ((c₄ ") * a))),c₄) is Element of the carrier of G
[((c₄ ") * ((H * c₄) * ((c₄ ") * a))),c₄] is set
{((c₄ ") * ((H * c₄) * ((c₄ ") * a))),c₄} is non empty finite set
{((c₄ ") * ((H * c₄) * ((c₄ ") * a)))} is non empty finite set
{{((c₄ ") * ((H * c₄) * ((c₄ ") * a))),c₄},{((c₄ ") * ((H * c₄) * ((c₄ ") * a)))}} is non empty finite V45() set
the multF of G . [((c₄ ") * ((H * c₄) * ((c₄ ") * a))),c₄] is set
(c₄ ") * (H * c₄) is Element of the carrier of G
the multF of G . ((c₄ "),(H * c₄)) is Element of the carrier of G
[(c₄ "),(H * c₄)] is set
{(c₄ "),(H * c₄)} is non empty finite set
{{(c₄ "),(H * c₄)},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),(H * c₄)] is set
((c₄ ") * (H * c₄)) * ((c₄ ") * a) is Element of the carrier of G
the multF of G . (((c₄ ") * (H * c₄)),((c₄ ") * a)) is Element of the carrier of G
[((c₄ ") * (H * c₄)),((c₄ ") * a)] is set
{((c₄ ") * (H * c₄)),((c₄ ") * a)} is non empty finite set
{((c₄ ") * (H * c₄))} is non empty finite set
{{((c₄ ") * (H * c₄)),((c₄ ") * a)},{((c₄ ") * (H * c₄))}} is non empty finite V45() set
the multF of G . [((c₄ ") * (H * c₄)),((c₄ ") * a)] is set
(((c₄ ") * (H * c₄)) * ((c₄ ") * a)) * c₄ is Element of the carrier of G
the multF of G . ((((c₄ ") * (H * c₄)) * ((c₄ ") * a)),c₄) is Element of the carrier of G
[(((c₄ ") * (H * c₄)) * ((c₄ ") * a)),c₄] is set
{(((c₄ ") * (H * c₄)) * ((c₄ ") * a)),c₄} is non empty finite set
{(((c₄ ") * (H * c₄)) * ((c₄ ") * a))} is non empty finite set
{{(((c₄ ") * (H * c₄)) * ((c₄ ") * a)),c₄},{(((c₄ ") * (H * c₄)) * ((c₄ ") * a))}} is non empty finite V45() set
the multF of G . [(((c₄ ") * (H * c₄)) * ((c₄ ") * a)),c₄] is set
(G,H,c₄) * ((c₄ ") * a) is Element of the carrier of G
the multF of G . ((G,H,c₄),((c₄ ") * a)) is Element of the carrier of G
[(G,H,c₄),((c₄ ") * a)] is set
{(G,H,c₄),((c₄ ") * a)} is non empty finite set
{{(G,H,c₄),((c₄ ") * a)},{(G,H,c₄)}} is non empty finite V45() set
the multF of G . [(G,H,c₄),((c₄ ") * a)] is set
((G,H,c₄) * ((c₄ ") * a)) * c₄ is Element of the carrier of G
the multF of G . (((G,H,c₄) * ((c₄ ") * a)),c₄) is Element of the carrier of G
[((G,H,c₄) * ((c₄ ") * a)),c₄] is set
{((G,H,c₄) * ((c₄ ") * a)),c₄} is non empty finite set
{((G,H,c₄) * ((c₄ ") * a))} is non empty finite set
{{((G,H,c₄) * ((c₄ ") * a)),c₄},{((G,H,c₄) * ((c₄ ") * a))}} is non empty finite V45() set
the multF of G . [((G,H,c₄) * ((c₄ ") * a)),c₄] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
c₄ is Element of the carrier of G
(G,(G,H,a),c₄) is Element of the carrier of G
c₄ " is Element of the carrier of G
(c₄ ") * (G,H,a) is Element of the carrier of G
the multF of G . ((c₄ "),(G,H,a)) is Element of the carrier of G
[(c₄ "),(G,H,a)] is set
{(c₄ "),(G,H,a)} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),(G,H,a)},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),(G,H,a)] is set
((c₄ ") * (G,H,a)) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * (G,H,a)),c₄) is Element of the carrier of G
[((c₄ ") * (G,H,a)),c₄] is set
{((c₄ ") * (G,H,a)),c₄} is non empty finite set
{((c₄ ") * (G,H,a))} is non empty finite set
{{((c₄ ") * (G,H,a)),c₄},{((c₄ ") * (G,H,a))}} is non empty finite V45() set
the multF of G . [((c₄ ") * (G,H,a)),c₄] is set
a * c₄ is Element of the carrier of G
the multF of G . (a,c₄) is Element of the carrier of G
[a,c₄] is set
{a,c₄} is non empty finite set
{a} is non empty finite set
{{a,c₄},{a}} is non empty finite V45() set
the multF of G . [a,c₄] is set
(G,H,(a * c₄)) is Element of the carrier of G
(a * c₄) " is Element of the carrier of G
((a * c₄) ") * H is Element of the carrier of G
the multF of G . (((a * c₄) "),H) is Element of the carrier of G
[((a * c₄) "),H] is set
{((a * c₄) "),H} is non empty finite set
{((a * c₄) ")} is non empty finite set
{{((a * c₄) "),H},{((a * c₄) ")}} is non empty finite V45() set
the multF of G . [((a * c₄) "),H] is set
(((a * c₄) ") * H) * (a * c₄) is Element of the carrier of G
the multF of G . ((((a * c₄) ") * H),(a * c₄)) is Element of the carrier of G
[(((a * c₄) ") * H),(a * c₄)] is set
{(((a * c₄) ") * H),(a * c₄)} is non empty finite set
{(((a * c₄) ") * H)} is non empty finite set
{{(((a * c₄) ") * H),(a * c₄)},{(((a * c₄) ") * H)}} is non empty finite V45() set
the multF of G . [(((a * c₄) ") * H),(a * c₄)] is set
(c₄ ") * ((a ") * H) is Element of the carrier of G
the multF of G . ((c₄ "),((a ") * H)) is Element of the carrier of G
[(c₄ "),((a ") * H)] is set
{(c₄ "),((a ") * H)} is non empty finite set
{{(c₄ "),((a ") * H)},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),((a ") * H)] is set
((c₄ ") * ((a ") * H)) * a is Element of the carrier of G
the multF of G . (((c₄ ") * ((a ") * H)),a) is Element of the carrier of G
[((c₄ ") * ((a ") * H)),a] is set
{((c₄ ") * ((a ") * H)),a} is non empty finite set
{((c₄ ") * ((a ") * H))} is non empty finite set
{{((c₄ ") * ((a ") * H)),a},{((c₄ ") * ((a ") * H))}} is non empty finite V45() set
the multF of G . [((c₄ ") * ((a ") * H)),a] is set
(((c₄ ") * ((a ") * H)) * a) * c₄ is Element of the carrier of G
the multF of G . ((((c₄ ") * ((a ") * H)) * a),c₄) is Element of the carrier of G
[(((c₄ ") * ((a ") * H)) * a),c₄] is set
{(((c₄ ") * ((a ") * H)) * a),c₄} is non empty finite set
{(((c₄ ") * ((a ") * H)) * a)} is non empty finite set
{{(((c₄ ") * ((a ") * H)) * a),c₄},{(((c₄ ") * ((a ") * H)) * a)}} is non empty finite V45() set
the multF of G . [(((c₄ ") * ((a ") * H)) * a),c₄] is set
(c₄ ") * (a ") is Element of the carrier of G
the multF of G . ((c₄ "),(a ")) is Element of the carrier of G
[(c₄ "),(a ")] is set
{(c₄ "),(a ")} is non empty finite set
{{(c₄ "),(a ")},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),(a ")] is set
((c₄ ") * (a ")) * H is Element of the carrier of G
the multF of G . (((c₄ ") * (a ")),H) is Element of the carrier of G
[((c₄ ") * (a ")),H] is set
{((c₄ ") * (a ")),H} is non empty finite set
{((c₄ ") * (a "))} is non empty finite set
{{((c₄ ") * (a ")),H},{((c₄ ") * (a "))}} is non empty finite V45() set
the multF of G . [((c₄ ") * (a ")),H] is set
(((c₄ ") * (a ")) * H) * a is Element of the carrier of G
the multF of G . ((((c₄ ") * (a ")) * H),a) is Element of the carrier of G
[(((c₄ ") * (a ")) * H),a] is set
{(((c₄ ") * (a ")) * H),a} is non empty finite set
{(((c₄ ") * (a ")) * H)} is non empty finite set
{{(((c₄ ") * (a ")) * H),a},{(((c₄ ") * (a ")) * H)}} is non empty finite V45() set
the multF of G . [(((c₄ ") * (a ")) * H),a] is set
((((c₄ ") * (a ")) * H) * a) * c₄ is Element of the carrier of G
the multF of G . (((((c₄ ") * (a ")) * H) * a),c₄) is Element of the carrier of G
[((((c₄ ") * (a ")) * H) * a),c₄] is set
{((((c₄ ") * (a ")) * H) * a),c₄} is non empty finite set
{((((c₄ ") * (a ")) * H) * a)} is non empty finite set
{{((((c₄ ") * (a ")) * H) * a),c₄},{((((c₄ ") * (a ")) * H) * a)}} is non empty finite V45() set
the multF of G . [((((c₄ ") * (a ")) * H) * a),c₄] is set
(((a * c₄) ") * H) * a is Element of the carrier of G
the multF of G . ((((a * c₄) ") * H),a) is Element of the carrier of G
[(((a * c₄) ") * H),a] is set
{(((a * c₄) ") * H),a} is non empty finite set
{{(((a * c₄) ") * H),a},{(((a * c₄) ") * H)}} is non empty finite V45() set
the multF of G . [(((a * c₄) ") * H),a] is set
((((a * c₄) ") * H) * a) * c₄ is Element of the carrier of G
the multF of G . (((((a * c₄) ") * H) * a),c₄) is Element of the carrier of G
[((((a * c₄) ") * H) * a),c₄] is set
{((((a * c₄) ") * H) * a),c₄} is non empty finite set
{((((a * c₄) ") * H) * a)} is non empty finite set
{{((((a * c₄) ") * H) * a),c₄},{((((a * c₄) ") * H) * a)}} is non empty finite V45() set
the multF of G . [((((a * c₄) ") * H) * a),c₄] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
(G,(G,H,a),(a ")) is Element of the carrier of G
(a ") " is Element of the carrier of G
((a ") ") * (G,H,a) is Element of the carrier of G
the multF of G . (((a ") "),(G,H,a)) is Element of the carrier of G
[((a ") "),(G,H,a)] is set
{((a ") "),(G,H,a)} is non empty finite set
{((a ") ")} is non empty finite set
{{((a ") "),(G,H,a)},{((a ") ")}} is non empty finite V45() set
the multF of G . [((a ") "),(G,H,a)] is set
(((a ") ") * (G,H,a)) * (a ") is Element of the carrier of G
the multF of G . ((((a ") ") * (G,H,a)),(a ")) is Element of the carrier of G
[(((a ") ") * (G,H,a)),(a ")] is set
{(((a ") ") * (G,H,a)),(a ")} is non empty finite set
{(((a ") ") * (G,H,a))} is non empty finite set
{{(((a ") ") * (G,H,a)),(a ")},{(((a ") ") * (G,H,a))}} is non empty finite V45() set
the multF of G . [(((a ") ") * (G,H,a)),(a ")] is set
(G,H,(a ")) is Element of the carrier of G
((a ") ") * H is Element of the carrier of G
the multF of G . (((a ") "),H) is Element of the carrier of G
[((a ") "),H] is set
{((a ") "),H} is non empty finite set
{{((a ") "),H},{((a ") ")}} is non empty finite V45() set
the multF of G . [((a ") "),H] is set
(((a ") ") * H) * (a ") is Element of the carrier of G
the multF of G . ((((a ") ") * H),(a ")) is Element of the carrier of G
[(((a ") ") * H),(a ")] is set
{(((a ") ") * H),(a ")} is non empty finite set
{(((a ") ") * H)} is non empty finite set
{{(((a ") ") * H),(a ")},{(((a ") ") * H)}} is non empty finite V45() set
the multF of G . [(((a ") ") * H),(a ")] is set
(G,(G,H,(a ")),a) is Element of the carrier of G
(a ") * (G,H,(a ")) is Element of the carrier of G
the multF of G . ((a "),(G,H,(a "))) is Element of the carrier of G
[(a "),(G,H,(a "))] is set
{(a "),(G,H,(a "))} is non empty finite set
{{(a "),(G,H,(a "))},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(G,H,(a "))] is set
((a ") * (G,H,(a "))) * a is Element of the carrier of G
the multF of G . (((a ") * (G,H,(a "))),a) is Element of the carrier of G
[((a ") * (G,H,(a "))),a] is set
{((a ") * (G,H,(a "))),a} is non empty finite set
{((a ") * (G,H,(a ")))} is non empty finite set
{{((a ") * (G,H,(a "))),a},{((a ") * (G,H,(a ")))}} is non empty finite V45() set
the multF of G . [((a ") * (G,H,(a "))),a] is set
a * (a ") is Element of the carrier of G
the multF of G . (a,(a ")) is Element of the carrier of G
[a,(a ")] is set
{a,(a ")} is non empty finite set
{a} is non empty finite set
{{a,(a ")},{a}} is non empty finite V45() set
the multF of G . [a,(a ")] is set
(G,H,(a * (a "))) is Element of the carrier of G
(a * (a ")) " is Element of the carrier of G
((a * (a ")) ") * H is Element of the carrier of G
the multF of G . (((a * (a ")) "),H) is Element of the carrier of G
[((a * (a ")) "),H] is set
{((a * (a ")) "),H} is non empty finite set
{((a * (a ")) ")} is non empty finite set
{{((a * (a ")) "),H},{((a * (a ")) ")}} is non empty finite V45() set
the multF of G . [((a * (a ")) "),H] is set
(((a * (a ")) ") * H) * (a * (a ")) is Element of the carrier of G
the multF of G . ((((a * (a ")) ") * H),(a * (a "))) is Element of the carrier of G
[(((a * (a ")) ") * H),(a * (a "))] is set
{(((a * (a ")) ") * H),(a * (a "))} is non empty finite set
{(((a * (a ")) ") * H)} is non empty finite set
{{(((a * (a ")) ") * H),(a * (a "))},{(((a * (a ")) ") * H)}} is non empty finite V45() set
the multF of G . [(((a * (a ")) ") * H),(a * (a "))] is set
1_ G is non being_of_order_0 Element of the carrier of G
(G,H,(1_ G)) is Element of the carrier of G
(1_ G) " is Element of the carrier of G
((1_ G) ") * H is Element of the carrier of G
the multF of G . (((1_ G) "),H) is Element of the carrier of G
[((1_ G) "),H] is set
{((1_ G) "),H} is non empty finite set
{((1_ G) ")} is non empty finite set
{{((1_ G) "),H},{((1_ G) ")}} is non empty finite V45() set
the multF of G . [((1_ G) "),H] is set
(((1_ G) ") * H) * (1_ G) is Element of the carrier of G
the multF of G . ((((1_ G) ") * H),(1_ G)) is Element of the carrier of G
[(((1_ G) ") * H),(1_ G)] is set
{(((1_ G) ") * H),(1_ G)} is non empty finite set
{(((1_ G) ") * H)} is non empty finite set
{{(((1_ G) ") * H),(1_ G)},{(((1_ G) ") * H)}} is non empty finite V45() set
the multF of G . [(((1_ G) ") * H),(1_ G)] is set
(a ") * a is Element of the carrier of G
the multF of G . ((a "),a) is Element of the carrier of G
[(a "),a] is set
{(a "),a} is non empty finite set
{{(a "),a},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),a] is set
(G,H,((a ") * a)) is Element of the carrier of G
((a ") * a) " is Element of the carrier of G
(((a ") * a) ") * H is Element of the carrier of G
the multF of G . ((((a ") * a) "),H) is Element of the carrier of G
[(((a ") * a) "),H] is set
{(((a ") * a) "),H} is non empty finite set
{(((a ") * a) ")} is non empty finite set
{{(((a ") * a) "),H},{(((a ") * a) ")}} is non empty finite V45() set
the multF of G . [(((a ") * a) "),H] is set
((((a ") * a) ") * H) * ((a ") * a) is Element of the carrier of G
the multF of G . (((((a ") * a) ") * H),((a ") * a)) is Element of the carrier of G
[((((a ") * a) ") * H),((a ") * a)] is set
{((((a ") * a) ") * H),((a ") * a)} is non empty finite set
{((((a ") * a) ") * H)} is non empty finite set
{{((((a ") * a) ") * H),((a ") * a)},{((((a ") * a) ") * H)}} is non empty finite V45() set
the multF of G . [((((a ") * a) ") * H),((a ") * a)] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
H " is Element of the carrier of G
a is Element of the carrier of G
(G,(H "),a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * (H ") 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),(H ")) is Element of the carrier of G
[(a "),(H ")] is set
{(a "),(H ")} is non empty finite set
{(a ")} is non empty finite set
{{(a "),(H ")},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(H ")] is set
((a ") * (H ")) * a is Element of the carrier of G
the multF of G . (((a ") * (H ")),a) is Element of the carrier of G
[((a ") * (H ")),a] is set
{((a ") * (H ")),a} is non empty finite set
{((a ") * (H "))} is non empty finite set
{{((a ") * (H ")),a},{((a ") * (H "))}} is non empty finite V45() set
the multF of G . [((a ") * (H ")),a] is set
(G,H,a) is Element of the carrier of G
(a ") * H is Element of the carrier of G
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
(G,H,a) " is Element of the carrier of G
H * a is Element of the carrier of G
the multF of G . (H,a) is Element of the carrier of G
[H,a] is set
{H,a} is non empty finite set
{H} is non empty finite set
{{H,a},{H}} is non empty finite V45() set
the multF of G . [H,a] is set
(H * a) " is Element of the carrier of G
((H * a) ") * a is Element of the carrier of G
the multF of G . (((H * a) "),a) is Element of the carrier of G
[((H * a) "),a] is set
{((H * a) "),a} is non empty finite set
{((H * a) ")} is non empty finite set
{{((H * a) "),a},{((H * a) ")}} is non empty finite V45() set
the multF of G . [((H * a) "),a] is set
(a ") " is Element of the carrier of G
((H * a) ") * ((a ") ") is Element of the carrier of G
the multF of G . (((H * a) "),((a ") ")) is Element of the carrier of G
[((H * a) "),((a ") ")] is set
{((H * a) "),((a ") ")} is non empty finite set
{{((H * a) "),((a ") ")},{((H * a) ")}} is non empty finite V45() set
the multF of G . [((H * a) "),((a ") ")] is set
(a ") * (H * a) is Element of the carrier of G
the multF of G . ((a "),(H * a)) is Element of the carrier of G
[(a "),(H * a)] is set
{(a "),(H * a)} is non empty finite set
{{(a "),(H * a)},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(H * a)] is set
((a ") * (H * a)) " is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
H |^ {} is Element of the carrier of G
power G is Relation-like [: the carrier of G,K113():] -defined the carrier of G -valued Function-like V28([: the carrier of G,K113():], the carrier of G) Element of bool [:[: the carrier of G,K113():], the carrier of G:]
[: the carrier of G,K113():] is non empty set
[:[: the carrier of G,K113():], the carrier of G:] is non empty set
bool [:[: the carrier of G,K113():], the carrier of G:] is non empty set
(power G) . (H,{}) is set
[H,{}] is set
{H,{}} is non empty finite set
{H} is non empty finite set
{{H,{}},{H}} is non empty finite V45() set
(power G) . [H,{}] is set
a is Element of the carrier of G
(G,(H |^ {}),a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * (H |^ {}) 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),(H |^ {})) is Element of the carrier of G
[(a "),(H |^ {})] is set
{(a "),(H |^ {})} is non empty finite set
{(a ")} is non empty finite set
{{(a "),(H |^ {})},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(H |^ {})] is set
((a ") * (H |^ {})) * a is Element of the carrier of G
the multF of G . (((a ") * (H |^ {})),a) is Element of the carrier of G
[((a ") * (H |^ {})),a] is set
{((a ") * (H |^ {})),a} is non empty finite set
{((a ") * (H |^ {}))} is non empty finite set
{{((a ") * (H |^ {})),a},{((a ") * (H |^ {}))}} is non empty finite V45() set
the multF of G . [((a ") * (H |^ {})),a] is set
1_ G is non being_of_order_0 Element of the carrier of G
(G,(1_ G),a) is Element of the carrier of G
(a ") * (1_ G) is Element of the carrier of G
the multF of G . ((a "),(1_ G)) is Element of the carrier of G
[(a "),(1_ G)] is set
{(a "),(1_ G)} is non empty finite set
{{(a "),(1_ G)},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(1_ G)] is set
((a ") * (1_ G)) * a is Element of the carrier of G
the multF of G . (((a ") * (1_ G)),a) is Element of the carrier of G
[((a ") * (1_ G)),a] is set
{((a ") * (1_ G)),a} is non empty finite set
{((a ") * (1_ G))} is non empty finite set
{{((a ") * (1_ G)),a},{((a ") * (1_ G))}} is non empty finite V45() set
the multF of G . [((a ") * (1_ G)),a] is set
(G,H,a) is Element of the carrier of G
(a ") * H is Element of the carrier of G
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
(G,H,a) |^ {} is Element of the carrier of G
(power G) . ((G,H,a),{}) is set
[(G,H,a),{}] is set
{(G,H,a),{}} is non empty finite set
{(G,H,a)} is non empty finite set
{{(G,H,a),{}},{(G,H,a)}} is non empty finite V45() set
(power G) . [(G,H,a),{}] is set
G is V12() V13() V14() V18() ext-real V37() V38() integer set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
G + 1 is V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
a is Element of the carrier of H
a |^ (G + 1) is Element of the carrier of H
power H is Relation-like [: the carrier of H,K113():] -defined the carrier of H -valued Function-like V28([: the carrier of H,K113():], the carrier of H) Element of bool [:[: the carrier of H,K113():], the carrier of H:]
[: the carrier of H,K113():] is non empty set
[:[: the carrier of H,K113():], the carrier of H:] is non empty set
bool [:[: the carrier of H,K113():], the carrier of H:] is non empty set
(power H) . (a,(G + 1)) is set
[a,(G + 1)] is set
{a,(G + 1)} is non empty finite set
{a} is non empty finite set
{{a,(G + 1)},{a}} is non empty finite V45() set
(power H) . [a,(G + 1)] is set
c₄ is Element of the carrier of H
(H,(a |^ (G + 1)),c₄) is Element of the carrier of H
c₄ " is Element of the carrier of H
(c₄ ") * (a |^ (G + 1)) is Element of the carrier of H
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
the multF of H . ((c₄ "),(a |^ (G + 1))) is Element of the carrier of H
[(c₄ "),(a |^ (G + 1))] is set
{(c₄ "),(a |^ (G + 1))} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),(a |^ (G + 1))},{(c₄ ")}} is non empty finite V45() set
the multF of H . [(c₄ "),(a |^ (G + 1))] is set
((c₄ ") * (a |^ (G + 1))) * c₄ is Element of the carrier of H
the multF of H . (((c₄ ") * (a |^ (G + 1))),c₄) is Element of the carrier of H
[((c₄ ") * (a |^ (G + 1))),c₄] is set
{((c₄ ") * (a |^ (G + 1))),c₄} is non empty finite set
{((c₄ ") * (a |^ (G + 1)))} is non empty finite set
{{((c₄ ") * (a |^ (G + 1))),c₄},{((c₄ ") * (a |^ (G + 1)))}} is non empty finite V45() set
the multF of H . [((c₄ ") * (a |^ (G + 1))),c₄] is set
a |^ G is Element of the carrier of H
(power H) . (a,G) is set
[a,G] is set
{a,G} is non empty finite set
{{a,G},{a}} is non empty finite V45() set
(power H) . [a,G] is set
(a |^ G) * a is Element of the carrier of H
the multF of H . ((a |^ G),a) is Element of the carrier of H
[(a |^ G),a] is set
{(a |^ G),a} is non empty finite set
{(a |^ G)} is non empty finite set
{{(a |^ G),a},{(a |^ G)}} is non empty finite V45() set
the multF of H . [(a |^ G),a] is set
(H,((a |^ G) * a),c₄) is Element of the carrier of H
(c₄ ") * ((a |^ G) * a) is Element of the carrier of H
the multF of H . ((c₄ "),((a |^ G) * a)) is Element of the carrier of H
[(c₄ "),((a |^ G) * a)] is set
{(c₄ "),((a |^ G) * a)} is non empty finite set
{{(c₄ "),((a |^ G) * a)},{(c₄ ")}} is non empty finite V45() set
the multF of H . [(c₄ "),((a |^ G) * a)] is set
((c₄ ") * ((a |^ G) * a)) * c₄ is Element of the carrier of H
the multF of H . (((c₄ ") * ((a |^ G) * a)),c₄) is Element of the carrier of H
[((c₄ ") * ((a |^ G) * a)),c₄] is set
{((c₄ ") * ((a |^ G) * a)),c₄} is non empty finite set
{((c₄ ") * ((a |^ G) * a))} is non empty finite set
{{((c₄ ") * ((a |^ G) * a)),c₄},{((c₄ ") * ((a |^ G) * a))}} is non empty finite V45() set
the multF of H . [((c₄ ") * ((a |^ G) * a)),c₄] is set
(H,(a |^ G),c₄) is Element of the carrier of H
(c₄ ") * (a |^ G) is Element of the carrier of H
the multF of H . ((c₄ "),(a |^ G)) is Element of the carrier of H
[(c₄ "),(a |^ G)] is set
{(c₄ "),(a |^ G)} is non empty finite set
{{(c₄ "),(a |^ G)},{(c₄ ")}} is non empty finite V45() set
the multF of H . [(c₄ "),(a |^ G)] is set
((c₄ ") * (a |^ G)) * c₄ is Element of the carrier of H
the multF of H . (((c₄ ") * (a |^ G)),c₄) is Element of the carrier of H
[((c₄ ") * (a |^ G)),c₄] is set
{((c₄ ") * (a |^ G)),c₄} is non empty finite set
{((c₄ ") * (a |^ G))} is non empty finite set
{{((c₄ ") * (a |^ G)),c₄},{((c₄ ") * (a |^ G))}} is non empty finite V45() set
the multF of H . [((c₄ ") * (a |^ G)),c₄] is set
(H,a,c₄) is Element of the carrier of H
(c₄ ") * a is Element of the carrier of H
the multF of H . ((c₄ "),a) is Element of the carrier of H
[(c₄ "),a] is set
{(c₄ "),a} is non empty finite set
{{(c₄ "),a},{(c₄ ")}} is non empty finite V45() set
the multF of H . [(c₄ "),a] is set
((c₄ ") * a) * c₄ is Element of the carrier of H
the multF of H . (((c₄ ") * a),c₄) is Element of the carrier of H
[((c₄ ") * a),c₄] is set
{((c₄ ") * a),c₄} is non empty finite set
{((c₄ ") * a)} is non empty finite set
{{((c₄ ") * a),c₄},{((c₄ ") * a)}} is non empty finite V45() set
the multF of H . [((c₄ ") * a),c₄] is set
(H,(a |^ G),c₄) * (H,a,c₄) is Element of the carrier of H
the multF of H . ((H,(a |^ G),c₄),(H,a,c₄)) is Element of the carrier of H
[(H,(a |^ G),c₄),(H,a,c₄)] is set
{(H,(a |^ G),c₄),(H,a,c₄)} is non empty finite set
{(H,(a |^ G),c₄)} is non empty finite set
{{(H,(a |^ G),c₄),(H,a,c₄)},{(H,(a |^ G),c₄)}} is non empty finite V45() set
the multF of H . [(H,(a |^ G),c₄),(H,a,c₄)] is set
(H,a,c₄) |^ G is Element of the carrier of H
(power H) . ((H,a,c₄),G) is set
[(H,a,c₄),G] is set
{(H,a,c₄),G} is non empty finite set
{(H,a,c₄)} is non empty finite set
{{(H,a,c₄),G},{(H,a,c₄)}} is non empty finite V45() set
(power H) . [(H,a,c₄),G] is set
((H,a,c₄) |^ G) * (H,a,c₄) is Element of the carrier of H
the multF of H . (((H,a,c₄) |^ G),(H,a,c₄)) is Element of the carrier of H
[((H,a,c₄) |^ G),(H,a,c₄)] is set
{((H,a,c₄) |^ G),(H,a,c₄)} is non empty finite set
{((H,a,c₄) |^ G)} is non empty finite set
{{((H,a,c₄) |^ G),(H,a,c₄)},{((H,a,c₄) |^ G)}} is non empty finite V45() set
the multF of H . [((H,a,c₄) |^ G),(H,a,c₄)] is set
(H,a,c₄) |^ (G + 1) is Element of the carrier of H
(power H) . ((H,a,c₄),(G + 1)) is set
[(H,a,c₄),(G + 1)] is set
{(H,a,c₄),(G + 1)} is non empty finite set
{{(H,a,c₄),(G + 1)},{(H,a,c₄)}} is non empty finite V45() set
(power H) . [(H,a,c₄),(G + 1)] is set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
G is V12() V13() V14() V18() ext-real V37() V38() integer set
G + 1 is V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
a is Element of the carrier of H
a |^ (G + 1) is Element of the carrier of H
power H is Relation-like [: the carrier of H,K113():] -defined the carrier of H -valued Function-like V28([: the carrier of H,K113():], the carrier of H) Element of bool [:[: the carrier of H,K113():], the carrier of H:]
[: the carrier of H,K113():] is non empty set
[:[: the carrier of H,K113():], the carrier of H:] is non empty set
bool [:[: the carrier of H,K113():], the carrier of H:] is non empty set
(power H) . (a,(G + 1)) is set
[a,(G + 1)] is set
{a,(G + 1)} is non empty finite set
{a} is non empty finite set
{{a,(G + 1)},{a}} is non empty finite V45() set
(power H) . [a,(G + 1)] is set
c₄ is Element of the carrier of H
(H,(a |^ (G + 1)),c₄) is Element of the carrier of H
c₄ " is Element of the carrier of H
(c₄ ") * (a |^ (G + 1)) is Element of the carrier of H
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
the multF of H . ((c₄ "),(a |^ (G + 1))) is Element of the carrier of H
[(c₄ "),(a |^ (G + 1))] is set
{(c₄ "),(a |^ (G + 1))} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),(a |^ (G + 1))},{(c₄ ")}} is non empty finite V45() set
the multF of H . [(c₄ "),(a |^ (G + 1))] is set
((c₄ ") * (a |^ (G + 1))) * c₄ is Element of the carrier of H
the multF of H . (((c₄ ") * (a |^ (G + 1))),c₄) is Element of the carrier of H
[((c₄ ") * (a |^ (G + 1))),c₄] is set
{((c₄ ") * (a |^ (G + 1))),c₄} is non empty finite set
{((c₄ ") * (a |^ (G + 1)))} is non empty finite set
{{((c₄ ") * (a |^ (G + 1))),c₄},{((c₄ ") * (a |^ (G + 1)))}} is non empty finite V45() set
the multF of H . [((c₄ ") * (a |^ (G + 1))),c₄] is set
(H,a,c₄) is Element of the carrier of H
(c₄ ") * a is Element of the carrier of H
the multF of H . ((c₄ "),a) is Element of the carrier of H
[(c₄ "),a] is set
{(c₄ "),a} is non empty finite set
{{(c₄ "),a},{(c₄ ")}} is non empty finite V45() set
the multF of H . [(c₄ "),a] is set
((c₄ ") * a) * c₄ is Element of the carrier of H
the multF of H . (((c₄ ") * a),c₄) is Element of the carrier of H
[((c₄ ") * a),c₄] is set
{((c₄ ") * a),c₄} is non empty finite set
{((c₄ ") * a)} is non empty finite set
{{((c₄ ") * a),c₄},{((c₄ ") * a)}} is non empty finite V45() set
the multF of H . [((c₄ ") * a),c₄] is set
(H,a,c₄) |^ (G + 1) is Element of the carrier of H
(power H) . ((H,a,c₄),(G + 1)) is set
[(H,a,c₄),(G + 1)] is set
{(H,a,c₄),(G + 1)} is non empty finite set
{(H,a,c₄)} is non empty finite set
{{(H,a,c₄),(G + 1)},{(H,a,c₄)}} is non empty finite V45() set
(power H) . [(H,a,c₄),(G + 1)] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
H |^ {} is Element of the carrier of G
power G is Relation-like [: the carrier of G,K113():] -defined the carrier of G -valued Function-like V28([: the carrier of G,K113():], the carrier of G) Element of bool [:[: the carrier of G,K113():], the carrier of G:]
[: the carrier of G,K113():] is non empty set
[:[: the carrier of G,K113():], the carrier of G:] is non empty set
bool [:[: the carrier of G,K113():], the carrier of G:] is non empty set
(power G) . (H,{}) is set
[H,{}] is set
{H,{}} is non empty finite set
{H} is non empty finite set
{{H,{}},{H}} is non empty finite V45() set
(power G) . [H,{}] is set
a is Element of the carrier of G
(G,(H |^ {}),a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * (H |^ {}) 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),(H |^ {})) is Element of the carrier of G
[(a "),(H |^ {})] is set
{(a "),(H |^ {})} is non empty finite set
{(a ")} is non empty finite set
{{(a "),(H |^ {})},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(H |^ {})] is set
((a ") * (H |^ {})) * a is Element of the carrier of G
the multF of G . (((a ") * (H |^ {})),a) is Element of the carrier of G
[((a ") * (H |^ {})),a] is set
{((a ") * (H |^ {})),a} is non empty finite set
{((a ") * (H |^ {}))} is non empty finite set
{{((a ") * (H |^ {})),a},{((a ") * (H |^ {}))}} is non empty finite V45() set
the multF of G . [((a ") * (H |^ {})),a] is set
(G,H,a) is Element of the carrier of G
(a ") * H is Element of the carrier of G
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
(G,H,a) |^ {} is Element of the carrier of G
(power G) . ((G,H,a),{}) is set
[(G,H,a),{}] is set
{(G,H,a),{}} is non empty finite set
{(G,H,a)} is non empty finite set
{{(G,H,a),{}},{(G,H,a)}} is non empty finite V45() set
(power G) . [(G,H,a),{}] is set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
G is V12() V13() V14() V18() ext-real V37() V38() integer set
a is Element of the carrier of H
a |^ G is Element of the carrier of H
power H is Relation-like [: the carrier of H,K113():] -defined the carrier of H -valued Function-like V28([: the carrier of H,K113():], the carrier of H) Element of bool [:[: the carrier of H,K113():], the carrier of H:]
[: the carrier of H,K113():] is non empty set
[:[: the carrier of H,K113():], the carrier of H:] is non empty set
bool [:[: the carrier of H,K113():], the carrier of H:] is non empty set
(power H) . (a,G) is set
[a,G] is set
{a,G} is non empty finite set
{a} is non empty finite set
{{a,G},{a}} is non empty finite V45() set
(power H) . [a,G] is set
c₄ is Element of the carrier of H
(H,(a |^ G),c₄) is Element of the carrier of H
c₄ " is Element of the carrier of H
(c₄ ") * (a |^ G) is Element of the carrier of H
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
the multF of H . ((c₄ "),(a |^ G)) is Element of the carrier of H
[(c₄ "),(a |^ G)] is set
{(c₄ "),(a |^ G)} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),(a |^ G)},{(c₄ ")}} is non empty finite V45() set
the multF of H . [(c₄ "),(a |^ G)] is set
((c₄ ") * (a |^ G)) * c₄ is Element of the carrier of H
the multF of H . (((c₄ ") * (a |^ G)),c₄) is Element of the carrier of H
[((c₄ ") * (a |^ G)),c₄] is set
{((c₄ ") * (a |^ G)),c₄} is non empty finite set
{((c₄ ") * (a |^ G))} is non empty finite set
{{((c₄ ") * (a |^ G)),c₄},{((c₄ ") * (a |^ G))}} is non empty finite V45() set
the multF of H . [((c₄ ") * (a |^ G)),c₄] is set
(H,a,c₄) is Element of the carrier of H
(c₄ ") * a is Element of the carrier of H
the multF of H . ((c₄ "),a) is Element of the carrier of H
[(c₄ "),a] is set
{(c₄ "),a} is non empty finite set
{{(c₄ "),a},{(c₄ ")}} is non empty finite V45() set
the multF of H . [(c₄ "),a] is set
((c₄ ") * a) * c₄ is Element of the carrier of H
the multF of H . (((c₄ ") * a),c₄) is Element of the carrier of H
[((c₄ ") * a),c₄] is set
{((c₄ ") * a),c₄} is non empty finite set
{((c₄ ") * a)} is non empty finite set
{{((c₄ ") * a),c₄},{((c₄ ") * a)}} is non empty finite V45() set
the multF of H . [((c₄ ") * a),c₄] is set
(H,a,c₄) |^ G is Element of the carrier of H
(power H) . ((H,a,c₄),G) is set
[(H,a,c₄),G] is set
{(H,a,c₄),G} is non empty finite set
{(H,a,c₄)} is non empty finite set
{{(H,a,c₄),G},{(H,a,c₄)}} is non empty finite V45() set
(power H) . [(H,a,c₄),G] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
c₄ is V12() V13() V14() V18() ext-real V37() V38() integer set
H is Element of the carrier of G
H |^ c₄ is Element of the carrier of G
power G is Relation-like [: the carrier of G,K113():] -defined the carrier of G -valued Function-like V28([: the carrier of G,K113():], the carrier of G) Element of bool [:[: the carrier of G,K113():], the carrier of G:]
[: the carrier of G,K113():] is non empty set
[:[: the carrier of G,K113():], the carrier of G:] is non empty set
bool [:[: the carrier of G,K113():], the carrier of G:] is non empty set
(power G) . (H,c₄) is set
[H,c₄] is set
{H,c₄} is non empty finite set
{H} is non empty finite set
{{H,c₄},{H}} is non empty finite V45() set
(power G) . [H,c₄] is set
a is Element of the carrier of G
(G,(H |^ c₄),a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * (H |^ c₄) 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),(H |^ c₄)) is Element of the carrier of G
[(a "),(H |^ c₄)] is set
{(a "),(H |^ c₄)} is non empty finite set
{(a ")} is non empty finite set
{{(a "),(H |^ c₄)},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(H |^ c₄)] is set
((a ") * (H |^ c₄)) * a is Element of the carrier of G
the multF of G . (((a ") * (H |^ c₄)),a) is Element of the carrier of G
[((a ") * (H |^ c₄)),a] is set
{((a ") * (H |^ c₄)),a} is non empty finite set
{((a ") * (H |^ c₄))} is non empty finite set
{{((a ") * (H |^ c₄)),a},{((a ") * (H |^ c₄))}} is non empty finite V45() set
the multF of G . [((a ") * (H |^ c₄)),a] is set
(G,H,a) is Element of the carrier of G
(a ") * H is Element of the carrier of G
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
(G,H,a) |^ c₄ is Element of the carrier of G
(power G) . ((G,H,a),c₄) is set
[(G,H,a),c₄] is set
{(G,H,a),c₄} is non empty finite set
{(G,H,a)} is non empty finite set
{{(G,H,a),c₄},{(G,H,a)}} is non empty finite V45() set
(power G) . [(G,H,a),c₄] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
c₄ is ext-real V37() V38() integer set
H |^ c₄ is Element of the carrier of G
(G,(H |^ c₄),a) is Element of the carrier of G
(a ") * (H |^ c₄) is Element of the carrier of G
the multF of G . ((a "),(H |^ c₄)) is Element of the carrier of G
[(a "),(H |^ c₄)] is set
{(a "),(H |^ c₄)} is non empty finite set
{{(a "),(H |^ c₄)},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(H |^ c₄)] is set
((a ") * (H |^ c₄)) * a is Element of the carrier of G
the multF of G . (((a ") * (H |^ c₄)),a) is Element of the carrier of G
[((a ") * (H |^ c₄)),a] is set
{((a ") * (H |^ c₄)),a} is non empty finite set
{((a ") * (H |^ c₄))} is non empty finite set
{{((a ") * (H |^ c₄)),a},{((a ") * (H |^ c₄))}} is non empty finite V45() set
the multF of G . [((a ") * (H |^ c₄)),a] is set
(G,H,a) |^ c₄ is Element of the carrier of G
abs c₄ is V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
abs c₄ is V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
H |^ (abs c₄) is Element of the carrier of G
power G is Relation-like [: the carrier of G,K113():] -defined the carrier of G -valued Function-like V28([: the carrier of G,K113():], the carrier of G) Element of bool [:[: the carrier of G,K113():], the carrier of G:]
[: the carrier of G,K113():] is non empty set
[:[: the carrier of G,K113():], the carrier of G:] is non empty set
bool [:[: the carrier of G,K113():], the carrier of G:] is non empty set
(power G) . (H,(abs c₄)) is set
[H,(abs c₄)] is set
{H,(abs c₄)} is non empty finite set
{H} is non empty finite set
{{H,(abs c₄)},{H}} is non empty finite V45() set
(power G) . [H,(abs c₄)] is set
(H |^ (abs c₄)) " is Element of the carrier of G
(G,((H |^ (abs c₄)) "),a) is Element of the carrier of G
(a ") * ((H |^ (abs c₄)) ") is Element of the carrier of G
the multF of G . ((a "),((H |^ (abs c₄)) ")) is Element of the carrier of G
[(a "),((H |^ (abs c₄)) ")] is set
{(a "),((H |^ (abs c₄)) ")} is non empty finite set
{{(a "),((H |^ (abs c₄)) ")},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),((H |^ (abs c₄)) ")] is set
((a ") * ((H |^ (abs c₄)) ")) * a is Element of the carrier of G
the multF of G . (((a ") * ((H |^ (abs c₄)) ")),a) is Element of the carrier of G
[((a ") * ((H |^ (abs c₄)) ")),a] is set
{((a ") * ((H |^ (abs c₄)) ")),a} is non empty finite set
{((a ") * ((H |^ (abs c₄)) "))} is non empty finite set
{{((a ") * ((H |^ (abs c₄)) ")),a},{((a ") * ((H |^ (abs c₄)) "))}} is non empty finite V45() set
the multF of G . [((a ") * ((H |^ (abs c₄)) ")),a] is set
(G,(H |^ (abs c₄)),a) is Element of the carrier of G
(a ") * (H |^ (abs c₄)) is Element of the carrier of G
the multF of G . ((a "),(H |^ (abs c₄))) is Element of the carrier of G
[(a "),(H |^ (abs c₄))] is set
{(a "),(H |^ (abs c₄))} is non empty finite set
{{(a "),(H |^ (abs c₄))},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(H |^ (abs c₄))] is set
((a ") * (H |^ (abs c₄))) * a is Element of the carrier of G
the multF of G . (((a ") * (H |^ (abs c₄))),a) is Element of the carrier of G
[((a ") * (H |^ (abs c₄))),a] is set
{((a ") * (H |^ (abs c₄))),a} is non empty finite set
{((a ") * (H |^ (abs c₄)))} is non empty finite set
{{((a ") * (H |^ (abs c₄))),a},{((a ") * (H |^ (abs c₄)))}} is non empty finite V45() set
the multF of G . [((a ") * (H |^ (abs c₄))),a] is set
(G,(H |^ (abs c₄)),a) " is Element of the carrier of G
(G,H,a) |^ (abs c₄) is Element of the carrier of G
(power G) . ((G,H,a),(abs c₄)) is set
[(G,H,a),(abs c₄)] is set
{(G,H,a),(abs c₄)} is non empty finite set
{(G,H,a)} is non empty finite set
{{(G,H,a),(abs c₄)},{(G,H,a)}} is non empty finite V45() set
(power G) . [(G,H,a),(abs c₄)] is set
((G,H,a) |^ (abs c₄)) " is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
H * (a ") is Element of the carrier of G
the multF of G . (H,(a ")) is Element of the carrier of G
[H,(a ")] is set
{H,(a ")} is non empty finite set
{H} is non empty finite set
{{H,(a ")},{H}} is non empty finite V45() set
the multF of G . [H,(a ")] is set
(H * (a ")) * a is Element of the carrier of G
the multF of G . ((H * (a ")),a) is Element of the carrier of G
[(H * (a ")),a] is set
{(H * (a ")),a} is non empty finite set
{(H * (a "))} is non empty finite set
{{(H * (a ")),a},{(H * (a "))}} is non empty finite V45() set
the multF of G . [(H * (a ")),a] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
H * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H,a) is Element of the carrier of G
[H,a] is set
{H,a} is non empty finite set
{H} is non empty finite set
{{H,a},{H}} is non empty finite V45() set
the multF of G . [H,a] is set
a * H is Element of the carrier of G
the multF of G . (a,H) is Element of the carrier of G
[a,H] is set
{a,H} is non empty finite set
{a} is non empty finite set
{{a,H},{a}} is non empty finite V45() set
the multF of G . [a,H] is set
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H is Element of the carrier of G
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] 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 non empty set
H is Element of bool the carrier of G
a is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in a ) } is set
y is set
H1 is Element of the carrier of G
a is Element of the carrier of G
(G,H1,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H1) is Element of the carrier of G
[(a "),H1] is set
{(a "),H1} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H1},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H1] is set
((a ") * H1) * a is Element of the carrier of G
the multF of G . (((a ") * H1),a) is Element of the carrier of G
[((a ") * H1),a] is set
{((a ") * H1),a} is non empty finite set
{((a ") * H1)} is non empty finite set
{{((a ") * H1),a},{((a ") * H1)}} is non empty finite V45() set
the multF of G . [((a ") * H1),a] is set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
bool the carrier of H is non empty set
H1 is non empty unital Group-like associative multMagma
the carrier of H1 is non empty set
bool the carrier of H1 is non empty set
G is set
a is Element of bool the carrier of H
c₄ is Element of bool the carrier of H
(H,a,c₄) is Element of bool the carrier of H
{ (H,b₁,b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in a & b₂ in c₄ ) } is set
y is set
c is Element of the carrier of H1
c is Element of the carrier of H1
(H1,c,c) is Element of the carrier of H1
c " is Element of the carrier of H1
(c ") * c is Element of the carrier of H1
the multF of H1 is Relation-like [: the carrier of H1, the carrier of H1:] -defined the carrier of H1 -valued Function-like V28([: the carrier of H1, the carrier of H1:], the carrier of H1) associative Element of bool [:[: the carrier of H1, the carrier of H1:], the carrier of H1:]
[: the carrier of H1, the carrier of H1:] is non empty set
[:[: the carrier of H1, the carrier of H1:], the carrier of H1:] is non empty set
bool [:[: the carrier of H1, the carrier of H1:], the carrier of H1:] is non empty set
the multF of H1 . ((c "),c) is Element of the carrier of H1
[(c "),c] is set
{(c "),c} is non empty finite set
{(c ")} is non empty finite set
{{(c "),c},{(c ")}} is non empty finite V45() set
the multF of H1 . [(c "),c] is set
((c ") * c) * c is Element of the carrier of H1
the multF of H1 . (((c ") * c),c) is Element of the carrier of H1
[((c ") * c),c] is set
{((c ") * c),c} is non empty finite set
{((c ") * c)} is non empty finite set
{{((c ") * c),c},{((c ") * c)}} is non empty finite V45() set
the multF of H1 . [((c ") * c),c] is set
a is Element of bool the carrier of H1
b is Element of bool the carrier of H1
(H1,a,b) is Element of bool the carrier of H1
{ (H1,b₁,b₂) where b₁, b₂ is Element of the carrier of H1 : ( b₁ in a & b₂ in b ) } 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 non empty set
H is Element of bool the carrier of G
a is Element of bool the carrier of G
(G,H,a) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in a ) } is set
the Element of H is Element of H
the Element of a is Element of a
the Element of (G,H,a) is Element of (G,H,a)
a is Element of the carrier of G
b is Element of the carrier of G
(G,a,b) is Element of the carrier of G
b " is Element of the carrier of G
(b ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((b "),a) is Element of the carrier of G
[(b "),a] is set
{(b "),a} is non empty finite set
{(b ")} is non empty finite set
{{(b "),a},{(b ")}} is non empty finite V45() set
the multF of G . [(b "),a] is set
((b ") * a) * b is Element of the carrier of G
the multF of G . (((b ") * a),b) is Element of the carrier of G
[((b ") * a),b] is set
{((b ") * a),b} is non empty finite set
{((b ") * a)} is non empty finite set
{{((b ") * a),b},{((b ") * a)}} is non empty finite V45() set
the multF of G . [((b ") * a),b] is set
H1 is Element of the carrier of G
a is Element of the carrier of G
(G,H1,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H1) is Element of the carrier of G
[(a "),H1] is set
{(a "),H1} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H1},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H1] is set
((a ") * H1) * a is Element of the carrier of G
the multF of G . (((a ") * H1),a) is Element of the carrier of G
[((a ") * H1),a] is set
{((a ") * H1),a} is non empty finite set
{((a ") * H1)} is non empty finite set
{{((a ") * H1),a},{((a ") * H1)}} is non empty finite V45() set
the multF of G . [((a ") * H1),a] 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 non empty set
H is Element of bool the carrier of G
a is Element of bool the carrier of G
(G,H,a) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in a ) } is set
a " is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in a } is set
(a ") * H is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a " & b₂ in H ) } is set
((a ") * H) * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (a ") * H & b₂ in a ) } is set
c₄ is set
y is Element of the carrier of G
H1 is Element of the carrier of G
(G,y,H1) is Element of the carrier of G
H1 " is Element of the carrier of G
(H1 ") * y 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H1 "),y) is Element of the carrier of G
[(H1 "),y] is set
{(H1 "),y} is non empty finite set
{(H1 ")} is non empty finite set
{{(H1 "),y},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),y] is set
((H1 ") * y) * H1 is Element of the carrier of G
the multF of G . (((H1 ") * y),H1) is Element of the carrier of G
[((H1 ") * y),H1] is set
{((H1 ") * y),H1} is non empty finite set
{((H1 ") * y)} is non empty finite set
{{((H1 ") * y),H1},{((H1 ") * y)}} is non empty finite V45() set
the multF of G . [((H1 ") * y),H1] 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 non empty set
H is Element of bool the carrier of G
a is Element of bool the carrier of G
H * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in a ) } is set
c₄ is Element of bool the carrier of G
(G,(H * a),c₄) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H * a & b₂ in c₄ ) } is set
(G,H,c₄) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in c₄ ) } is set
(G,a,c₄) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in c₄ ) } is set
(G,H,c₄) * (G,a,c₄) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,H,c₄) & b₂ in (G,a,c₄) ) } is set
y is set
H1 is Element of the carrier of G
a is Element of the carrier of G
(G,H1,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H1) is Element of the carrier of G
[(a "),H1] is set
{(a "),H1} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H1},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H1] is set
((a ") * H1) * a is Element of the carrier of G
the multF of G . (((a ") * H1),a) is Element of the carrier of G
[((a ") * H1),a] is set
{((a ") * H1),a} is non empty finite set
{((a ") * H1)} is non empty finite set
{{((a ") * H1),a},{((a ") * H1)}} is non empty finite V45() set
the multF of G . [((a ") * H1),a] is set
b is Element of the carrier of G
c is Element of the carrier of G
b * c is Element of the carrier of G
the multF of G . (b,c) is Element of the carrier of G
[b,c] is set
{b,c} is non empty finite set
{b} is non empty finite set
{{b,c},{b}} is non empty finite V45() set
the multF of G . [b,c] is set
(G,c,a) is Element of the carrier of G
(a ") * c is Element of the carrier of G
the multF of G . ((a "),c) is Element of the carrier of G
[(a "),c] is set
{(a "),c} is non empty finite set
{{(a "),c},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),c] is set
((a ") * c) * a is Element of the carrier of G
the multF of G . (((a ") * c),a) is Element of the carrier of G
[((a ") * c),a] is set
{((a ") * c),a} is non empty finite set
{((a ") * c)} is non empty finite set
{{((a ") * c),a},{((a ") * c)}} is non empty finite V45() set
the multF of G . [((a ") * c),a] is set
(G,b,a) is Element of the carrier of G
(a ") * b is Element of the carrier of G
the multF of G . ((a "),b) is Element of the carrier of G
[(a "),b] is set
{(a "),b} is non empty finite set
{{(a "),b},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),b] is set
((a ") * b) * a is Element of the carrier of G
the multF of G . (((a ") * b),a) is Element of the carrier of G
[((a ") * b),a] is set
{((a ") * b),a} is non empty finite set
{((a ") * b)} is non empty finite set
{{((a ") * b),a},{((a ") * b)}} is non empty finite V45() set
the multF of G . [((a ") * b),a] is set
(G,b,a) * (G,c,a) is Element of the carrier of G
the multF of G . ((G,b,a),(G,c,a)) is Element of the carrier of G
[(G,b,a),(G,c,a)] is set
{(G,b,a),(G,c,a)} is non empty finite set
{(G,b,a)} is non empty finite set
{{(G,b,a),(G,c,a)},{(G,b,a)}} is non empty finite V45() set
the multF of G . [(G,b,a),(G,c,a)] 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 non empty set
H is Element of bool the carrier of G
a is Element of bool the carrier of G
(G,H,a) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in a ) } is set
c₄ is Element of bool the carrier of G
(G,(G,H,a),c₄) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,H,a) & b₂ in c₄ ) } is set
a * c₄ is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in c₄ ) } is set
(G,H,(a * c₄)) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in a * c₄ ) } is set
y is set
H1 is Element of the carrier of G
a is Element of the carrier of G
(G,H1,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H1) is Element of the carrier of G
[(a "),H1] is set
{(a "),H1} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H1},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H1] is set
((a ") * H1) * a is Element of the carrier of G
the multF of G . (((a ") * H1),a) is Element of the carrier of G
[((a ") * H1),a] is set
{((a ") * H1),a} is non empty finite set
{((a ") * H1)} is non empty finite set
{{((a ") * H1),a},{((a ") * H1)}} is non empty finite V45() set
the multF of G . [((a ") * H1),a] is set
b is Element of the carrier of G
c is Element of the carrier of G
(G,b,c) is Element of the carrier of G
c " is Element of the carrier of G
(c ") * b is Element of the carrier of G
the multF of G . ((c "),b) is Element of the carrier of G
[(c "),b] is set
{(c "),b} is non empty finite set
{(c ")} is non empty finite set
{{(c "),b},{(c ")}} is non empty finite V45() set
the multF of G . [(c "),b] is set
((c ") * b) * c is Element of the carrier of G
the multF of G . (((c ") * b),c) is Element of the carrier of G
[((c ") * b),c] is set
{((c ") * b),c} is non empty finite set
{((c ") * b)} is non empty finite set
{{((c ") * b),c},{((c ") * b)}} is non empty finite V45() set
the multF of G . [((c ") * b),c] is set
c * a is Element of the carrier of G
the multF of G . (c,a) is Element of the carrier of G
[c,a] is set
{c,a} is non empty finite set
{c} is non empty finite set
{{c,a},{c}} is non empty finite V45() set
the multF of G . [c,a] is set
(G,b,(c * a)) is Element of the carrier of G
(c * a) " is Element of the carrier of G
((c * a) ") * b is Element of the carrier of G
the multF of G . (((c * a) "),b) is Element of the carrier of G
[((c * a) "),b] is set
{((c * a) "),b} is non empty finite set
{((c * a) ")} is non empty finite set
{{((c * a) "),b},{((c * a) ")}} is non empty finite V45() set
the multF of G . [((c * a) "),b] is set
(((c * a) ") * b) * (c * a) is Element of the carrier of G
the multF of G . ((((c * a) ") * b),(c * a)) is Element of the carrier of G
[(((c * a) ") * b),(c * a)] is set
{(((c * a) ") * b),(c * a)} is non empty finite set
{(((c * a) ") * b)} is non empty finite set
{{(((c * a) ") * b),(c * a)},{(((c * a) ") * b)}} is non empty finite V45() set
the multF of G . [(((c * a) ") * b),(c * a)] is set
y is set
H1 is Element of the carrier of G
a is Element of the carrier of G
(G,H1,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H1) is Element of the carrier of G
[(a "),H1] is set
{(a "),H1} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H1},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H1] is set
((a ") * H1) * a is Element of the carrier of G
the multF of G . (((a ") * H1),a) is Element of the carrier of G
[((a ") * H1),a] is set
{((a ") * H1),a} is non empty finite set
{((a ") * H1)} is non empty finite set
{{((a ") * H1),a},{((a ") * H1)}} is non empty finite V45() set
the multF of G . [((a ") * H1),a] is set
b is Element of the carrier of G
c is Element of the carrier of G
b * c is Element of the carrier of G
the multF of G . (b,c) is Element of the carrier of G
[b,c] is set
{b,c} is non empty finite set
{b} is non empty finite set
{{b,c},{b}} is non empty finite V45() set
the multF of G . [b,c] is set
(G,H1,b) is Element of the carrier of G
b " is Element of the carrier of G
(b ") * H1 is Element of the carrier of G
the multF of G . ((b "),H1) is Element of the carrier of G
[(b "),H1] is set
{(b "),H1} is non empty finite set
{(b ")} is non empty finite set
{{(b "),H1},{(b ")}} is non empty finite V45() set
the multF of G . [(b "),H1] is set
((b ") * H1) * b is Element of the carrier of G
the multF of G . (((b ") * H1),b) is Element of the carrier of G
[((b ") * H1),b] is set
{((b ") * H1),b} is non empty finite set
{((b ") * H1)} is non empty finite set
{{((b ") * H1),b},{((b ") * H1)}} is non empty finite V45() set
the multF of G . [((b ") * H1),b] is set
(G,(G,H1,b),c) is Element of the carrier of G
c " is Element of the carrier of G
(c ") * (G,H1,b) is Element of the carrier of G
the multF of G . ((c "),(G,H1,b)) is Element of the carrier of G
[(c "),(G,H1,b)] is set
{(c "),(G,H1,b)} is non empty finite set
{(c ")} is non empty finite set
{{(c "),(G,H1,b)},{(c ")}} is non empty finite V45() set
the multF of G . [(c "),(G,H1,b)] is set
((c ") * (G,H1,b)) * c is Element of the carrier of G
the multF of G . (((c ") * (G,H1,b)),c) is Element of the carrier of G
[((c ") * (G,H1,b)),c] is set
{((c ") * (G,H1,b)),c} is non empty finite set
{((c ") * (G,H1,b))} is non empty finite set
{{((c ") * (G,H1,b)),c},{((c ") * (G,H1,b))}} is non empty finite V45() set
the multF of G . [((c ") * (G,H1,b)),c] 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 non empty set
H is Element of bool the carrier of G
H " is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in H } is set
a is Element of bool the carrier of G
(G,(H "),a) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H " & b₂ in a ) } is set
(G,H,a) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in a ) } is set
(G,H,a) " is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in (G,H,a) } is set
c₄ is set
y is Element of the carrier of G
H1 is Element of the carrier of G
(G,y,H1) is Element of the carrier of G
H1 " is Element of the carrier of G
(H1 ") * y 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H1 "),y) is Element of the carrier of G
[(H1 "),y] is set
{(H1 "),y} is non empty finite set
{(H1 ")} is non empty finite set
{{(H1 "),y},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),y] is set
((H1 ") * y) * H1 is Element of the carrier of G
the multF of G . (((H1 ") * y),H1) is Element of the carrier of G
[((H1 ") * y),H1] is set
{((H1 ") * y),H1} is non empty finite set
{((H1 ") * y)} is non empty finite set
{{((H1 ") * y),H1},{((H1 ") * y)}} is non empty finite V45() set
the multF of G . [((H1 ") * y),H1] is set
a is Element of the carrier of G
a " is Element of the carrier of G
(G,a,H1) is Element of the carrier of G
(H1 ") * a is Element of the carrier of G
the multF of G . ((H1 "),a) is Element of the carrier of G
[(H1 "),a] is set
{(H1 "),a} is non empty finite set
{{(H1 "),a},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),a] is set
((H1 ") * a) * H1 is Element of the carrier of G
the multF of G . (((H1 ") * a),H1) is Element of the carrier of G
[((H1 ") * a),H1] is set
{((H1 ") * a),H1} is non empty finite set
{((H1 ") * a)} is non empty finite set
{{((H1 ") * a),H1},{((H1 ") * a)}} is non empty finite V45() set
the multF of G . [((H1 ") * a),H1] is set
(G,a,H1) " is Element of the carrier of G
c₄ is set
y is Element of the carrier of G
y " is Element of the carrier of G
H1 is Element of the carrier of G
a is Element of the carrier of G
(G,H1,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H1) is Element of the carrier of G
[(a "),H1] is set
{(a "),H1} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H1},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H1] is set
((a ") * H1) * a is Element of the carrier of G
the multF of G . (((a ") * H1),a) is Element of the carrier of G
[((a ") * H1),a] is set
{((a ") * H1),a} is non empty finite set
{((a ") * H1)} is non empty finite set
{{((a ") * H1),a},{((a ") * H1)}} is non empty finite V45() set
the multF of G . [((a ") * H1),a] is set
H1 " is Element of the carrier of G
(G,(H1 "),a) is Element of the carrier of G
(a ") * (H1 ") is Element of the carrier of G
the multF of G . ((a "),(H1 ")) is Element of the carrier of G
[(a "),(H1 ")] is set
{(a "),(H1 ")} is non empty finite set
{{(a "),(H1 ")},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(H1 ")] is set
((a ") * (H1 ")) * a is Element of the carrier of G
the multF of G . (((a ") * (H1 ")),a) is Element of the carrier of G
[((a ") * (H1 ")),a] is set
{((a ") * (H1 ")),a} is non empty finite set
{((a ") * (H1 "))} is non empty finite set
{{((a ") * (H1 ")),a},{((a ") * (H1 "))}} is non empty finite V45() set
the multF of G . [((a ") * (H1 ")),a] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
{H} is non empty finite Element of bool the carrier of G
bool the carrier of G is non empty set
a is Element of the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,{H},{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in {a} ) } is set
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
{(G,H,a)} is non empty finite Element of bool the carrier of G
{a} " is non empty Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in {a} } is set
({a} ") * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} " & b₂ in {H} ) } is set
(({a} ") * {H}) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ({a} ") * {H} & b₂ in {a} ) } is set
{(a ")} is non empty finite Element of bool the carrier of G
{(a ")} * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(a ")} & b₂ in {H} ) } is set
({(a ")} * {H}) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(a ")} * {H} & b₂ in {a} ) } is set
{((a ") * H)} is non empty finite Element of bool the carrier of G
{((a ") * H)} * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((a ") * H)} & b₂ in {a} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
{H} is non empty finite Element of bool the carrier of G
bool the carrier of G is non empty set
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
c₄ is Element of the carrier of G
{a,c₄} is non empty finite Element of bool the carrier of G
(G,{H},{a,c₄}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in {a,c₄} ) } is set
(G,H,c₄) is Element of the carrier of G
c₄ " is Element of the carrier of G
(c₄ ") * H is Element of the carrier of G
the multF of G . ((c₄ "),H) is Element of the carrier of G
[(c₄ "),H] is set
{(c₄ "),H} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),H},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),H] is set
((c₄ ") * H) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * H),c₄) is Element of the carrier of G
[((c₄ ") * H),c₄] is set
{((c₄ ") * H),c₄} is non empty finite set
{((c₄ ") * H)} is non empty finite set
{{((c₄ ") * H),c₄},{((c₄ ") * H)}} is non empty finite V45() set
the multF of G . [((c₄ ") * H),c₄] is set
{(G,H,a),(G,H,c₄)} is non empty finite Element of bool the carrier of G
y is set
H1 is Element of the carrier of G
a is Element of the carrier of G
(G,H1,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H1 is Element of the carrier of G
the multF of G . ((a "),H1) is Element of the carrier of G
[(a "),H1] is set
{(a "),H1} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H1},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H1] is set
((a ") * H1) * a is Element of the carrier of G
the multF of G . (((a ") * H1),a) is Element of the carrier of G
[((a ") * H1),a] is set
{((a ") * H1),a} is non empty finite set
{((a ") * H1)} is non empty finite set
{{((a ") * H1),a},{((a ") * H1)}} is non empty finite V45() set
the multF of G . [((a ") * H1),a] is set
y is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
{H,a} is non empty finite Element of bool the carrier of G
bool the carrier of G is non empty set
c₄ is Element of the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
(G,{H,a},{c₄}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H,a} & b₂ in {c₄} ) } is set
(G,H,c₄) is Element of the carrier of G
c₄ " is Element of the carrier of G
(c₄ ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((c₄ "),H) is Element of the carrier of G
[(c₄ "),H] is set
{(c₄ "),H} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),H},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),H] is set
((c₄ ") * H) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * H),c₄) is Element of the carrier of G
[((c₄ ") * H),c₄] is set
{((c₄ ") * H),c₄} is non empty finite set
{((c₄ ") * H)} is non empty finite set
{{((c₄ ") * H),c₄},{((c₄ ") * H)}} is non empty finite V45() set
the multF of G . [((c₄ ") * H),c₄] is set
(G,a,c₄) is Element of the carrier of G
(c₄ ") * a is Element of the carrier of G
the multF of G . ((c₄ "),a) is Element of the carrier of G
[(c₄ "),a] is set
{(c₄ "),a} is non empty finite set
{{(c₄ "),a},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),a] is set
((c₄ ") * a) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * a),c₄) is Element of the carrier of G
[((c₄ ") * a),c₄] is set
{((c₄ ") * a),c₄} is non empty finite set
{((c₄ ") * a)} is non empty finite set
{{((c₄ ") * a),c₄},{((c₄ ") * a)}} is non empty finite V45() set
the multF of G . [((c₄ ") * a),c₄] is set
{(G,H,c₄),(G,a,c₄)} is non empty finite Element of bool the carrier of G
y is set
H1 is Element of the carrier of G
a is Element of the carrier of G
(G,H1,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H1 is Element of the carrier of G
the multF of G . ((a "),H1) is Element of the carrier of G
[(a "),H1] is set
{(a "),H1} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H1},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H1] is set
((a ") * H1) * a is Element of the carrier of G
the multF of G . (((a ") * H1),a) is Element of the carrier of G
[((a ") * H1),a] is set
{((a ") * H1),a} is non empty finite set
{((a ") * H1)} is non empty finite set
{{((a ") * H1),a},{((a ") * H1)}} is non empty finite V45() set
the multF of G . [((a ") * H1),a] is set
y is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
{H,a} is non empty finite Element of bool the carrier of G
bool the carrier of G is non empty set
c₄ is Element of the carrier of G
(G,H,c₄) is Element of the carrier of G
c₄ " is Element of the carrier of G
(c₄ ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((c₄ "),H) is Element of the carrier of G
[(c₄ "),H] is set
{(c₄ "),H} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),H},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),H] is set
((c₄ ") * H) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * H),c₄) is Element of the carrier of G
[((c₄ ") * H),c₄] is set
{((c₄ ") * H),c₄} is non empty finite set
{((c₄ ") * H)} is non empty finite set
{{((c₄ ") * H),c₄},{((c₄ ") * H)}} is non empty finite V45() set
the multF of G . [((c₄ ") * H),c₄] is set
(G,a,c₄) is Element of the carrier of G
(c₄ ") * a is Element of the carrier of G
the multF of G . ((c₄ "),a) is Element of the carrier of G
[(c₄ "),a] is set
{(c₄ "),a} is non empty finite set
{{(c₄ "),a},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),a] is set
((c₄ ") * a) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * a),c₄) is Element of the carrier of G
[((c₄ ") * a),c₄] is set
{((c₄ ") * a),c₄} is non empty finite set
{((c₄ ") * a)} is non empty finite set
{{((c₄ ") * a),c₄},{((c₄ ") * a)}} is non empty finite V45() set
the multF of G . [((c₄ ") * a),c₄] is set
y is Element of the carrier of G
{c₄,y} is non empty finite Element of bool the carrier of G
(G,{H,a},{c₄,y}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H,a} & b₂ in {c₄,y} ) } is set
(G,H,y) is Element of the carrier of G
y " is Element of the carrier of G
(y ") * H is Element of the carrier of G
the multF of G . ((y "),H) is Element of the carrier of G
[(y "),H] is set
{(y "),H} is non empty finite set
{(y ")} is non empty finite set
{{(y "),H},{(y ")}} is non empty finite V45() set
the multF of G . [(y "),H] is set
((y ") * H) * y is Element of the carrier of G
the multF of G . (((y ") * H),y) is Element of the carrier of G
[((y ") * H),y] is set
{((y ") * H),y} is non empty finite set
{((y ") * H)} is non empty finite set
{{((y ") * H),y},{((y ") * H)}} is non empty finite V45() set
the multF of G . [((y ") * H),y] is set
(G,a,y) is Element of the carrier of G
(y ") * a is Element of the carrier of G
the multF of G . ((y "),a) is Element of the carrier of G
[(y "),a] is set
{(y "),a} is non empty finite set
{{(y "),a},{(y ")}} is non empty finite V45() set
the multF of G . [(y "),a] is set
((y ") * a) * y is Element of the carrier of G
the multF of G . (((y ") * a),y) is Element of the carrier of G
[((y ") * a),y] is set
{((y ") * a),y} is non empty finite set
{((y ") * a)} is non empty finite set
{{((y ") * a),y},{((y ") * a)}} is non empty finite V45() set
the multF of G . [((y ") * a),y] is set
{(G,H,c₄),(G,H,y),(G,a,c₄),(G,a,y)} is non empty finite Element of bool the carrier of G
H1 is set
a is Element of the carrier of G
b is Element of the carrier of G
(G,a,b) is Element of the carrier of G
b " is Element of the carrier of G
(b ") * a is Element of the carrier of G
the multF of G . ((b "),a) is Element of the carrier of G
[(b "),a] is set
{(b "),a} is non empty finite set
{(b ")} is non empty finite set
{{(b "),a},{(b ")}} is non empty finite V45() set
the multF of G . [(b "),a] is set
((b ") * a) * b is Element of the carrier of G
the multF of G . (((b ") * a),b) is Element of the carrier of G
[((b ") * a),b] is set
{((b ") * a),b} is non empty finite set
{((b ") * a)} is non empty finite set
{{((b ") * a),b},{((b ") * a)}} is non empty finite V45() set
the multF of G . [((b ") * a),b] is set
H1 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 non empty set
H is Element of bool the carrier of G
a is Element of the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,H,{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {a} ) } is set
(G,{a},H) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in H ) } is set
G is set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
bool the carrier of H is non empty set
a is Element of the carrier of H
c₄ is Element of bool the carrier of H
(H,c₄,a) is Element of bool the carrier of H
{a} is non empty finite Element of bool the carrier of H
(H,c₄,{a}) is Element of bool the carrier of H
{ (H,b₁,b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in c₄ & b₂ in {a} ) } is set
y is Element of the carrier of H
H1 is Element of the carrier of H
(H,y,H1) is Element of the carrier of H
H1 " is Element of the carrier of H
(H1 ") * y is Element of the carrier of H
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
the multF of H . ((H1 "),y) is Element of the carrier of H
[(H1 "),y] is set
{(H1 "),y} is non empty finite set
{(H1 ")} is non empty finite set
{{(H1 "),y},{(H1 ")}} is non empty finite V45() set
the multF of H . [(H1 "),y] is set
((H1 ") * y) * H1 is Element of the carrier of H
the multF of H . (((H1 ") * y),H1) is Element of the carrier of H
[((H1 ") * y),H1] is set
{((H1 ") * y),H1} is non empty finite set
{((H1 ") * y)} is non empty finite set
{{((H1 ") * y),H1},{((H1 ") * y)}} is non empty finite V45() set
the multF of H . [((H1 ") * y),H1] is set
y is Element of the carrier of H
(H,y,a) is Element of the carrier of H
a " is Element of the carrier of H
(a ") * y is Element of the carrier of H
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
the multF of H . ((a "),y) is Element of the carrier of H
[(a "),y] is set
{(a "),y} is non empty finite set
{(a ")} is non empty finite set
{{(a "),y},{(a ")}} is non empty finite V45() set
the multF of H . [(a "),y] is set
((a ") * y) * a is Element of the carrier of H
the multF of H . (((a ") * y),a) is Element of the carrier of H
[((a ") * y),a] is set
{((a ") * y),a} is non empty finite set
{((a ") * y)} is non empty finite set
{{((a ") * y),a},{((a ") * y)}} is non empty finite V45() set
the multF of H . [((a ") * y),a] is set
G is set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
bool the carrier of H is non empty set
a is Element of the carrier of H
c₄ is Element of bool the carrier of H
(H,c₄,a) is Element of bool the carrier of H
{a} is non empty finite Element of bool the carrier of H
(H,{a},c₄) is Element of bool the carrier of H
{ (H,b₁,b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in {a} & b₂ in c₄ ) } is set
y is Element of the carrier of H
H1 is Element of the carrier of H
(H,y,H1) is Element of the carrier of H
H1 " is Element of the carrier of H
(H1 ") * y is Element of the carrier of H
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
the multF of H . ((H1 "),y) is Element of the carrier of H
[(H1 "),y] is set
{(H1 "),y} is non empty finite set
{(H1 ")} is non empty finite set
{{(H1 "),y},{(H1 ")}} is non empty finite V45() set
the multF of H . [(H1 "),y] is set
((H1 ") * y) * H1 is Element of the carrier of H
the multF of H . (((H1 ") * y),H1) is Element of the carrier of H
[((H1 ") * y),H1] is set
{((H1 ") * y),H1} is non empty finite set
{((H1 ") * y)} is non empty finite set
{{((H1 ") * y),H1},{((H1 ") * y)}} is non empty finite V45() set
the multF of H . [((H1 ") * y),H1] is set
y is Element of the carrier of H
(H,a,y) is Element of the carrier of H
y " is Element of the carrier of H
(y ") * a is Element of the carrier of H
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
the multF of H . ((y "),a) is Element of the carrier of H
[(y "),a] is set
{(y "),a} is non empty finite set
{(y ")} is non empty finite set
{{(y "),a},{(y ")}} is non empty finite V45() set
the multF of H . [(y "),a] is set
((y ") * a) * y is Element of the carrier of H
the multF of H . (((y ") * a),y) is Element of the carrier of H
[((y ") * a),y] is set
{((y ") * a),y} is non empty finite set
{((y ") * a)} is non empty finite set
{{((y ") * a),y},{((y ") * a)}} is non empty finite V45() set
the multF of H . [((y ") * a),y] 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 non empty set
H is Element of the carrier of G
a is Element of bool the carrier of G
(G,a,H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},a) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in a ) } is set
a " is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in a } is set
(a ") * H is Element of bool the carrier of G
(a ") * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a " & b₂ in {H} ) } is set
((a ") * H) * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (a ") * H & b₂ in a ) } is set
c₄ is set
y is Element of the carrier of G
(G,H,y) is Element of the carrier of G
y " is Element of the carrier of G
(y ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((y "),H) is Element of the carrier of G
[(y "),H] is set
{(y "),H} is non empty finite set
{(y ")} is non empty finite set
{{(y "),H},{(y ")}} is non empty finite V45() set
the multF of G . [(y "),H] is set
((y ") * H) * y is Element of the carrier of G
the multF of G . (((y ") * H),y) is Element of the carrier of G
[((y ") * H),y] is set
{((y ") * H),y} is non empty finite set
{((y ") * H)} is non empty finite set
{{((y ") * H),y},{((y ") * H)}} is non empty finite V45() set
the multF of G . [((y ") * H),y] 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 non empty set
a is Element of bool the carrier of G
c₄ is Element of bool the carrier of G
(G,a,c₄) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in c₄ ) } is set
H is Element of the carrier of G
(G,(G,a,c₄),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,(G,a,c₄),{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,a,c₄) & b₂ in {H} ) } is set
c₄ * H is Element of bool the carrier of G
c₄ * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in c₄ & b₂ in {H} ) } is set
(G,a,(c₄ * H)) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in c₄ * H ) } 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 non empty set
a is Element of bool the carrier of G
H is Element of the carrier of G
(G,a,H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,a,{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {H} ) } is set
c₄ is Element of bool the carrier of G
(G,(G,a,H),c₄) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,a,H) & b₂ in c₄ ) } is set
H * c₄ is Element of bool the carrier of G
{H} * c₄ is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in c₄ ) } is set
(G,a,(H * c₄)) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in H * c₄ ) } 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 non empty set
a is Element of bool the carrier of G
H is Element of the carrier of G
(G,a,H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},a) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in a ) } is set
c₄ is Element of bool the carrier of G
(G,(G,a,H),c₄) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,a,H) & b₂ in c₄ ) } is set
a * c₄ is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in c₄ ) } is set
(G,(a * c₄),H) is Element of bool the carrier of G
(G,{H},(a * c₄)) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in a * c₄ ) } 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 non empty set
H is Element of the carrier of G
a is Element of the carrier of G
H * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H,a) is Element of the carrier of G
[H,a] is set
{H,a} is non empty finite set
{H} is non empty finite set
{{H,a},{H}} is non empty finite V45() set
the multF of G . [H,a] is set
c₄ is Element of bool the carrier of G
(G,c₄,H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,c₄,{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in c₄ & b₂ in {H} ) } is set
(G,(G,c₄,H),a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,(G,c₄,H),{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,c₄,H) & b₂ in {a} ) } is set
(G,c₄,(H * a)) is Element of bool the carrier of G
{(H * a)} is non empty finite Element of bool the carrier of G
(G,c₄,{(H * a)}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in c₄ & b₂ in {(H * a)} ) } is set
H * {a} is Element of bool the carrier of G
{H} * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in {a} ) } is set
(G,c₄,(H * {a})) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in c₄ & b₂ in H * {a} ) } 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 non empty set
c₄ is Element of bool the carrier of G
H is Element of the carrier of G
(G,c₄,H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},c₄) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in c₄ ) } is set
a is Element of the carrier of G
(G,(G,c₄,H),a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,(G,c₄,H),{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,c₄,H) & b₂ in {a} ) } is set
c₄ * a is Element of bool the carrier of G
c₄ * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in c₄ & b₂ in {a} ) } is set
(G,(c₄ * a),H) is Element of bool the carrier of G
(G,{H},(c₄ * a)) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in c₄ * a ) } 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 non empty set
H is Element of the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
c₄ is Element of bool the carrier of G
(G,c₄,(G,H,a)) is Element of bool the carrier of G
{(G,H,a)} is non empty finite Element of bool the carrier of G
(G,{(G,H,a)},c₄) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(G,H,a)} & b₂ in c₄ ) } is set
a * c₄ is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
{a} * c₄ is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in c₄ ) } is set
(G,(a * c₄),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},(a * c₄)) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in a * c₄ ) } is set
(G,{H},{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in {a} ) } is set
(G,(G,{H},{a}),c₄) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,{H},{a}) & b₂ in c₄ ) } 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 non empty set
H is Element of the carrier of G
H " is Element of the carrier of G
a is Element of bool the carrier of G
(G,a,H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,a,{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {H} ) } is set
(H ") * a is Element of bool the carrier of G
{(H ")} is non empty finite Element of bool the carrier of G
{(H ")} * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(H ")} & b₂ in a ) } is set
((H ") * a) * H is Element of bool the carrier of G
((H ") * a) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (H ") * a & b₂ in {H} ) } is set
{H} " is non empty Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in {H} } is set
({H} ") * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} " & b₂ in a ) } is set
(({H} ") * a) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ({H} ") * a & b₂ in {H} ) } is set
c₄ is set
y is Element of the carrier of G
y * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (y,H) is Element of the carrier of G
[y,H] is set
{y,H} is non empty finite set
{y} is non empty finite set
{{y,H},{y}} is non empty finite V45() set
the multF of G . [y,H] is set
H1 is Element of the carrier of G
(H ") * H1 is Element of the carrier of G
the multF of G . ((H "),H1) is Element of the carrier of G
[(H "),H1] is set
{(H "),H1} is non empty finite set
{(H ")} is non empty finite set
{{(H "),H1},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),H1] is set
(G,H1,H) is Element of the carrier of G
((H ") * H1) * H is Element of the carrier of G
the multF of G . (((H ") * H1),H) is Element of the carrier of G
[((H ") * H1),H] is set
{((H ") * H1),H} is non empty finite set
{((H ") * H1)} is non empty finite set
{{((H ") * H1),H},{((H ") * H1)}} is non empty finite V45() set
the multF of G . [((H ") * H1),H] 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 non empty set
a is Element of bool the carrier of G
c₄ is Element of bool the carrier of G
a * c₄ is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in c₄ ) } is set
H is Element of the carrier of G
(G,(a * c₄),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,(a * c₄),{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a * c₄ & b₂ in {H} ) } is set
(G,a,H) is Element of bool the carrier of G
(G,a,{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {H} ) } is set
(G,c₄,H) is Element of bool the carrier of G
(G,c₄,{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in c₄ & b₂ in {H} ) } is set
(G,a,H) * (G,c₄,H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,a,H) & b₂ in (G,c₄,H) ) } 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 non empty set
1_ G is non being_of_order_0 Element of the carrier of G
H is Element of bool the carrier of G
(G,H,(1_ G)) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
(G,H,{(1_ G)}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {(1_ G)} ) } is set
(1_ G) " is Element of the carrier of G
((1_ G) ") * H is Element of bool the carrier of G
{((1_ G) ")} is non empty finite Element of bool the carrier of G
{((1_ G) ")} * H is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((1_ G) ")} & b₂ in H ) } is set
(((1_ G) ") * H) * (1_ G) is Element of bool the carrier of G
(((1_ G) ") * H) * {(1_ G)} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((1_ G) ") * H & b₂ in {(1_ G)} ) } is set
(1_ G) * H is Element of bool the carrier of G
{(1_ G)} * H is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(1_ G)} & b₂ in H ) } 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 non empty set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
H is Element of bool the carrier of G
(G,H,(1_ G)) is Element of bool the carrier of G
(G,{(1_ G)},H) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(1_ G)} & b₂ in H ) } is set
the Element of H is Element of H
y is set
H1 is Element of the carrier of G
(G,(1_ G),H1) is Element of the carrier of G
H1 " is Element of the carrier of G
(H1 ") * (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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H1 "),(1_ G)) is Element of the carrier of G
[(H1 "),(1_ G)] is set
{(H1 "),(1_ G)} is non empty finite set
{(H1 ")} is non empty finite set
{{(H1 "),(1_ G)},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),(1_ G)] is set
((H1 ") * (1_ G)) * H1 is Element of the carrier of G
the multF of G . (((H1 ") * (1_ G)),H1) is Element of the carrier of G
[((H1 ") * (1_ G)),H1] is set
{((H1 ") * (1_ G)),H1} is non empty finite set
{((H1 ") * (1_ G))} is non empty finite set
{{((H1 ") * (1_ G)),H1},{((H1 ") * (1_ G))}} is non empty finite V45() set
the multF of G . [((H1 ") * (1_ G)),H1] is set
y is set
c₄ is Element of the carrier of G
(G,(1_ G),c₄) is Element of the carrier of G
c₄ " is Element of the carrier of G
(c₄ ") * (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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((c₄ "),(1_ G)) is Element of the carrier of G
[(c₄ "),(1_ G)] is set
{(c₄ "),(1_ G)} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),(1_ G)},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),(1_ G)] is set
((c₄ ") * (1_ G)) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * (1_ G)),c₄) is Element of the carrier of G
[((c₄ ") * (1_ G)),c₄] is set
{((c₄ ") * (1_ G)),c₄} is non empty finite set
{((c₄ ") * (1_ G))} is non empty finite set
{{((c₄ ") * (1_ G)),c₄},{((c₄ ") * (1_ G))}} is non empty finite V45() set
the multF of G . [((c₄ ") * (1_ G)),c₄] 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 non empty set
H is Element of the carrier of G
H " is Element of the carrier of G
a is Element of bool the carrier of G
(G,a,H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,a,{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {H} ) } is set
(G,(G,a,H),(H ")) is Element of bool the carrier of G
{(H ")} is non empty finite Element of bool the carrier of G
(G,(G,a,H),{(H ")}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,a,H) & b₂ in {(H ")} ) } is set
(G,a,(H ")) is Element of bool the carrier of G
(G,a,{(H ")}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {(H ")} ) } is set
(G,(G,a,(H ")),H) is Element of bool the carrier of G
(G,(G,a,(H ")),{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,a,(H ")) & b₂ in {H} ) } is set
H * (H ") 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H,(H ")) is Element of the carrier of G
[H,(H ")] is set
{H,(H ")} is non empty finite set
{H} is non empty finite set
{{H,(H ")},{H}} is non empty finite V45() set
the multF of G . [H,(H ")] is set
(G,a,(H * (H "))) is Element of bool the carrier of G
{(H * (H "))} is non empty finite Element of bool the carrier of G
(G,a,{(H * (H "))}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {(H * (H "))} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(G,a,(1_ G)) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
(G,a,{(1_ G)}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {(1_ G)} ) } is set
(H ") * H is Element of the carrier of G
the multF of G . ((H "),H) is Element of the carrier of G
[(H "),H] is set
{(H "),H} is non empty finite set
{(H ")} is non empty finite set
{{(H "),H},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),H] is set
(G,a,((H ") * H)) is Element of bool the carrier of G
{((H ") * H)} is non empty finite Element of bool the carrier of G
(G,a,{((H ") * H)}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {((H ") * H)} ) } 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 non empty set
H is Element of bool the carrier of G
a is Element of bool the carrier of G
(G,H,a) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in a ) } is set
the Element of a is Element of a
H1 is set
a is Element of the carrier of G
b is Element of the carrier of G
(G,a,b) is Element of the carrier of G
b " is Element of the carrier of G
(b ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((b "),a) is Element of the carrier of G
[(b "),a] is set
{(b "),a} is non empty finite set
{(b ")} is non empty finite set
{{(b "),a},{(b ")}} is non empty finite V45() set
the multF of G . [(b "),a] is set
((b ") * a) * b is Element of the carrier of G
the multF of G . (((b ") * a),b) is Element of the carrier of G
[((b ") * a),b] is set
{((b ") * a),b} is non empty finite set
{((b ") * a)} is non empty finite set
{{((b ") * a),b},{((b ") * a)}} is non empty finite V45() set
the multF of G . [((b ") * a),b] is set
H1 is set
a is Element of the carrier of G
y is Element of the carrier of G
(G,a,y) is Element of the carrier of G
y " is Element of the carrier of G
(y ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((y "),a) is Element of the carrier of G
[(y "),a] is set
{(y "),a} is non empty finite set
{(y ")} is non empty finite set
{{(y "),a},{(y ")}} is non empty finite V45() set
the multF of G . [(y "),a] is set
((y ") * a) * y is Element of the carrier of G
the multF of G . (((y ") * a),y) is Element of the carrier of G
[((y ") * a),y] is set
{((y ") * a),y} is non empty finite set
{((y ") * a)} is non empty finite set
{{((y ") * a),y},{((y ") * a)}} is non empty finite V45() set
the multF of G . [((y ") * a),y] is set
H is Element of the carrier of G
{H} is non empty finite Element of bool the carrier of G
a is Element of the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,{H},{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in {a} ) } is set
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
{(G,H,a)} is non empty finite Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
H is Element of bool the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,H,{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {a} ) } is set
H is Element of the carrier of G
{H} is non empty finite Element of bool the carrier of G
a is Element of the carrier of G
(G,{H},a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,{H},{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in {a} ) } is set
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
{(G,H,a)} is non empty finite Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
H is Element of bool the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,{a},H) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in H ) } is set
H is Element of the carrier of G
{H} is non empty finite Element of bool the carrier of G
a is Element of the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,{a},H) is Element of bool the carrier of G
(G,{H},{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in {a} ) } is set
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
{(G,H,a)} is non empty finite Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is non empty unital Group-like associative Subgroup of G
carr H is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of H is non empty set
a is Element of the carrier of G
(G,(carr H),a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,(carr H),{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {a} ) } is set
a " is Element of the carrier of G
(a ") " is Element of the carrier of G
(a ") * H is Element of bool the carrier of G
(a ") * (carr H) is Element of bool the carrier of G
{(a ")} is non empty finite Element of bool the carrier of G
{(a ")} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(a ")} & b₂ in carr H ) } is set
((a ") * H) * ((a ") ") is Element of bool the carrier of G
{((a ") ")} is non empty finite Element of bool the carrier of G
((a ") * H) * {((a ") ")} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (a ") * H & b₂ in {((a ") ")} ) } is set
c₄ is non empty strict unital Group-like associative Subgroup of G
the carrier of c₄ is non empty set
((a ") * (carr H)) * a is Element of bool the carrier of G
((a ") * (carr H)) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (a ") * (carr H) & b₂ in {a} ) } is set
c₄ is non empty strict unital Group-like associative Subgroup of G
the carrier of c₄ is non empty set
y is non empty strict unital Group-like associative Subgroup of G
the carrier of y is non empty set
G is set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
a is Element of the carrier of H
c₄ is non empty unital Group-like associative Subgroup of H
(H,c₄,a) is non empty strict unital Group-like associative Subgroup of H
the carrier of (H,c₄,a) is non empty set
carr c₄ is Element of bool the carrier of H
bool the carrier of H is non empty set
the carrier of c₄ is non empty set
(H,(carr c₄),a) is Element of bool the carrier of H
{a} is non empty finite Element of bool the carrier of H
(H,(carr c₄),{a}) is Element of bool the carrier of H
{ (H,b₁,b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in carr c₄ & b₂ in {a} ) } is set
y is Element of the carrier of H
(H,y,a) is Element of the carrier of H
a " is Element of the carrier of H
(a ") * y is Element of the carrier of H
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
the multF of H . ((a "),y) is Element of the carrier of H
[(a "),y] is set
{(a "),y} is non empty finite set
{(a ")} is non empty finite set
{{(a "),y},{(a ")}} is non empty finite V45() set
the multF of H . [(a "),y] is set
((a ") * y) * a is Element of the carrier of H
the multF of H . (((a ") * y),a) is Element of the carrier of H
[((a ") * y),a] is set
{((a ") * y),a} is non empty finite set
{((a ") * y)} is non empty finite set
{{((a ") * y),a},{((a ") * y)}} is non empty finite V45() set
the multF of H . [((a ") * y),a] is set
y is Element of the carrier of H
(H,y,a) is Element of the carrier of H
a " is Element of the carrier of H
(a ") * y is Element of the carrier of H
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
the multF of H . ((a "),y) is Element of the carrier of H
[(a "),y] is set
{(a "),y} is non empty finite set
{(a ")} is non empty finite set
{{(a "),y},{(a ")}} is non empty finite V45() set
the multF of H . [(a "),y] is set
((a ") * y) * a is Element of the carrier of H
the multF of H . (((a ") * y),a) is Element of the carrier of H
[((a ") * y),a] is set
{((a ") * y),a} is non empty finite set
{((a ") * y)} is non empty finite set
{{((a ") * y),a},{((a ") * y)}} is non empty finite V45() set
the multF of H . [((a ") * y),a] is set
carr c₄ is Element of bool the carrier of H
bool the carrier of H is non empty set
the carrier of c₄ is non empty set
(H,(carr c₄),a) is Element of bool the carrier of H
{a} is non empty finite Element of bool the carrier of H
(H,(carr c₄),{a}) is Element of bool the carrier of H
{ (H,b₁,b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in carr c₄ & b₂ in {a} ) } is set
the carrier of (H,c₄,a) is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
H " is Element of the carrier of G
a is non empty unital Group-like associative Subgroup of G
(G,a,H) is non empty strict unital Group-like associative Subgroup of G
the carrier of (G,a,H) is non empty set
(H ") * a is Element of bool the carrier of G
bool the carrier of G is non empty set
carr a is Element of bool the carrier of G
the carrier of a is non empty set
(H ") * (carr a) is Element of bool the carrier of G
{(H ")} is non empty finite Element of bool the carrier of G
{(H ")} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(H ")} & b₂ in carr a ) } is set
((H ") * a) * H is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
((H ") * a) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (H ") * a & b₂ in {H} ) } is set
(G,(carr a),H) is Element of bool the carrier of G
(G,(carr a),{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in {H} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
H * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H,a) is Element of the carrier of G
[H,a] is set
{H,a} is non empty finite set
{H} is non empty finite set
{{H,a},{H}} is non empty finite V45() set
the multF of G . [H,a] is set
c₄ is non empty unital Group-like associative Subgroup of G
(G,c₄,H) is non empty strict unital Group-like associative Subgroup of G
(G,(G,c₄,H),a) is non empty strict unital Group-like associative Subgroup of G
(G,c₄,(H * a)) is non empty strict unital Group-like associative Subgroup of G
the carrier of (G,(G,c₄,H),a) is non empty set
carr (G,c₄,H) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of (G,c₄,H) is non empty set
(G,(carr (G,c₄,H)),a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,(carr (G,c₄,H)),{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr (G,c₄,H) & b₂ in {a} ) } is set
carr c₄ is Element of bool the carrier of G
the carrier of c₄ is non empty set
(G,(carr c₄),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,(carr c₄),{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr c₄ & b₂ in {H} ) } is set
(G,(G,(carr c₄),H),a) is Element of bool the carrier of G
(G,(G,(carr c₄),H),{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,(carr c₄),H) & b₂ in {a} ) } is set
(G,(carr c₄),(H * a)) is Element of bool the carrier of G
{(H * a)} is non empty finite Element of bool the carrier of G
(G,(carr c₄),{(H * a)}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr c₄ & b₂ in {(H * a)} ) } is set
the carrier of (G,c₄,(H * a)) is non empty set
G is non empty unital Group-like associative multMagma
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
H is non empty strict unital Group-like associative Subgroup of G
(G,H,(1_ G)) is non empty strict unital Group-like associative Subgroup of G
the carrier of (G,H,(1_ G)) is non empty set
carr H is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of H is non empty set
(G,(carr H),(1_ G)) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
(G,(carr H),{(1_ G)}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {(1_ G)} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
H " is Element of the carrier of G
a is non empty strict unital Group-like associative Subgroup of G
(G,a,H) is non empty strict unital Group-like associative Subgroup of G
(G,(G,a,H),(H ")) is non empty strict unital Group-like associative Subgroup of G
(G,a,(H ")) is non empty strict unital Group-like associative Subgroup of G
(G,(G,a,(H ")),H) is non empty strict unital Group-like associative Subgroup of G
H * (H ") 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H,(H ")) is Element of the carrier of G
[H,(H ")] is set
{H,(H ")} is non empty finite set
{H} is non empty finite set
{{H,(H ")},{H}} is non empty finite V45() set
the multF of G . [H,(H ")] is set
(G,a,(H * (H "))) is non empty strict unital Group-like associative Subgroup of G
1_ G is non being_of_order_0 Element of the carrier of G
(G,a,(1_ G)) is non empty strict unital Group-like associative Subgroup of G
(H ") * H is Element of the carrier of G
the multF of G . ((H "),H) is Element of the carrier of G
[(H "),H] is set
{(H "),H} is non empty finite set
{(H ")} is non empty finite set
{{(H "),H},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),H] is set
(G,a,((H ") * H)) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is non empty unital Group-like associative Subgroup of G
(G,a,H) is non empty strict unital Group-like associative Subgroup of G
c₄ is non empty unital Group-like associative Subgroup of G
a /\ c₄ is non empty strict unital Group-like associative Subgroup of G
(G,(a /\ c₄),H) is non empty strict unital Group-like associative Subgroup of G
(G,c₄,H) is non empty strict unital Group-like associative Subgroup of G
(G,a,H) /\ (G,c₄,H) is non empty strict unital Group-like associative Subgroup of G
y is Element of the carrier of G
H1 is Element of the carrier of G
(G,H1,H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),H1) is Element of the carrier of G
[(H "),H1] is set
{(H "),H1} is non empty finite set
{(H ")} is non empty finite set
{{(H "),H1},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),H1] is set
((H ") * H1) * H is Element of the carrier of G
the multF of G . (((H ") * H1),H) is Element of the carrier of G
[((H ") * H1),H] is set
{((H ") * H1),H} is non empty finite set
{((H ") * H1)} is non empty finite set
{{((H ") * H1),H},{((H ") * H1)}} is non empty finite V45() set
the multF of G . [((H ") * H1),H] is set
H1 is Element of the carrier of G
(G,H1,H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),H1) is Element of the carrier of G
[(H "),H1] is set
{(H "),H1} is non empty finite set
{(H ")} is non empty finite set
{{(H "),H1},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),H1] is set
((H ") * H1) * H is Element of the carrier of G
the multF of G . (((H ") * H1),H) is Element of the carrier of G
[((H ") * H1),H] is set
{((H ") * H1),H} is non empty finite set
{((H ") * H1)} is non empty finite set
{{((H ") * H1),H},{((H ") * H1)}} is non empty finite V45() set
the multF of G . [((H ") * H1),H] is set
a is Element of the carrier of G
(G,a,H) is Element of the carrier of G
(H ") * a is Element of the carrier of G
the multF of G . ((H "),a) is Element of the carrier of G
[(H "),a] is set
{(H "),a} is non empty finite set
{{(H "),a},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),a] is set
((H ") * a) * H is Element of the carrier of G
the multF of G . (((H ") * a),H) is Element of the carrier of G
[((H ") * a),H] is set
{((H ") * a),H} is non empty finite set
{((H ") * a)} is non empty finite set
{{((H ") * a),H},{((H ") * a)}} is non empty finite V45() set
the multF of G . [((H ") * a),H] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is non empty unital Group-like associative Subgroup of G
card a is cardinal set
the carrier of a is non empty set
card the carrier of a is cardinal set
(G,a,H) is non empty strict unital Group-like associative Subgroup of G
card (G,a,H) is cardinal set
the carrier of (G,a,H) is non empty set
card the carrier of (G,a,H) is cardinal set
[: the carrier of G, the carrier of G:] is non empty set
bool [: the carrier of G, the carrier of G:] is non empty set
c₄ is Relation-like the carrier of G -defined the carrier of G -valued Function-like V28( the carrier of G, the carrier of G) Element of bool [: the carrier of G, the carrier of G:]
c₄ | the carrier of a is Relation-like Function-like set
dom c₄ is set
dom (c₄ | the carrier of a) is set
rng (c₄ | the carrier of a) is set
H1 is set
a is set
(c₄ | the carrier of a) . a is set
b is Element of the carrier of a
c is Element of the carrier of G
c₄ . c is Element of the carrier of G
(c₄ | the carrier of a) . c is set
(G,c,H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * c 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),c) is Element of the carrier of G
[(H "),c] is set
{(H "),c} is non empty finite set
{(H ")} is non empty finite set
{{(H "),c},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),c] is set
((H ") * c) * H is Element of the carrier of G
the multF of G . (((H ") * c),H) is Element of the carrier of G
[((H ") * c),H] is set
{((H ") * c),H} is non empty finite set
{((H ") * c)} is non empty finite set
{{((H ") * c),H},{((H ") * c)}} is non empty finite V45() set
the multF of G . [((H ") * c),H] is set
carr a is Element of bool the carrier of G
bool the carrier of G is non empty set
(G,(carr a),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,(carr a),{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in {H} ) } is set
H1 is set
carr a is Element of bool the carrier of G
bool the carrier of G is non empty set
(G,(carr a),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,(carr a),{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in {H} ) } is set
a is Element of the carrier of G
(G,a,H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),a) is Element of the carrier of G
[(H "),a] is set
{(H "),a} is non empty finite set
{(H ")} is non empty finite set
{{(H "),a},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),a] is set
((H ") * a) * H is Element of the carrier of G
the multF of G . (((H ") * a),H) is Element of the carrier of G
[((H ") * a),H] is set
{((H ") * a),H} is non empty finite set
{((H ") * a)} is non empty finite set
{{((H ") * a),H},{((H ") * a)}} is non empty finite V45() set
the multF of G . [((H ") * a),H] is set
c₄ . a is Element of the carrier of G
(c₄ | the carrier of a) . a is set
H1 is set
(c₄ | the carrier of a) . H1 is set
a is set
(c₄ | the carrier of a) . a is set
b is Element of the carrier of a
c is Element of the carrier of a
c₄ . H1 is set
c is Element of the carrier of G
(G,c,H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * c 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),c) is Element of the carrier of G
[(H "),c] is set
{(H "),c} is non empty finite set
{(H ")} is non empty finite set
{{(H "),c},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),c] is set
((H ") * c) * H is Element of the carrier of G
the multF of G . (((H ") * c),H) is Element of the carrier of G
[((H ") * c),H] is set
{((H ") * c),H} is non empty finite set
{((H ") * c)} is non empty finite set
{{((H ") * c),H},{((H ") * c)}} is non empty finite V45() set
the multF of G . [((H ") * c),H] is set
c₄ . a is set
c is Element of the carrier of G
(G,c,H) is Element of the carrier of G
(H ") * c is Element of the carrier of G
the multF of G . ((H "),c) is Element of the carrier of G
[(H "),c] is set
{(H "),c} is non empty finite set
{{(H "),c},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),c] is set
((H ") * c) * H is Element of the carrier of G
the multF of G . (((H ") * c),H) is Element of the carrier of G
[((H ") * c),H] is set
{((H ") * c),H} is non empty finite set
{((H ") * c)} is non empty finite set
{{((H ") * c),H},{((H ") * c)}} is non empty finite V45() set
the multF of G . [((H ") * c),H] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is non empty unital Group-like associative Subgroup of G
(G,a,H) is non empty strict unital Group-like associative Subgroup of G
card a is cardinal set
the carrier of a is non empty set
card the carrier of a is cardinal set
card (G,a,H) is cardinal set
the carrier of (G,a,H) is non empty set
card the carrier of (G,a,H) is cardinal set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
a is non empty finite unital Group-like associative Subgroup of G
H is Element of the carrier of G
(G,a,H) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
a is non empty finite unital Group-like associative Subgroup of G
card a is non empty V12() V13() V14() V18() ext-real V37() V38() integer cardinal Element of K113()
the carrier of a is non empty finite set
card the carrier of a is cardinal set
H is Element of the carrier of G
(G,a,H) is non empty finite strict unital Group-like associative Subgroup of G
card (G,a,H) is non empty V12() V13() V14() V18() ext-real V37() V38() integer cardinal Element of K113()
the carrier of (G,a,H) is non empty finite set
card the carrier of (G,a,H) is cardinal set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
(1). G is non empty finite strict unital Group-like associative Subgroup of G
H is Element of the carrier of G
(G,((1). G),H) is non empty finite strict unital Group-like associative 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
{(1_ G)} is non empty finite Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of (G,((1). G),H) is non empty finite set
carr ((1). G) is Element of bool the carrier of G
(G,(carr ((1). G)),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,(carr ((1). G)),{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr ((1). G) & b₂ in {H} ) } is set
(G,(1_ G),H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * (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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),(1_ G)) is Element of the carrier of G
[(H "),(1_ G)] is set
{(H "),(1_ G)} is non empty finite set
{(H ")} is non empty finite set
{{(H "),(1_ G)},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),(1_ G)] is set
((H ") * (1_ G)) * H is Element of the carrier of G
the multF of G . (((H ") * (1_ G)),H) is Element of the carrier of G
[((H ") * (1_ G)),H] is set
{((H ") * (1_ G)),H} is non empty finite set
{((H ") * (1_ G))} is non empty finite set
{{((H ") * (1_ G)),H},{((H ") * (1_ G))}} is non empty finite V45() set
the multF of G . [((H ") * (1_ G)),H] is set
{(G,(1_ G),H)} is non empty finite Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
(1). G is non empty finite strict unital Group-like associative Subgroup of G
H is Element of the carrier of G
a is non empty strict unital Group-like associative Subgroup of G
(G,a,H) is non empty strict unital Group-like associative Subgroup of G
card ((1). G) is non empty V12() V13() V14() V18() ext-real V37() V38() integer cardinal Element of K113()
the carrier of ((1). G) is non empty finite set
card the carrier of ((1). G) is cardinal set
c₄ is non empty finite unital Group-like associative Subgroup of G
card c₄ is non empty V12() V13() V14() V18() ext-real V37() V38() integer cardinal Element of K113()
the carrier of c₄ is non empty finite set
card the carrier of c₄ is cardinal set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
H is Element of the carrier of G
(G,((Omega). G),H) is non empty strict unital Group-like associative Subgroup of G
a is Element of the carrier of G
H " is Element of the carrier of G
(G,a,(H ")) is Element of the carrier of G
(H ") " is Element of the carrier of G
((H ") ") * a is Element of the carrier of G
the multF of G . (((H ") "),a) is Element of the carrier of G
[((H ") "),a] is set
{((H ") "),a} is non empty finite set
{((H ") ")} is non empty finite set
{{((H ") "),a},{((H ") ")}} is non empty finite V45() set
the multF of G . [((H ") "),a] is set
(((H ") ") * a) * (H ") is Element of the carrier of G
the multF of G . ((((H ") ") * a),(H ")) is Element of the carrier of G
[(((H ") ") * a),(H ")] is set
{(((H ") ") * a),(H ")} is non empty finite set
{(((H ") ") * a)} is non empty finite set
{{(((H ") ") * a),(H ")},{(((H ") ") * a)}} is non empty finite V45() set
the multF of G . [(((H ") ") * a),(H ")] is set
(G,(G,a,(H ")),H) is Element of the carrier of G
(H ") * (G,a,(H ")) is Element of the carrier of G
the multF of G . ((H "),(G,a,(H "))) is Element of the carrier of G
[(H "),(G,a,(H "))] is set
{(H "),(G,a,(H "))} is non empty finite set
{(H ")} is non empty finite set
{{(H "),(G,a,(H "))},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),(G,a,(H "))] is set
((H ") * (G,a,(H "))) * H is Element of the carrier of G
the multF of G . (((H ") * (G,a,(H "))),H) is Element of the carrier of G
[((H ") * (G,a,(H "))),H] is set
{((H ") * (G,a,(H "))),H} is non empty finite set
{((H ") * (G,a,(H ")))} is non empty finite set
{{((H ") * (G,a,(H "))),H},{((H ") * (G,a,(H ")))}} is non empty finite V45() set
the multF of G . [((H ") * (G,a,(H "))),H] is set
(H ") * H is Element of the carrier of G
the multF of G . ((H "),H) is Element of the carrier of G
[(H "),H] is set
{(H "),H} is non empty finite set
{{(H "),H},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),H] is set
(G,a,((H ") * H)) is Element of the carrier of G
((H ") * H) " is Element of the carrier of G
(((H ") * H) ") * a is Element of the carrier of G
the multF of G . ((((H ") * H) "),a) is Element of the carrier of G
[(((H ") * H) "),a] is set
{(((H ") * H) "),a} is non empty finite set
{(((H ") * H) ")} is non empty finite set
{{(((H ") * H) "),a},{(((H ") * H) ")}} is non empty finite V45() set
the multF of G . [(((H ") * H) "),a] is set
((((H ") * H) ") * a) * ((H ") * H) is Element of the carrier of G
the multF of G . (((((H ") * H) ") * a),((H ") * H)) is Element of the carrier of G
[((((H ") * H) ") * a),((H ") * H)] is set
{((((H ") * H) ") * a),((H ") * H)} is non empty finite set
{((((H ") * H) ") * a)} is non empty finite set
{{((((H ") * H) ") * a),((H ") * H)},{((((H ") * H) ") * a)}} is non empty finite V45() set
the multF of G . [((((H ") * H) ") * a),((H ") * H)] is set
1_ G is non being_of_order_0 Element of the carrier of G
(G,a,(1_ G)) is Element of the carrier of G
(1_ G) " is Element of the carrier of G
((1_ G) ") * a is Element of the carrier of G
the multF of G . (((1_ G) "),a) is Element of the carrier of G
[((1_ G) "),a] is set
{((1_ G) "),a} is non empty finite set
{((1_ G) ")} is non empty finite set
{{((1_ G) "),a},{((1_ G) ")}} is non empty finite V45() set
the multF of G . [((1_ G) "),a] is set
(((1_ G) ") * a) * (1_ G) is Element of the carrier of G
the multF of G . ((((1_ G) ") * a),(1_ G)) is Element of the carrier of G
[(((1_ G) ") * a),(1_ G)] is set
{(((1_ G) ") * a),(1_ G)} is non empty finite set
{(((1_ G) ") * a)} is non empty finite set
{{(((1_ G) ") * a),(1_ G)},{(((1_ G) ") * a)}} is non empty finite V45() set
the multF of G . [(((1_ G) ") * a),(1_ G)] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is non empty strict unital Group-like associative Subgroup of G
(G,a,H) is non empty strict unital Group-like associative Subgroup of G
c₄ is Element of the carrier of G
(G,c₄,H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * c₄ 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),c₄) is Element of the carrier of G
[(H "),c₄] is set
{(H "),c₄} is non empty finite set
{(H ")} is non empty finite set
{{(H "),c₄},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),c₄] is set
((H ") * c₄) * H is Element of the carrier of G
the multF of G . (((H ") * c₄),H) is Element of the carrier of G
[((H ") * c₄),H] is set
{((H ") * c₄),H} is non empty finite set
{((H ") * c₄)} is non empty finite set
{{((H ") * c₄),H},{((H ") * c₄)}} is non empty finite V45() set
the multF of G . [((H ") * c₄),H] is set
y is Element of the carrier of G
(G,y,H) is Element of the carrier of G
(H ") * y is Element of the carrier of G
the multF of G . ((H "),y) is Element of the carrier of G
[(H "),y] is set
{(H "),y} is non empty finite set
{{(H "),y},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),y] is set
((H ") * y) * H is Element of the carrier of G
the multF of G . (((H ") * y),H) is Element of the carrier of G
[((H ") * y),H] is set
{((H ") * y),H} is non empty finite set
{((H ") * y)} is non empty finite set
{{((H ") * y),H},{((H ") * y)}} is non empty finite V45() set
the multF of G . [((H ") * y),H] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is non empty unital Group-like associative Subgroup of G
Index a is cardinal set
Left_Cosets a is Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
card (Left_Cosets a) is cardinal set
(G,a,H) is non empty strict unital Group-like associative Subgroup of G
Index (G,a,H) is cardinal set
Left_Cosets (G,a,H) is Element of bool (bool the carrier of G)
card (Left_Cosets (G,a,H)) is cardinal set
c₄ is set
y is Element of the carrier of G
y * a is Element of bool the carrier of G
carr a is Element of bool the carrier of G
the carrier of a is non empty set
y * (carr a) is Element of bool the carrier of G
{y} is non empty finite Element of bool the carrier of G
{y} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {y} & b₂ in carr a ) } is set
(G,y,H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * y 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),y) is Element of the carrier of G
[(H "),y] is set
{(H "),y} is non empty finite set
{(H ")} is non empty finite set
{{(H "),y},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),y] is set
((H ") * y) * H is Element of the carrier of G
the multF of G . (((H ") * y),H) is Element of the carrier of G
[((H ") * y),H] is set
{((H ") * y),H} is non empty finite set
{((H ") * y)} is non empty finite set
{{((H ") * y),H},{((H ") * y)}} is non empty finite V45() set
the multF of G . [((H ") * y),H] is set
(G,y,H) * (G,a,H) is Element of bool the carrier of G
carr (G,a,H) is Element of bool the carrier of G
the carrier of (G,a,H) is non empty set
(G,y,H) * (carr (G,a,H)) is Element of bool the carrier of G
{(G,y,H)} is non empty finite Element of bool the carrier of G
{(G,y,H)} * (carr (G,a,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(G,y,H)} & b₂ in carr (G,a,H) ) } is set
H1 is set
c₄ is Relation-like Function-like set
dom c₄ is set
carr a is Element of bool the carrier of G
the carrier of a is non empty set
H1 is set
a is set
b is set
c is Element of the carrier of G
c * a is Element of bool the carrier of G
c * (carr a) is Element of bool the carrier of G
{c} is non empty finite Element of bool the carrier of G
{c} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {c} & b₂ in carr a ) } is set
(G,c,H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * c 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),c) is Element of the carrier of G
[(H "),c] is set
{(H "),c} is non empty finite set
{(H ")} is non empty finite set
{{(H "),c},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),c] is set
((H ") * c) * H is Element of the carrier of G
the multF of G . (((H ") * c),H) is Element of the carrier of G
[((H ") * c),H] is set
{((H ") * c),H} is non empty finite set
{((H ") * c)} is non empty finite set
{{((H ") * c),H},{((H ") * c)}} is non empty finite V45() set
the multF of G . [((H ") * c),H] is set
(G,c,H) * (G,a,H) is Element of bool the carrier of G
carr (G,a,H) is Element of bool the carrier of G
the carrier of (G,a,H) is non empty set
(G,c,H) * (carr (G,a,H)) is Element of bool the carrier of G
{(G,c,H)} is non empty finite Element of bool the carrier of G
{(G,c,H)} * (carr (G,a,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(G,c,H)} & b₂ in carr (G,a,H) ) } is set
c is Element of the carrier of G
c * a is Element of bool the carrier of G
c * (carr a) is Element of bool the carrier of G
{c} is non empty finite Element of bool the carrier of G
{c} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {c} & b₂ in carr a ) } is set
(G,c,H) is Element of the carrier of G
(H ") * c is Element of the carrier of G
the multF of G . ((H "),c) is Element of the carrier of G
[(H "),c] is set
{(H "),c} is non empty finite set
{{(H "),c},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),c] is set
((H ") * c) * H is Element of the carrier of G
the multF of G . (((H ") * c),H) is Element of the carrier of G
[((H ") * c),H] is set
{((H ") * c),H} is non empty finite set
{((H ") * c)} is non empty finite set
{{((H ") * c),H},{((H ") * c)}} is non empty finite V45() set
the multF of G . [((H ") * c),H] is set
(G,c,H) * (G,a,H) is Element of bool the carrier of G
(G,c,H) * (carr (G,a,H)) is Element of bool the carrier of G
{(G,c,H)} is non empty finite Element of bool the carrier of G
{(G,c,H)} * (carr (G,a,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(G,c,H)} & b₂ in carr (G,a,H) ) } is set
(H ") * a is Element of bool the carrier of G
(H ") * (carr a) is Element of bool the carrier of G
{(H ")} is non empty finite Element of bool the carrier of G
{(H ")} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(H ")} & b₂ in carr a ) } is set
((H ") * a) * H is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
((H ") * a) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (H ") * a & b₂ in {H} ) } is set
(((H ") * c) * H) * (((H ") * a) * H) is Element of bool the carrier of G
{(((H ") * c) * H)} is non empty finite Element of bool the carrier of G
{(((H ") * c) * H)} * (((H ") * a) * H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(((H ") * c) * H)} & b₂ in ((H ") * a) * H ) } is set
(((H ") * c) * H) * ((H ") * (carr a)) is Element of bool the carrier of G
{(((H ") * c) * H)} * ((H ") * (carr a)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(((H ") * c) * H)} & b₂ in (H ") * (carr a) ) } is set
((((H ") * c) * H) * ((H ") * (carr a))) * H is Element of bool the carrier of G
((((H ") * c) * H) * ((H ") * (carr a))) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (((H ") * c) * H) * ((H ") * (carr a)) & b₂ in {H} ) } is set
H * ((H ") * (carr a)) is Element of bool the carrier of G
{H} * ((H ") * (carr a)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in (H ") * (carr a) ) } is set
((H ") * c) * (H * ((H ") * (carr a))) is Element of bool the carrier of G
{((H ") * c)} is non empty finite Element of bool the carrier of G
{((H ") * c)} * (H * ((H ") * (carr a))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((H ") * c)} & b₂ in H * ((H ") * (carr a)) ) } is set
(((H ") * c) * (H * ((H ") * (carr a)))) * H is Element of bool the carrier of G
(((H ") * c) * (H * ((H ") * (carr a)))) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((H ") * c) * (H * ((H ") * (carr a))) & b₂ in {H} ) } is set
H * (H ") is Element of the carrier of G
the multF of G . (H,(H ")) is Element of the carrier of G
[H,(H ")] is set
{H,(H ")} is non empty finite set
{H} is non empty finite set
{{H,(H ")},{H}} is non empty finite V45() set
the multF of G . [H,(H ")] is set
(H * (H ")) * (carr a) is Element of bool the carrier of G
{(H * (H "))} is non empty finite Element of bool the carrier of G
{(H * (H "))} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(H * (H "))} & b₂ in carr a ) } is set
((H ") * c) * ((H * (H ")) * (carr a)) is Element of bool the carrier of G
{((H ") * c)} * ((H * (H ")) * (carr a)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((H ") * c)} & b₂ in (H * (H ")) * (carr a) ) } is set
(((H ") * c) * ((H * (H ")) * (carr a))) * H is Element of bool the carrier of G
(((H ") * c) * ((H * (H ")) * (carr a))) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((H ") * c) * ((H * (H ")) * (carr a)) & b₂ in {H} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * (carr a) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
{(1_ G)} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(1_ G)} & b₂ in carr a ) } is set
((H ") * c) * ((1_ G) * (carr a)) is Element of bool the carrier of G
{((H ") * c)} * ((1_ G) * (carr a)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((H ") * c)} & b₂ in (1_ G) * (carr a) ) } is set
(((H ") * c) * ((1_ G) * (carr a))) * H is Element of bool the carrier of G
(((H ") * c) * ((1_ G) * (carr a))) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((H ") * c) * ((1_ G) * (carr a)) & b₂ in {H} ) } is set
((H ") * c) * (carr a) is Element of bool the carrier of G
{((H ") * c)} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((H ") * c)} & b₂ in carr a ) } is set
(((H ") * c) * (carr a)) * H is Element of bool the carrier of G
(((H ") * c) * (carr a)) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((H ") * c) * (carr a) & b₂ in {H} ) } is set
(H ") * (c * a) is Element of bool the carrier of G
{(H ")} * (c * a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(H ")} & b₂ in c * a ) } is set
((H ") * (c * a)) * H is Element of bool the carrier of G
((H ") * (c * a)) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (H ") * (c * a) & b₂ in {H} ) } is set
((H ") * c) * (carr a) is Element of bool the carrier of G
{((H ") * c)} is non empty finite Element of bool the carrier of G
{((H ") * c)} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((H ") * c)} & b₂ in carr a ) } is set
(((H ") * c) * (carr a)) * H is Element of bool the carrier of G
(((H ") * c) * (carr a)) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((H ") * c) * (carr a) & b₂ in {H} ) } is set
((H ") * c) * ((1_ G) * (carr a)) is Element of bool the carrier of G
{((H ") * c)} * ((1_ G) * (carr a)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((H ") * c)} & b₂ in (1_ G) * (carr a) ) } is set
(((H ") * c) * ((1_ G) * (carr a))) * H is Element of bool the carrier of G
(((H ") * c) * ((1_ G) * (carr a))) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((H ") * c) * ((1_ G) * (carr a)) & b₂ in {H} ) } is set
((H ") * c) * ((H * (H ")) * (carr a)) is Element of bool the carrier of G
{((H ") * c)} * ((H * (H ")) * (carr a)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((H ") * c)} & b₂ in (H * (H ")) * (carr a) ) } is set
(((H ") * c) * ((H * (H ")) * (carr a))) * H is Element of bool the carrier of G
(((H ") * c) * ((H * (H ")) * (carr a))) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((H ") * c) * ((H * (H ")) * (carr a)) & b₂ in {H} ) } is set
((H ") * c) * (H * ((H ") * (carr a))) is Element of bool the carrier of G
{((H ") * c)} * (H * ((H ") * (carr a))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((H ") * c)} & b₂ in H * ((H ") * (carr a)) ) } is set
(((H ") * c) * (H * ((H ") * (carr a)))) * H is Element of bool the carrier of G
(((H ") * c) * (H * ((H ") * (carr a)))) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((H ") * c) * (H * ((H ") * (carr a))) & b₂ in {H} ) } is set
(((H ") * c) * H) * ((H ") * (carr a)) is Element of bool the carrier of G
{(((H ") * c) * H)} is non empty finite Element of bool the carrier of G
{(((H ") * c) * H)} * ((H ") * (carr a)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(((H ") * c) * H)} & b₂ in (H ") * (carr a) ) } is set
((((H ") * c) * H) * ((H ") * (carr a))) * H is Element of bool the carrier of G
((((H ") * c) * H) * ((H ") * (carr a))) * {H} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (((H ") * c) * H) * ((H ") * (carr a)) & b₂ in {H} ) } is set
(((H ") * c) * H) * (((H ") * a) * H) is Element of bool the carrier of G
{(((H ") * c) * H)} * (((H ") * a) * H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(((H ") * c) * H)} & b₂ in ((H ") * a) * H ) } is set
rng c₄ is set
y is set
H1 is set
c₄ . H1 is set
a is Element of the carrier of G
a * a is Element of bool the carrier of G
carr a is Element of bool the carrier of G
the carrier of a is non empty set
a * (carr a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
{a} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr a ) } is set
(G,a,H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),a) is Element of the carrier of G
[(H "),a] is set
{(H "),a} is non empty finite set
{(H ")} is non empty finite set
{{(H "),a},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),a] is set
((H ") * a) * H is Element of the carrier of G
the multF of G . (((H ") * a),H) is Element of the carrier of G
[((H ") * a),H] is set
{((H ") * a),H} is non empty finite set
{((H ") * a)} is non empty finite set
{{((H ") * a),H},{((H ") * a)}} is non empty finite V45() set
the multF of G . [((H ") * a),H] is set
(G,a,H) * (G,a,H) is Element of bool the carrier of G
carr (G,a,H) is Element of bool the carrier of G
the carrier of (G,a,H) is non empty set
(G,a,H) * (carr (G,a,H)) is Element of bool the carrier of G
{(G,a,H)} is non empty finite Element of bool the carrier of G
{(G,a,H)} * (carr (G,a,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(G,a,H)} & b₂ in carr (G,a,H) ) } is set
y is set
H1 is Element of the carrier of G
H1 * (G,a,H) is Element of bool the carrier of G
carr (G,a,H) is Element of bool the carrier of G
the carrier of (G,a,H) is non empty set
H1 * (carr (G,a,H)) is Element of bool the carrier of G
{H1} is non empty finite Element of bool the carrier of G
{H1} * (carr (G,a,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H1} & b₂ in carr (G,a,H) ) } is set
H " is Element of the carrier of G
(G,H1,(H ")) is Element of the carrier of G
(H ") " is Element of the carrier of G
((H ") ") * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (((H ") "),H1) is Element of the carrier of G
[((H ") "),H1] is set
{((H ") "),H1} is non empty finite set
{((H ") ")} is non empty finite set
{{((H ") "),H1},{((H ") ")}} is non empty finite V45() set
the multF of G . [((H ") "),H1] is set
(((H ") ") * H1) * (H ") is Element of the carrier of G
the multF of G . ((((H ") ") * H1),(H ")) is Element of the carrier of G
[(((H ") ") * H1),(H ")] is set
{(((H ") ") * H1),(H ")} is non empty finite set
{(((H ") ") * H1)} is non empty finite set
{{(((H ") ") * H1),(H ")},{(((H ") ") * H1)}} is non empty finite V45() set
the multF of G . [(((H ") ") * H1),(H ")] is set
(G,(G,H1,(H ")),H) is Element of the carrier of G
(H ") * (G,H1,(H ")) is Element of the carrier of G
the multF of G . ((H "),(G,H1,(H "))) is Element of the carrier of G
[(H "),(G,H1,(H "))] is set
{(H "),(G,H1,(H "))} is non empty finite set
{(H ")} is non empty finite set
{{(H "),(G,H1,(H "))},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),(G,H1,(H "))] is set
((H ") * (G,H1,(H "))) * H is Element of the carrier of G
the multF of G . (((H ") * (G,H1,(H "))),H) is Element of the carrier of G
[((H ") * (G,H1,(H "))),H] is set
{((H ") * (G,H1,(H "))),H} is non empty finite set
{((H ") * (G,H1,(H ")))} is non empty finite set
{{((H ") * (G,H1,(H "))),H},{((H ") * (G,H1,(H ")))}} is non empty finite V45() set
the multF of G . [((H ") * (G,H1,(H "))),H] is set
(G,(G,H1,(H ")),H) * (G,a,H) is Element of bool the carrier of G
(G,(G,H1,(H ")),H) * (carr (G,a,H)) is Element of bool the carrier of G
{(G,(G,H1,(H ")),H)} is non empty finite Element of bool the carrier of G
{(G,(G,H1,(H ")),H)} * (carr (G,a,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(G,(G,H1,(H ")),H)} & b₂ in carr (G,a,H) ) } is set
(G,H1,(H ")) * a is Element of bool the carrier of G
carr a is Element of bool the carrier of G
the carrier of a is non empty set
(G,H1,(H ")) * (carr a) is Element of bool the carrier of G
{(G,H1,(H "))} is non empty finite Element of bool the carrier of G
{(G,H1,(H "))} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(G,H1,(H "))} & b₂ in carr a ) } is set
c₄ . ((G,H1,(H ")) * a) is set
b is Element of the carrier of G
b * a is Element of bool the carrier of G
b * (carr a) is Element of bool the carrier of G
{b} is non empty finite Element of bool the carrier of G
{b} * (carr a) 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 carr a ) } is set
(G,b,H) is Element of the carrier of G
(H ") * b is Element of the carrier of G
the multF of G . ((H "),b) is Element of the carrier of G
[(H "),b] is set
{(H "),b} is non empty finite set
{{(H "),b},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),b] is set
((H ") * b) * H is Element of the carrier of G
the multF of G . (((H ") * b),H) is Element of the carrier of G
[((H ") * b),H] is set
{((H ") * b),H} is non empty finite set
{((H ") * b)} is non empty finite set
{{((H ") * b),H},{((H ") * b)}} is non empty finite V45() set
the multF of G . [((H ") * b),H] is set
(G,b,H) * (G,a,H) is Element of bool the carrier of G
(G,b,H) * (carr (G,a,H)) is Element of bool the carrier of G
{(G,b,H)} is non empty finite Element of bool the carrier of G
{(G,b,H)} * (carr (G,a,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(G,b,H)} & b₂ in carr (G,a,H) ) } is set
y is set
c₄ . y is set
H1 is set
c₄ . H1 is set
a is Element of the carrier of G
a * a is Element of bool the carrier of G
carr a is Element of bool the carrier of G
the carrier of a is non empty set
a * (carr a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
{a} * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr a ) } is set
(G,a,H) is Element of the carrier of G
H " is Element of the carrier of G
(H ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((H "),a) is Element of the carrier of G
[(H "),a] is set
{(H "),a} is non empty finite set
{(H ")} is non empty finite set
{{(H "),a},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),a] is set
((H ") * a) * H is Element of the carrier of G
the multF of G . (((H ") * a),H) is Element of the carrier of G
[((H ") * a),H] is set
{((H ") * a),H} is non empty finite set
{((H ") * a)} is non empty finite set
{{((H ") * a),H},{((H ") * a)}} is non empty finite V45() set
the multF of G . [((H ") * a),H] is set
(G,a,H) * (G,a,H) is Element of bool the carrier of G
carr (G,a,H) is Element of bool the carrier of G
the carrier of (G,a,H) is non empty set
(G,a,H) * (carr (G,a,H)) is Element of bool the carrier of G
{(G,a,H)} is non empty finite Element of bool the carrier of G
{(G,a,H)} * (carr (G,a,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(G,a,H)} & b₂ in carr (G,a,H) ) } is set
b is Element of the carrier of G
b * a is Element of bool the carrier of G
b * (carr a) is Element of bool the carrier of G
{b} is non empty finite Element of bool the carrier of G
{b} * (carr a) 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 carr a ) } is set
(G,b,H) is Element of the carrier of G
(H ") * b is Element of the carrier of G
the multF of G . ((H "),b) is Element of the carrier of G
[(H "),b] is set
{(H "),b} is non empty finite set
{{(H "),b},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),b] is set
((H ") * b) * H is Element of the carrier of G
the multF of G . (((H ") * b),H) is Element of the carrier of G
[((H ") * b),H] is set
{((H ") * b),H} is non empty finite set
{((H ") * b)} is non empty finite set
{{((H ") * b),H},{((H ") * b)}} is non empty finite V45() set
the multF of G . [((H ") * b),H] is set
(G,b,H) * (G,a,H) is Element of bool the carrier of G
(G,b,H) * (carr (G,a,H)) is Element of bool the carrier of G
{(G,b,H)} is non empty finite Element of bool the carrier of G
{(G,b,H)} * (carr (G,a,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(G,b,H)} & b₂ in carr (G,a,H) ) } is set
(G,a,H) " is Element of the carrier of G
((G,a,H) ") * (G,b,H) is Element of the carrier of G
the multF of G . (((G,a,H) "),(G,b,H)) is Element of the carrier of G
[((G,a,H) "),(G,b,H)] is set
{((G,a,H) "),(G,b,H)} is non empty finite set
{((G,a,H) ")} is non empty finite set
{{((G,a,H) "),(G,b,H)},{((G,a,H) ")}} is non empty finite V45() set
the multF of G . [((G,a,H) "),(G,b,H)] is set
a " is Element of the carrier of G
(G,(a "),H) is Element of the carrier of G
(H ") * (a ") is Element of the carrier of G
the multF of G . ((H "),(a ")) is Element of the carrier of G
[(H "),(a ")] is set
{(H "),(a ")} is non empty finite set
{{(H "),(a ")},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),(a ")] is set
((H ") * (a ")) * H is Element of the carrier of G
the multF of G . (((H ") * (a ")),H) is Element of the carrier of G
[((H ") * (a ")),H] is set
{((H ") * (a ")),H} is non empty finite set
{((H ") * (a "))} is non empty finite set
{{((H ") * (a ")),H},{((H ") * (a "))}} is non empty finite V45() set
the multF of G . [((H ") * (a ")),H] is set
(G,(a "),H) * (G,b,H) is Element of the carrier of G
the multF of G . ((G,(a "),H),(G,b,H)) is Element of the carrier of G
[(G,(a "),H),(G,b,H)] is set
{(G,(a "),H),(G,b,H)} is non empty finite set
{(G,(a "),H)} is non empty finite set
{{(G,(a "),H),(G,b,H)},{(G,(a "),H)}} is non empty finite V45() set
the multF of G . [(G,(a "),H),(G,b,H)] is set
(a ") * b is Element of the carrier of G
the multF of G . ((a "),b) is Element of the carrier of G
[(a "),b] is set
{(a "),b} is non empty finite set
{(a ")} is non empty finite set
{{(a "),b},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),b] is set
(G,((a ") * b),H) is Element of the carrier of G
(H ") * ((a ") * b) is Element of the carrier of G
the multF of G . ((H "),((a ") * b)) is Element of the carrier of G
[(H "),((a ") * b)] is set
{(H "),((a ") * b)} is non empty finite set
{{(H "),((a ") * b)},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),((a ") * b)] is set
((H ") * ((a ") * b)) * H is Element of the carrier of G
the multF of G . (((H ") * ((a ") * b)),H) is Element of the carrier of G
[((H ") * ((a ") * b)),H] is set
{((H ") * ((a ") * b)),H} is non empty finite set
{((H ") * ((a ") * b))} is non empty finite set
{{((H ") * ((a ") * b)),H},{((H ") * ((a ") * b))}} is non empty finite V45() set
the multF of G . [((H ") * ((a ") * b)),H] is set
c is Element of the carrier of G
(G,c,H) is Element of the carrier of G
(H ") * c is Element of the carrier of G
the multF of G . ((H "),c) is Element of the carrier of G
[(H "),c] is set
{(H "),c} is non empty finite set
{{(H "),c},{(H ")}} is non empty finite V45() set
the multF of G . [(H "),c] is set
((H ") * c) * H is Element of the carrier of G
the multF of G . (((H ") * c),H) is Element of the carrier of G
[((H ") * c),H] is set
{((H ") * c),H} is non empty finite set
{((H ") * c)} is non empty finite set
{{((H ") * c),H},{((H ") * c)}} is non empty finite V45() set
the multF of G . [((H ") * c),H] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is non empty unital Group-like associative Subgroup of G
Left_Cosets a is Element of bool (bool the carrier of G)
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
index a is V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
(G,a,H) is non empty strict unital Group-like associative Subgroup of G
index (G,a,H) is V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
Index a is cardinal set
card (Left_Cosets a) is cardinal set
Index (G,a,H) is cardinal set
Left_Cosets (G,a,H) is Element of bool (bool the carrier of G)
card (Left_Cosets (G,a,H)) is cardinal set
c₄ is finite set
card c₄ is V12() V13() V14() V18() ext-real V37() V38() integer Element of K46()
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is non empty strict unital Group-like associative Subgroup of G
a is Element of the carrier of G
(G,H,a) is non empty strict unital Group-like associative Subgroup of G
the carrier of (G,H,a) is non empty set
a " is Element of the carrier of G
(a ") * H is Element of bool the carrier of G
bool the carrier of G is non empty set
carr H is Element of bool the carrier of G
the carrier of H is non empty set
(a ") * (carr H) is Element of bool the carrier of G
{(a ")} is non empty finite Element of bool the carrier of G
{(a ")} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(a ")} & b₂ in carr H ) } is set
((a ") * H) * a is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
((a ") * H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (a ") * H & b₂ in {a} ) } is set
H * (a ") is Element of bool the carrier of G
(carr H) * (a ") is Element of bool the carrier of G
(carr H) * {(a ")} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {(a ")} ) } is set
(H * (a ")) * a is Element of bool the carrier of G
(H * (a ")) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H * (a ") & b₂ in {a} ) } is set
(a ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),a) is Element of the carrier of G
[(a "),a] is set
{(a "),a} is non empty finite set
{(a ")} is non empty finite set
{{(a "),a},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),a] is set
H * ((a ") * a) is Element of bool the carrier of G
(carr H) * ((a ") * a) is Element of bool the carrier of G
{((a ") * a)} is non empty finite Element of bool the carrier of G
(carr H) * {((a ") * a)} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {((a ") * a)} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
H * (1_ G) is Element of bool the carrier of G
(carr H) * (1_ G) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
(carr H) * {(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 H & b₂ in {(1_ G)} ) } is set
G is non empty unital Group-like associative multMagma
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
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
c₄ is Element of the carrier of G
(G,a,c₄) is Element of the carrier of G
c₄ " is Element of the carrier of G
(c₄ ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((c₄ "),a) is Element of the carrier of G
[(c₄ "),a] is set
{(c₄ "),a} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),a},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),a] is set
((c₄ ") * a) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * a),c₄) is Element of the carrier of G
[((c₄ ") * a),c₄] is set
{((c₄ ") * a),c₄} is non empty finite set
{((c₄ ") * a)} is non empty finite set
{{((c₄ ") * a),c₄},{((c₄ ") * a)}} is non empty finite V45() set
the multF of G . [((c₄ ") * a),c₄] is set
(G,H,(c₄ ")) is Element of the carrier of G
(c₄ ") " is Element of the carrier of G
((c₄ ") ") * H is Element of the carrier of G
the multF of G . (((c₄ ") "),H) is Element of the carrier of G
[((c₄ ") "),H] is set
{((c₄ ") "),H} is non empty finite set
{((c₄ ") ")} is non empty finite set
{{((c₄ ") "),H},{((c₄ ") ")}} is non empty finite V45() set
the multF of G . [((c₄ ") "),H] is set
(((c₄ ") ") * H) * (c₄ ") is Element of the carrier of G
the multF of G . ((((c₄ ") ") * H),(c₄ ")) is Element of the carrier of G
[(((c₄ ") ") * H),(c₄ ")] is set
{(((c₄ ") ") * H),(c₄ ")} is non empty finite set
{(((c₄ ") ") * H)} is non empty finite set
{{(((c₄ ") ") * H),(c₄ ")},{(((c₄ ") ") * H)}} is non empty finite V45() set
the multF of G . [(((c₄ ") ") * H),(c₄ ")] is set
c₄ is Element of the carrier of G
(G,H,c₄) is Element of the carrier of G
c₄ " is Element of the carrier of G
(c₄ ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((c₄ "),H) is Element of the carrier of G
[(c₄ "),H] is set
{(c₄ "),H} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),H},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),H] is set
((c₄ ") * H) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * H),c₄) is Element of the carrier of G
[((c₄ ") * H),c₄] is set
{((c₄ ") * H),c₄} is non empty finite set
{((c₄ ") * H)} is non empty finite set
{{((c₄ ") * H),c₄},{((c₄ ") * H)}} is non empty finite V45() set
the multF of G . [((c₄ ") * H),c₄] is set
(G,a,(c₄ ")) is Element of the carrier of G
(c₄ ") " is Element of the carrier of G
((c₄ ") ") * a is Element of the carrier of G
the multF of G . (((c₄ ") "),a) is Element of the carrier of G
[((c₄ ") "),a] is set
{((c₄ ") "),a} is non empty finite set
{((c₄ ") ")} is non empty finite set
{{((c₄ ") "),a},{((c₄ ") ")}} is non empty finite V45() set
the multF of G . [((c₄ ") "),a] is set
(((c₄ ") ") * a) * (c₄ ") is Element of the carrier of G
the multF of G . ((((c₄ ") ") * a),(c₄ ")) is Element of the carrier of G
[(((c₄ ") ") * a),(c₄ ")] is set
{(((c₄ ") ") * a),(c₄ ")} is non empty finite set
{(((c₄ ") ") * a)} is non empty finite set
{{(((c₄ ") ") * a),(c₄ ")},{(((c₄ ") ") * a)}} is non empty finite V45() set
the multF of G . [(((c₄ ") ") * a),(c₄ ")] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
(G,H,(1_ G)) is Element of the carrier of G
(1_ G) " is Element of the carrier of G
((1_ G) ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (((1_ G) "),H) is Element of the carrier of G
[((1_ G) "),H] is set
{((1_ G) "),H} is non empty finite set
{((1_ G) ")} is non empty finite set
{{((1_ G) "),H},{((1_ G) ")}} is non empty finite V45() set
the multF of G . [((1_ G) "),H] is set
(((1_ G) ") * H) * (1_ G) is Element of the carrier of G
the multF of G . ((((1_ G) ") * H),(1_ G)) is Element of the carrier of G
[(((1_ G) ") * H),(1_ G)] is set
{(((1_ G) ") * H),(1_ G)} is non empty finite set
{(((1_ G) ") * H)} is non empty finite set
{{(((1_ G) ") * H),(1_ G)},{(((1_ G) ") * H)}} is non empty finite V45() set
the multF of G . [(((1_ G) ") * H),(1_ G)] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
c₄ is Element of the carrier of G
(G,a,c₄) is Element of the carrier of G
c₄ " is Element of the carrier of G
(c₄ ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((c₄ "),a) is Element of the carrier of G
[(c₄ "),a] is set
{(c₄ "),a} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),a},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),a] is set
((c₄ ") * a) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * a),c₄) is Element of the carrier of G
[((c₄ ") * a),c₄] is set
{((c₄ ") * a),c₄} is non empty finite set
{((c₄ ") * a)} is non empty finite set
{{((c₄ ") * a),c₄},{((c₄ ") * a)}} is non empty finite V45() set
the multF of G . [((c₄ ") * a),c₄] is set
(G,H,(c₄ ")) is Element of the carrier of G
(c₄ ") " is Element of the carrier of G
((c₄ ") ") * H is Element of the carrier of G
the multF of G . (((c₄ ") "),H) is Element of the carrier of G
[((c₄ ") "),H] is set
{((c₄ ") "),H} is non empty finite set
{((c₄ ") ")} is non empty finite set
{{((c₄ ") "),H},{((c₄ ") ")}} is non empty finite V45() set
the multF of G . [((c₄ ") "),H] is set
(((c₄ ") ") * H) * (c₄ ") is Element of the carrier of G
the multF of G . ((((c₄ ") ") * H),(c₄ ")) is Element of the carrier of G
[(((c₄ ") ") * H),(c₄ ")] is set
{(((c₄ ") ") * H),(c₄ ")} is non empty finite set
{(((c₄ ") ") * H)} is non empty finite set
{{(((c₄ ") ") * H),(c₄ ")},{(((c₄ ") ") * H)}} is non empty finite V45() set
the multF of G . [(((c₄ ") ") * H),(c₄ ")] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
c₄ is Element of the carrier of G
c₄ is Element of the carrier of G
y is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
c₄ is Element of the carrier of G
y is Element of the carrier of G
(G,a,y) is Element of the carrier of G
y " is Element of the carrier of G
(y ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((y "),a) is Element of the carrier of G
[(y "),a] is set
{(y "),a} is non empty finite set
{(y ")} is non empty finite set
{{(y "),a},{(y ")}} is non empty finite V45() set
the multF of G . [(y "),a] is set
((y ") * a) * y is Element of the carrier of G
the multF of G . (((y ") * a),y) is Element of the carrier of G
[((y ") * a),y] is set
{((y ") * a),y} is non empty finite set
{((y ") * a)} is non empty finite set
{{((y ") * a),y},{((y ") * a)}} is non empty finite V45() set
the multF of G . [((y ") * a),y] is set
H1 is Element of the carrier of G
(G,c₄,H1) is Element of the carrier of G
H1 " is Element of the carrier of G
(H1 ") * c₄ is Element of the carrier of G
the multF of G . ((H1 "),c₄) is Element of the carrier of G
[(H1 "),c₄] is set
{(H1 "),c₄} is non empty finite set
{(H1 ")} is non empty finite set
{{(H1 "),c₄},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),c₄] is set
((H1 ") * c₄) * H1 is Element of the carrier of G
the multF of G . (((H1 ") * c₄),H1) is Element of the carrier of G
[((H1 ") * c₄),H1] is set
{((H1 ") * c₄),H1} is non empty finite set
{((H1 ") * c₄)} is non empty finite set
{{((H1 ") * c₄),H1},{((H1 ") * c₄)}} is non empty finite V45() set
the multF of G . [((H1 ") * c₄),H1] is set
H1 * y is Element of the carrier of G
the multF of G . (H1,y) is Element of the carrier of G
[H1,y] is set
{H1,y} is non empty finite set
{H1} is non empty finite set
{{H1,y},{H1}} is non empty finite V45() set
the multF of G . [H1,y] is set
(G,c₄,(H1 * y)) is Element of the carrier of G
(H1 * y) " is Element of the carrier of G
((H1 * y) ") * c₄ is Element of the carrier of G
the multF of G . (((H1 * y) "),c₄) is Element of the carrier of G
[((H1 * y) "),c₄] is set
{((H1 * y) "),c₄} is non empty finite set
{((H1 * y) ")} is non empty finite set
{{((H1 * y) "),c₄},{((H1 * y) ")}} is non empty finite V45() set
the multF of G . [((H1 * y) "),c₄] is set
(((H1 * y) ") * c₄) * (H1 * y) is Element of the carrier of G
the multF of G . ((((H1 * y) ") * c₄),(H1 * y)) is Element of the carrier of G
[(((H1 * y) ") * c₄),(H1 * y)] is set
{(((H1 * y) ") * c₄),(H1 * y)} is non empty finite set
{(((H1 * y) ") * c₄)} is non empty finite set
{{(((H1 * y) ") * c₄),(H1 * y)},{(((H1 * y) ") * c₄)}} is non empty finite V45() set
the multF of G . [(((H1 * y) ") * c₄),(H1 * y)] 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
H is Element of the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of ((Omega). G) is non empty set
H is Element of the carrier of G
(G,(carr ((Omega). G)),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr ((Omega). G) ) } is set
{ b₁ where b₁ is Element of the carrier of G : (G,H,b₁) } is set
y is set
H1 is Element of the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H is Element of the carrier of G
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
y is set
H1 is Element of the carrier of G
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H is Element of the carrier of G
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of ((Omega). G) is non empty set
H is Element of the carrier of G
(G,(carr ((Omega). G)),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr ((Omega). G) ) } is set
G is set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
a is Element of the carrier of H
(H,a) is Element of bool the carrier of H
bool the carrier of H is non empty set
(Omega). H is non empty strict unital Group-like associative Subgroup of H
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
carr ((Omega). H) is Element of bool the carrier of H
the carrier of ((Omega). H) is non empty set
(H,(carr ((Omega). H)),a) is Element of bool the carrier of H
{a} is non empty finite Element of bool the carrier of H
(H,{a},(carr ((Omega). H))) is Element of bool the carrier of H
{ (H,b₁,b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in {a} & b₂ in carr ((Omega). H) ) } is set
{ b₁ where b₁ is Element of the carrier of H : (H,a,b₁) } is set
c₄ is Element of the carrier of H
{ b₁ where b₁ is Element of the carrier of H : (H,a,b₁) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is Element of the carrier of G
(G,a) is Element of bool the carrier of G
bool the carrier of G is non empty set
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,{a},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr ((Omega). G) ) } is set
c₄ is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
(G,H) is Element of bool the carrier of G
bool the carrier of G is non empty set
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr ((Omega). G) ) } is set
a is Element of the carrier of G
(G,H,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H is Element of the carrier of G
the multF of G . ((a "),H) is Element of the carrier of G
[(a "),H] is set
{(a "),H} is non empty finite set
{(a ")} is non empty finite set
{{(a "),H},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),H] is set
((a ") * H) * a is Element of the carrier of G
the multF of G . (((a ") * H),a) is Element of the carrier of G
[((a ") * H),a] is set
{((a ") * H),a} is non empty finite set
{((a ") * H)} is non empty finite set
{{((a ") * H),a},{((a ") * H)}} is non empty finite V45() set
the multF of G . [((a ") * H),a] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
(G,H) is Element of bool the carrier of G
bool the carrier of G is non empty set
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr ((Omega). G) ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
(G,H) is Element of bool the carrier of G
bool the carrier of G is non empty set
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr ((Omega). G) ) } is set
a is Element of the carrier of G
(G,a) is Element of bool the carrier of G
(G,(carr ((Omega). G)),a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,{a},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr ((Omega). G) ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
(G,H) is Element of bool the carrier of G
bool the carrier of G is non empty set
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr ((Omega). G) ) } is set
a is Element of the carrier of G
(G,a) is Element of bool the carrier of G
(G,(carr ((Omega). G)),a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,{a},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr ((Omega). G) ) } is set
c₄ is set
y is Element of the carrier of G
H1 is set
a is Element of the carrier of G
H1 is set
a is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
bool the carrier of G is non empty set
H is Element of the carrier of G
(G,H) is Element of bool the carrier of G
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr ((Omega). G) ) } is set
a is set
c₄ is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
H is Element of the carrier of G
(G,H) is Element of bool the carrier of G
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr ((Omega). G) ) } is set
a is Element of bool the carrier of G
(G,H) * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,H) & b₂ in a ) } is set
a * (G,H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in (G,H) ) } is set
c₄ is set
y is Element of the carrier of G
H1 is Element of the carrier of G
y * H1 is Element of the carrier of G
the multF of G . (y,H1) is Element of the carrier of G
[y,H1] is set
{y,H1} is non empty finite set
{y} is non empty finite set
{{y,H1},{y}} is non empty finite V45() set
the multF of G . [y,H1] is set
b is Element of the carrier of G
(G,H,b) is Element of the carrier of G
b " is Element of the carrier of G
(b ") * H is Element of the carrier of G
the multF of G . ((b "),H) is Element of the carrier of G
[(b "),H] is set
{(b "),H} is non empty finite set
{(b ")} is non empty finite set
{{(b "),H},{(b ")}} is non empty finite V45() set
the multF of G . [(b "),H] is set
((b ") * H) * b is Element of the carrier of G
the multF of G . (((b ") * H),b) is Element of the carrier of G
[((b ") * H),b] is set
{((b ") * H),b} is non empty finite set
{((b ") * H)} is non empty finite set
{{((b ") * H),b},{((b ") * H)}} is non empty finite V45() set
the multF of G . [((b ") * H),b] is set
a is Element of the carrier of G
H1 " is Element of the carrier of G
(G,a,(H1 ")) is Element of the carrier of G
(H1 ") " is Element of the carrier of G
((H1 ") ") * a is Element of the carrier of G
the multF of G . (((H1 ") "),a) is Element of the carrier of G
[((H1 ") "),a] is set
{((H1 ") "),a} is non empty finite set
{((H1 ") ")} is non empty finite set
{{((H1 ") "),a},{((H1 ") ")}} is non empty finite V45() set
the multF of G . [((H1 ") "),a] is set
(((H1 ") ") * a) * (H1 ") is Element of the carrier of G
the multF of G . ((((H1 ") ") * a),(H1 ")) is Element of the carrier of G
[(((H1 ") ") * a),(H1 ")] is set
{(((H1 ") ") * a),(H1 ")} is non empty finite set
{(((H1 ") ") * a)} is non empty finite set
{{(((H1 ") ") * a),(H1 ")},{(((H1 ") ") * a)}} is non empty finite V45() set
the multF of G . [(((H1 ") ") * a),(H1 ")] is set
(G,H,b) * H1 is Element of the carrier of G
the multF of G . ((G,H,b),H1) is Element of the carrier of G
[(G,H,b),H1] is set
{(G,H,b),H1} is non empty finite set
{(G,H,b)} is non empty finite set
{{(G,H,b),H1},{(G,H,b)}} is non empty finite V45() set
the multF of G . [(G,H,b),H1] is set
H1 * ((G,H,b) * H1) is Element of the carrier of G
the multF of G . (H1,((G,H,b) * H1)) is Element of the carrier of G
[H1,((G,H,b) * H1)] is set
{H1,((G,H,b) * H1)} is non empty finite set
{H1} is non empty finite set
{{H1,((G,H,b) * H1)},{H1}} is non empty finite V45() set
the multF of G . [H1,((G,H,b) * H1)] is set
(H1 * ((G,H,b) * H1)) * (H1 ") is Element of the carrier of G
the multF of G . ((H1 * ((G,H,b) * H1)),(H1 ")) is Element of the carrier of G
[(H1 * ((G,H,b) * H1)),(H1 ")] is set
{(H1 * ((G,H,b) * H1)),(H1 ")} is non empty finite set
{(H1 * ((G,H,b) * H1))} is non empty finite set
{{(H1 * ((G,H,b) * H1)),(H1 ")},{(H1 * ((G,H,b) * H1))}} is non empty finite V45() set
the multF of G . [(H1 * ((G,H,b) * H1)),(H1 ")] is set
((G,H,b) * H1) * (H1 ") is Element of the carrier of G
the multF of G . (((G,H,b) * H1),(H1 ")) is Element of the carrier of G
[((G,H,b) * H1),(H1 ")] is set
{((G,H,b) * H1),(H1 ")} is non empty finite set
{((G,H,b) * H1)} is non empty finite set
{{((G,H,b) * H1),(H1 ")},{((G,H,b) * H1)}} is non empty finite V45() set
the multF of G . [((G,H,b) * H1),(H1 ")] is set
H1 * (((G,H,b) * H1) * (H1 ")) is Element of the carrier of G
the multF of G . (H1,(((G,H,b) * H1) * (H1 "))) is Element of the carrier of G
[H1,(((G,H,b) * H1) * (H1 "))] is set
{H1,(((G,H,b) * H1) * (H1 "))} is non empty finite set
{{H1,(((G,H,b) * H1) * (H1 "))},{H1}} is non empty finite V45() set
the multF of G . [H1,(((G,H,b) * H1) * (H1 "))] is set
H1 * (G,H,b) is Element of the carrier of G
the multF of G . (H1,(G,H,b)) is Element of the carrier of G
[H1,(G,H,b)] is set
{H1,(G,H,b)} is non empty finite set
{{H1,(G,H,b)},{H1}} is non empty finite V45() set
the multF of G . [H1,(G,H,b)] is set
(G,(H1 * (G,H,b)),H1) is Element of the carrier of G
(H1 ") * (H1 * (G,H,b)) is Element of the carrier of G
the multF of G . ((H1 "),(H1 * (G,H,b))) is Element of the carrier of G
[(H1 "),(H1 * (G,H,b))] is set
{(H1 "),(H1 * (G,H,b))} is non empty finite set
{(H1 ")} is non empty finite set
{{(H1 "),(H1 * (G,H,b))},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),(H1 * (G,H,b))] is set
((H1 ") * (H1 * (G,H,b))) * H1 is Element of the carrier of G
the multF of G . (((H1 ") * (H1 * (G,H,b))),H1) is Element of the carrier of G
[((H1 ") * (H1 * (G,H,b))),H1] is set
{((H1 ") * (H1 * (G,H,b))),H1} is non empty finite set
{((H1 ") * (H1 * (G,H,b)))} is non empty finite set
{{((H1 ") * (H1 * (G,H,b))),H1},{((H1 ") * (H1 * (G,H,b)))}} is non empty finite V45() set
the multF of G . [((H1 ") * (H1 * (G,H,b))),H1] is set
(G,H1,H1) is Element of the carrier of G
(H1 ") * H1 is Element of the carrier of G
the multF of G . ((H1 "),H1) is Element of the carrier of G
[(H1 "),H1] is set
{(H1 "),H1} is non empty finite set
{{(H1 "),H1},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),H1] is set
((H1 ") * H1) * H1 is Element of the carrier of G
the multF of G . (((H1 ") * H1),H1) is Element of the carrier of G
[((H1 ") * H1),H1] is set
{((H1 ") * H1),H1} is non empty finite set
{((H1 ") * H1)} is non empty finite set
{{((H1 ") * H1),H1},{((H1 ") * H1)}} is non empty finite V45() set
the multF of G . [((H1 ") * H1),H1] is set
(G,(G,H,b),H1) is Element of the carrier of G
(H1 ") * (G,H,b) is Element of the carrier of G
the multF of G . ((H1 "),(G,H,b)) is Element of the carrier of G
[(H1 "),(G,H,b)] is set
{(H1 "),(G,H,b)} is non empty finite set
{{(H1 "),(G,H,b)},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),(G,H,b)] is set
((H1 ") * (G,H,b)) * H1 is Element of the carrier of G
the multF of G . (((H1 ") * (G,H,b)),H1) is Element of the carrier of G
[((H1 ") * (G,H,b)),H1] is set
{((H1 ") * (G,H,b)),H1} is non empty finite set
{((H1 ") * (G,H,b))} is non empty finite set
{{((H1 ") * (G,H,b)),H1},{((H1 ") * (G,H,b))}} is non empty finite V45() set
the multF of G . [((H1 ") * (G,H,b)),H1] is set
(G,H1,H1) * (G,(G,H,b),H1) is Element of the carrier of G
the multF of G . ((G,H1,H1),(G,(G,H,b),H1)) is Element of the carrier of G
[(G,H1,H1),(G,(G,H,b),H1)] is set
{(G,H1,H1),(G,(G,H,b),H1)} is non empty finite set
{(G,H1,H1)} is non empty finite set
{{(G,H1,H1),(G,(G,H,b),H1)},{(G,H1,H1)}} is non empty finite V45() set
the multF of G . [(G,H1,H1),(G,(G,H,b),H1)] is set
H1 * (G,(G,H,b),H1) is Element of the carrier of G
the multF of G . (H1,(G,(G,H,b),H1)) is Element of the carrier of G
[H1,(G,(G,H,b),H1)] is set
{H1,(G,(G,H,b),H1)} is non empty finite set
{{H1,(G,(G,H,b),H1)},{H1}} is non empty finite V45() set
the multF of G . [H1,(G,(G,H,b),H1)] is set
b * H1 is Element of the carrier of G
the multF of G . (b,H1) is Element of the carrier of G
[b,H1] is set
{b,H1} is non empty finite set
{b} is non empty finite set
{{b,H1},{b}} is non empty finite V45() set
the multF of G . [b,H1] is set
(G,H,(b * H1)) is Element of the carrier of G
(b * H1) " is Element of the carrier of G
((b * H1) ") * H is Element of the carrier of G
the multF of G . (((b * H1) "),H) is Element of the carrier of G
[((b * H1) "),H] is set
{((b * H1) "),H} is non empty finite set
{((b * H1) ")} is non empty finite set
{{((b * H1) "),H},{((b * H1) ")}} is non empty finite V45() set
the multF of G . [((b * H1) "),H] is set
(((b * H1) ") * H) * (b * H1) is Element of the carrier of G
the multF of G . ((((b * H1) ") * H),(b * H1)) is Element of the carrier of G
[(((b * H1) ") * H),(b * H1)] is set
{(((b * H1) ") * H),(b * H1)} is non empty finite set
{(((b * H1) ") * H)} is non empty finite set
{{(((b * H1) ") * H),(b * H1)},{(((b * H1) ") * H)}} is non empty finite V45() set
the multF of G . [(((b * H1) ") * H),(b * H1)] is set
H1 * (G,H,(b * H1)) is Element of the carrier of G
the multF of G . (H1,(G,H,(b * H1))) is Element of the carrier of G
[H1,(G,H,(b * H1))] is set
{H1,(G,H,(b * H1))} is non empty finite set
{{H1,(G,H,(b * H1))},{H1}} is non empty finite V45() set
the multF of G . [H1,(G,H,(b * H1))] is set
c₄ is set
y is Element of the carrier of G
H1 is Element of the carrier of G
y * H1 is Element of the carrier of G
the multF of G . (y,H1) is Element of the carrier of G
[y,H1] is set
{y,H1} is non empty finite set
{y} is non empty finite set
{{y,H1},{y}} is non empty finite V45() set
the multF of G . [y,H1] is set
b is Element of the carrier of G
(G,H,b) is Element of the carrier of G
b " is Element of the carrier of G
(b ") * H is Element of the carrier of G
the multF of G . ((b "),H) is Element of the carrier of G
[(b "),H] is set
{(b "),H} is non empty finite set
{(b ")} is non empty finite set
{{(b "),H},{(b ")}} is non empty finite V45() set
the multF of G . [(b "),H] is set
((b ") * H) * b is Element of the carrier of G
the multF of G . (((b ") * H),b) is Element of the carrier of G
[((b ") * H),b] is set
{((b ") * H),b} is non empty finite set
{((b ") * H)} is non empty finite set
{{((b ") * H),b},{((b ") * H)}} is non empty finite V45() set
the multF of G . [((b ") * H),b] is set
a is Element of the carrier of G
(G,a,y) is Element of the carrier of G
y " is Element of the carrier of G
(y ") * a is Element of the carrier of G
the multF of G . ((y "),a) is Element of the carrier of G
[(y "),a] is set
{(y "),a} is non empty finite set
{(y ")} is non empty finite set
{{(y "),a},{(y ")}} is non empty finite V45() set
the multF of G . [(y "),a] is set
((y ") * a) * y is Element of the carrier of G
the multF of G . (((y ") * a),y) is Element of the carrier of G
[((y ") * a),y] is set
{((y ") * a),y} is non empty finite set
{((y ") * a)} is non empty finite set
{{((y ") * a),y},{((y ") * a)}} is non empty finite V45() set
the multF of G . [((y ") * a),y] is set
(G,H,b) * y is Element of the carrier of G
the multF of G . ((G,H,b),y) is Element of the carrier of G
[(G,H,b),y] is set
{(G,H,b),y} is non empty finite set
{(G,H,b)} is non empty finite set
{{(G,H,b),y},{(G,H,b)}} is non empty finite V45() set
the multF of G . [(G,H,b),y] is set
(G,((G,H,b) * y),(y ")) is Element of the carrier of G
(y ") " is Element of the carrier of G
((y ") ") * ((G,H,b) * y) is Element of the carrier of G
the multF of G . (((y ") "),((G,H,b) * y)) is Element of the carrier of G
[((y ") "),((G,H,b) * y)] is set
{((y ") "),((G,H,b) * y)} is non empty finite set
{((y ") ")} is non empty finite set
{{((y ") "),((G,H,b) * y)},{((y ") ")}} is non empty finite V45() set
the multF of G . [((y ") "),((G,H,b) * y)] is set
(((y ") ") * ((G,H,b) * y)) * (y ") is Element of the carrier of G
the multF of G . ((((y ") ") * ((G,H,b) * y)),(y ")) is Element of the carrier of G
[(((y ") ") * ((G,H,b) * y)),(y ")] is set
{(((y ") ") * ((G,H,b) * y)),(y ")} is non empty finite set
{(((y ") ") * ((G,H,b) * y))} is non empty finite set
{{(((y ") ") * ((G,H,b) * y)),(y ")},{(((y ") ") * ((G,H,b) * y))}} is non empty finite V45() set
the multF of G . [(((y ") ") * ((G,H,b) * y)),(y ")] is set
(G,(G,H,b),(y ")) is Element of the carrier of G
((y ") ") * (G,H,b) is Element of the carrier of G
the multF of G . (((y ") "),(G,H,b)) is Element of the carrier of G
[((y ") "),(G,H,b)] is set
{((y ") "),(G,H,b)} is non empty finite set
{{((y ") "),(G,H,b)},{((y ") ")}} is non empty finite V45() set
the multF of G . [((y ") "),(G,H,b)] is set
(((y ") ") * (G,H,b)) * (y ") is Element of the carrier of G
the multF of G . ((((y ") ") * (G,H,b)),(y ")) is Element of the carrier of G
[(((y ") ") * (G,H,b)),(y ")] is set
{(((y ") ") * (G,H,b)),(y ")} is non empty finite set
{(((y ") ") * (G,H,b))} is non empty finite set
{{(((y ") ") * (G,H,b)),(y ")},{(((y ") ") * (G,H,b))}} is non empty finite V45() set
the multF of G . [(((y ") ") * (G,H,b)),(y ")] is set
(G,y,(y ")) is Element of the carrier of G
((y ") ") * y is Element of the carrier of G
the multF of G . (((y ") "),y) is Element of the carrier of G
[((y ") "),y] is set
{((y ") "),y} is non empty finite set
{{((y ") "),y},{((y ") ")}} is non empty finite V45() set
the multF of G . [((y ") "),y] is set
(((y ") ") * y) * (y ") is Element of the carrier of G
the multF of G . ((((y ") ") * y),(y ")) is Element of the carrier of G
[(((y ") ") * y),(y ")] is set
{(((y ") ") * y),(y ")} is non empty finite set
{(((y ") ") * y)} is non empty finite set
{{(((y ") ") * y),(y ")},{(((y ") ") * y)}} is non empty finite V45() set
the multF of G . [(((y ") ") * y),(y ")] is set
(G,(G,H,b),(y ")) * (G,y,(y ")) is Element of the carrier of G
the multF of G . ((G,(G,H,b),(y ")),(G,y,(y "))) is Element of the carrier of G
[(G,(G,H,b),(y ")),(G,y,(y "))] is set
{(G,(G,H,b),(y ")),(G,y,(y "))} is non empty finite set
{(G,(G,H,b),(y "))} is non empty finite set
{{(G,(G,H,b),(y ")),(G,y,(y "))},{(G,(G,H,b),(y "))}} is non empty finite V45() set
the multF of G . [(G,(G,H,b),(y ")),(G,y,(y "))] is set
b * (y ") is Element of the carrier of G
the multF of G . (b,(y ")) is Element of the carrier of G
[b,(y ")] is set
{b,(y ")} is non empty finite set
{b} is non empty finite set
{{b,(y ")},{b}} is non empty finite V45() set
the multF of G . [b,(y ")] is set
(G,H,(b * (y "))) is Element of the carrier of G
(b * (y ")) " is Element of the carrier of G
((b * (y ")) ") * H is Element of the carrier of G
the multF of G . (((b * (y ")) "),H) is Element of the carrier of G
[((b * (y ")) "),H] is set
{((b * (y ")) "),H} is non empty finite set
{((b * (y ")) ")} is non empty finite set
{{((b * (y ")) "),H},{((b * (y ")) ")}} is non empty finite V45() set
the multF of G . [((b * (y ")) "),H] is set
(((b * (y ")) ") * H) * (b * (y ")) is Element of the carrier of G
the multF of G . ((((b * (y ")) ") * H),(b * (y "))) is Element of the carrier of G
[(((b * (y ")) ") * H),(b * (y "))] is set
{(((b * (y ")) ") * H),(b * (y "))} is non empty finite set
{(((b * (y ")) ") * H)} is non empty finite set
{{(((b * (y ")) ") * H),(b * (y "))},{(((b * (y ")) ") * H)}} is non empty finite V45() set
the multF of G . [(((b * (y ")) ") * H),(b * (y "))] is set
(G,H,(b * (y "))) * (G,y,(y ")) is Element of the carrier of G
the multF of G . ((G,H,(b * (y "))),(G,y,(y "))) is Element of the carrier of G
[(G,H,(b * (y "))),(G,y,(y "))] is set
{(G,H,(b * (y "))),(G,y,(y "))} is non empty finite set
{(G,H,(b * (y ")))} is non empty finite set
{{(G,H,(b * (y "))),(G,y,(y "))},{(G,H,(b * (y ")))}} is non empty finite V45() set
the multF of G . [(G,H,(b * (y "))),(G,y,(y "))] is set
(G,H,(b * (y "))) * y is Element of the carrier of G
the multF of G . ((G,H,(b * (y "))),y) is Element of the carrier of G
[(G,H,(b * (y "))),y] is set
{(G,H,(b * (y "))),y} is non empty finite set
{{(G,H,(b * (y "))),y},{(G,H,(b * (y ")))}} is non empty finite V45() set
the multF of G . [(G,H,(b * (y "))),y] 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 non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
H is Element of bool the carrier of G
a is Element of bool the carrier of G
c₄ is Element of the carrier of G
(G,a,c₄) is Element of bool the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
(G,a,{c₄}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {c₄} ) } is set
c₄ " is Element of the carrier of G
(G,H,(c₄ ")) is Element of bool the carrier of G
{(c₄ ")} is non empty finite Element of bool the carrier of G
(G,H,{(c₄ ")}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {(c₄ ")} ) } is set
c₄ is Element of the carrier of G
(G,H,c₄) is Element of bool the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
(G,H,{c₄}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {c₄} ) } is set
c₄ " is Element of the carrier of G
(G,a,(c₄ ")) is Element of bool the carrier of G
{(c₄ ")} is non empty finite Element of bool the carrier of G
(G,a,{(c₄ ")}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {(c₄ ")} ) } 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 non empty set
H is Element of bool the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
(G,H,(1_ G)) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
(G,H,{(1_ G)}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {(1_ 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 non empty set
H is Element of bool the carrier of G
a is Element of bool the carrier of G
c₄ is Element of the carrier of G
(G,a,c₄) is Element of bool the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
(G,a,{c₄}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {c₄} ) } is set
c₄ " is Element of the carrier of G
(G,H,(c₄ ")) is Element of bool the carrier of G
{(c₄ ")} is non empty finite Element of bool the carrier of G
(G,H,{(c₄ ")}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {(c₄ ")} ) } 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 non empty set
c₄ is Element of bool the carrier of G
c₄ is Element of bool the carrier of G
y is Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
H is Element of bool the carrier of G
a is Element of bool the carrier of G
c₄ is Element of bool the carrier of G
y is Element of the carrier of G
(G,a,y) is Element of bool the carrier of G
{y} is non empty finite Element of bool the carrier of G
(G,a,{y}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {y} ) } is set
H1 is Element of the carrier of G
(G,c₄,H1) is Element of bool the carrier of G
{H1} is non empty finite Element of bool the carrier of G
(G,c₄,{H1}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in c₄ & b₂ in {H1} ) } is set
H1 * y 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H1,y) is Element of the carrier of G
[H1,y] is set
{H1,y} is non empty finite set
{H1} is non empty finite set
{{H1,y},{H1}} is non empty finite V45() set
the multF of G . [H1,y] is set
(G,c₄,(H1 * y)) is Element of bool the carrier of G
{(H1 * y)} is non empty finite Element of bool the carrier of G
(G,c₄,{(H1 * y)}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in c₄ & b₂ in {(H1 * y)} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
{H} is non empty finite Element of bool the carrier of G
bool the carrier of G is non empty set
a is Element of the carrier of G
{a} is non empty finite Element of bool the carrier of G
c₄ is Element of the carrier of G
(G,{H},c₄) is Element of bool the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
(G,{H},{c₄}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in {c₄} ) } is set
(G,H,c₄) is Element of the carrier of G
c₄ " is Element of the carrier of G
(c₄ ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((c₄ "),H) is Element of the carrier of G
[(c₄ "),H] is set
{(c₄ "),H} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),H},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),H] is set
((c₄ ") * H) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * H),c₄) is Element of the carrier of G
[((c₄ ") * H),c₄] is set
{((c₄ ") * H),c₄} is non empty finite set
{((c₄ ") * H)} is non empty finite set
{{((c₄ ") * H),c₄},{((c₄ ") * H)}} is non empty finite V45() set
the multF of G . [((c₄ ") * H),c₄] is set
{(G,H,c₄)} is non empty finite Element of bool the carrier of G
c₄ is Element of the carrier of G
(G,H,c₄) is Element of the carrier of G
c₄ " is Element of the carrier of G
(c₄ ") * H 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((c₄ "),H) is Element of the carrier of G
[(c₄ "),H] is set
{(c₄ "),H} is non empty finite set
{(c₄ ")} is non empty finite set
{{(c₄ "),H},{(c₄ ")}} is non empty finite V45() set
the multF of G . [(c₄ "),H] is set
((c₄ ") * H) * c₄ is Element of the carrier of G
the multF of G . (((c₄ ") * H),c₄) is Element of the carrier of G
[((c₄ ") * H),c₄] is set
{((c₄ ") * H),c₄} is non empty finite set
{((c₄ ") * H)} is non empty finite set
{{((c₄ ") * H),c₄},{((c₄ ") * H)}} is non empty finite V45() set
the multF of G . [((c₄ ") * H),c₄] is set
(G,{H},c₄) is Element of bool the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
(G,{H},{c₄}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in {c₄} ) } 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 non empty set
H is Element of bool the carrier of G
a is non empty unital Group-like associative Subgroup of G
carr a is Element of bool the carrier of G
the carrier of a is non empty set
c₄ is Element of the carrier of G
(G,(carr a),c₄) is Element of bool the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
(G,(carr a),{c₄}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in {c₄} ) } is set
(G,a,c₄) is non empty strict unital Group-like associative Subgroup of G
the carrier of (G,a,c₄) is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
H is Element of bool the carrier of G
{ b₁ where b₁ is Element of bool the carrier of G : (G,H,b₁) } is set
bool (bool the carrier of G) is non empty set
c₄ is set
y is Element of bool the carrier of G
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
bool the carrier of H is non empty set
y is non empty unital Group-like associative multMagma
the carrier of y is non empty set
bool the carrier of y is non empty set
G is set
a is Element of bool the carrier of H
(H,a) is Element of bool (bool the carrier of H)
bool (bool the carrier of H) is non empty set
{ b₁ where b₁ is Element of bool the carrier of H : (H,a,b₁) } is set
c₄ is set
a is Element of bool the carrier of y
H1 is Element of bool the carrier of y
(y,H1) is Element of bool (bool the carrier of y)
bool (bool the carrier of y) is non empty set
{ b₁ where b₁ is Element of bool the carrier of y : (y,H1,b₁) } 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 non empty set
H is Element of bool the carrier of G
a is Element of bool the carrier of G
(G,a) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
{ b₁ where b₁ is Element of bool the carrier of G : (G,a,b₁) } is set
c₄ is Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
H is Element of the carrier of G
a is Element of bool the carrier of G
(G,a,H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,a,{H}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in {H} ) } is set
(G,a) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
{ b₁ where b₁ is Element of bool the carrier of G : (G,a,b₁) } 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 non empty set
H is Element of bool the carrier of G
(G,H) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
{ b₁ where b₁ is Element of bool the carrier of G : (G,H,b₁) } 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 non empty set
H is Element of bool the carrier of G
(G,H) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
{ b₁ where b₁ is Element of bool the carrier of G : (G,H,b₁) } is set
a is Element of bool the carrier of G
(G,a) is Element of bool (bool the carrier of G)
{ b₁ where b₁ is Element of bool the carrier of G : (G,a,b₁) } 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 non empty set
H is Element of bool the carrier of G
(G,H) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
{ b₁ where b₁ is Element of bool the carrier of G : (G,H,b₁) } is set
a is Element of bool the carrier of G
(G,a) is Element of bool (bool the carrier of G)
{ b₁ where b₁ is Element of bool the carrier of G : (G,a,b₁) } is set
c₄ is set
y is Element of bool the carrier of G
H1 is set
a is Element of bool the carrier of G
H1 is set
a is Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
{H} is non empty finite Element of bool the carrier of G
bool the carrier of G is non empty set
(G,{H}) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
{ b₁ where b₁ is Element of bool the carrier of G : (G,{H},b₁) } is set
(G,H) is Element of bool the carrier of G
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),H) is Element of bool the carrier of G
(G,{H},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr ((Omega). G) ) } is set
{ {b₁} where b₁ is Element of the carrier of G : b₁ in (G,H) } is set
c₄ is set
y is Element of bool the carrier of G
H1 is Element of the carrier of G
(G,{H},H1) is Element of bool the carrier of G
{H1} is non empty finite Element of bool the carrier of G
(G,{H},{H1}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in {H1} ) } is set
(G,H,H1) is Element of the carrier of G
H1 " is Element of the carrier of G
(H1 ") * H is Element of the carrier of G
the multF of G . ((H1 "),H) is Element of the carrier of G
[(H1 "),H] is set
{(H1 "),H} is non empty finite set
{(H1 ")} is non empty finite set
{{(H1 "),H},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),H] is set
((H1 ") * H) * H1 is Element of the carrier of G
the multF of G . (((H1 ") * H),H1) is Element of the carrier of G
[((H1 ") * H),H1] is set
{((H1 ") * H),H1} is non empty finite set
{((H1 ") * H)} is non empty finite set
{{((H1 ") * H),H1},{((H1 ") * H)}} is non empty finite V45() set
the multF of G . [((H1 ") * H),H1] is set
{(G,H,H1)} is non empty finite Element of bool the carrier of G
c₄ is set
y is Element of the carrier of G
{y} is non empty finite Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
H is Element of bool the carrier of G
(G,H) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
{ b₁ where b₁ is Element of bool the carrier of G : (G,H,b₁) } is set
G is non empty unital Group-like associative multMagma
G is non empty unital Group-like associative multMagma
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is non empty strict unital Group-like associative Subgroup of G
a is non empty strict unital Group-like associative Subgroup of G
the carrier of H is non empty set
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
c₄ is Element of the carrier of G
(G,a,c₄) is non empty strict unital Group-like associative Subgroup of G
c₄ " is Element of the carrier of G
(G,H,(c₄ ")) is non empty strict unital Group-like associative Subgroup of G
c₄ is Element of the carrier of G
(G,H,c₄) is non empty strict unital Group-like associative Subgroup of G
c₄ " is Element of the carrier of G
(G,a,(c₄ ")) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative Subgroup of G
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
the carrier of H is non empty set
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
(G,H,(1_ G)) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative Subgroup of G
a is non empty strict unital Group-like associative Subgroup of G
the carrier of G is non empty set
the carrier of H is non empty set
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
c₄ is Element of the carrier of G
(G,a,c₄) is non empty strict unital Group-like associative Subgroup of G
c₄ " is Element of the carrier of G
(G,H,(c₄ ")) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
c₄ is non empty strict unital Group-like associative Subgroup of G
c₄ is non empty strict unital Group-like associative Subgroup of G
y is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative Subgroup of G
a is non empty strict unital Group-like associative Subgroup of G
c₄ is non empty strict unital Group-like associative Subgroup of G
the carrier of G is non empty set
the carrier of a is non empty set
the multF of a is Relation-like [: the carrier of a, the carrier of a:] -defined the carrier of a -valued Function-like V28([: the carrier of a, the carrier of a:], the carrier of a) associative Element of bool [:[: the carrier of a, the carrier of a:], the carrier of a:]
[: the carrier of a, the carrier of a:] is non empty set
[:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
bool [:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
multMagma(# the carrier of a, the multF of a #) is strict multMagma
y is Element of the carrier of G
(G,c₄,y) is non empty strict unital Group-like associative Subgroup of G
the carrier of c₄ is non empty set
the multF of c₄ is Relation-like [: the carrier of c₄, the carrier of c₄:] -defined the carrier of c₄ -valued Function-like V28([: the carrier of c₄, the carrier of c₄:], the carrier of c₄) associative Element of bool [:[: the carrier of c₄, the carrier of c₄:], the carrier of c₄:]
[: the carrier of c₄, the carrier of c₄:] is non empty set
[:[: the carrier of c₄, the carrier of c₄:], the carrier of c₄:] is non empty set
bool [:[: the carrier of c₄, the carrier of c₄:], the carrier of c₄:] is non empty set
multMagma(# the carrier of c₄, the multF of c₄ #) is strict multMagma
H1 is Element of the carrier of G
(G,H,H1) is non empty strict unital Group-like associative Subgroup of G
H1 * y 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H1,y) is Element of the carrier of G
[H1,y] is set
{H1,y} is non empty finite set
{H1} is non empty finite set
{{H1,y},{H1}} is non empty finite V45() set
the multF of G . [H1,y] is set
(G,H,(H1 * y)) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
(G) is non empty set
bool (G) is non empty set
H is non empty unital Group-like associative Subgroup of G
a is Element of bool (G)
c₄ is set
y is non empty strict unital Group-like associative Subgroup of G
a is Element of bool (G)
c₄ is Element of bool (G)
G is set
H is non empty unital Group-like associative multMagma
a is non empty unital Group-like associative Subgroup of H
(H,a) is Element of bool (H)
(H) is non empty set
bool (H) is non empty set
c₄ is non empty strict unital Group-like associative Subgroup of H
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative Subgroup of G
a is non empty strict unital Group-like associative Subgroup of G
(G,a) is Element of bool (G)
(G) is non empty set
bool (G) is non empty set
c₄ is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
a is non empty strict unital Group-like associative Subgroup of G
(G,a,H) is non empty strict unital Group-like associative Subgroup of G
(G,a) is Element of bool (G)
(G) is non empty set
bool (G) is non empty set
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative Subgroup of G
(G,H) is Element of bool (G)
(G) is non empty set
bool (G) is non empty set
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative Subgroup of G
a is non empty strict unital Group-like associative Subgroup of G
(G,a) is Element of bool (G)
(G) is non empty set
bool (G) is non empty set
(G,H) is Element of bool (G)
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative Subgroup of G
(G,H) is Element of bool (G)
(G) is non empty set
bool (G) is non empty set
a is non empty strict unital Group-like associative Subgroup of G
(G,a) is Element of bool (G)
c₄ is set
y is non empty strict unital Group-like associative Subgroup of G
H1 is set
a is non empty strict unital Group-like associative Subgroup of G
H1 is set
a is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative Subgroup of G
(G,H) is Element of bool (G)
(G) is non empty set
bool (G) is non empty set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative Subgroup of G
carr H is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of H is non empty set
a is non empty strict unital Group-like associative Subgroup of G
carr a is Element of bool the carrier of G
the carrier of a is non empty set
the multF of a is Relation-like [: the carrier of a, the carrier of a:] -defined the carrier of a -valued Function-like V28([: the carrier of a, the carrier of a:], the carrier of a) associative Element of bool [:[: the carrier of a, the carrier of a:], the carrier of a:]
[: the carrier of a, the carrier of a:] is non empty set
[:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
bool [:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
multMagma(# the carrier of a, the multF of a #) is strict multMagma
c₄ is Element of the carrier of G
(G,H,c₄) is non empty strict unital Group-like associative Subgroup of G
(G,(carr H),c₄) is Element of bool the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
(G,(carr H),{c₄}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {c₄} ) } is set
c₄ is Element of the carrier of G
(G,(carr H),c₄) is Element of bool the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
(G,(carr H),{c₄}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {c₄} ) } is set
(G,H,c₄) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
(1). G is non empty finite strict unital Group-like associative Subgroup of G
H is Element of the carrier of G
(G,((1). G),H) is non empty finite strict unital Group-like associative Subgroup of G
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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
the carrier of ((1). G) is non empty finite set
the multF of ((1). G) is Relation-like [: the carrier of ((1). G), the carrier of ((1). G):] -defined the carrier of ((1). G) -valued Function-like V28([: the carrier of ((1). G), the carrier of ((1). G):], the carrier of ((1). G)) associative finite Element of bool [:[: the carrier of ((1). G), the carrier of ((1). G):], the carrier of ((1). G):]
[: the carrier of ((1). G), the carrier of ((1). G):] is non empty finite set
[:[: the carrier of ((1). G), the carrier of ((1). G):], the carrier of ((1). G):] is non empty finite set
bool [:[: the carrier of ((1). G), the carrier of ((1). G):], the carrier of ((1). G):] is non empty finite V45() set
multMagma(# the carrier of ((1). G), the multF of ((1). G) #) is strict multMagma
H is Element of the carrier of G
(G,((1). G),H) is non empty finite strict unital Group-like associative Subgroup of G
the carrier of ((Omega). G) is non empty set
the multF of ((Omega). G) is Relation-like [: the carrier of ((Omega). G), the carrier of ((Omega). G):] -defined the carrier of ((Omega). G) -valued Function-like V28([: the carrier of ((Omega). G), the carrier of ((Omega). G):], the carrier of ((Omega). G)) associative Element of bool [:[: the carrier of ((Omega). G), the carrier of ((Omega). G):], the carrier of ((Omega). G):]
[: the carrier of ((Omega). G), the carrier of ((Omega). G):] is non empty set
[:[: the carrier of ((Omega). G), the carrier of ((Omega). G):], the carrier of ((Omega). G):] is non empty set
bool [:[: the carrier of ((Omega). G), the carrier of ((Omega). G):], the carrier of ((Omega). G):] is non empty set
multMagma(# the carrier of ((Omega). G), the multF of ((Omega). G) #) is strict multMagma
H is Element of the carrier of G
(G,((Omega). G),H) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative (G) Subgroup of G
a is non empty strict unital Group-like associative (G) Subgroup of G
H /\ a is non empty strict unital Group-like associative Subgroup of G
the carrier of G is non empty set
c₄ is Element of the carrier of G
(G,(H /\ a),c₄) is non empty strict unital Group-like associative Subgroup of G
the carrier of (H /\ a) is non empty set
the multF of (H /\ a) is Relation-like [: the carrier of (H /\ a), the carrier of (H /\ a):] -defined the carrier of (H /\ a) -valued Function-like V28([: the carrier of (H /\ a), the carrier of (H /\ a):], the carrier of (H /\ a)) associative Element of bool [:[: the carrier of (H /\ a), the carrier of (H /\ a):], the carrier of (H /\ a):]
[: the carrier of (H /\ a), the carrier of (H /\ a):] is non empty set
[:[: the carrier of (H /\ a), the carrier of (H /\ a):], the carrier of (H /\ a):] is non empty set
bool [:[: the carrier of (H /\ a), the carrier of (H /\ a):], the carrier of (H /\ a):] is non empty set
multMagma(# the carrier of (H /\ a), the multF of (H /\ a) #) is strict multMagma
(G,H,c₄) is non empty strict unital Group-like associative Subgroup of G
(G,a,c₄) is non empty strict unital Group-like associative Subgroup of G
(G,H,c₄) /\ (G,a,c₄) is non empty strict unital Group-like associative Subgroup of G
H /\ (G,a,c₄) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative Subgroup of G
the carrier of G is non empty set
the carrier of H is non empty set
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
a is Element of the carrier of G
(G,H,a) is non empty strict unital Group-like associative Subgroup of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is non empty unital Group-like associative Subgroup of G
a is Element of the carrier of G
a * H is Element of bool the carrier of G
bool the carrier of G is non empty set
carr H is Element of bool the carrier of G
the carrier of H is non empty set
a * (carr H) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
{a} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr H ) } is set
H * a is Element of bool the carrier of G
(carr H) * a is Element of bool the carrier of G
(carr H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {a} ) } is set
(G,H,a) is non empty strict unital Group-like associative Subgroup of G
carr (G,H,a) is Element of bool the carrier of G
the carrier of (G,H,a) is non empty set
a " is Element of the carrier of G
(a ") * H is Element of bool the carrier of G
(a ") * (carr H) is Element of bool the carrier of G
{(a ")} is non empty finite Element of bool the carrier of G
{(a ")} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(a ")} & b₂ in carr H ) } is set
((a ") * H) * a is Element of bool the carrier of G
((a ") * H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (a ") * H & b₂ in {a} ) } is set
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
the carrier of multMagma(# the carrier of H, the multF of H #) is set
a * ((a ") * H) is Element of bool the carrier of G
{a} * ((a ") * H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in (a ") * H ) } is set
(a * ((a ") * H)) * a is Element of bool the carrier of G
(a * ((a ") * H)) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a * ((a ") * H) & b₂ in {a} ) } is set
a * (a ") 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (a,(a ")) is Element of the carrier of G
[a,(a ")] is set
{a,(a ")} is non empty finite set
{a} is non empty finite set
{{a,(a ")},{a}} is non empty finite V45() set
the multF of G . [a,(a ")] is set
(a * (a ")) * H is Element of bool the carrier of G
(a * (a ")) * (carr H) is Element of bool the carrier of G
{(a * (a "))} is non empty finite Element of bool the carrier of G
{(a * (a "))} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(a * (a "))} & b₂ in carr H ) } is set
((a * (a ")) * H) * a is Element of bool the carrier of G
((a * (a ")) * H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (a * (a ")) * H & b₂ in {a} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * H is Element of bool the carrier of G
(1_ G) * (carr H) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
{(1_ G)} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(1_ G)} & b₂ in carr H ) } is set
((1_ G) * H) * a is Element of bool the carrier of G
((1_ G) * H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (1_ G) * H & b₂ in {a} ) } is set
a is Element of the carrier of G
(G,H,a) is non empty strict unital Group-like associative Subgroup of G
the carrier of H is non empty set
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
the carrier of (G,H,a) is non empty set
a " is Element of the carrier of G
(a ") * H is Element of bool the carrier of G
bool the carrier of G is non empty set
carr H is Element of bool the carrier of G
(a ") * (carr H) is Element of bool the carrier of G
{(a ")} is non empty finite Element of bool the carrier of G
{(a ")} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(a ")} & b₂ in carr H ) } is set
((a ") * H) * a is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
((a ") * H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (a ") * H & b₂ in {a} ) } is set
H * (a ") is Element of bool the carrier of G
(carr H) * (a ") is Element of bool the carrier of G
(carr H) * {(a ")} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {(a ")} ) } is set
(H * (a ")) * a is Element of bool the carrier of G
(H * (a ")) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H * (a ") & b₂ in {a} ) } is set
(a ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),a) is Element of the carrier of G
[(a "),a] is set
{(a "),a} is non empty finite set
{(a ")} is non empty finite set
{{(a "),a},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),a] is set
H * ((a ") * a) is Element of bool the carrier of G
(carr H) * ((a ") * a) is Element of bool the carrier of G
{((a ") * a)} is non empty finite Element of bool the carrier of G
(carr H) * {((a ") * a)} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {((a ") * a)} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
H * (1_ G) is Element of bool the carrier of G
(carr H) * (1_ G) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
(carr H) * {(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 H & b₂ in {(1_ G)} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is non empty unital Group-like associative Subgroup of G
a is Element of the carrier of G
a * H is Element of bool the carrier of G
bool the carrier of G is non empty set
carr H is Element of bool the carrier of G
the carrier of H is non empty set
a * (carr H) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
{a} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr H ) } is set
H * a is Element of bool the carrier of G
(carr H) * a is Element of bool the carrier of G
(carr H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {a} ) } is set
a is Element of the carrier of G
a " is Element of the carrier of G
(a ") * H is Element of bool the carrier of G
bool the carrier of G is non empty set
carr H is Element of bool the carrier of G
the carrier of H is non empty set
(a ") * (carr H) is Element of bool the carrier of G
{(a ")} is non empty finite Element of bool the carrier of G
{(a ")} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(a ")} & b₂ in carr H ) } is set
H * (a ") is Element of bool the carrier of G
(carr H) * (a ") is Element of bool the carrier of G
(carr H) * {(a ")} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {(a ")} ) } is set
a * ((a ") * H) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
{a} * ((a ") * H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in (a ") * H ) } is set
a * (H * (a ")) is Element of bool the carrier of G
{a} * (H * (a ")) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in H * (a ") ) } is set
a * (a ") 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (a,(a ")) is Element of the carrier of G
[a,(a ")] is set
{a,(a ")} is non empty finite set
{a} is non empty finite set
{{a,(a ")},{a}} is non empty finite V45() set
the multF of G . [a,(a ")] is set
(a * (a ")) * H is Element of bool the carrier of G
(a * (a ")) * (carr H) is Element of bool the carrier of G
{(a * (a "))} is non empty finite Element of bool the carrier of G
{(a * (a "))} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(a * (a "))} & b₂ in carr H ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * H is Element of bool the carrier of G
(1_ G) * (carr H) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
{(1_ G)} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(1_ G)} & b₂ in carr H ) } is set
a * H is Element of bool the carrier of G
a * (carr H) is Element of bool the carrier of G
{a} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr H ) } is set
(a * H) * (a ") is Element of bool the carrier of G
(a * H) * {(a ")} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a * H & b₂ in {(a ")} ) } is set
(carr H) * a is Element of bool the carrier of G
(carr H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {a} ) } is set
((a * H) * (a ")) * a is Element of bool the carrier of G
((a * H) * (a ")) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (a * H) * (a ") & b₂ in {a} ) } is set
H * a is Element of bool the carrier of G
(a ") * a is Element of the carrier of G
the multF of G . ((a "),a) is Element of the carrier of G
[(a "),a] is set
{(a "),a} is non empty finite set
{(a ")} is non empty finite set
{{(a "),a},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),a] is set
(a * H) * ((a ") * a) is Element of bool the carrier of G
{((a ") * a)} is non empty finite Element of bool the carrier of G
(a * H) * {((a ") * a)} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a * H & b₂ in {((a ") * a)} ) } is set
(a * H) * (1_ G) is Element of bool the carrier of G
(a * H) * {(1_ G)} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a * H & b₂ in {(1_ G)} ) } is set
a is Element of the carrier of G
a * H is Element of bool the carrier of G
a * (carr H) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
{a} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr H ) } is set
H * a is Element of bool the carrier of G
(carr H) * a is Element of bool the carrier of G
(carr H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {a} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is non empty unital Group-like associative Subgroup of G
a is Element of the carrier of G
H * a is Element of bool the carrier of G
bool the carrier of G is non empty set
carr H is Element of bool the carrier of G
the carrier of H is non empty set
(carr H) * a is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(carr H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {a} ) } is set
a * H is Element of bool the carrier of G
a * (carr H) is Element of bool the carrier of G
{a} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr H ) } is set
a is Element of the carrier of G
a " is Element of the carrier of G
H * (a ") is Element of bool the carrier of G
bool the carrier of G is non empty set
carr H is Element of bool the carrier of G
the carrier of H is non empty set
(carr H) * (a ") is Element of bool the carrier of G
{(a ")} is non empty finite Element of bool the carrier of G
(carr H) * {(a ")} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {(a ")} ) } is set
(a ") * H is Element of bool the carrier of G
(a ") * (carr H) is Element of bool the carrier of G
{(a ")} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(a ")} & b₂ in carr H ) } is set
a * (H * (a ")) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
{a} * (H * (a ")) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in H * (a ") ) } is set
a * ((a ") * H) is Element of bool the carrier of G
{a} * ((a ") * H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in (a ") * H ) } is set
a * (a ") 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (a,(a ")) is Element of the carrier of G
[a,(a ")] is set
{a,(a ")} is non empty finite set
{a} is non empty finite set
{{a,(a ")},{a}} is non empty finite V45() set
the multF of G . [a,(a ")] is set
(a * (a ")) * H is Element of bool the carrier of G
(a * (a ")) * (carr H) is Element of bool the carrier of G
{(a * (a "))} is non empty finite Element of bool the carrier of G
{(a * (a "))} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(a * (a "))} & b₂ in carr H ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * H is Element of bool the carrier of G
(1_ G) * (carr H) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
{(1_ G)} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(1_ G)} & b₂ in carr H ) } is set
a * H is Element of bool the carrier of G
a * (carr H) is Element of bool the carrier of G
{a} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr H ) } is set
(a * H) * (a ") is Element of bool the carrier of G
(a * H) * {(a ")} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a * H & b₂ in {(a ")} ) } is set
((a * H) * (a ")) * a is Element of bool the carrier of G
((a * H) * (a ")) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (a * H) * (a ") & b₂ in {a} ) } is set
(carr H) * a is Element of bool the carrier of G
(carr H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {a} ) } is set
(a ") * a is Element of the carrier of G
the multF of G . ((a "),a) is Element of the carrier of G
[(a "),a] is set
{(a "),a} is non empty finite set
{(a ")} is non empty finite set
{{(a "),a},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),a] is set
(a * H) * ((a ") * a) is Element of bool the carrier of G
{((a ") * a)} is non empty finite Element of bool the carrier of G
(a * H) * {((a ") * a)} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a * H & b₂ in {((a ") * a)} ) } is set
H * a is Element of bool the carrier of G
(a * H) * (1_ G) is Element of bool the carrier of G
(a * H) * {(1_ G)} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a * H & b₂ in {(1_ G)} ) } is set
a is Element of the carrier of G
a * H is Element of bool the carrier of G
a * (carr H) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
{a} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr H ) } is set
H * a is Element of bool the carrier of G
(carr H) * a is Element of bool the carrier of G
(carr H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {a} ) } 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 non empty set
H is non empty unital Group-like associative Subgroup of G
a is Element of bool the carrier of G
a * H is Element of bool the carrier of G
carr H is Element of bool the carrier of G
the carrier of H is non empty set
a * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in a & b₂ in carr H ) } is set
H * a is Element of bool the carrier of G
(carr H) * a is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in a ) } is set
c₄ is set
y is Element of the carrier of G
H1 is Element of the carrier of G
y * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (y,H1) is Element of the carrier of G
[y,H1] is set
{y,H1} is non empty finite set
{y} is non empty finite set
{{y,H1},{y}} is non empty finite V45() set
the multF of G . [y,H1] is set
y * H is Element of bool the carrier of G
y * (carr H) is Element of bool the carrier of G
{y} is non empty finite Element of bool the carrier of G
{y} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {y} & b₂ in carr H ) } is set
H * y is Element of bool the carrier of G
(carr H) * y is Element of bool the carrier of G
(carr H) * {y} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {y} ) } is set
a is Element of the carrier of G
a * y is Element of the carrier of G
the multF of G . (a,y) is Element of the carrier of G
[a,y] is set
{a,y} is non empty finite set
{a} is non empty finite set
{{a,y},{a}} is non empty finite V45() set
the multF of G . [a,y] is set
c₄ is set
y is Element of the carrier of G
H1 is Element of the carrier of G
y * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (y,H1) is Element of the carrier of G
[y,H1] is set
{y,H1} is non empty finite set
{y} is non empty finite set
{{y,H1},{y}} is non empty finite V45() set
the multF of G . [y,H1] is set
H * H1 is Element of bool the carrier of G
(carr H) * H1 is Element of bool the carrier of G
{H1} is non empty finite Element of bool the carrier of G
(carr H) * {H1} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {H1} ) } is set
H1 * H is Element of bool the carrier of G
H1 * (carr H) is Element of bool the carrier of G
{H1} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H1} & b₂ in carr H ) } is set
a is Element of the carrier of G
H1 * a is Element of the carrier of G
the multF of G . (H1,a) is Element of the carrier of G
[H1,a] is set
{H1,a} is non empty finite set
{H1} is non empty finite set
{{H1,a},{H1}} is non empty finite V45() set
the multF of G . [H1,a] is set
a is Element of the carrier of G
a * H is Element of bool the carrier of G
carr H is Element of bool the carrier of G
the carrier of H is non empty set
a * (carr H) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
{a} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr H ) } is set
{a} * H is Element of bool the carrier of G
H * {a} is Element of bool the carrier of G
(carr H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {a} ) } is set
H * a is Element of bool the carrier of G
(carr H) * a is Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is non empty strict unital Group-like associative Subgroup of G
a is Element of the carrier of G
(G,H,a) is non empty strict unital Group-like associative Subgroup of G
the carrier of H is non empty set
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
a is Element of the carrier of G
a " is Element of the carrier of G
(G,H,(a ")) is non empty strict unital Group-like associative Subgroup of G
(G,H,(a ")) * a is Element of bool the carrier of G
bool the carrier of G is non empty set
carr (G,H,(a ")) is Element of bool the carrier of G
the carrier of (G,H,(a ")) is non empty set
(carr (G,H,(a "))) * a is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(carr (G,H,(a "))) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr (G,H,(a ")) & b₂ in {a} ) } is set
(a ") " is Element of the carrier of G
((a ") ") * H is Element of bool the carrier of G
carr H is Element of bool the carrier of G
the carrier of H is non empty set
((a ") ") * (carr H) is Element of bool the carrier of G
{((a ") ")} is non empty finite Element of bool the carrier of G
{((a ") ")} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((a ") ")} & b₂ in carr H ) } is set
(((a ") ") * H) * (a ") is Element of bool the carrier of G
{(a ")} is non empty finite Element of bool the carrier of G
(((a ") ") * H) * {(a ")} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((a ") ") * H & b₂ in {(a ")} ) } is set
((((a ") ") * H) * (a ")) * a is Element of bool the carrier of G
((((a ") ") * H) * (a ")) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (((a ") ") * H) * (a ") & b₂ in {a} ) } is set
(a ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),a) is Element of the carrier of G
[(a "),a] is set
{(a "),a} is non empty finite set
{(a ")} is non empty finite set
{{(a "),a},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),a] is set
(((a ") ") * H) * ((a ") * a) is Element of bool the carrier of G
{((a ") * a)} is non empty finite Element of bool the carrier of G
(((a ") ") * H) * {((a ") * a)} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((a ") ") * H & b₂ in {((a ") * a)} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(((a ") ") * H) * (1_ G) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
(((a ") ") * H) * {(1_ G)} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((a ") ") * H & b₂ in {(1_ G)} ) } is set
a * H is Element of bool the carrier of G
a * (carr H) is Element of bool the carrier of G
{a} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr H ) } is set
H * a is Element of bool the carrier of G
(carr H) * a is Element of bool the carrier of G
(carr H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {a} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is non empty strict unital Group-like associative Subgroup of G
a is Element of the carrier of G
(G,H,a) is non empty strict unital Group-like associative Subgroup of G
the carrier of H is non empty set
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
a is Element of the carrier of G
a " is Element of the carrier of G
(G,H,(a ")) is non empty strict unital Group-like associative Subgroup of G
(G,H,(a ")) * a is Element of bool the carrier of G
bool the carrier of G is non empty set
carr (G,H,(a ")) is Element of bool the carrier of G
the carrier of (G,H,(a ")) is non empty set
(carr (G,H,(a "))) * a is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(carr (G,H,(a "))) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr (G,H,(a ")) & b₂ in {a} ) } is set
(a ") " is Element of the carrier of G
((a ") ") * H is Element of bool the carrier of G
carr H is Element of bool the carrier of G
the carrier of H is non empty set
((a ") ") * (carr H) is Element of bool the carrier of G
{((a ") ")} is non empty finite Element of bool the carrier of G
{((a ") ")} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((a ") ")} & b₂ in carr H ) } is set
(((a ") ") * H) * (a ") is Element of bool the carrier of G
{(a ")} is non empty finite Element of bool the carrier of G
(((a ") ") * H) * {(a ")} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((a ") ") * H & b₂ in {(a ")} ) } is set
((((a ") ") * H) * (a ")) * a is Element of bool the carrier of G
((((a ") ") * H) * (a ")) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (((a ") ") * H) * (a ") & b₂ in {a} ) } is set
(a ") * a 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),a) is Element of the carrier of G
[(a "),a] is set
{(a "),a} is non empty finite set
{(a ")} is non empty finite set
{{(a "),a},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),a] is set
(((a ") ") * H) * ((a ") * a) is Element of bool the carrier of G
{((a ") * a)} is non empty finite Element of bool the carrier of G
(((a ") ") * H) * {((a ") * a)} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((a ") ") * H & b₂ in {((a ") * a)} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(((a ") ") * H) * (1_ G) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
(((a ") ") * H) * {(1_ G)} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in ((a ") ") * H & b₂ in {(1_ G)} ) } is set
a * H is Element of bool the carrier of G
a * (carr H) is Element of bool the carrier of G
{a} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr H ) } is set
H * a is Element of bool the carrier of G
(carr H) * a is Element of bool the carrier of G
(carr H) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {a} ) } is set
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative Subgroup of G
(G,H) is Element of bool (G)
(G) is non empty set
bool (G) is non empty set
{H} is non empty finite set
a is set
c₄ is non empty strict unital Group-like associative Subgroup of G
the carrier of G is non empty set
y is Element of the carrier of G
(G,H,y) is non empty strict unital Group-like associative Subgroup of G
the carrier of G is non empty set
a is Element of the carrier of G
(G,H,a) is non empty strict unital Group-like associative Subgroup of G
the carrier of H is non empty set
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is non empty strict unital Group-like associative Subgroup of G
carr H is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of H is non empty set
a is Element of the carrier of G
(G,a) is Element of bool the carrier of G
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,{a},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {a} & b₂ in carr ((Omega). G) ) } is set
c₄ is set
y is Element of the carrier of G
H1 is Element of the carrier of G
(G,a,H1) is Element of the carrier of G
H1 " is Element of the carrier of G
(H1 ") * a is Element of the carrier of G
the multF of G . ((H1 "),a) is Element of the carrier of G
[(H1 "),a] is set
{(H1 "),a} is non empty finite set
{(H1 ")} is non empty finite set
{{(H1 "),a},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),a] is set
((H1 ") * a) * H1 is Element of the carrier of G
the multF of G . (((H1 ") * a),H1) is Element of the carrier of G
[((H1 ") * a),H1] is set
{((H1 ") * a),H1} is non empty finite set
{((H1 ") * a)} is non empty finite set
{{((H1 ") * a),H1},{((H1 ") * a)}} is non empty finite V45() set
the multF of G . [((H1 ") * a),H1] is set
(G,H,H1) is non empty strict unital Group-like associative Subgroup of G
a is Element of the carrier of G
(G,H,a) is non empty strict unital Group-like associative Subgroup of G
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
c₄ is Element of the carrier of G
y is Element of the carrier of G
(G,y,a) is Element of the carrier of G
a " is Element of the carrier of G
(a ") * y 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . ((a "),y) is Element of the carrier of G
[(a "),y] is set
{(a "),y} is non empty finite set
{(a ")} is non empty finite set
{{(a "),y},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),y] is set
((a ") * y) * a is Element of the carrier of G
the multF of G . (((a ") * y),a) is Element of the carrier of G
[((a ") * y),a] is set
{((a ") * y),a} is non empty finite set
{((a ") * y)} is non empty finite set
{{((a ") * y),a},{((a ") * y)}} is non empty finite V45() set
the multF of G . [((a ") * y),a] is set
(G,y) is Element of bool the carrier of G
(Omega). G is non empty strict unital Group-like associative Subgroup of G
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),y) is Element of bool the carrier of G
{y} is non empty finite Element of bool the carrier of G
(G,{y},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {y} & b₂ in carr ((Omega). G) ) } is set
(G,c₄) is Element of bool the carrier of G
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),c₄) is Element of bool the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
(G,{c₄},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {c₄} & b₂ in carr ((Omega). G) ) } is set
a " is Element of the carrier of G
(G,c₄,(a ")) is Element of the carrier of G
(a ") " is Element of the carrier of G
((a ") ") * c₄ is Element of the carrier of G
the multF of G . (((a ") "),c₄) is Element of the carrier of G
[((a ") "),c₄] is set
{((a ") "),c₄} is non empty finite set
{((a ") ")} is non empty finite set
{{((a ") "),c₄},{((a ") ")}} is non empty finite V45() set
the multF of G . [((a ") "),c₄] is set
(((a ") ") * c₄) * (a ") is Element of the carrier of G
the multF of G . ((((a ") ") * c₄),(a ")) is Element of the carrier of G
[(((a ") ") * c₄),(a ")] is set
{(((a ") ") * c₄),(a ")} is non empty finite set
{(((a ") ") * c₄)} is non empty finite set
{{(((a ") ") * c₄),(a ")},{(((a ") ") * c₄)}} is non empty finite V45() set
the multF of G . [(((a ") ") * c₄),(a ")] is set
(G,(G,c₄,(a ")),a) is Element of the carrier of G
(a ") * (G,c₄,(a ")) is Element of the carrier of G
the multF of G . ((a "),(G,c₄,(a "))) is Element of the carrier of G
[(a "),(G,c₄,(a "))] is set
{(a "),(G,c₄,(a "))} is non empty finite set
{(a ")} is non empty finite set
{{(a "),(G,c₄,(a "))},{(a ")}} is non empty finite V45() set
the multF of G . [(a "),(G,c₄,(a "))] is set
((a ") * (G,c₄,(a "))) * a is Element of the carrier of G
the multF of G . (((a ") * (G,c₄,(a "))),a) is Element of the carrier of G
[((a ") * (G,c₄,(a "))),a] is set
{((a ") * (G,c₄,(a "))),a} is non empty finite set
{((a ") * (G,c₄,(a ")))} is non empty finite set
{{((a ") * (G,c₄,(a "))),a},{((a ") * (G,c₄,(a ")))}} is non empty finite V45() set
the multF of G . [((a ") * (G,c₄,(a "))),a] is set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G) Subgroup of G
carr H is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of H is non empty set
a is non empty strict unital Group-like associative (G) Subgroup of G
carr a is Element of bool the carrier of G
the carrier of a is non empty set
(carr a) * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in carr H ) } is set
(carr H) * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in carr a ) } is set
c₄ is set
y is Element of the carrier of G
H1 is Element of the carrier of G
y * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (y,H1) is Element of the carrier of G
[y,H1] is set
{y,H1} is non empty finite set
{y} is non empty finite set
{{y,H1},{y}} is non empty finite V45() set
the multF of G . [y,H1] is set
(G,a,H1) is non empty strict unital Group-like associative Subgroup of G
(G,y,H1) is Element of the carrier of G
H1 " is Element of the carrier of G
(H1 ") * y is Element of the carrier of G
the multF of G . ((H1 "),y) is Element of the carrier of G
[(H1 "),y] is set
{(H1 "),y} is non empty finite set
{(H1 ")} is non empty finite set
{{(H1 "),y},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),y] is set
((H1 ") * y) * H1 is Element of the carrier of G
the multF of G . (((H1 ") * y),H1) is Element of the carrier of G
[((H1 ") * y),H1] is set
{((H1 ") * y),H1} is non empty finite set
{((H1 ") * y)} is non empty finite set
{{((H1 ") * y),H1},{((H1 ") * y)}} is non empty finite V45() set
the multF of G . [((H1 ") * y),H1] is set
the multF of a is Relation-like [: the carrier of a, the carrier of a:] -defined the carrier of a -valued Function-like V28([: the carrier of a, the carrier of a:], the carrier of a) associative Element of bool [:[: the carrier of a, the carrier of a:], the carrier of a:]
[: the carrier of a, the carrier of a:] is non empty set
[:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
bool [:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
multMagma(# the carrier of a, the multF of a #) is strict multMagma
H1 * (G,y,H1) is Element of the carrier of G
the multF of G . (H1,(G,y,H1)) is Element of the carrier of G
[H1,(G,y,H1)] is set
{H1,(G,y,H1)} is non empty finite set
{H1} is non empty finite set
{{H1,(G,y,H1)},{H1}} is non empty finite V45() set
the multF of G . [H1,(G,y,H1)] is set
(H1 ") * (y * H1) is Element of the carrier of G
the multF of G . ((H1 "),(y * H1)) is Element of the carrier of G
[(H1 "),(y * H1)] is set
{(H1 "),(y * H1)} is non empty finite set
{{(H1 "),(y * H1)},{(H1 ")}} is non empty finite V45() set
the multF of G . [(H1 "),(y * H1)] is set
H1 * ((H1 ") * (y * H1)) is Element of the carrier of G
the multF of G . (H1,((H1 ") * (y * H1))) is Element of the carrier of G
[H1,((H1 ") * (y * H1))] is set
{H1,((H1 ") * (y * H1))} is non empty finite set
{{H1,((H1 ") * (y * H1))},{H1}} is non empty finite V45() set
the multF of G . [H1,((H1 ") * (y * H1))] is set
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative (G) Subgroup of G
carr H is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of H is non empty set
a is non empty strict unital Group-like associative (G) Subgroup of G
carr a is Element of bool the carrier of G
the carrier of a is non empty set
(carr H) * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in carr a ) } is set
(carr a) * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in carr H ) } is set
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative (G) Subgroup of G
carr H is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of H is non empty set
a is non empty strict unital Group-like associative (G) Subgroup of G
carr a is Element of bool the carrier of G
the carrier of a is non empty set
(carr H) * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in carr a ) } is set
(carr a) * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in carr H ) } is set
a is non empty strict unital Group-like associative Subgroup of G
the carrier of a is non empty set
b is Element of the carrier of G
b * a is Element of bool the carrier of G
carr a is Element of bool the carrier of G
b * (carr a) is Element of bool the carrier of G
{b} is non empty finite Element of bool the carrier of G
{b} * (carr a) 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 carr a ) } is set
b * H is Element of bool the carrier of G
b * (carr H) is Element of bool the carrier of G
{b} * (carr H) 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 carr H ) } is set
(b * H) * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in b * H & b₂ in carr a ) } is set
H * b is Element of bool the carrier of G
(carr H) * b is Element of bool the carrier of G
(carr H) * {b} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {b} ) } is set
(H * b) * (carr a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H * b & b₂ in carr a ) } is set
b * a is Element of bool the carrier of G
b * (carr a) is Element of bool the carrier of G
{b} * (carr a) 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 carr a ) } is set
(carr H) * (b * a) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in b * a ) } is set
a * b is Element of bool the carrier of G
(carr a) * b is Element of bool the carrier of G
(carr a) * {b} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in {b} ) } is set
(carr H) * (a * b) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in a * b ) } is set
a * b is Element of bool the carrier of G
(carr a) * b is Element of bool the carrier of G
(carr a) * {b} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr a & b₂ in {b} ) } is set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G) Subgroup of G
Left_Cosets H is Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
Right_Cosets H is Element of bool (bool the carrier of G)
a is set
c₄ is Element of the carrier of G
c₄ * H is Element of bool the carrier of G
carr H is Element of bool the carrier of G
the carrier of H is non empty set
c₄ * (carr H) is Element of bool the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
{c₄} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {c₄} & b₂ in carr H ) } is set
H * c₄ is Element of bool the carrier of G
(carr H) * c₄ is Element of bool the carrier of G
(carr H) * {c₄} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {c₄} ) } is set
a is set
c₄ is Element of the carrier of G
H * c₄ is Element of bool the carrier of G
carr H is Element of bool the carrier of G
the carrier of H is non empty set
(carr H) * c₄ is Element of bool the carrier of G
{c₄} is non empty finite Element of bool the carrier of G
(carr H) * {c₄} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {c₄} ) } is set
c₄ * H is Element of bool the carrier of G
c₄ * (carr H) is Element of bool the carrier of G
{c₄} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {c₄} & b₂ in carr H ) } is set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative Subgroup of G
Left_Cosets H is Element of bool (bool the carrier of G)
the carrier of G is non empty set
bool the carrier of G is non empty set
bool (bool the carrier of G) is non empty set
index H is V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
a is finite set
card a is V12() V13() V14() V18() ext-real V37() V38() integer Element of K46()
a is set
c₄ is set
{a,c₄} is non empty finite set
carr H is Element of bool the carrier of G
the carrier of H is non empty set
{a,(carr H)} is non empty finite set
{(carr H),c₄} is non empty finite set
y is set
{(carr H),y} is non empty finite set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * H is Element of bool the carrier of G
(1_ G) * (carr H) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
{(1_ G)} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(1_ G)} & b₂ in carr H ) } is set
H1 is Element of the carrier of G
H1 * H is Element of bool the carrier of G
H1 * (carr H) is Element of bool the carrier of G
{H1} is non empty finite Element of bool the carrier of G
{H1} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H1} & b₂ in carr H ) } is set
{(carr H)} is non empty finite Element of bool (bool the carrier of G)
union (Left_Cosets H) is set
(carr H) \/ y is set
Right_Cosets H is Element of bool (bool the carrier of G)
union (Right_Cosets H) is set
the carrier of G \ (carr H) is Element of bool the carrier of G
H1 is finite set
card H1 is V12() V13() V14() V18() ext-real V37() V38() integer Element of K46()
H1 is set
a is set
{H1,a} is non empty finite set
{H1,(carr H)} is non empty finite set
{(carr H),a} is non empty finite set
b is set
{(carr H),b} is non empty finite set
1_ G is non being_of_order_0 Element of the carrier of G
H * (1_ G) is Element of bool the carrier of G
(carr H) * (1_ G) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
(carr H) * {(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 H & b₂ in {(1_ G)} ) } is set
c is Element of the carrier of G
H * c is Element of bool the carrier of G
(carr H) * c is Element of bool the carrier of G
{c} is non empty finite Element of bool the carrier of G
(carr H) * {c} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {c} ) } is set
{(carr H)} is non empty finite Element of bool (bool the carrier of G)
(carr H) \/ b is set
c is Element of the carrier of G
c * H is Element of bool the carrier of G
c * (carr H) is Element of bool the carrier of G
{c} is non empty finite Element of bool the carrier of G
{c} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {c} & b₂ in carr H ) } is set
H * c is Element of bool the carrier of G
(carr H) * c is Element of bool the carrier of G
(carr H) * {c} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {c} ) } is set
c is Element of the carrier of G
c * H is Element of bool the carrier of G
c * (carr H) is Element of bool the carrier of G
{c} is non empty finite Element of bool the carrier of G
{c} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {c} & b₂ in carr H ) } is set
H * c is Element of bool the carrier of G
(carr H) * c is Element of bool the carrier of G
(carr H) * {c} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {c} ) } is set
c is Element of the carrier of G
c * H is Element of bool the carrier of G
c * (carr H) is Element of bool the carrier of G
{c} is non empty finite Element of bool the carrier of G
{c} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {c} & b₂ in carr H ) } is set
H * c is Element of bool the carrier of G
(carr H) * c is Element of bool the carrier of G
(carr H) * {c} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {c} ) } is set
c is Element of the carrier of G
c * H is Element of bool the carrier of G
c * (carr H) is Element of bool the carrier of G
{c} is non empty finite Element of bool the carrier of G
{c} * (carr H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {c} & b₂ in carr H ) } is set
H * c is Element of bool the carrier of G
(carr H) * c is Element of bool the carrier of G
(carr H) * {c} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr H & b₂ in {c} ) } 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 non empty set
H is Element of bool the carrier of G
{ b₁ where b₁ is Element of the carrier of G : (G,H,b₁) = H } is set
c₄ is set
y is Element of the carrier of G
(G,H,y) is Element of bool the carrier of G
{y} is non empty finite Element of bool the carrier of G
(G,H,{y}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {y} ) } is set
y is Element of the carrier of G
c₄ is Element of bool the carrier of G
H1 is Element of the carrier of G
y * H1 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (y,H1) is Element of the carrier of G
[y,H1] is set
{y,H1} is non empty finite set
{y} is non empty finite set
{{y,H1},{y}} is non empty finite V45() set
the multF of G . [y,H1] is set
(G,H,(y * H1)) is Element of bool the carrier of G
{(y * H1)} is non empty finite Element of bool the carrier of G
(G,H,{(y * H1)}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {(y * H1)} ) } is set
a is Element of the carrier of G
(G,H,a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,H,{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {a} ) } is set
b is Element of the carrier of G
(G,H,b) is Element of bool the carrier of G
{b} is non empty finite Element of bool the carrier of G
(G,H,{b}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {b} ) } is set
y is Element of the carrier of G
y " is Element of the carrier of G
(G,H,(y ")) is Element of bool the carrier of G
{(y ")} is non empty finite Element of bool the carrier of G
(G,H,{(y ")}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {(y ")} ) } is set
H1 is Element of the carrier of G
(G,H,H1) is Element of bool the carrier of G
{H1} is non empty finite Element of bool the carrier of G
(G,H,{H1}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {H1} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(G,H,(1_ G)) is Element of bool the carrier of G
{(1_ G)} is non empty finite Element of bool the carrier of G
(G,H,{(1_ G)}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {(1_ G)} ) } is set
a is non empty strict unital Group-like associative Subgroup of G
the carrier of a is non empty set
c₄ is non empty strict unital Group-like associative Subgroup of G
the carrier of c₄ is non empty set
G is set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
bool the carrier of H is non empty set
a is Element of bool the carrier of H
(H,a) is non empty strict unital Group-like associative Subgroup of H
the carrier of (H,a) is non empty set
{ b₁ where b₁ is Element of the carrier of H : (H,a,b₁) = a } is set
c₄ is Element of the carrier of H
(H,a,c₄) is Element of bool the carrier of H
{c₄} is non empty finite Element of bool the carrier of H
(H,a,{c₄}) is Element of bool the carrier of H
{ (H,b₁,b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in a & b₂ in {c₄} ) } is set
{ b₁ where b₁ is Element of the carrier of H : (H,a,b₁) = a } is set
the carrier of (H,a) is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
H is Element of bool the carrier of G
(G,H) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
{ b₁ where b₁ is Element of bool the carrier of G : (G,H,b₁) } is set
card (G,H) is cardinal set
(G,H) is non empty strict unital Group-like associative Subgroup of G
Index (G,H) is cardinal set
Left_Cosets (G,H) is Element of bool (bool the carrier of G)
card (Left_Cosets (G,H)) is cardinal set
a is set
c₄ is Element of bool the carrier of G
y is Element of the carrier of G
(G,H,y) is Element of bool the carrier of G
{y} is non empty finite Element of bool the carrier of G
(G,H,{y}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {y} ) } is set
(G,H) * y is Element of bool the carrier of G
carr (G,H) is Element of bool the carrier of G
the carrier of (G,H) is non empty set
(carr (G,H)) * y is Element of bool the carrier of G
(carr (G,H)) * {y} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr (G,H) & b₂ in {y} ) } is set
H1 is set
a is Relation-like Function-like set
dom a is set
c₄ is set
y is set
H1 is set
a is Element of the carrier of G
(G,H,a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,H,{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {a} ) } is set
(G,H) * a is Element of bool the carrier of G
carr (G,H) is Element of bool the carrier of G
the carrier of (G,H) is non empty set
(carr (G,H)) * a is Element of bool the carrier of G
(carr (G,H)) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr (G,H) & b₂ in {a} ) } is set
b is Element of the carrier of G
(G,H,b) is Element of bool the carrier of G
{b} is non empty finite Element of bool the carrier of G
(G,H,{b}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {b} ) } is set
(G,H) * b is Element of bool the carrier of G
(carr (G,H)) * b is Element of bool the carrier of G
(carr (G,H)) * {b} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr (G,H) & b₂ in {b} ) } is set
a " is Element of the carrier of G
(G,(G,H,b),(a ")) is Element of bool the carrier of G
{(a ")} is non empty finite Element of bool the carrier of G
(G,(G,H,b),{(a ")}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,H,b) & b₂ in {(a ")} ) } is set
b * (a ") 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (b,(a ")) is Element of the carrier of G
[b,(a ")] is set
{b,(a ")} is non empty finite set
{b} is non empty finite set
{{b,(a ")},{b}} is non empty finite V45() set
the multF of G . [b,(a ")] is set
(G,H,(b * (a "))) is Element of bool the carrier of G
{(b * (a "))} is non empty finite Element of bool the carrier of G
(G,H,{(b * (a "))}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {(b * (a "))} ) } is set
rng a is set
Right_Cosets (G,H) is Element of bool (bool the carrier of G)
c₄ is set
y is set
a . y is set
H1 is Element of the carrier of G
(G,H,H1) is Element of bool the carrier of G
{H1} is non empty finite Element of bool the carrier of G
(G,H,{H1}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {H1} ) } is set
(G,H) * H1 is Element of bool the carrier of G
carr (G,H) is Element of bool the carrier of G
the carrier of (G,H) is non empty set
(carr (G,H)) * H1 is Element of bool the carrier of G
(carr (G,H)) * {H1} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr (G,H) & b₂ in {H1} ) } is set
c₄ is set
y is Element of the carrier of G
(G,H) * y is Element of bool the carrier of G
carr (G,H) is Element of bool the carrier of G
the carrier of (G,H) is non empty set
(carr (G,H)) * y is Element of bool the carrier of G
{y} is non empty finite Element of bool the carrier of G
(carr (G,H)) * {y} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr (G,H) & b₂ in {y} ) } is set
(G,H,y) is Element of bool the carrier of G
(G,H,{y}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {y} ) } is set
a . (G,H,y) is set
a is Element of the carrier of G
(G,H,a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,H,{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {a} ) } is set
(G,H) * a is Element of bool the carrier of G
(carr (G,H)) * a is Element of bool the carrier of G
(carr (G,H)) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr (G,H) & b₂ in {a} ) } is set
c₄ is set
a . c₄ is set
y is set
a . y is set
H1 is Element of the carrier of G
(G,H,H1) is Element of bool the carrier of G
{H1} is non empty finite Element of bool the carrier of G
(G,H,{H1}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {H1} ) } is set
(G,H) * H1 is Element of bool the carrier of G
carr (G,H) is Element of bool the carrier of G
the carrier of (G,H) is non empty set
(carr (G,H)) * H1 is Element of bool the carrier of G
(carr (G,H)) * {H1} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr (G,H) & b₂ in {H1} ) } is set
a is Element of the carrier of G
(G,H,a) is Element of bool the carrier of G
{a} is non empty finite Element of bool the carrier of G
(G,H,{a}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {a} ) } is set
(G,H) * a is Element of bool the carrier of G
(carr (G,H)) * a is Element of bool the carrier of G
(carr (G,H)) * {a} is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in carr (G,H) & b₂ in {a} ) } is set
a " is Element of the carrier of G
H1 * (a ") 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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
the multF of G . (H1,(a ")) is Element of the carrier of G
[H1,(a ")] is set
{H1,(a ")} is non empty finite set
{H1} is non empty finite set
{{H1,(a ")},{H1}} is non empty finite V45() set
the multF of G . [H1,(a ")] is set
(G,(G,H,H1),(a ")) is Element of bool the carrier of G
{(a ")} is non empty finite Element of bool the carrier of G
(G,(G,H,H1),{(a ")}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,H,H1) & b₂ in {(a ")} ) } is set
b is Element of the carrier of G
(G,H,b) is Element of bool the carrier of G
{b} is non empty finite Element of bool the carrier of G
(G,H,{b}) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in {b} ) } is set
card (Right_Cosets (G,H)) is cardinal set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty set
H is Element of bool the carrier of G
(G,H) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
{ b₁ where b₁ is Element of bool the carrier of G : (G,H,b₁) } is set
(G,H) is non empty strict unital Group-like associative Subgroup of G
Left_Cosets (G,H) is Element of bool (bool the carrier of G)
index (G,H) is V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
card (G,H) is cardinal set
Index (G,H) is cardinal set
card (Left_Cosets (G,H)) is cardinal set
a is finite set
card a is V12() V13() V14() V18() ext-real V37() V38() integer Element of K46()
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
(G,H) is Element of bool the carrier of G
bool the carrier of G is non empty set
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr ((Omega). G) ) } is set
card (G,H) is cardinal set
(G,{H}) is non empty strict unital Group-like associative Subgroup of G
Index (G,{H}) is cardinal set
Left_Cosets (G,{H}) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
card (Left_Cosets (G,{H})) is cardinal set
a is Relation-like Function-like set
dom a is set
rng a is set
(G,{H}) is Element of bool (bool the carrier of G)
{ b₁ where b₁ is Element of bool the carrier of G : (G,{H},b₁) } is set
c₄ is set
y is set
a . y is set
H1 is Element of the carrier of G
a . H1 is set
{H1} is non empty finite Element of bool the carrier of G
{ {b₁} where b₁ is Element of the carrier of G : b₁ in (G,H) } is set
c₄ is set
{ {b₁} where b₁ is Element of the carrier of G : b₁ in (G,H) } is set
y is Element of the carrier of G
{y} is non empty finite Element of bool the carrier of G
a . y is set
c₄ is set
a . c₄ is set
y is set
a . y is set
{c₄} is non empty finite set
{y} is non empty finite set
card (G,{H}) is cardinal set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
(G,H) is Element of bool the carrier of G
bool the carrier of G is non empty set
(Omega). G is non empty strict unital Group-like associative Subgroup of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
carr ((Omega). G) is Element of bool the carrier of G
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G)),H) is Element of bool the carrier of G
{H} is non empty finite Element of bool the carrier of G
(G,{H},(carr ((Omega). G))) is Element of bool the carrier of G
{ (G,b₁,b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H} & b₂ in carr ((Omega). G) ) } is set
(G,{H}) is non empty strict unital Group-like associative Subgroup of G
Left_Cosets (G,{H}) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
index (G,{H}) is V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
card (G,H) is cardinal set
Index (G,{H}) is cardinal set
card (Left_Cosets (G,{H})) is cardinal set
a is finite set
card a is V12() V13() V14() V18() ext-real V37() V38() integer Element of K46()
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative Subgroup of G
carr H is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of H is non empty set
(G,(carr H)) is non empty strict unital Group-like associative Subgroup of G
G is set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
a is non empty strict unital Group-like associative Subgroup of H
(H,a) is non empty strict unital Group-like associative Subgroup of H
carr a is Element of bool the carrier of H
bool the carrier of H is non empty set
the carrier of a is non empty set
(H,(carr a)) is non empty strict unital Group-like associative Subgroup of H
c₄ is Element of the carrier of H
(H,(carr a),c₄) is Element of bool the carrier of H
{c₄} is non empty finite Element of bool the carrier of H
(H,(carr a),{c₄}) is Element of bool the carrier of H
{ (H,b₁,b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in carr a & b₂ in {c₄} ) } is set
(H,a,c₄) is non empty strict unital Group-like associative Subgroup of H
c₄ is Element of the carrier of H
(H,a,c₄) is non empty strict unital Group-like associative Subgroup of H
(H,(carr a),c₄) is Element of bool the carrier of H
{c₄} is non empty finite Element of bool the carrier of H
(H,(carr a),{c₄}) is Element of bool the carrier of H
{ (H,b₁,b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in carr a & b₂ in {c₄} ) } is set
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative Subgroup of G
(G,H) is Element of bool (G)
(G) is non empty set
bool (G) is non empty set
card (G,H) is cardinal set
(G,H) is non empty strict unital Group-like associative Subgroup of G
carr H is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of H is non empty set
(G,(carr H)) is non empty strict unital Group-like associative Subgroup of G
Index (G,H) is cardinal set
Left_Cosets (G,H) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
card (Left_Cosets (G,H)) is cardinal set
a is set
c₄ is non empty strict unital Group-like associative Subgroup of G
carr c₄ is Element of bool the carrier of G
the carrier of c₄ is non empty set
y is set
a is Relation-like Function-like set
dom a is set
rng a is set
(G,(carr H)) is Element of bool (bool the carrier of G)
{ b₁ where b₁ is Element of bool the carrier of G : (G, carr H,b₁) } is set
c₄ is set
y is set
a . y is set
H1 is non empty strict unital Group-like associative Subgroup of G
carr H1 is Element of bool the carrier of G
the carrier of H1 is non empty set
c₄ is set
y is Element of bool the carrier of G
H1 is non empty strict unital Group-like associative Subgroup of G
the carrier of H1 is non empty set
carr H1 is Element of bool the carrier of G
a . H1 is set
a is non empty strict unital Group-like associative Subgroup of G
carr a is Element of bool the carrier of G
the carrier of a is non empty set
c₄ is set
a . c₄ is set
y is set
a . y is set
H1 is non empty strict unital Group-like associative Subgroup of G
carr H1 is Element of bool the carrier of G
the carrier of H1 is non empty set
a is non empty strict unital Group-like associative Subgroup of G
carr a is Element of bool the carrier of G
the carrier of a is non empty set
card (G,(carr H)) is cardinal set
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative Subgroup of G
(G,H) is Element of bool (G)
(G) is non empty set
bool (G) is non empty set
(G,H) is non empty strict unital Group-like associative Subgroup of G
carr H is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of H is non empty set
(G,(carr H)) is non empty strict unital Group-like associative Subgroup of G
Left_Cosets (G,H) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty set
index (G,H) is V12() V13() V14() V18() ext-real V37() V38() integer Element of K113()
card (G,H) is cardinal set
Index (G,H) is cardinal set
card (Left_Cosets (G,H)) is cardinal set
a is finite set
card a is V12() V13() V14() V18() ext-real V37() V38() integer Element of K46()
G is non empty strict unital Group-like associative multMagma
H is non empty strict unital Group-like associative Subgroup of G
(G,H) is non empty strict unital Group-like associative Subgroup of G
carr H is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of H is non empty set
(G,(carr H)) is non empty strict unital Group-like associative Subgroup of G
a is Element of the carrier of G
(G,H,a) is non empty strict unital Group-like associative Subgroup of G
a is Element of the carrier of G
(G,H,a) is non empty strict unital Group-like associative Subgroup of G
the multF of H is Relation-like [: the carrier of H, the carrier of H:] -defined the carrier of H -valued Function-like V28([: the carrier of H, the carrier of H:], the carrier of H) associative Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[: the carrier of H, the carrier of H:] is non empty set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty set
multMagma(# the carrier of H, the multF of H #) is strict multMagma
c₄ is Element of the carrier of G
(G,H,c₄) is non empty strict unital Group-like associative Subgroup of G
G is non empty strict unital Group-like associative multMagma
(1). G is non empty finite strict unital Group-like associative Subgroup of G
(G,((1). G)) is non empty strict unital Group-like associative Subgroup of G
carr ((1). G) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty set
the carrier of ((1). G) is non empty finite set
(G,(carr ((1). G))) is non empty strict unital Group-like associative Subgroup of G
G is non empty strict unital Group-like associative multMagma
(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 V28([: the carrier of G, the carrier of G:], the carrier of G) 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 non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
multMagma(# the carrier of G, the multF of G #) is strict multMagma
(G,((Omega). G)) is non empty strict unital Group-like associative Subgroup of G
carr ((Omega). G) is Element of bool the carrier of G
bool the carrier of G is non empty set
the carrier of ((Omega). G) is non empty set
(G,(carr ((Omega). G))) is non empty strict unital Group-like associative Subgroup of G