:: GROUP_2 semantic presentation

REAL is V46() V47() V48() V52() set
NAT is non trivial V20() V46() V47() V48() V49() V50() V51() V52() non finite cardinal limit_cardinal Element of bool REAL
bool REAL is non empty cup-closed diff-closed preBoolean set
{} is empty V20() V24() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element set
the empty V20() V24() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element set is empty V20() V24() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element set
COMPLEX is V46() V52() set
NAT is non trivial V20() V46() V47() V48() V49() V50() V51() V52() non finite cardinal limit_cardinal set
bool NAT is non empty non trivial cup-closed diff-closed preBoolean non finite set
1 is non empty V20() V24() ext-real positive V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
card {} is empty V20() V24() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element set
2 is non empty V20() V24() ext-real positive V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
union {} is finite set
0 is empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element Element of NAT
H is non empty 1-sorted
the carrier of H is non empty set
bool the carrier of H is non empty cup-closed diff-closed preBoolean set
G is set
x is Element of bool the carrier of H
G is non empty 1-sorted
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
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 cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in H } is set
y is set
y is Element of the carrier of G
y " is Element of the carrier of G
x is Element of bool the carrier of G
y is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in y } is set
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in x } is set
y is set
B is Element of the carrier of G
B " is Element of the carrier of G
(B ") " is Element of the carrier of G
y is set
B is Element of the carrier of G
B " is Element of the carrier of G
B is Element of the carrier of G
B " is Element of 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 cup-closed diff-closed preBoolean 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 cup-closed diff-closed preBoolean set
G is set
x is Element of bool the carrier of H
(H,x) is Element of bool the carrier of H
{ (b₁ ") where b₁ is Element of the carrier of H : b₁ in x } is set
y is set
B is Element of the carrier of y
B " is Element of the carrier of y
B is Element of bool the carrier of y
(y,B) is Element of bool the carrier of y
{ (b₁ ") where b₁ is Element of the carrier of y : b₁ in B } 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 trivial finite 1 -element Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(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
H " is Element of the carrier of G
{(H ")} is non empty trivial finite 1 -element Element of bool the carrier of G
x is set
y is Element of the carrier of G
y " is Element of the carrier of G
x 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
x is Element of the carrier of G
{H,x} is non empty finite Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,{H,x}) is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in {H,x} } is set
x " is Element of the carrier of G
{(H "),(x ")} is non empty finite Element of bool the carrier of G
y is set
y is Element of the carrier of G
y " is Element of the carrier of G
y is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
{} the carrier of G is empty proper V20() V24() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,({} the carrier of G)) is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in {} the carrier of G } is set
H is set
x is Element of the carrier of G
x " is Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
[#] the carrier of G is non empty non proper Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,([#] the carrier of G)) is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in [#] the carrier of G } is set
H is set
x is Element of the carrier of G
x " is Element of the carrier of G
(x ") " 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 cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
(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
the Element of (G,H) is Element of (G,H)
the Element of H is Element of H
y is Element of the carrier of G
y " is Element of the carrier of G
y 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
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is empty proper V20() V24() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element Element of bool the carrier of G
(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
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 cup-closed diff-closed preBoolean set
H is non empty Element of bool the carrier of G
(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
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H & b₂ in x ) } is set
y is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
G is non empty commutative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
y is Element of bool the carrier of G
y is Element of bool the carrier of G
(G,y,y) 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 y ) } is set
(G,y,y) 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 y ) } is set
B is set
B is Element of the carrier of G
a 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) 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{B,a},{B}} is non empty finite V57() set
the multF of G . [B,a] is set
B is set
B is Element of the carrier of G
a 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) 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{B,a},{B}} is non empty finite V57() set
the multF of G . [B,a] is set
H is non empty multMagma
the carrier of H is non empty set
bool the carrier of H is non empty cup-closed diff-closed preBoolean set
B is non empty multMagma
the carrier of B is non empty set
bool the carrier of B is non empty cup-closed diff-closed preBoolean set
G is set
x is Element of bool the carrier of H
y is Element of bool the carrier of H
(H,x,y) is Element of bool the carrier of H
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in x & b₂ in y ) } is set
y is set
Y is Element of the carrier of B
y is Element of the carrier of B
Y * y is Element of the carrier of B
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V28([: the carrier of B, the carrier of B:], the carrier of B) Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty cup-closed diff-closed preBoolean set
the multF of B . (Y,y) is Element of the carrier of B
[Y,y] is set
{Y,y} is non empty finite set
{Y} is non empty trivial finite 1 -element set
{{Y,y},{Y}} is non empty finite V57() set
the multF of B . [Y,y] is set
B is Element of bool the carrier of B
a is Element of bool the carrier of B
(B,B,a) is Element of bool the carrier of B
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of B : ( b₁ in B & b₂ in a ) } is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of bool the carrier of G
(G,H,x) 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 x ) } is set
the Element of H is Element of H
the Element of x is Element of x
y is Element of the carrier of G
y is Element of the carrier of G
y * 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) 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 cup-closed diff-closed preBoolean set
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} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of G . [y,y] is set
the Element of (G,H,x) is Element of (G,H,x)
y is Element of the carrier of G
B is Element of the carrier of G
y * 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) 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 cup-closed diff-closed preBoolean set
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of bool the carrier of G
(G,H,x) 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 x ) } is set
y is Element of bool the carrier of G
(G,(G,H,x),y) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,H,x) & b₂ in y ) } is set
(G,x,y) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in x & b₂ in y ) } is set
(G,H,(G,x,y)) 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 (G,x,y) ) } is set
y is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
a is Element of the carrier of G
Y 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 trivial finite 1 -element set
{{a,Y},{a}} is non empty finite V57() set
the multF of G . [a,Y] is set
Y * B is Element of the carrier of G
the multF of G . (Y,B) is Element of the carrier of G
[Y,B] is set
{Y,B} is non empty finite set
{Y} is non empty trivial finite 1 -element set
{{Y,B},{Y}} is non empty finite V57() set
the multF of G . [Y,B] is set
a * (Y * B) is Element of the carrier of G
the multF of G . (a,(Y * B)) is Element of the carrier of G
[a,(Y * B)] is set
{a,(Y * B)} is non empty finite set
{{a,(Y * B)},{a}} is non empty finite V57() set
the multF of G . [a,(Y * B)] is set
y is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
a is Element of the carrier of G
Y 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 trivial finite 1 -element set
{{a,Y},{a}} is non empty finite V57() set
the multF of G . [a,Y] is set
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,a},{B}} is non empty finite V57() set
the multF of G . [B,a] is set
(B * a) * Y is Element of the carrier of G
the multF of G . ((B * a),Y) is Element of the carrier of G
[(B * a),Y] is set
{(B * a),Y} is non empty finite set
{(B * a)} is non empty trivial finite 1 -element set
{{(B * a),Y},{(B * a)}} is non empty finite V57() set
the multF of G . [(B * 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 cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of bool the carrier of G
(G,H,x) 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 x ) } is set
(G,(G,H,x)) is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in (G,H,x) } is set
(G,x) is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in x } is set
(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
(G,(G,x),(G,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in (G,H) ) } is set
y is set
y is Element of the carrier of G
y " is Element of the carrier of G
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
B " is Element of the carrier of G
B " is Element of the carrier of G
(B ") * (B ") is Element of the carrier of G
the multF of G . ((B "),(B ")) is Element of the carrier of G
[(B "),(B ")] is set
{(B "),(B ")} is non empty finite set
{(B ")} is non empty trivial finite 1 -element set
{{(B "),(B ")},{(B ")}} is non empty finite V57() set
the multF of G . [(B "),(B ")] is set
y is set
y is Element of the carrier of G
B is Element of the carrier of G
y * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] is set
B is Element of the carrier of G
B " is Element of the carrier of G
a is Element of the carrier of G
a " 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 trivial finite 1 -element set
{{B,a},{B}} is non empty finite V57() set
the multF of G . [B,a] is set
(B * a) " is Element of the carrier of G
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of bool the carrier of G
(G,H,x) 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 x ) } is set
y is Element of bool the carrier of G
x \/ y is Element of bool the carrier of G
(G,H,(x \/ y)) 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 x \/ y ) } is set
(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 H & b₂ in y ) } is set
(G,H,x) \/ (G,H,y) is Element of bool the carrier of G
y is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
y is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of bool the carrier of G
H \/ x is Element of bool the carrier of G
y is Element of bool the carrier of G
(G,(H \/ x),y) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H \/ x & b₂ in y ) } is set
(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 H & b₂ in y ) } is set
(G,x,y) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in x & b₂ in y ) } is set
(G,H,y) \/ (G,x,y) is Element of bool the carrier of G
y is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
y is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of bool the carrier of G
(G,H,x) 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 x ) } is set
y is Element of bool the carrier of G
x /\ y is Element of bool the carrier of G
(G,H,(x /\ y)) 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 x /\ y ) } is set
(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 H & b₂ in y ) } is set
(G,H,x) /\ (G,H,y) is Element of bool the carrier of G
y is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of bool the carrier of G
H /\ x is Element of bool the carrier of G
y is Element of bool the carrier of G
(G,(H /\ x),y) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in H /\ x & b₂ in y ) } is set
(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 H & b₂ in y ) } is set
(G,x,y) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in x & b₂ in y ) } is set
(G,H,y) /\ (G,x,y) is Element of bool the carrier of G
y is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
{} the carrier of G is empty proper V20() V24() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element Element of bool the carrier of G
H is Element of bool the carrier of G
(G,({} the carrier of G),H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {} the carrier of G & b₂ in H ) } is set
(G,H,({} the carrier of G)) 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 {} the carrier of G ) } is set
the Element of (G,H,({} the carrier of G)) is Element of (G,H,({} the carrier of G))
y is Element of the carrier of G
y is Element of the carrier of G
y * 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) 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 cup-closed diff-closed preBoolean set
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} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of G . [y,y] is set
the Element of (G,({} the carrier of G),H) is Element of (G,({} the carrier of G),H)
y is Element of the carrier of G
y is Element of the carrier of G
y * 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) 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 cup-closed diff-closed preBoolean set
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} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of G . [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 cup-closed diff-closed preBoolean set
[#] the carrier of G is non empty non proper Element of bool the carrier of G
H is Element of bool the carrier of G
(G,([#] the carrier of G),H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in [#] the carrier of G & b₂ in H ) } is set
(G,H,([#] the carrier of G)) 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 [#] the carrier of G ) } is set
the Element of H is Element of H
B is set
a is Element of the carrier of G
B 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{a,(B ")},{a}} is non empty finite V57() set
the multF of G . [a,(B ")] is set
(a * (B ")) * B is Element of the carrier of G
the multF of G . ((a * (B ")),B) is Element of the carrier of G
[(a * (B ")),B] is set
{(a * (B ")),B} is non empty finite set
{(a * (B "))} is non empty trivial finite 1 -element set
{{(a * (B ")),B},{(a * (B "))}} is non empty finite V57() set
the multF of G . [(a * (B ")),B] is set
(B ") * B is Element of the carrier of G
the multF of G . ((B "),B) is Element of the carrier of G
[(B "),B] is set
{(B "),B} is non empty finite set
{(B ")} is non empty trivial finite 1 -element set
{{(B "),B},{(B ")}} is non empty finite V57() set
the multF of G . [(B "),B] is set
a * ((B ") * B) is Element of the carrier of G
the multF of G . (a,((B ") * B)) is Element of the carrier of G
[a,((B ") * B)] is set
{a,((B ") * B)} is non empty finite set
{{a,((B ") * B)},{a}} is non empty finite V57() set
the multF of G . [a,((B ") * B)] is set
1_ G is non being_of_order_0 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 V57() set
the multF of G . [a,(1_ G)] is set
y is set
y is Element of the carrier of G
y " is Element of the carrier of G
B is Element of the carrier of G
(y ") * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . ((y "),B) is Element of the carrier of G
[(y "),B] is set
{(y "),B} is non empty finite set
{(y ")} is non empty trivial finite 1 -element set
{{(y "),B},{(y ")}} is non empty finite V57() set
the multF of G . [(y "),B] is set
y * ((y ") * B) is Element of the carrier of G
the multF of G . (y,((y ") * B)) is Element of the carrier of G
[y,((y ") * B)] is set
{y,((y ") * B)} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,((y ") * B)},{y}} is non empty finite V57() set
the multF of G . [y,((y ") * B)] is set
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 V57() set
the multF of G . [y,(y ")] is set
(y * (y ")) * B is Element of the carrier of G
the multF of G . ((y * (y ")),B) is Element of the carrier of G
[(y * (y ")),B] is set
{(y * (y ")),B} is non empty finite set
{(y * (y "))} is non empty trivial finite 1 -element set
{{(y * (y ")),B},{(y * (y "))}} is non empty finite V57() set
the multF of G . [(y * (y ")),B] is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * B is Element of the carrier of G
the multF of G . ((1_ G),B) is Element of the carrier of G
[(1_ G),B] is set
{(1_ G),B} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),B},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),B] is set
G is non empty multMagma
the carrier of G is non empty set
H is Element of the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
x is Element of the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},{x}) 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 {x} ) } is set
H * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (H,x) is Element of the carrier of G
[H,x] is set
{H,x} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,x},{H}} is non empty finite V57() set
the multF of G . [H,x] is set
{(H * x)} is non empty trivial finite 1 -element Element of bool the carrier of G
y is set
y is Element of the carrier of G
B is Element of the carrier of G
y * B is Element of the carrier of G
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] is set
y is set
G is non empty multMagma
the carrier of G is non empty set
H is Element of the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
x is Element of the carrier of G
H * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) Element of bool [:[: the carrier of G, the carrier of G:], the carrier of G:]
[: the carrier of G, the carrier of G:] is non empty set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G . (H,x) is Element of the carrier of G
[H,x] is set
{H,x} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,x},{H}} is non empty finite V57() set
the multF of G . [H,x] is set
y is Element of the carrier of G
{x,y} is non empty finite Element of bool the carrier of G
(G,{H},{x,y}) 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 {x,y} ) } is set
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 V57() set
the multF of G . [H,y] is set
{(H * x),(H * y)} is non empty finite Element of bool the carrier of G
y is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
y is set
G is non empty multMagma
the carrier of G is non empty set
H is Element of the carrier of G
x is Element of the carrier of G
{H,x} is non empty finite Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
y is Element of the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H,x},{y}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H,x} & b₂ in {y} ) } is set
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) 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{H,y},{H}} is non empty finite V57() set
the multF of G . [H,y] is set
x * y is Element of the carrier of G
the multF of G . (x,y) is Element of the carrier of G
[x,y] is set
{x,y} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,y},{x}} is non empty finite V57() set
the multF of G . [x,y] is set
{(H * y),(x * y)} is non empty finite Element of bool the carrier of G
y is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
y is set
G is non empty multMagma
the carrier of G is non empty set
H is Element of the carrier of G
x is Element of the carrier of G
{H,x} is non empty finite Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
y 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) 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{H,y},{H}} is non empty finite V57() set
the multF of G . [H,y] is set
x * y is Element of the carrier of G
the multF of G . (x,y) is Element of the carrier of G
[x,y] is set
{x,y} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,y},{x}} is non empty finite V57() set
the multF of G . [x,y] is set
y is Element of the carrier of G
{y,y} is non empty finite Element of bool the carrier of G
(G,{H,x},{y,y}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {H,x} & b₂ in {y,y} ) } is set
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 V57() set
the multF of G . [H,y] is set
x * y is Element of the carrier of G
the multF of G . (x,y) is Element of the carrier of G
[x,y] is set
{x,y} is non empty finite set
{{x,y},{x}} is non empty finite V57() set
the multF of G . [x,y] is set
{(H * y),(H * y),(x * y),(x * y)} is non empty finite Element of bool the carrier of G
a is set
Y is Element of the carrier of G
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} is non empty trivial finite 1 -element set
{{Y,y},{Y}} is non empty finite V57() set
the multF of G . [Y,y] is set
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 cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
(G,H,H) 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 ) } is set
x is set
y is Element of the carrier of G
y is Element of the carrier of G
y * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of G . [y,y] is set
x is set
y is Element of the carrier of G
y " is Element of the carrier of G
(y ") * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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 ")} is non empty trivial finite 1 -element set
{{(y "),y},{(y ")}} is non empty finite V57() set
the multF of G . [(y "),y] is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * y is Element of the carrier of G
the multF of G . ((1_ G),y) is Element of the carrier of G
[(1_ G),y] is set
{(1_ G),y} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),y},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),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 cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
(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
x is set
y is Element of the carrier of G
y " is Element of the carrier of G
x is set
y is Element of the carrier of G
y " is Element of the carrier of G
(y ") " is Element of the carrier of G
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of bool the carrier of G
(G,H,x) 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 x ) } is set
(G,x,H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in x & b₂ in H ) } is set
y is set
y is Element of the carrier of G
B is Element of the carrier of G
y * 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) 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 cup-closed diff-closed preBoolean set
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] 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 trivial finite 1 -element set
{{B,y},{B}} is non empty finite V57() set
the multF of G . [B,y] is set
y is set
y is Element of the carrier of G
B is Element of the carrier of G
y * 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) 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 cup-closed diff-closed preBoolean set
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] 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 trivial finite 1 -element set
{{B,y},{B}} is non empty finite V57() set
the multF of G . [B,y] 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
x is Element of the carrier of G
H * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (H,x) is Element of the carrier of G
[H,x] is set
{H,x} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,x},{H}} is non empty finite V57() set
the multF of G . [H,x] is set
x * H is Element of the carrier of G
the multF of G . (x,H) is Element of the carrier of G
[x,H] is set
{x,H} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,H},{x}} is non empty finite V57() set
the multF of G . [x,H] is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of bool the carrier of G
(G,H,x) 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 x ) } is set
(G,x,H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in x & b₂ in H ) } is set
y is set
y is Element of the carrier of G
B is Element of the carrier of G
y * 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) 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 cup-closed diff-closed preBoolean set
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] 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 trivial finite 1 -element set
{{B,y},{B}} is non empty finite V57() set
the multF of G . [B,y] is set
y is set
y is Element of the carrier of G
B is Element of the carrier of G
y * 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) 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 cup-closed diff-closed preBoolean set
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] 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 trivial finite 1 -element set
{{B,y},{B}} is non empty finite V57() set
the multF of G . [B,y] is set
G is non empty unital Group-like associative commutative multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of bool the carrier of G
(G,H,x) 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 x ) } is set
(G,(G,H,x)) is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in (G,H,x) } is set
(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
(G,x) is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in x } is set
(G,(G,H),(G,x)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,H) & b₂ in (G,x) ) } is set
y is set
y is Element of the carrier of G
y " is Element of the carrier of G
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
B " is Element of the carrier of G
B " is Element of the carrier of G
(B ") * (B ") is Element of the carrier of G
the multF of G . ((B "),(B ")) is Element of the carrier of G
[(B "),(B ")] is set
{(B "),(B ")} is non empty finite set
{(B ")} is non empty trivial finite 1 -element set
{{(B "),(B ")},{(B ")}} is non empty finite V57() set
the multF of G . [(B "),(B ")] is set
y is set
y is Element of the carrier of G
B is Element of the carrier of G
y * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] is set
B is Element of the carrier of G
B " is Element of the carrier of G
a is Element of the carrier of G
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} is non empty trivial finite 1 -element set
{{a,B},{a}} is non empty finite V57() set
the multF of G . [a,B] is set
(a * B) " is Element of the carrier of G
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
x is Element of bool the carrier of G
(G,{H},x) 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 x ) } is set
(G,x,{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in x & b₂ in {H} ) } is set
G is set
H is non empty multMagma
the carrier of H is non empty set
bool the carrier of H is non empty cup-closed diff-closed preBoolean set
x is Element of bool the carrier of H
y is Element of the carrier of H
(H,y,x) is Element of bool the carrier of H
{y} is non empty trivial finite 1 -element Element of bool the carrier of H
(H,{y},x) is Element of bool the carrier of H
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in {y} & b₂ in x ) } is set
y is Element of the carrier of H
B is Element of the carrier of H
y * B 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) 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 cup-closed diff-closed preBoolean set
the multF of H . (y,B) is Element of the carrier of H
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of H . [y,B] is set
y is Element of the carrier of H
y * 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) 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 cup-closed diff-closed preBoolean set
the multF of H . (y,y) is Element of the carrier of H
[y,y] is set
{y,y} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of H . [y,y] is set
G is set
H is non empty multMagma
the carrier of H is non empty set
bool the carrier of H is non empty cup-closed diff-closed preBoolean set
x is Element of bool the carrier of H
y is Element of the carrier of H
(H,y,x) is Element of bool the carrier of H
{y} is non empty trivial finite 1 -element Element of bool the carrier of H
(H,x,{y}) is Element of bool the carrier of H
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in x & b₂ in {y} ) } is set
y is Element of the carrier of H
B is Element of the carrier of H
y * B 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) 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 cup-closed diff-closed preBoolean set
the multF of H . (y,B) is Element of the carrier of H
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of H . [y,B] is set
y is Element of the carrier of H
y * 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) 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 cup-closed diff-closed preBoolean set
the multF of H . (y,y) is Element of the carrier of H
[y,y] is set
{y,y} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of H . [y,y] is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
y is Element of the carrier of G
H is Element of bool the carrier of G
(G,y,H) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{y},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 H ) } is set
x is Element of bool the carrier of G
(G,(G,y,H),x) 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 x ) } is set
(G,H,x) 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 x ) } is set
(G,y,(G,H,x)) is Element of bool the carrier of G
(G,{y},(G,H,x)) 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 (G,H,x) ) } is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
y is Element of the carrier of G
H is Element of bool the carrier of G
(G,y,H) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(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 H & b₂ in {y} ) } is set
x is Element of bool the carrier of G
(G,(G,y,H),x) 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 x ) } is set
(G,y,x) is Element of bool the carrier of G
(G,{y},x) 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 x ) } is set
(G,H,(G,y,x)) 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 (G,y,x) ) } is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
y is Element of the carrier of G
H is Element of bool the carrier of G
x is Element of bool the carrier of G
(G,H,x) 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 x ) } is set
(G,y,(G,H,x)) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,H,x),{y}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,H,x) & b₂ in {y} ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,{y}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in x & b₂ in {y} ) } is set
(G,H,(G,y,x)) 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 (G,y,x) ) } is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of the carrier of G
y is Element of the carrier of G
x * 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) 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 cup-closed diff-closed preBoolean set
the multF of G . (x,y) is Element of the carrier of G
[x,y] is set
{x,y} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,y},{x}} is non empty finite V57() set
the multF of G . [x,y] is set
(G,(x * y),H) is Element of bool the carrier of G
{(x * y)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(x * y)},H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(x * y)} & b₂ in H ) } is set
(G,y,H) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{y},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 H ) } is set
(G,x,(G,y,H)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{x},(G,y,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in (G,y,H) ) } is set
(G,{x},{y}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in {y} ) } is set
(G,(G,{x},{y}),H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,{x},{y}) & b₂ in H ) } is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
y is Element of the carrier of G
x is Element of the carrier of G
H is Element of bool the carrier of G
(G,x,H) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{x},H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in H ) } is set
(G,y,(G,x,H)) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x,H),{y}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,H) & b₂ in {y} ) } is set
(G,y,H) is Element of bool the carrier of G
(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 H & b₂ in {y} ) } is set
(G,x,(G,y,H)) is Element of bool the carrier of G
(G,{x},(G,y,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in (G,y,H) ) } is set
G is non empty multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of bool the carrier of G
x is Element of the carrier of G
(G,x,H) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,H,{x}) 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 {x} ) } is set
y is Element of the carrier of G
(G,y,(G,x,H)) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x,H),{y}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,H) & b₂ in {y} ) } is set
x * 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) 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 cup-closed diff-closed preBoolean set
the multF of G . (x,y) is Element of the carrier of G
[x,y] is set
{x,y} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,y},{x}} is non empty finite V57() set
the multF of G . [x,y] is set
(G,(x * y),H) is Element of bool the carrier of G
{(x * y)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,H,{(x * y)}) 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 {(x * y)} ) } is set
(G,{x},{y}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in {y} ) } is set
(G,H,(G,{x},{y})) 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 (G,{x},{y}) ) } is set
G is non empty multMagma
the carrier of G is non empty set
x is non empty multMagma
the carrier of x is non empty set
H is Element of the carrier of G
{} the carrier of G is empty proper V20() V24() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,H,({} the carrier of G)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,({} the carrier of G),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {} the carrier of G & b₂ in {H} ) } is set
y is Element of the carrier of x
{} the carrier of x is empty proper V20() V24() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element Element of bool the carrier of x
bool the carrier of x is non empty cup-closed diff-closed preBoolean set
(x,y,({} the carrier of x)) is Element of bool the carrier of x
{y} is non empty trivial finite 1 -element Element of bool the carrier of x
(x,{y},({} the carrier of x)) is Element of bool the carrier of x
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of x : ( b₁ in {y} & b₂ in {} the carrier of x ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
x is non empty unital Group-like associative multMagma
the carrier of x is non empty set
H is Element of the carrier of G
[#] the carrier of G is non empty non proper Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,H,([#] the carrier of G)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,([#] the carrier of G),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in [#] the carrier of G & b₂ in {H} ) } is set
y is Element of the carrier of x
[#] the carrier of x is non empty non proper Element of bool the carrier of x
bool the carrier of x is non empty cup-closed diff-closed preBoolean set
(x,y,([#] the carrier of x)) is Element of bool the carrier of x
{y} is non empty trivial finite 1 -element Element of bool the carrier of x
(x,{y},([#] the carrier of x)) is Element of bool the carrier of x
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of x : ( b₁ in {y} & b₂ in [#] the carrier of x ) } is set
G is non empty unital Group-like multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
1_ G is Element of the carrier of G
H is Element of bool the carrier of G
(G,(1_ G),H) is Element of bool the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
(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,(1_ G),H) is Element of bool the carrier of G
(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 H & b₂ in {(1_ G)} ) } is set
x is set
y is Element of the carrier of G
(1_ G) * 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) having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . ((1_ G),y) is Element of the carrier of G
[(1_ G),y] is set
{(1_ G),y} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),y},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),y] is set
x is set
y is Element of the carrier of G
(1_ G) * 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) having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . ((1_ G),y) is Element of the carrier of G
[(1_ G),y] is set
{(1_ G),y} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),y},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),y] is set
x is set
y is Element of the carrier of G
y * (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) having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (y,(1_ G)) is Element of the carrier of G
[y,(1_ G)] is set
{y,(1_ G)} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,(1_ G)},{y}} is non empty finite V57() set
the multF of G . [y,(1_ G)] is set
x is set
y is Element of the carrier of G
y * (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) having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (y,(1_ G)) is Element of the carrier of G
[y,(1_ G)] is set
{y,(1_ G)} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,(1_ G)},{y}} is non empty finite V57() set
the multF of G . [y,(1_ G)] is set
G is non empty unital Group-like multMagma
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of the carrier of G
x is Element of bool the carrier of G
(G,H,x) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},x) 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 x ) } is set
(G,H,x) is Element of bool the carrier of G
(G,x,{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in x & b₂ in {H} ) } is set
G is non empty unital Group-like 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) having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G || the carrier of G is Relation-like Function-like set
the multF of G | [: the carrier of G, the carrier of G:] is Relation-like set
dom the multF of G is Relation-like the carrier of G -defined the carrier of G -valued Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty cup-closed diff-closed preBoolean set
G is non empty unital Group-like multMagma
H is non empty unital Group-like (G)
the carrier of H is non empty set
the carrier of G is non empty set
G is set
H is non empty unital Group-like multMagma
x is non empty unital Group-like (H)
the carrier of x is non empty set
the carrier of H is non empty set
G is non empty unital Group-like multMagma
H is non empty unital Group-like (G)
the carrier of H is non empty set
x is Element of the carrier of H
G is non empty unital Group-like multMagma
the carrier of G is non empty set
H is non empty unital Group-like (G)
the carrier of H is non empty set
x is Element of the carrier of H
G is non empty unital Group-like multMagma
the carrier of G is non empty set
H is Element of the carrier of G
x is Element of the carrier of G
H * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (H,x) is Element of the carrier of G
[H,x] is set
{H,x} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,x},{H}} is non empty finite V57() set
the multF of G . [H,x] is set
y is non empty unital Group-like (G)
the carrier of y is non empty set
y is Element of the carrier of y
B is Element of the carrier of y
y * B is Element of the carrier of y
the multF of y is Relation-like [: the carrier of y, the carrier of y:] -defined the carrier of y -valued Function-like V28([: the carrier of y, the carrier of y:], the carrier of y) having_a_unity Element of bool [:[: the carrier of y, the carrier of y:], the carrier of y:]
[: the carrier of y, the carrier of y:] is non empty set
[:[: the carrier of y, the carrier of y:], the carrier of y:] is non empty set
bool [:[: the carrier of y, the carrier of y:], the carrier of y:] is non empty cup-closed diff-closed preBoolean set
the multF of y . (y,B) is Element of the carrier of y
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of y . [y,B] is set
the multF of G || the carrier of y is Relation-like Function-like set
the multF of G | [: the carrier of y, the carrier of y:] is Relation-like set
[y,B] is Element of [: the carrier of y, the carrier of y:]
( the multF of G || the carrier of y) . [y,B] is set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like (G)
the carrier of H is non empty set
x is Element of the carrier of H
y is Element of the carrier of H
x * 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) having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of H . (x,y) is Element of the carrier of H
[x,y] is set
{x,y} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,y},{x}} is non empty finite V57() set
the multF of H . [x,y] is set
y is Element of the carrier of H
(x * y) * y is Element of the carrier of H
the multF of H . ((x * y),y) is Element of the carrier of H
[(x * y),y] is set
{(x * y),y} is non empty finite set
{(x * y)} is non empty trivial finite 1 -element set
{{(x * y),y},{(x * y)}} is non empty finite V57() set
the multF of H . [(x * y),y] is set
y * y is Element of the carrier of H
the multF of H . (y,y) is Element of the carrier of H
[y,y] is set
{y,y} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of H . [y,y] is set
x * (y * y) is Element of the carrier of H
the multF of H . (x,(y * y)) is Element of the carrier of H
[x,(y * y)] is set
{x,(y * y)} is non empty finite set
{{x,(y * y)},{x}} is non empty finite V57() set
the multF of H . [x,(y * y)] is set
the carrier of G is non empty set
Y is Element of the carrier of G
a 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{Y,a},{Y}} is non empty finite V57() set
the multF of G . [Y,a] is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
(B * B) * a is Element of the carrier of G
the multF of G . ((B * B),a) is Element of the carrier of G
[(B * B),a] is set
{(B * B),a} is non empty finite set
{(B * B)} is non empty trivial finite 1 -element set
{{(B * B),a},{(B * B)}} is non empty finite V57() set
the multF of G . [(B * B),a] is set
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 trivial finite 1 -element set
{{B,a},{B}} is non empty finite V57() set
the multF of G . [B,a] is set
B * (B * a) is Element of the carrier of G
the multF of G . (B,(B * a)) is Element of the carrier of G
[B,(B * a)] is set
{B,(B * a)} is non empty finite set
{{B,(B * a)},{B}} is non empty finite V57() set
the multF of G . [B,(B * a)] is set
y is Element of the carrier of G
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,y},{B}} is non empty finite V57() set
the multF of G . [B,y] is 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 unital Group-like associative (G)
1_ H is non being_of_order_0 Element of the carrier of H
the carrier of H is non empty set
the Element of the carrier of H is Element of the carrier of H
the Element of the carrier of H * (1_ H) 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of H . ( the Element of the carrier of H,(1_ H)) is Element of the carrier of H
[ the Element of the carrier of H,(1_ H)] is set
{ the Element of the carrier of H,(1_ H)} is non empty finite set
{ the Element of the carrier of H} is non empty trivial finite 1 -element set
{{ the Element of the carrier of H,(1_ H)},{ the Element of the carrier of H}} is non empty finite V57() set
the multF of H . [ the Element of the carrier of H,(1_ H)] is set
y is Element of the carrier of G
y is Element of the carrier of G
y * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of G . [y,y] is set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
1_ H is non being_of_order_0 Element of the carrier of H
the carrier of H is non empty set
x is non empty unital Group-like associative (G)
1_ x is non being_of_order_0 Element of the carrier of x
the carrier of x is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G 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 unital Group-like associative (G)
1_ H is non being_of_order_0 Element of the carrier of H
the carrier of H is non empty set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
1_ H is non being_of_order_0 Element of the carrier of H
the carrier of H is non empty set
x is non empty unital Group-like associative (G)
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
H is Element of the carrier of G
H " is Element of the carrier of G
x is non empty unital Group-like associative (G)
the carrier of x is non empty set
y is Element of the carrier of x
y " is Element of the carrier of x
y * (y ") is Element of the carrier of x
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V28([: the carrier of x, the carrier of x:], the carrier of x) associative having_a_unity Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty cup-closed diff-closed preBoolean set
the multF of x . (y,(y ")) is Element of the carrier of x
[y,(y ")] is set
{y,(y ")} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,(y ")},{y}} is non empty finite V57() set
the multF of x . [y,(y ")] is set
1_ x is non being_of_order_0 Element of the carrier of x
y 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{H,y},{H}} is non empty finite V57() set
the multF of G . [H,y] is set
1_ G is non being_of_order_0 Element of the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
inverse_op G 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:]
[: the carrier of G, the carrier of G:] is non empty set
bool [: the carrier of G, the carrier of G:] is non empty cup-closed diff-closed preBoolean set
H is non empty unital Group-like associative (G)
inverse_op H is Relation-like the carrier of H -defined the carrier of H -valued Function-like V28( the carrier of H, the carrier of H) Element of bool [: the carrier of H, the carrier of H:]
the carrier of H is non empty set
[: the carrier of H, the carrier of H:] is non empty set
bool [: the carrier of H, the carrier of H:] is non empty cup-closed diff-closed preBoolean set
(inverse_op G) | the carrier of H is Relation-like the carrier of G -defined the carrier of G -valued Element of bool [: the carrier of G, the carrier of G:]
the carrier of G /\ the carrier of H is set
x is set
dom (inverse_op H) is Element of bool the carrier of H
bool the carrier of H is non empty cup-closed diff-closed preBoolean set
y is Element of the carrier of H
(inverse_op H) . x is set
y " is Element of the carrier of H
y is Element of the carrier of G
y " is Element of the carrier of G
(inverse_op G) . x is set
dom (inverse_op G) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean 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
x is Element of the carrier of G
H * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (H,x) is Element of the carrier of G
[H,x] is set
{H,x} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,x},{H}} is non empty finite V57() set
the multF of G . [H,x] is set
y is non empty unital Group-like associative (G)
the carrier of y is non empty set
y is Element of the carrier of y
B is Element of the carrier of y
y * B is Element of the carrier of y
the multF of y is Relation-like [: the carrier of y, the carrier of y:] -defined the carrier of y -valued Function-like V28([: the carrier of y, the carrier of y:], the carrier of y) associative having_a_unity Element of bool [:[: the carrier of y, the carrier of y:], the carrier of y:]
[: the carrier of y, the carrier of y:] is non empty set
[:[: the carrier of y, the carrier of y:], the carrier of y:] is non empty set
bool [:[: the carrier of y, the carrier of y:], the carrier of y:] is non empty cup-closed diff-closed preBoolean set
the multF of y . (y,B) is Element of the carrier of y
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of y . [y,B] 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
x is non empty unital Group-like associative (G)
the carrier of x is non empty set
y is Element of the carrier of x
y " is Element of the carrier of x
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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
the carrier of multMagma(# the carrier of G, the multF of G #) is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
x is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
y is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
y * x is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
the multF of multMagma(# the carrier of G, the multF of G #) is Relation-like [: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):] -defined the carrier of multMagma(# the carrier of G, the multF of G #) -valued Function-like V28([: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):], the carrier of multMagma(# the carrier of G, the multF of G #)) Element of bool [:[: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):], the carrier of multMagma(# the carrier of G, the multF of G #):]
[: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):] is non empty set
[:[: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):], the carrier of multMagma(# the carrier of G, the multF of G #):] is non empty set
bool [:[: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):], the carrier of multMagma(# the carrier of G, the multF of G #):] is non empty cup-closed diff-closed preBoolean set
the multF of multMagma(# the carrier of G, the multF of G #) . (y,x) is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
[y,x] is set
{y,x} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,x},{y}} is non empty finite V57() set
the multF of multMagma(# the carrier of G, the multF of G #) . [y,x] is set
x * y is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
the multF of multMagma(# the carrier of G, the multF of G #) . (x,y) is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
[x,y] is set
{x,y} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,y},{x}} is non empty finite V57() set
the multF of multMagma(# the carrier of G, the multF of G #) . [x,y] is set
y is Element of the carrier of G
y * (1_ G) is Element of the carrier of G
the multF of G . (y,(1_ G)) is Element of the carrier of G
[y,(1_ G)] is set
{y,(1_ G)} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,(1_ G)},{y}} is non empty finite V57() set
the multF of G . [y,(1_ G)] is set
(1_ G) * y is Element of the carrier of G
the multF of G . ((1_ G),y) is Element of the carrier of G
[(1_ G),y] is set
{(1_ G),y} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),y},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),y] is set
y " is Element of the carrier of G
B is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
y * B is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
the multF of multMagma(# the carrier of G, the multF of G #) . (y,B) is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
[y,B] is set
{y,B} is non empty finite set
{{y,B},{y}} is non empty finite V57() set
the multF of multMagma(# the carrier of G, the multF of G #) . [y,B] is set
B * y is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
the multF of multMagma(# the carrier of G, the multF of G #) . (B,y) is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
[B,y] is set
{B,y} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,y},{B}} is non empty finite V57() set
the multF of multMagma(# the carrier of G, the multF of G #) . [B,y] is set
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 V57() set
the multF of G . [y,(y ")] is set
(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 ")} is non empty trivial finite 1 -element set
{{(y "),y},{(y ")}} is non empty finite V57() set
the multF of G . [(y "),y] is set
x is non empty unital Group-like multMagma
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V28([: the carrier of x, the carrier of x:], the carrier of x) having_a_unity Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty cup-closed diff-closed preBoolean set
the multF of G || the carrier of x is Relation-like Function-like set
the multF of G | [: the carrier of x, the carrier of x:] is Relation-like 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 cup-closed diff-closed preBoolean set
H is Element of bool 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G || H is Relation-like Function-like set
[:H,H:] is set
the multF of G | [:H,H:] is Relation-like set
dom ( the multF of G || H) is set
bool [: the carrier of G, the carrier of G:] is non empty cup-closed diff-closed preBoolean set
dom the multF of G is Relation-like the carrier of G -defined the carrier of G -valued Element of bool [: the carrier of G, the carrier of G:]
[:H,H:] is Relation-like the carrier of G -defined the carrier of G -valued Element of bool [: the carrier of G, the carrier of G:]
(dom the multF of G) /\ [:H,H:] is Relation-like the carrier of G -defined the carrier of G -valued Element of bool [: the carrier of G, the carrier of G:]
rng ( the multF of G || H) is set
y is set
B is set
( the multF of G || H) . B is set
B is set
a is set
[B,a] is set
{B,a} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,a},{B}} is non empty finite V57() set
Y is Element of the carrier of G
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} is non empty trivial finite 1 -element set
{{Y,y},{Y}} is non empty finite V57() set
the multF of G . [Y,y] is set
x is non empty set
[:x,x:] is non empty set
[:[:x,x:],x:] is non empty set
bool [:[:x,x:],x:] is non empty cup-closed diff-closed preBoolean set
y is Relation-like [:x,x:] -defined x -valued Function-like V28([:x,x:],x) Element of bool [:[:x,x:],x:]
multMagma(# x,y #) is non empty strict multMagma
the carrier of multMagma(# x,y #) is non empty set
Y is Element of the carrier of multMagma(# x,y #)
y is Element of the carrier of multMagma(# x,y #)
Y * y is Element of the carrier of multMagma(# x,y #)
the multF of multMagma(# x,y #) is Relation-like [: the carrier of multMagma(# x,y #), the carrier of multMagma(# x,y #):] -defined the carrier of multMagma(# x,y #) -valued Function-like V28([: the carrier of multMagma(# x,y #), the carrier of multMagma(# x,y #):], the carrier of multMagma(# x,y #)) Element of bool [:[: the carrier of multMagma(# x,y #), the carrier of multMagma(# x,y #):], the carrier of multMagma(# x,y #):]
[: the carrier of multMagma(# x,y #), the carrier of multMagma(# x,y #):] is non empty set
[:[: the carrier of multMagma(# x,y #), the carrier of multMagma(# x,y #):], the carrier of multMagma(# x,y #):] is non empty set
bool [:[: the carrier of multMagma(# x,y #), the carrier of multMagma(# x,y #):], the carrier of multMagma(# x,y #):] is non empty cup-closed diff-closed preBoolean set
the multF of multMagma(# x,y #) . (Y,y) is Element of the carrier of multMagma(# x,y #)
[Y,y] is set
{Y,y} is non empty finite set
{Y} is non empty trivial finite 1 -element set
{{Y,y},{Y}} is non empty finite V57() set
the multF of multMagma(# x,y #) . [Y,y] is set
[Y,y] is Element of [: the carrier of multMagma(# x,y #), the carrier of multMagma(# x,y #):]
( the multF of G || H) . [Y,y] is set
B is Element of the carrier of G
a 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 trivial finite 1 -element set
{{B,a},{B}} is non empty finite V57() set
the multF of G . [B,a] is set
the Element of the carrier of multMagma(# x,y #) is Element of the carrier of multMagma(# x,y #)
a is Element of the carrier of G
a " is Element of 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 trivial finite 1 -element set
{{a,(a ")},{a}} is non empty finite V57() set
the multF of G . [a,(a ")] is set
1_ G is non being_of_order_0 Element of the carrier of G
Y is Element of the carrier of multMagma(# x,y #)
y is Element of the carrier of multMagma(# x,y #)
y * Y is Element of the carrier of multMagma(# x,y #)
the multF of multMagma(# x,y #) . (y,Y) is Element of the carrier of multMagma(# x,y #)
[y,Y] is set
{y,Y} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,Y},{y}} is non empty finite V57() set
the multF of multMagma(# x,y #) . [y,Y] is set
Y * y is Element of the carrier of multMagma(# x,y #)
the multF of multMagma(# x,y #) . (Y,y) is Element of the carrier of multMagma(# x,y #)
[Y,y] is set
{Y,y} is non empty finite set
{Y} is non empty trivial finite 1 -element set
{{Y,y},{Y}} is non empty finite V57() set
the multF of multMagma(# x,y #) . [Y,y] is set
Z is Element of the carrier of G
Z * (1_ G) is Element of the carrier of G
the multF of G . (Z,(1_ G)) is Element of the carrier of G
[Z,(1_ G)] is set
{Z,(1_ G)} is non empty finite set
{Z} is non empty trivial finite 1 -element set
{{Z,(1_ G)},{Z}} is non empty finite V57() set
the multF of G . [Z,(1_ G)] is set
(1_ G) * Z is Element of the carrier of G
the multF of G . ((1_ G),Z) is Element of the carrier of G
[(1_ G),Z] is set
{(1_ G),Z} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),Z},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),Z] is set
Z " is Element of the carrier of G
c is Element of the carrier of multMagma(# x,y #)
y * c is Element of the carrier of multMagma(# x,y #)
the multF of multMagma(# x,y #) . (y,c) is Element of the carrier of multMagma(# x,y #)
[y,c] is set
{y,c} is non empty finite set
{{y,c},{y}} is non empty finite V57() set
the multF of multMagma(# x,y #) . [y,c] is set
c * y is Element of the carrier of multMagma(# x,y #)
the multF of multMagma(# x,y #) . (c,y) is Element of the carrier of multMagma(# x,y #)
[c,y] is set
{c,y} is non empty finite set
{c} is non empty trivial finite 1 -element set
{{c,y},{c}} is non empty finite V57() set
the multF of multMagma(# x,y #) . [c,y] is set
Z * (Z ") is Element of the carrier of G
the multF of G . (Z,(Z ")) is Element of the carrier of G
[Z,(Z ")] is set
{Z,(Z ")} is non empty finite set
{{Z,(Z ")},{Z}} is non empty finite V57() set
the multF of G . [Z,(Z ")] is set
(Z ") * Z is Element of the carrier of G
the multF of G . ((Z "),Z) is Element of the carrier of G
[(Z "),Z] is set
{(Z "),Z} is non empty finite set
{(Z ")} is non empty trivial finite 1 -element set
{{(Z "),Z},{(Z ")}} is non empty finite V57() set
the multF of G . [(Z "),Z] is set
B is non empty unital Group-like multMagma
a is non empty strict unital Group-like associative (G)
the carrier of a is non empty set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
the carrier of H is non empty set
x is Element of the carrier of H
y is Element of the carrier of H
x * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of H . (x,y) is Element of the carrier of H
[x,y] is set
{x,y} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,y},{x}} is non empty finite V57() set
the multF of H . [x,y] is set
y * x is Element of the carrier of H
the multF of H . (y,x) is Element of the carrier of H
[y,x] is set
{y,x} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,x},{y}} is non empty finite V57() set
the multF of H . [y,x] is set
the carrier of G is non empty set
y is Element of the carrier of G
B is Element of the carrier of G
y * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] 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 trivial finite 1 -element set
{{B,y},{B}} is non empty finite V57() set
the multF of G . [B,y] is set
G is non empty unital Group-like associative commutative multMagma
H is non empty unital Group-like associative (G)
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
[: the carrier 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 having_a_unity 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 cup-closed diff-closed preBoolean set
dom the multF of G is Relation-like the carrier of G -defined the carrier of G -valued Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty cup-closed diff-closed preBoolean set
the multF of G || the carrier of G is Relation-like Function-like set
the multF of G | [: the carrier of G, the carrier of G:] is Relation-like 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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
H is non empty unital Group-like associative multMagma
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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of H, the multF of H #) is non empty strict multMagma
the multF of H || the carrier of G is Relation-like Function-like set
the multF of H | [: the carrier of G, the carrier of G:] is Relation-like set
the multF of G || the carrier of H is Relation-like Function-like set
the multF of G | [: the carrier of H, the carrier of H:] is Relation-like set
( the multF of G || the carrier of H) || the carrier of G is Relation-like Function-like set
( the multF of G || the carrier of H) | [: the carrier of G, the carrier of G:] is Relation-like set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative multMagma
x is non empty unital Group-like associative multMagma
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V28([: the carrier of x, the carrier of x:], the carrier of x) associative having_a_unity Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty cup-closed diff-closed preBoolean set
the carrier of H is non empty set
the carrier of G is non empty set
[: the carrier of G, the carrier of G:] is non empty set
[: the carrier of H, the carrier of H:] 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 having_a_unity Element of bool [:[: the carrier of H, the carrier of H:], the carrier of H:]
[:[: 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 cup-closed diff-closed preBoolean set
the multF of H || the carrier of G is Relation-like Function-like set
the multF of H | [: the carrier of G, the carrier of G:] is Relation-like set
the multF of x || the carrier of H is Relation-like Function-like set
the multF of x | [: the carrier of H, the carrier of H:] is Relation-like set
( the multF of x || the carrier of H) || the carrier of G is Relation-like Function-like set
( the multF of x || the carrier of H) | [: the carrier of G, the carrier of G:] is Relation-like set
the multF of x || the carrier of G is Relation-like Function-like set
the multF of x | [: the carrier of G, the carrier of G:] is Relation-like set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
the carrier of H is non empty set
x is non empty unital Group-like associative (G)
the carrier of x 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 having_a_unity Element of bool [:[: 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:] 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 cup-closed diff-closed preBoolean 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V28([: the carrier of x, the carrier of x:], the carrier of x) associative having_a_unity Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty cup-closed diff-closed preBoolean set
the multF of x || the carrier of H is Relation-like Function-like set
the multF of x | [: the carrier of H, the carrier of H:] is Relation-like set
the multF of G || the carrier of H is Relation-like Function-like set
the multF of G | [: the carrier of H, the carrier of H:] is Relation-like set
the multF of G || the carrier of x is Relation-like Function-like set
the multF of G | [: the carrier of x, the carrier of x:] is Relation-like 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 (G)
x is non empty unital Group-like associative (G)
the carrier of H is non empty set
the carrier of x is non empty set
y is set
y is Element of the carrier of H
B is Element of the carrier of G
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of H, the multF of H #) is non empty strict multMagma
x is non empty unital Group-like associative (G)
the carrier of x is non empty set
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V28([: the carrier of x, the carrier of x:], the carrier of x) associative having_a_unity Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty cup-closed diff-closed preBoolean set
multMagma(# the carrier of x, the multF of x #) is non empty strict multMagma
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 (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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of H, the multF of H #) is non empty strict multMagma
x is non empty unital Group-like associative (G)
the carrier of x is non empty set
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V28([: the carrier of x, the carrier of x:], the carrier of x) associative having_a_unity Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty cup-closed diff-closed preBoolean set
multMagma(# the carrier of x, the multF of x #) is non empty strict multMagma
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative (G)
x is non empty strict unital Group-like associative (G)
the carrier of G is non empty set
y 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
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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
H is non empty unital Group-like associative (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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of H, the multF of H #) is non empty strict multMagma
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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
H is non empty unital Group-like associative (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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of H, the multF of H #) is non empty strict multMagma
x 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
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
H is Element of the carrier of G
H " is Element of the carrier of G
H is Element of the carrier of G
x is Element of the carrier of G
H * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (H,x) is Element of the carrier of G
[H,x] is set
{H,x} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,x},{H}} is non empty finite V57() set
the multF of G . [H,x] is set
H is non empty strict unital Group-like associative (G)
the carrier of H is non empty set
x is non empty strict unital Group-like associative (G)
the carrier of x is non empty 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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
x is Element of the carrier of G
the carrier of multMagma(# the carrier of G, the multF of G #) is non empty set
y is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
y is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
y * y is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
the multF of multMagma(# the carrier of G, the multF of G #) is Relation-like [: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):] -defined the carrier of multMagma(# the carrier of G, the multF of G #) -valued Function-like V28([: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):], the carrier of multMagma(# the carrier of G, the multF of G #)) Element of bool [:[: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):], the carrier of multMagma(# the carrier of G, the multF of G #):]
[: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):] is non empty set
[:[: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):], the carrier of multMagma(# the carrier of G, the multF of G #):] is non empty set
bool [:[: the carrier of multMagma(# the carrier of G, the multF of G #), the carrier of multMagma(# the carrier of G, the multF of G #):], the carrier of multMagma(# the carrier of G, the multF of G #):] is non empty cup-closed diff-closed preBoolean set
the multF of multMagma(# the carrier of G, the multF of G #) . (y,y) is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
[y,y] is set
{y,y} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of multMagma(# the carrier of G, the multF of G #) . [y,y] is set
y * y is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
the multF of multMagma(# the carrier of G, the multF of G #) . (y,y) is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
[y,y] is set
{y,y} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of multMagma(# the carrier of G, the multF of G #) . [y,y] is set
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
B * B is Element of the carrier of G
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
B * x is Element of the carrier of G
the multF of G . (B,x) is Element of the carrier of G
[B,x] is set
{B,x} is non empty finite set
{{B,x},{B}} is non empty finite V57() set
the multF of G . [B,x] is set
x * B is Element of the carrier of G
the multF of G . (x,B) is Element of the carrier of G
[x,B] is set
{x,B} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,B},{x}} is non empty finite V57() set
the multF of G . [x,B] is set
a is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
y * a is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
the multF of multMagma(# the carrier of G, the multF of G #) . (y,a) is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
[y,a] is set
{y,a} is non empty finite set
{{y,a},{y}} is non empty finite V57() set
the multF of multMagma(# the carrier of G, the multF of G #) . [y,a] is set
a * y is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
the multF of multMagma(# the carrier of G, the multF of G #) . (a,y) is Element of the carrier of multMagma(# the carrier of G, the multF of G #)
[a,y] is set
{a,y} is non empty finite set
{a} is non empty trivial finite 1 -element set
{{a,y},{a}} is non empty finite V57() set
the multF of multMagma(# the carrier of G, the multF of G #) . [a,y] is set
dom the multF of G is Relation-like the carrier of G -defined the carrier of G -valued Element of bool [: the carrier of G, the carrier of G:]
bool [: the carrier of G, the carrier of G:] is non empty cup-closed diff-closed preBoolean set
x is non empty unital Group-like multMagma
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V28([: the carrier of x, the carrier of x:], the carrier of x) having_a_unity Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty cup-closed diff-closed preBoolean set
the multF of G || the carrier of x is Relation-like Function-like set
the multF of G | [: the carrier of x, the carrier of x:] is Relation-like set
H is non empty strict unital Group-like associative (G)
x is non empty unital Group-like associative multMagma
the carrier of x is non empty set
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V28([: the carrier of x, the carrier of x:], the carrier of x) associative having_a_unity Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty cup-closed diff-closed preBoolean set
multMagma(# the carrier of x, the multF of x #) is non empty strict multMagma
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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of H, the multF of H #) is non empty strict multMagma
G is non empty unital Group-like associative multMagma
(G) is non empty strict unital Group-like associative (G)
H is non empty unital Group-like associative (G)
(H) is non empty strict unital Group-like associative (H)
1_ H is non being_of_order_0 Element of the carrier of H
the carrier of H is non empty set
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
{(1_ H)} is non empty trivial finite 1 -element Element of bool the carrier of H
bool the carrier of H is non empty cup-closed diff-closed preBoolean set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(H) is non empty strict unital Group-like associative (H)
x is non empty unital Group-like associative (G)
(x) is non empty strict unital Group-like associative (x)
(G) is non empty strict unital Group-like associative (G)
G is non empty unital Group-like associative multMagma
(G) is non empty strict unital Group-like associative (G)
H is non empty unital Group-like associative (G)
(H) is non empty strict unital Group-like associative (H)
G is non empty strict unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G) is non empty strict unital Group-like associative (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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
G is non empty strict unital Group-like associative multMagma
(G) is non empty strict unital Group-like associative (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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
G is non empty unital Group-like associative multMagma
(G) is non empty strict unital Group-like associative (G)
the carrier of (G) is non empty 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 trivial finite 1 -element Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
G is non empty unital Group-like associative multMagma
(G) is non empty strict unital Group-like associative (G)
the non empty unital Group-like associative multMagma is non empty unital Group-like associative multMagma
( the non empty unital Group-like associative multMagma ) is non empty finite strict unital Group-like associative ( the non empty unital Group-like associative multMagma )
G is non empty finite unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
G is non empty unital Group-like associative multMagma
(G) is non empty finite strict unital Group-like associative (G)
card (G) is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of (G) is non empty finite set
card the carrier of (G) is non empty V20() V24() finite cardinal 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 trivial finite 1 -element Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
G is non empty unital Group-like associative multMagma
(G) is non empty finite strict unital Group-like associative (G)
H is non empty finite strict unital Group-like associative (G)
card H is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of H is non empty finite set
card the carrier of H is non empty V20() V24() finite cardinal set
x is set
{x} is non empty trivial finite 1 -element set
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
G is non empty unital Group-like associative multMagma
card G is V20() cardinal set
the carrier of G is non empty set
card the carrier of G is non empty V20() cardinal set
H is non empty unital Group-like associative (G)
card H is V20() cardinal set
the carrier of H is non empty set
card the carrier of H is non empty V20() cardinal set
G is non empty finite unital Group-like associative multMagma
card G is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of G is non empty finite set
card the carrier of G is non empty V20() V24() finite cardinal set
H is non empty finite unital Group-like associative (G)
card H is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of H is non empty finite set
card the carrier of H is non empty V20() V24() finite cardinal set
G is non empty finite unital Group-like associative multMagma
card G is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of G is non empty finite set
card the carrier of G is non empty V20() V24() finite cardinal 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 having_a_unity finite 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 finite set
[:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty finite set
bool [:[: the carrier of G, the carrier of G:], the carrier of G:] is non empty cup-closed diff-closed preBoolean finite V57() set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
H is non empty finite unital Group-like associative (G)
card H is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of H is non empty finite set
card the carrier of H is non empty V20() V24() finite cardinal 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 having_a_unity finite 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 finite set
[:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty finite set
bool [:[: the carrier of H, the carrier of H:], the carrier of H:] is non empty cup-closed diff-closed preBoolean finite V57() set
multMagma(# the carrier of H, the multF of H #) is non empty strict multMagma
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
the carrier of H is non empty set
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean 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
x is Element of the carrier of G
H * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (H,x) is Element of the carrier of G
[H,x] is set
{H,x} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,x},{H}} is non empty finite V57() set
the multF of G . [H,x] is set
y is non empty unital Group-like associative (G)
(G,y) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
the carrier of y 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
x is non empty unital Group-like associative (G)
(G,x) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
the carrier of x is non empty set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G,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 cup-closed diff-closed preBoolean set
the carrier of H is non empty set
(G,(G,H),(G,H)) 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 (G,H) ) } is set
x is Element of the carrier of G
x " is Element of the carrier of G
x is Element of the carrier of G
y is Element of the carrier of G
x * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (x,y) is Element of the carrier of G
[x,y] is set
{x,y} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,y},{x}} is non empty finite V57() set
the multF of G . [x,y] is set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G,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 cup-closed diff-closed preBoolean set
the carrier of H is non empty set
(G,(G,H)) is Element of bool the carrier of G
{ (b₁ ") where b₁ is Element of the carrier of G : b₁ in (G,H) } is set
x is Element of the carrier of G
x " is Element of the carrier of G
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G,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 cup-closed diff-closed preBoolean set
the carrier of H is non empty set
x is non empty unital Group-like associative (G)
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,(G,H),(G,x)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,H) & b₂ in (G,x) ) } is set
(G,(G,x),(G,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in (G,H) ) } is set
y is Element of the carrier of G
y is Element of the carrier of G
B is Element of the carrier of G
y * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] is set
B is Element of the carrier of G
a 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 trivial finite 1 -element set
{{B,a},{B}} is non empty finite V57() set
the multF of G . [B,a] is set
a " is Element of the carrier of G
B " is Element of the carrier of G
y " 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 ")} is non empty trivial finite 1 -element set
{{(a "),(B ")},{(a ")}} is non empty finite V57() set
the multF of G . [(a "),(B ")] is set
y is Element of the carrier of G
y is Element of the carrier of G
B is Element of the carrier of G
B is Element of the carrier of G
B * B is Element of the carrier of G
the multF of G . (B,B) is Element of the carrier of G
[B,B] is set
{B,B} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,B},{B}} is non empty finite V57() set
the multF of G . [B,B] is set
a is Element of the carrier of G
Y 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 trivial finite 1 -element set
{{a,Y},{a}} is non empty finite V57() set
the multF of G . [a,Y] is set
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 trivial finite 1 -element set
{{B,a},{B}} is non empty finite V57() set
the multF of G . [B,a] is set
y is Element of the carrier of G
Z is Element of the carrier of G
y * Z is Element of the carrier of G
the multF of G . (y,Z) is Element of the carrier of G
[y,Z] is set
{y,Z} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,Z},{y}} is non empty finite V57() set
the multF of G . [y,Z] is set
Z * Y is Element of the carrier of G
the multF of G . (Z,Y) is Element of the carrier of G
[Z,Y] is set
{Z,Y} is non empty finite set
{Z} is non empty trivial finite 1 -element set
{{Z,Y},{Z}} is non empty finite V57() set
the multF of G . [Z,Y] is set
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} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of G . [y,y] is set
(B * B) * a is Element of the carrier of G
the multF of G . ((B * B),a) is Element of the carrier of G
[(B * B),a] is set
{(B * B),a} is non empty finite set
{(B * B)} is non empty trivial finite 1 -element set
{{(B * B),a},{(B * B)}} is non empty finite V57() set
the multF of G . [(B * B),a] is set
((B * B) * a) * Y is Element of the carrier of G
the multF of G . (((B * B) * a),Y) is Element of the carrier of G
[((B * B) * a),Y] is set
{((B * B) * a),Y} is non empty finite set
{((B * B) * a)} is non empty trivial finite 1 -element set
{{((B * B) * a),Y},{((B * B) * a)}} is non empty finite V57() set
the multF of G . [((B * B) * a),Y] is set
B * (B * a) is Element of the carrier of G
the multF of G . (B,(B * a)) is Element of the carrier of G
[B,(B * a)] is set
{B,(B * a)} is non empty finite set
{{B,(B * a)},{B}} is non empty finite V57() set
the multF of G . [B,(B * a)] is set
(B * (B * a)) * Y is Element of the carrier of G
the multF of G . ((B * (B * a)),Y) is Element of the carrier of G
[(B * (B * a)),Y] is set
{(B * (B * a)),Y} is non empty finite set
{(B * (B * a))} is non empty trivial finite 1 -element set
{{(B * (B * a)),Y},{(B * (B * a))}} is non empty finite V57() set
the multF of G . [(B * (B * a)),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,y},{B}} is non empty finite V57() set
the multF of G . [B,y] is set
(B * y) * Z is Element of the carrier of G
the multF of G . ((B * y),Z) is Element of the carrier of G
[(B * y),Z] is set
{(B * y),Z} is non empty finite set
{(B * y)} is non empty trivial finite 1 -element set
{{(B * y),Z},{(B * y)}} is non empty finite V57() set
the multF of G . [(B * y),Z] is set
((B * y) * Z) * Y is Element of the carrier of G
the multF of G . (((B * y) * Z),Y) is Element of the carrier of G
[((B * y) * Z),Y] is set
{((B * y) * Z),Y} is non empty finite set
{((B * y) * Z)} is non empty trivial finite 1 -element set
{{((B * y) * Z),Y},{((B * y) * Z)}} is non empty finite V57() set
the multF of G . [((B * y) * Z),Y] is set
(B * y) * (Z * Y) is Element of the carrier of G
the multF of G . ((B * y),(Z * Y)) is Element of the carrier of G
[(B * y),(Z * Y)] is set
{(B * y),(Z * Y)} is non empty finite set
{{(B * y),(Z * Y)},{(B * y)}} is non empty finite V57() set
the multF of G . [(B * y),(Z * Y)] is set
y is non empty unital Group-like associative (G)
the carrier of y is non empty set
y is set
B is Element of the carrier of G
B " is Element of the carrier of G
(G,y) is Element of bool the carrier of G
B is Element of the carrier of G
a 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{B,a},{B}} is non empty finite V57() set
the multF of G . [B,a] is set
(B ") " is Element of the carrier of G
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 . ((a "),(B ")) is Element of the carrier of G
[(a "),(B ")] is set
{(a "),(B ")} is non empty finite set
{(a ")} is non empty trivial finite 1 -element set
{{(a "),(B ")},{(a ")}} is non empty finite V57() set
the multF of G . [(a "),(B ")] is set
y is set
B is Element of the carrier of G
a 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{B,a},{B}} is non empty finite V57() set
the multF of G . [B,a] is set
a " is Element of the carrier of G
B is Element of the carrier of G
B " is Element of the carrier of G
(G,y) is Element of bool 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 . ((a "),(B ")) is Element of the carrier of G
[(a "),(B ")] is set
{(a "),(B ")} is non empty finite set
{(a ")} is non empty trivial finite 1 -element set
{{(a "),(B ")},{(a ")}} is non empty finite V57() set
the multF of G . [(a "),(B ")] is set
(B ") " is Element of the carrier of G
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G,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 cup-closed diff-closed preBoolean set
the carrier of H is non empty set
x is non empty unital Group-like associative (G)
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,(G,H),(G,x)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,H) & b₂ in (G,x) ) } is set
(G,(G,x),(G,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in (G,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 cup-closed diff-closed preBoolean set
H is non empty unital Group-like associative (G)
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
x is non empty unital Group-like associative (G)
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H) /\ (G,x) is Element of bool the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
y is Element of the carrier of G
B is Element of the carrier of G
y * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] is set
y is Element of the carrier of G
y " is Element of the carrier of G
y is non empty strict unital Group-like associative (G)
the carrier of y is non empty set
y is non empty strict unital Group-like associative (G)
the carrier of y is non empty set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
the carrier of H is non empty set
x is non empty unital Group-like associative (G)
(G,H,x) is non empty strict unital Group-like associative (G)
the carrier of x is non empty set
the carrier of H /\ the carrier of x is set
(G,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 cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
y is non empty unital Group-like associative (G)
the carrier of y is non empty set
(G,H) /\ (G,x) is Element of bool the carrier of G
y is non empty strict unital Group-like associative (G)
the carrier of y is non empty set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
x is non empty unital Group-like associative (G)
(G,H,x) is non empty strict unital Group-like associative (G)
(G,(G,H,x)) is Element of bool the carrier of G
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
the carrier of (G,H,x) is non empty set
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H) /\ (G,x) is Element of bool the carrier of G
G is set
H is non empty unital Group-like associative multMagma
x is non empty unital Group-like associative (H)
y is non empty unital Group-like associative (H)
(H,x,y) is non empty strict unital Group-like associative (H)
the carrier of (H,x,y) is non empty set
the carrier of H is non empty set
bool the carrier of H is non empty cup-closed diff-closed preBoolean set
(H,x) is Element of bool the carrier of H
the carrier of x is non empty set
(H,y) is Element of bool the carrier of H
the carrier of y is non empty set
(H,x) /\ (H,y) is Element of bool the carrier of H
(H,y) is Element of bool the carrier of H
the carrier of H is non empty set
bool the carrier of H is non empty cup-closed diff-closed preBoolean set
the carrier of y is non empty set
(H,x) is Element of bool the carrier of H
the carrier of x is non empty set
(H,x) /\ (H,y) is Element of bool the carrier of H
(H,(H,x,y)) is Element of bool the carrier of H
the carrier of (H,x,y) is non empty set
G is non empty unital Group-like associative multMagma
H is non empty strict unital Group-like associative (G)
(G,H,H) is non empty strict unital Group-like associative (G)
the carrier of (G,H,H) is non empty set
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,H) /\ (G,H) is Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
y is non empty unital Group-like associative (G)
y is non empty unital Group-like associative (G)
(G,y,y) is non empty strict unital Group-like associative (G)
(G,y,y) is non empty strict unital Group-like associative (G)
the carrier of (G,y,y) is non empty set
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,y) /\ (G,y) is Element of bool the carrier of G
the carrier of (G,y,y) is non empty set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
x is non empty unital Group-like associative (G)
(G,H,x) is non empty strict unital Group-like associative (G)
y is non empty unital Group-like associative (G)
(G,(G,H,x),y) is non empty strict unital Group-like associative (G)
(G,x,y) is non empty strict unital Group-like associative (G)
(G,H,(G,x,y)) is non empty strict unital Group-like associative (G)
the carrier of (G,(G,H,x),y) is non empty set
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,(G,H,x)) is Element of bool the carrier of G
the carrier of (G,H,x) is non empty set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,(G,H,x)) /\ (G,y) is Element of bool the carrier of G
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H) /\ (G,x) is Element of bool the carrier of G
((G,H) /\ (G,x)) /\ (G,y) is Element of bool the carrier of G
(G,x) /\ (G,y) is Element of bool the carrier of G
(G,H) /\ ((G,x) /\ (G,y)) is Element of bool the carrier of G
(G,(G,x,y)) is Element of bool the carrier of G
the carrier of (G,x,y) is non empty set
(G,H) /\ (G,(G,x,y)) is Element of bool the carrier of G
the carrier of (G,H,(G,x,y)) is non empty set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
x is non empty unital Group-like associative (G)
(G,x,H) is non empty strict unital Group-like associative (G)
the carrier of (G,x,H) is non empty set
the multF of (G,x,H) is Relation-like [: the carrier of (G,x,H), the carrier of (G,x,H):] -defined the carrier of (G,x,H) -valued Function-like V28([: the carrier of (G,x,H), the carrier of (G,x,H):], the carrier of (G,x,H)) associative having_a_unity Element of bool [:[: the carrier of (G,x,H), the carrier of (G,x,H):], the carrier of (G,x,H):]
[: the carrier of (G,x,H), the carrier of (G,x,H):] is non empty set
[:[: the carrier of (G,x,H), the carrier of (G,x,H):], the carrier of (G,x,H):] is non empty set
bool [:[: the carrier of (G,x,H), the carrier of (G,x,H):], the carrier of (G,x,H):] is non empty cup-closed diff-closed preBoolean set
multMagma(# the carrier of (G,x,H), the multF of (G,x,H) #) is non empty strict multMagma
the carrier of x is non empty set
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V28([: the carrier of x, the carrier of x:], the carrier of x) associative having_a_unity Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty cup-closed diff-closed preBoolean set
multMagma(# the carrier of x, the multF of x #) is non empty strict multMagma
the carrier of H is non empty set
the carrier of G is non empty set
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
(G,H) is Element of bool the carrier of G
(G,x) /\ (G,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 cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,x) /\ (G,H) is Element of bool the carrier of G
the carrier of x /\ the carrier of H is set
G is non empty unital Group-like associative multMagma
(G) is non empty finite strict unital Group-like associative (G)
H is non empty unital Group-like associative (G)
(G,(G),H) is non empty strict unital Group-like associative (G)
(G,H,(G)) is non empty strict unital Group-like associative (G)
G is non empty strict unital Group-like associative multMagma
x is non empty strict unital Group-like associative multMagma
H is non empty strict unital Group-like associative (G)
(G) is non empty strict unital Group-like associative (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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
(G,H,(G)) is non empty strict unital Group-like associative (G)
(x) is non empty strict unital Group-like associative (x)
the carrier of x is non empty set
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V28([: the carrier of x, the carrier of x:], the carrier of x) associative having_a_unity Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty cup-closed diff-closed preBoolean set
multMagma(# the carrier of x, the multF of x #) is non empty strict multMagma
y is non empty strict unital Group-like associative (x)
(x,(x),y) is non empty strict unital Group-like associative (x)
G is non empty strict unital Group-like associative multMagma
(G) is non empty strict unital Group-like associative (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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
(G,(G),(G)) is non empty strict unital Group-like associative (G)
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
x is non empty unital Group-like associative (G)
(G,H,x) is non empty strict unital Group-like associative (G)
the carrier of (G,H,x) is non empty set
the carrier of H is non empty set
the carrier of x is non empty set
the carrier of H /\ the carrier of x is set
G is non empty unital Group-like associative multMagma
y is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
x is non empty unital Group-like associative (G)
(G,H,x) is non empty strict unital Group-like associative (G)
y is non empty unital Group-like associative (y)
B is non empty unital Group-like associative (y)
(y,y,B) is non empty strict unital Group-like associative (y)
G is non empty unital Group-like associative multMagma
y is non empty unital Group-like associative multMagma
x is non empty unital Group-like associative (G)
H is non empty unital Group-like associative (G)
(G,x,H) is non empty strict unital Group-like associative (G)
the carrier of (G,x,H) is non empty set
the multF of (G,x,H) is Relation-like [: the carrier of (G,x,H), the carrier of (G,x,H):] -defined the carrier of (G,x,H) -valued Function-like V28([: the carrier of (G,x,H), the carrier of (G,x,H):], the carrier of (G,x,H)) associative having_a_unity Element of bool [:[: the carrier of (G,x,H), the carrier of (G,x,H):], the carrier of (G,x,H):]
[: the carrier of (G,x,H), the carrier of (G,x,H):] is non empty set
[:[: the carrier of (G,x,H), the carrier of (G,x,H):], the carrier of (G,x,H):] is non empty set
bool [:[: the carrier of (G,x,H), the carrier of (G,x,H):], the carrier of (G,x,H):] is non empty cup-closed diff-closed preBoolean set
multMagma(# the carrier of (G,x,H), the multF of (G,x,H) #) is non empty strict multMagma
the carrier of x is non empty set
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V28([: the carrier of x, the carrier of x:], the carrier of x) associative having_a_unity Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is non empty set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is non empty cup-closed diff-closed preBoolean set
multMagma(# the carrier of x, the multF of x #) is non empty strict multMagma
B is non empty unital Group-like associative (y)
y is non empty unital Group-like associative (y)
(y,B,y) is non empty strict unital Group-like associative (y)
the carrier of (y,B,y) is non empty set
the multF of (y,B,y) is Relation-like [: the carrier of (y,B,y), the carrier of (y,B,y):] -defined the carrier of (y,B,y) -valued Function-like V28([: the carrier of (y,B,y), the carrier of (y,B,y):], the carrier of (y,B,y)) associative having_a_unity Element of bool [:[: the carrier of (y,B,y), the carrier of (y,B,y):], the carrier of (y,B,y):]
[: the carrier of (y,B,y), the carrier of (y,B,y):] is non empty set
[:[: the carrier of (y,B,y), the carrier of (y,B,y):], the carrier of (y,B,y):] is non empty set
bool [:[: the carrier of (y,B,y), the carrier of (y,B,y):], the carrier of (y,B,y):] is non empty cup-closed diff-closed preBoolean set
multMagma(# the carrier of (y,B,y), the multF of (y,B,y) #) is non empty strict multMagma
the carrier of B is non empty set
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like V28([: the carrier of B, the carrier of B:], the carrier of B) associative having_a_unity Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty set
bool [:[: the carrier of B, the carrier of B:], the carrier of B:] is non empty cup-closed diff-closed preBoolean set
multMagma(# the carrier of B, the multF of B #) is non empty strict multMagma
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
x is non empty unital Group-like associative (G)
y is non empty unital Group-like associative (G)
(G,H,y) is non empty strict unital Group-like associative (G)
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
x is non empty unital Group-like associative (G)
y is non empty unital Group-like associative (G)
(G,x,y) is non empty strict unital Group-like associative (G)
the carrier of G is non empty set
y is Element of the carrier of G
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
x is non empty unital Group-like associative (G)
y is non empty unital Group-like associative (G)
(G,H,y) is non empty strict unital Group-like associative (G)
(G,x,y) is non empty strict unital Group-like associative (G)
the carrier of H is non empty set
the carrier of x is non empty set
the carrier of y is non empty set
the carrier of H /\ the carrier of y is set
the carrier of x /\ the carrier of y is set
the carrier of (G,H,y) is non empty set
the carrier of (G,x,y) is non empty set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
x is non empty unital Group-like associative (G)
(G,H,x) is non empty strict unital Group-like associative (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 cup-closed diff-closed preBoolean set
x is Element of bool the carrier of G
H is non empty unital Group-like associative (G)
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,x,(G,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in x & b₂ in (G,H) ) } is set
(G,(G,H),x) 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 x ) } 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 cup-closed diff-closed preBoolean set
x is Element of bool the carrier of H
y is non empty unital Group-like associative (H)
(H,y,x) is Element of bool the carrier of H
(H,y) is Element of bool the carrier of H
the carrier of y is non empty set
(H,x,(H,y)) is Element of bool the carrier of H
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in x & b₂ in (H,y) ) } is set
y is Element of the carrier of H
B is Element of the carrier of H
y * B 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of H . (y,B) is Element of the carrier of H
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of H . [y,B] is set
y is Element of the carrier of H
B is Element of the carrier of H
y * B 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of H . (y,B) is Element of the carrier of H
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of H . [y,B] 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 cup-closed diff-closed preBoolean set
x is Element of bool the carrier of H
y is non empty unital Group-like associative (H)
(H,y,x) is Element of bool the carrier of H
(H,y) is Element of bool the carrier of H
the carrier of y is non empty set
(H,(H,y),x) is Element of bool the carrier of H
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in (H,y) & b₂ in x ) } is set
y is Element of the carrier of H
B is Element of the carrier of H
y * B 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of H . (y,B) is Element of the carrier of H
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of H . [y,B] is set
y is Element of the carrier of H
B is Element of the carrier of H
y * B 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of H . (y,B) is Element of the carrier of H
[y,B] is set
{y,B} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,B},{y}} is non empty finite V57() set
the multF of H . [y,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 cup-closed diff-closed preBoolean set
y is non empty unital Group-like associative (G)
H is Element of bool the carrier of G
x is Element of bool the carrier of G
(G,H,x) 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 x ) } is set
(G,y,(G,H,x)) is Element of bool the carrier of G
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,(G,H,x),(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,H,x) & b₂ in (G,y) ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in x & b₂ in (G,y) ) } is set
(G,H,(G,y,x)) 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 (G,y,x) ) } 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 cup-closed diff-closed preBoolean set
y is non empty unital Group-like associative (G)
H is Element of bool the carrier of G
(G,y,H) is Element of bool the carrier of G
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) 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 (G,y) ) } is set
x is Element of bool the carrier of G
(G,(G,y,H),x) 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 x ) } is set
(G,y,x) is Element of bool the carrier of G
(G,(G,y),x) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & b₂ in x ) } is set
(G,H,(G,y,x)) 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 (G,y,x) ) } 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 cup-closed diff-closed preBoolean set
y is non empty unital Group-like associative (G)
H is Element of bool the carrier of G
(G,y,H) is Element of bool the carrier of G
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,(G,y),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) & b₂ in H ) } is set
x is Element of bool the carrier of G
(G,(G,y,H),x) 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 x ) } is set
(G,H,x) 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 x ) } is set
(G,y,(G,H,x)) is Element of bool the carrier of G
(G,(G,y),(G,H,x)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & b₂ in (G,H,x) ) } 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 cup-closed diff-closed preBoolean set
y is non empty unital Group-like associative (G)
x is non empty unital Group-like associative (G)
H is Element of bool the carrier of G
(G,x,H) is Element of bool the carrier of G
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H,(G,x)) 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 (G,x) ) } is set
(G,y,(G,x,H)) is Element of bool the carrier of G
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,(G,x,H),(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,H) & b₂ in (G,y) ) } is set
(G,x,(G,y)) is Element of bool the carrier of G
(G,(G,x),(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in (G,y) ) } is set
(G,H,(G,x,(G,y))) 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 (G,x,(G,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 cup-closed diff-closed preBoolean set
y is non empty unital Group-like associative (G)
x is non empty unital Group-like associative (G)
H is Element of bool the carrier of G
(G,x,H) is Element of bool the carrier of G
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,(G,x),H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in H ) } is set
(G,y,(G,x,H)) is Element of bool the carrier of G
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,(G,x,H),(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,H) & b₂ in (G,y) ) } is set
(G,y,H) is Element of bool the carrier of G
(G,H,(G,y)) 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 (G,y) ) } is set
(G,x,(G,y,H)) is Element of bool the carrier of G
(G,(G,x),(G,y,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in (G,y,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 cup-closed diff-closed preBoolean set
x is non empty unital Group-like associative (G)
y is non empty unital Group-like associative (G)
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,x,(G,y)) is Element of bool the carrier of G
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,(G,x),(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in (G,y) ) } is set
H is Element of bool the carrier of G
(G,(G,x,(G,y)),H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,(G,y)) & b₂ in H ) } is set
(G,y,H) is Element of bool the carrier of G
(G,(G,y),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) & b₂ in H ) } is set
(G,x,(G,y,H)) is Element of bool the carrier of G
(G,(G,x),(G,y,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in (G,y,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 cup-closed diff-closed preBoolean set
x is non empty unital Group-like associative (G)
H is Element of bool the carrier of G
(G,x,H) is Element of bool the carrier of G
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H,(G,x)) 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 (G,x) ) } is set
(G,x,H) is Element of bool the carrier of G
(G,(G,x),H) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in H ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
x is Element of the carrier of G
H is non empty unital Group-like associative (G)
(G,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
the carrier of H is non empty set
(G,x,(G,H)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{x},(G,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in (G,H) ) } is set
(G,x,(G,H)) is Element of bool the carrier of G
(G,(G,H),{x}) 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 {x} ) } is set
G is set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
x is Element of the carrier of H
y is non empty unital Group-like associative (H)
(H,y,x) is Element of bool the carrier of H
bool the carrier of H is non empty cup-closed diff-closed preBoolean set
(H,y) is Element of bool the carrier of H
the carrier of y is non empty set
(H,x,(H,y)) is Element of bool the carrier of H
{x} is non empty trivial finite 1 -element Element of bool the carrier of H
(H,{x},(H,y)) is Element of bool the carrier of H
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in {x} & b₂ in (H,y) ) } is set
y is Element of the carrier of H
x * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of H . (x,y) is Element of the carrier of H
[x,y] is set
{x,y} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,y},{x}} is non empty finite V57() set
the multF of H . [x,y] is set
y is Element of the carrier of H
x * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of H . (x,y) is Element of the carrier of H
[x,y] is set
{x,y} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,y},{x}} is non empty finite V57() set
the multF of H . [x,y] is set
G is set
H is non empty unital Group-like associative multMagma
the carrier of H is non empty set
x is Element of the carrier of H
y is non empty unital Group-like associative (H)
(H,y,x) is Element of bool the carrier of H
bool the carrier of H is non empty cup-closed diff-closed preBoolean set
(H,y) is Element of bool the carrier of H
the carrier of y is non empty set
(H,x,(H,y)) is Element of bool the carrier of H
{x} is non empty trivial finite 1 -element Element of bool the carrier of H
(H,(H,y),{x}) is Element of bool the carrier of H
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of H : ( b₁ in (H,y) & b₂ in {x} ) } is set
y is Element of the carrier of H
y * x 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of H . (y,x) is Element of the carrier of H
[y,x] is set
{y,x} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,x},{y}} is non empty finite V57() set
the multF of H . [y,x] is set
y is Element of the carrier of H
y * x 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of H . (y,x) is Element of the carrier of H
[y,x] is set
{y,x} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,x},{y}} is non empty finite V57() set
the multF of H . [y,x] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
y is non empty unital Group-like associative (G)
H is Element of the carrier of G
x is Element of the carrier of G
H * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (H,x) is Element of the carrier of G
[H,x] is set
{H,x} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,x},{H}} is non empty finite V57() set
the multF of G . [H,x] is set
(G,y,(H * x)) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,(H * x),(G,y)) is Element of bool the carrier of G
{(H * x)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(H * x)},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(H * x)} & b₂ in (G,y) ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{x},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in (G,y) ) } is set
(G,H,(G,y,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,y,x)) 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 (G,y,x) ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
x is Element of the carrier of G
y is non empty unital Group-like associative (G)
H is Element of the carrier of G
(G,y,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,y)) 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 (G,y) ) } is set
(G,x,(G,y,H)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y,H),{x}) 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 {x} ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
(G,(G,y),{x}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & b₂ in {x} ) } is set
(G,H,(G,y,x)) is Element of bool the carrier of G
(G,{H},(G,y,x)) 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 (G,y,x) ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
x is Element of the carrier of G
y is non empty unital Group-like associative (G)
H is Element of the carrier of G
(G,y,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{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) & b₂ in {H} ) } is set
(G,x,(G,y,H)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y,H),{x}) 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 {x} ) } is set
H * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (H,x) is Element of the carrier of G
[H,x] is set
{H,x} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,x},{H}} is non empty finite V57() set
the multF of G . [H,x] is set
(G,y,(H * x)) is Element of bool the carrier of G
(G,(H * x),(G,y)) is Element of bool the carrier of G
{(H * x)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{(H * x)}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & b₂ in {(H * x)} ) } 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
x is non empty unital Group-like associative (G)
(G,x,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,x)) 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 (G,x) ) } is set
(G,x,H) is Element of bool the carrier of G
(G,H,(G,x)) is Element of bool the carrier of G
(G,(G,x),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {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 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{(1_ G),H},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),H] is set
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} is non empty trivial finite 1 -element set
{{H,(1_ G)},{H}} is non empty finite V57() set
the multF of G . [H,(1_ G)] is set
G is non empty unital Group-like associative multMagma
x is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
1_ G is non being_of_order_0 Element of the carrier of G
the carrier of G is non empty set
(G,H,(1_ G)) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,(1_ G),(G,H)) is Element of bool the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(1_ G)},(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 (G,H) ) } is set
y is non empty unital Group-like associative (x)
1_ x is non being_of_order_0 Element of the carrier of x
the carrier of x is non empty set
(x,y,(1_ x)) is Element of bool the carrier of x
bool the carrier of x is non empty cup-closed diff-closed preBoolean set
(x,y) is Element of bool the carrier of x
the carrier of y is non empty set
(x,(1_ x),(x,y)) is Element of bool the carrier of x
{(1_ x)} is non empty trivial finite 1 -element Element of bool the carrier of x
(x,(x,y),{(1_ x)}) is Element of bool the carrier of x
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of x : ( b₁ in (x,y) & b₂ in {(1_ x)} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
(G) is non empty finite strict unital Group-like associative (G)
H is Element of the carrier of G
(G,(G),H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,(G)) is Element of bool the carrier of G
the carrier of (G) is non empty finite set
(G,H,(G,(G))) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,(G)),{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)) & b₂ in {H} ) } is set
(G,(G),H) is Element of bool the carrier of G
(G,H,(G,(G))) is Element of bool the carrier of G
(G,{H},(G,(G))) 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 (G,(G)) ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{(1_ G),H},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),H] is set
{((1_ G) * H)} is non empty trivial finite 1 -element Element of bool 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} is non empty trivial finite 1 -element set
{{H,(1_ G)},{H}} is non empty finite V57() set
the multF of G . [H,(1_ G)] is set
{(H * (1_ G))} is non empty trivial finite 1 -element Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
(G) is non empty strict unital Group-like associative (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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
H is Element of the carrier of G
(G,(G),H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,(G)) is Element of bool the carrier of G
the carrier of (G) is non empty set
(G,H,(G,(G))) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,(G))) 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 (G,(G)) ) } is set
(G,(G),H) is Element of bool the carrier of G
(G,H,(G,(G))) is Element of bool the carrier of G
(G,(G,(G)),{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)) & b₂ in {H} ) } is set
[#] the carrier of G is non empty non proper Element of bool the carrier of G
(G,H,([#] the carrier of G)) is Element of bool the carrier of G
(G,([#] the carrier of G),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in [#] the carrier of G & b₂ in {H} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
x is non empty unital Group-like associative (G)
H is Element of the carrier of G
(G,x,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,x)) 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 (G,x) ) } is set
(G,x,H) is Element of bool the carrier of G
(G,H,(G,x)) is Element of bool the carrier of G
(G,(G,x),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & 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
x is non empty unital Group-like associative (G)
(G,x,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,x)) 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 (G,x) ) } is set
y is set
y 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{H,y},{H}} is non empty finite V57() set
the multF of G . [H,y] is set
y is set
H " is Element of the carrier of G
y 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{(H "),y},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),y] is set
H * ((H ") * y) is Element of the carrier of G
the multF of G . (H,((H ") * y)) is Element of the carrier of G
[H,((H ") * y)] is set
{H,((H ") * y)} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,((H ") * y)},{H}} is non empty finite V57() set
the multF of G . [H,((H ") * y)] 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 V57() set
the multF of G . [H,(H ")] is set
(H * (H ")) * y is Element of the carrier of G
the multF of G . ((H * (H ")),y) is Element of the carrier of G
[(H * (H ")),y] is set
{(H * (H ")),y} is non empty finite set
{(H * (H "))} is non empty trivial finite 1 -element set
{{(H * (H ")),y},{(H * (H "))}} is non empty finite V57() set
the multF of G . [(H * (H ")),y] is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * y is Element of the carrier of G
the multF of G . ((1_ G),y) is Element of the carrier of G
[(1_ G),y] is set
{(1_ G),y} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),y},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),y] 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 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{H,(1_ G)},{H}} is non empty finite V57() set
the multF of 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
x is Element of the carrier of G
x " is Element of the carrier of G
(x ") * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . ((x "),H) is Element of the carrier of G
[(x "),H] is set
{(x "),H} is non empty finite set
{(x ")} is non empty trivial finite 1 -element set
{{(x "),H},{(x ")}} is non empty finite V57() set
the multF of G . [(x "),H] is set
y is non empty unital Group-like associative (G)
(G,y,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,y)) 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 (G,y) ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{x},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in (G,y) ) } is set
(G,y,((x ") * H)) is Element of bool the carrier of G
(G,((x ") * H),(G,y)) is Element of bool the carrier of G
{((x ") * H)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{((x ") * H)},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((x ") * H)} & b₂ in (G,y) ) } is set
(G,(x "),(G,y,H)) is Element of bool the carrier of G
{(x ")} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(x ")},(G,y,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(x ")} & b₂ in (G,y,H) ) } is set
(x ") * x is Element of the carrier of G
the multF of G . ((x "),x) is Element of the carrier of G
[(x "),x] is set
{(x "),x} is non empty finite set
{{(x "),x},{(x ")}} is non empty finite V57() set
the multF of G . [(x "),x] is set
(G,y,((x ") * x)) is Element of bool the carrier of G
(G,((x ") * x),(G,y)) is Element of bool the carrier of G
{((x ") * x)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{((x ") * x)},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((x ") * x)} & b₂ in (G,y) ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(G,y,(1_ G)) is Element of bool the carrier of G
(G,(1_ G),(G,y)) is Element of bool the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(1_ G)},(G,y)) 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 (G,y) ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(G,(1_ G),(G,y,H)) is Element of bool the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(1_ G)},(G,y,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 (G,y,H) ) } 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 trivial finite 1 -element set
{{(1_ G),H},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),H] is set
(G,y,((1_ G) * H)) is Element of bool the carrier of G
(G,((1_ G) * H),(G,y)) is Element of bool the carrier of G
{((1_ G) * H)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{((1_ G) * H)},(G,y)) 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 (G,y) ) } is set
x * (x ") is Element of the carrier of G
the multF of G . (x,(x ")) is Element of the carrier of G
[x,(x ")] is set
{x,(x ")} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,(x ")},{x}} is non empty finite V57() set
the multF of G . [x,(x ")] is set
(x * (x ")) * H is Element of the carrier of G
the multF of G . ((x * (x ")),H) is Element of the carrier of G
[(x * (x ")),H] is set
{(x * (x ")),H} is non empty finite set
{(x * (x "))} is non empty trivial finite 1 -element set
{{(x * (x ")),H},{(x * (x "))}} is non empty finite V57() set
the multF of G . [(x * (x ")),H] is set
(G,y,((x * (x ")) * H)) is Element of bool the carrier of G
(G,((x * (x ")) * H),(G,y)) is Element of bool the carrier of G
{((x * (x ")) * H)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{((x * (x ")) * H)},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((x * (x ")) * H)} & b₂ in (G,y) ) } is set
x * ((x ") * H) is Element of the carrier of G
the multF of G . (x,((x ") * H)) is Element of the carrier of G
[x,((x ") * H)] is set
{x,((x ") * H)} is non empty finite set
{{x,((x ") * H)},{x}} is non empty finite V57() set
the multF of G . [x,((x ") * H)] is set
(G,y,(x * ((x ") * H))) is Element of bool the carrier of G
(G,(x * ((x ") * H)),(G,y)) is Element of bool the carrier of G
{(x * ((x ") * H))} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(x * ((x ") * H))},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(x * ((x ") * H))} & b₂ in (G,y) ) } is set
(G,y,((x ") * H)) is Element of bool the carrier of G
(G,((x ") * H),(G,y)) is Element of bool the carrier of G
{((x ") * H)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{((x ") * H)},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {((x ") * H)} & b₂ in (G,y) ) } is set
(G,x,(G,y,((x ") * H))) is Element of bool the carrier of G
(G,{x},(G,y,((x ") * H))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in (G,y,((x ") * 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
x is Element of the carrier of G
y is non empty unital Group-like associative (G)
(G,y,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,y)) 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 (G,y) ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{x},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in (G,y) ) } is set
y is set
B is Element of the carrier of G
B is Element of the carrier of G
H * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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} is non empty trivial finite 1 -element set
{{H,B},{H}} is non empty finite V57() set
the multF of G . [H,B] is set
B " is Element of the carrier of G
a is Element of the carrier of G
x * a is Element of the carrier of G
the multF of G . (x,a) is Element of the carrier of G
[x,a] is set
{x,a} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,a},{x}} is non empty finite V57() set
the multF of G . [x,a] is set
(x * a) * (B ") is Element of the carrier of G
the multF of G . ((x * a),(B ")) is Element of the carrier of G
[(x * a),(B ")] is set
{(x * a),(B ")} is non empty finite set
{(x * a)} is non empty trivial finite 1 -element set
{{(x * a),(B ")},{(x * a)}} is non empty finite V57() set
the multF of G . [(x * a),(B ")] 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 trivial finite 1 -element set
{{a,(B ")},{a}} is non empty finite V57() set
the multF of G . [a,(B ")] is set
x * (a * (B ")) is Element of the carrier of G
the multF of G . (x,(a * (B "))) is Element of the carrier of G
[x,(a * (B "))] is set
{x,(a * (B "))} is non empty finite set
{{x,(a * (B "))},{x}} is non empty finite V57() set
the multF of G . [x,(a * (B "))] is set
x " is Element of the carrier of G
(x ") * H is Element of the carrier of G
the multF of G . ((x "),H) is Element of the carrier of G
[(x "),H] is set
{(x "),H} is non empty finite set
{(x ")} is non empty trivial finite 1 -element set
{{(x "),H},{(x ")}} is non empty finite V57() set
the multF of G . [(x "),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
x is Element of the carrier of G
H * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (H,x) is Element of the carrier of G
[H,x] is set
{H,x} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,x},{H}} is non empty finite V57() set
the multF of G . [H,x] is set
y is non empty unital Group-like associative (G)
(G,y,(H * x)) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,(H * x),(G,y)) is Element of bool the carrier of G
{(H * x)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(H * x)},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(H * x)} & b₂ in (G,y) ) } is set
(G,y,H) is Element of bool the carrier of G
(G,H,(G,y)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,y)) 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 (G,y) ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{x},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in (G,y) ) } is set
(G,(G,y,H),(G,y,x)) 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 (G,y,x) ) } is set
y is set
B is Element of the carrier of G
(H * x) * B is Element of the carrier of G
the multF of G . ((H * x),B) is Element of the carrier of G
[(H * x),B] is set
{(H * x),B} is non empty finite set
{(H * x)} is non empty trivial finite 1 -element set
{{(H * x),B},{(H * x)}} is non empty finite V57() set
the multF of G . [(H * x),B] 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 V57() set
the multF of G . [H,(1_ G)] is set
(H * (1_ G)) * x is Element of the carrier of G
the multF of G . ((H * (1_ G)),x) is Element of the carrier of G
[(H * (1_ G)),x] is set
{(H * (1_ G)),x} is non empty finite set
{(H * (1_ G))} is non empty trivial finite 1 -element set
{{(H * (1_ G)),x},{(H * (1_ G))}} is non empty finite V57() set
the multF of G . [(H * (1_ G)),x] is set
((H * (1_ G)) * x) * B is Element of the carrier of G
the multF of G . (((H * (1_ G)) * x),B) is Element of the carrier of G
[((H * (1_ G)) * x),B] is set
{((H * (1_ G)) * x),B} is non empty finite set
{((H * (1_ G)) * x)} is non empty trivial finite 1 -element set
{{((H * (1_ G)) * x),B},{((H * (1_ G)) * x)}} is non empty finite V57() set
the multF of G . [((H * (1_ G)) * x),B] is set
x * B is Element of the carrier of G
the multF of G . (x,B) is Element of the carrier of G
[x,B] is set
{x,B} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,B},{x}} is non empty finite V57() set
the multF of G . [x,B] is set
(H * (1_ G)) * (x * B) is Element of the carrier of G
the multF of G . ((H * (1_ G)),(x * B)) is Element of the carrier of G
[(H * (1_ G)),(x * B)] is set
{(H * (1_ G)),(x * B)} is non empty finite set
{{(H * (1_ G)),(x * B)},{(H * (1_ G))}} is non empty finite V57() set
the multF of G . [(H * (1_ G)),(x * B)] 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
x is non empty unital Group-like associative (G)
(G,x) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
the carrier of x is non empty set
(G,x,H) is Element of bool the carrier of G
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,x)) 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 (G,x) ) } is set
(G,x,(H ")) is Element of bool the carrier of G
(G,(H "),(G,x)) is Element of bool the carrier of G
{(H ")} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(H ")},(G,x)) 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 (G,x) ) } is set
(G,(G,x,H),(G,x,(H "))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,H) & b₂ in (G,x,(H ")) ) } is set
(G,(G,x,(H ")),(G,x,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,(H ")) & b₂ in (G,x,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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{(H "),H},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),H] is set
(G,x,((H ") * H)) is Element of bool the carrier of G
(G,((H ") * H),(G,x)) is Element of bool the carrier of G
{((H ") * H)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{((H ") * H)},(G,x)) 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 (G,x) ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(G,x,(1_ G)) is Element of bool the carrier of G
(G,(1_ G),(G,x)) is Element of bool the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(1_ G)},(G,x)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(1_ G)} & b₂ in (G,x) ) } 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 trivial finite 1 -element set
{{H,(H ")},{H}} is non empty finite V57() set
the multF of G . [H,(H ")] is set
(G,x,(H * (H "))) is Element of bool the carrier of G
(G,(H * (H ")),(G,x)) is Element of bool the carrier of G
{(H * (H "))} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(H * (H "))},(G,x)) 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 (G,x) ) } 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 |^ 2 is Element of the carrier of G
power G is Relation-like [: the carrier of G,NAT:] -defined the carrier of G -valued Function-like V28([: the carrier of G,NAT:], the carrier of G) Element of bool [:[: the carrier of G,NAT:], the carrier of G:]
[: the carrier of G,NAT:] is non trivial non finite set
[:[: the carrier of G,NAT:], the carrier of G:] is non trivial non finite set
bool [:[: the carrier of G,NAT:], the carrier of G:] is non empty non trivial cup-closed diff-closed preBoolean non finite set
(power G) . (H,2) is set
[H,2] is set
{H,2} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,2},{H}} is non empty finite V57() set
(power G) . [H,2] is set
x is non empty unital Group-like associative (G)
(G,x,(H |^ 2)) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,(H |^ 2),(G,x)) is Element of bool the carrier of G
{(H |^ 2)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(H |^ 2)},(G,x)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(H |^ 2)} & b₂ in (G,x) ) } is set
(G,x,H) is Element of bool the carrier of G
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,x)) 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 (G,x) ) } is set
(G,(G,x,H),(G,x,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,H) & b₂ in (G,x,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 having_a_unity 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 cup-closed diff-closed preBoolean 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,H},{H}} is non empty finite V57() set
the multF of G . [H,H] is set
(G,x,(H * H)) is Element of bool the carrier of G
(G,(H * H),(G,x)) is Element of bool the carrier of G
{(H * H)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(H * H)},(G,x)) 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 (G,x) ) } 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
x is non empty unital Group-like associative (G)
(G,x,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {H} ) } is set
y 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{y,H},{y}} is non empty finite V57() set
the multF of G . [y,H] is set
y is set
y is Element of the carrier of G
H " 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{y,(H ")},{y}} is non empty finite V57() set
the multF of G . [y,(H ")] is set
(y * (H ")) * H is Element of the carrier of G
the multF of G . ((y * (H ")),H) is Element of the carrier of G
[(y * (H ")),H] is set
{(y * (H ")),H} is non empty finite set
{(y * (H "))} is non empty trivial finite 1 -element set
{{(y * (H ")),H},{(y * (H "))}} is non empty finite V57() set
the multF of G . [(y * (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 ")} is non empty trivial finite 1 -element set
{{(H "),H},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),H] is set
y * ((H ") * H) is Element of the carrier of G
the multF of G . (y,((H ") * H)) is Element of the carrier of G
[y,((H ") * H)] is set
{y,((H ") * H)} is non empty finite set
{{y,((H ") * H)},{y}} is non empty finite V57() set
the multF of G . [y,((H ") * H)] is set
1_ G is non being_of_order_0 Element of the carrier of G
y * (1_ G) is Element of the carrier of G
the multF of G . (y,(1_ G)) is Element of the carrier of G
[y,(1_ G)] is set
{y,(1_ G)} is non empty finite set
{{y,(1_ G)},{y}} is non empty finite V57() set
the multF of G . [y,(1_ G)] 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 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{(1_ G),H},{(1_ G)}} is non empty finite V57() 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
H " is Element of the carrier of G
x is Element of the carrier of G
x * (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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (x,(H ")) is Element of the carrier of G
[x,(H ")] is set
{x,(H ")} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,(H ")},{x}} is non empty finite V57() set
the multF of G . [x,(H ")] is set
y is non empty unital Group-like associative (G)
(G,y,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{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) & b₂ in {H} ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{x}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & b₂ in {x} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(G,y,(1_ G)) is Element of bool the carrier of G
(G,(1_ G),(G,y)) is Element of bool the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{(1_ G)}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & 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 trivial finite 1 -element set
{{H,(H ")},{H}} is non empty finite V57() set
the multF of G . [H,(H ")] is set
(G,y,(H * (H "))) is Element of bool the carrier of G
(G,(H * (H ")),(G,y)) is Element of bool the carrier of G
{(H * (H "))} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{(H * (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) & b₂ in {(H * (H "))} ) } is set
(G,(H "),(G,y,x)) is Element of bool the carrier of G
{(H ")} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y,x),{(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,x) & b₂ in {(H ")} ) } is set
(G,y,(x * (H "))) is Element of bool the carrier of G
(G,(x * (H ")),(G,y)) is Element of bool the carrier of G
{(x * (H "))} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{(x * (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) & b₂ in {(x * (H "))} ) } is set
(G,y,(x * (H "))) is Element of bool the carrier of G
(G,(x * (H ")),(G,y)) is Element of bool the carrier of G
{(x * (H "))} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{(x * (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) & b₂ in {(x * (H "))} ) } is set
(G,H,(G,y,(x * (H ")))) is Element of bool the carrier of G
(G,(G,y,(x * (H "))),{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,(x * (H "))) & b₂ in {H} ) } is set
(x * (H ")) * H is Element of the carrier of G
the multF of G . ((x * (H ")),H) is Element of the carrier of G
[(x * (H ")),H] is set
{(x * (H ")),H} is non empty finite set
{(x * (H "))} is non empty trivial finite 1 -element set
{{(x * (H ")),H},{(x * (H "))}} is non empty finite V57() set
the multF of G . [(x * (H ")),H] is set
(G,y,((x * (H ")) * H)) is Element of bool the carrier of G
(G,((x * (H ")) * H),(G,y)) is Element of bool the carrier of G
{((x * (H ")) * H)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{((x * (H ")) * 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) & b₂ in {((x * (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 ")} is non empty trivial finite 1 -element set
{{(H "),H},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),H] is set
x * ((H ") * H) is Element of the carrier of G
the multF of G . (x,((H ") * H)) is Element of the carrier of G
[x,((H ") * H)] is set
{x,((H ") * H)} is non empty finite set
{{x,((H ") * H)},{x}} is non empty finite V57() set
the multF of G . [x,((H ") * H)] is set
(G,y,(x * ((H ") * H))) is Element of bool the carrier of G
(G,(x * ((H ") * H)),(G,y)) is Element of bool the carrier of G
{(x * ((H ") * H))} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{(x * ((H ") * 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) & b₂ in {(x * ((H ") * H))} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
x * (1_ G) is Element of the carrier of G
the multF of G . (x,(1_ G)) is Element of the carrier of G
[x,(1_ G)] is set
{x,(1_ G)} is non empty finite set
{{x,(1_ G)},{x}} is non empty finite V57() set
the multF of G . [x,(1_ G)] is set
(G,y,(x * (1_ G))) is Element of bool the carrier of G
(G,(x * (1_ G)),(G,y)) is Element of bool the carrier of G
{(x * (1_ G))} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{(x * (1_ G))}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & b₂ in {(x * (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
x is Element of the carrier of G
y is non empty unital Group-like associative (G)
(G,y,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{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) & b₂ in {H} ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{x}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & b₂ in {x} ) } is set
y is set
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 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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 trivial finite 1 -element set
{{B,H},{B}} is non empty finite V57() set
the multF of G . [B,H] is set
B " is Element of the carrier of G
a is Element of the carrier of G
a * x is Element of the carrier of G
the multF of G . (a,x) is Element of the carrier of G
[a,x] is set
{a,x} is non empty finite set
{a} is non empty trivial finite 1 -element set
{{a,x},{a}} is non empty finite V57() set
the multF of G . [a,x] is set
(B ") * (a * x) is Element of the carrier of G
the multF of G . ((B "),(a * x)) is Element of the carrier of G
[(B "),(a * x)] is set
{(B "),(a * x)} is non empty finite set
{(B ")} is non empty trivial finite 1 -element set
{{(B "),(a * x)},{(B ")}} is non empty finite V57() set
the multF of G . [(B "),(a * x)] is set
(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 "),a},{(B ")}} is non empty finite V57() set
the multF of G . [(B "),a] is set
((B ") * a) * x is Element of the carrier of G
the multF of G . (((B ") * a),x) is Element of the carrier of G
[((B ") * a),x] is set
{((B ") * a),x} is non empty finite set
{((B ") * a)} is non empty trivial finite 1 -element set
{{((B ") * a),x},{((B ") * a)}} is non empty finite V57() set
the multF of G . [((B ") * a),x] is set
x " is Element of the carrier of G
H * (x ") is Element of the carrier of G
the multF of G . (H,(x ")) is Element of the carrier of G
[H,(x ")] is set
{H,(x ")} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,(x ")},{H}} is non empty finite V57() set
the multF of G . [H,(x ")] 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
x is Element of the carrier of G
y is non empty unital Group-like associative (G)
(G,y,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{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) & b₂ in {H} ) } is set
(G,x,(G,y,H)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y,H),{x}) 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 {x} ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
(G,(G,y),{x}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & b₂ in {x} ) } is set
(G,(G,y,H),(G,y,x)) 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 (G,y,x) ) } is set
y is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . ((1_ G),x) is Element of the carrier of G
[(1_ G),x] is set
{(1_ G),x} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),x},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),x] is set
H * x is Element of the carrier of G
the multF of G . (H,x) is Element of the carrier of G
[H,x] is set
{H,x} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,x},{H}} is non empty finite V57() set
the multF of G . [H,x] is set
(G,y,(H * x)) is Element of bool the carrier of G
(G,(H * x),(G,y)) is Element of bool the carrier of G
{(H * x)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{(H * x)}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & b₂ in {(H * x)} ) } is set
B is Element of the carrier of G
B * (H * x) is Element of the carrier of G
the multF of G . (B,(H * x)) is Element of the carrier of G
[B,(H * x)] is set
{B,(H * x)} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,(H * x)},{B}} is non empty finite V57() set
the multF of G . [B,(H * x)] is set
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,H},{B}} is non empty finite V57() set
the multF of G . [B,H] is set
(B * H) * x is Element of the carrier of G
the multF of G . ((B * H),x) is Element of the carrier of G
[(B * H),x] is set
{(B * H),x} is non empty finite set
{(B * H)} is non empty trivial finite 1 -element set
{{(B * H),x},{(B * H)}} is non empty finite V57() set
the multF of G . [(B * H),x] is set
(B * H) * ((1_ G) * x) is Element of the carrier of G
the multF of G . ((B * H),((1_ G) * x)) is Element of the carrier of G
[(B * H),((1_ G) * x)] is set
{(B * H),((1_ G) * x)} is non empty finite set
{{(B * H),((1_ G) * x)},{(B * H)}} is non empty finite V57() set
the multF of G . [(B * H),((1_ G) * x)] 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
x is non empty unital Group-like associative (G)
(G,x) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
the carrier of x is non empty set
(G,x,H) is Element of bool the carrier of G
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {H} ) } is set
(G,x,(H ")) is Element of bool the carrier of G
(G,(H "),(G,x)) is Element of bool the carrier of G
{(H ")} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x),{(H ")}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {(H ")} ) } is set
(G,(G,x,H),(G,x,(H "))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,H) & b₂ in (G,x,(H ")) ) } is set
(G,(G,x,(H ")),(G,x,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,(H ")) & b₂ in (G,x,H) ) } is set
(G,H,(G,x,(H "))) is Element of bool the carrier of G
(G,(G,x,(H ")),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,(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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{(H "),H},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),H] is set
(G,x,((H ") * H)) is Element of bool the carrier of G
(G,((H ") * H),(G,x)) is Element of bool the carrier of G
{((H ") * H)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x),{((H ") * H)}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {((H ") * H)} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(G,x,(1_ G)) is Element of bool the carrier of G
(G,(1_ G),(G,x)) is Element of bool the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x),{(1_ G)}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {(1_ G)} ) } is set
(G,(H "),(G,x,H)) is Element of bool the carrier of G
(G,(G,x,H),{(H ")}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,H) & 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 trivial finite 1 -element set
{{H,(H ")},{H}} is non empty finite V57() set
the multF of G . [H,(H ")] is set
(G,x,(H * (H "))) is Element of bool the carrier of G
(G,(H * (H ")),(G,x)) is Element of bool the carrier of G
{(H * (H "))} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x),{(H * (H "))}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {(H * (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
H |^ 2 is Element of the carrier of G
power G is Relation-like [: the carrier of G,NAT:] -defined the carrier of G -valued Function-like V28([: the carrier of G,NAT:], the carrier of G) Element of bool [:[: the carrier of G,NAT:], the carrier of G:]
[: the carrier of G,NAT:] is non trivial non finite set
[:[: the carrier of G,NAT:], the carrier of G:] is non trivial non finite set
bool [:[: the carrier of G,NAT:], the carrier of G:] is non empty non trivial cup-closed diff-closed preBoolean non finite set
(power G) . (H,2) is set
[H,2] is set
{H,2} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,2},{H}} is non empty finite V57() set
(power G) . [H,2] is set
x is non empty unital Group-like associative (G)
(G,x,(H |^ 2)) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,(H |^ 2),(G,x)) is Element of bool the carrier of G
{(H |^ 2)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x),{(H |^ 2)}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {(H |^ 2)} ) } is set
(G,x,H) is Element of bool the carrier of G
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {H} ) } is set
(G,(G,x,H),(G,x,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,H) & b₂ in (G,x,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 having_a_unity 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 cup-closed diff-closed preBoolean 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,H},{H}} is non empty finite V57() set
the multF of G . [H,H] is set
(G,H,(G,x,H)) is Element of bool the carrier of G
(G,(G,x,H),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,H) & b₂ in {H} ) } is set
(G,x,(H * H)) is Element of bool the carrier of G
(G,(H * H),(G,x)) is Element of bool the carrier of G
{(H * H)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x),{(H * H)}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & 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 cup-closed diff-closed preBoolean set
H is Element of the carrier of G
x is non empty unital Group-like associative (G)
(G,x,H) is Element of bool the carrier of G
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,x)) 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 (G,x) ) } is set
y is non empty unital Group-like associative (G)
(G,x,y) is non empty strict unital Group-like associative (G)
(G,(G,x,y),H) is Element of bool the carrier of G
(G,(G,x,y)) is Element of bool the carrier of G
the carrier of (G,x,y) is non empty set
(G,H,(G,(G,x,y))) is Element of bool the carrier of G
(G,{H},(G,(G,x,y))) 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 (G,(G,x,y)) ) } is set
(G,y,H) is Element of bool the carrier of G
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
(G,{H},(G,y)) 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 (G,y) ) } is set
(G,x,H) /\ (G,y,H) is Element of bool the carrier of G
y is set
B is Element of the carrier of G
H * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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} is non empty trivial finite 1 -element set
{{H,B},{H}} is non empty finite V57() set
the multF of G . [H,B] is set
y is set
B is Element of the carrier of G
H * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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} is non empty trivial finite 1 -element set
{{H,B},{H}} is non empty finite V57() set
the multF of G . [H,B] is set
B 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 V57() set
the multF of G . [H,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 cup-closed diff-closed preBoolean set
H is Element of the carrier of G
x is non empty unital Group-like associative (G)
(G,x,H) is Element of bool the carrier of G
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {H} ) } is set
y is non empty unital Group-like associative (G)
(G,x,y) is non empty strict unital Group-like associative (G)
(G,(G,x,y),H) is Element of bool the carrier of G
(G,(G,x,y)) is Element of bool the carrier of G
the carrier of (G,x,y) is non empty set
(G,H,(G,(G,x,y))) is Element of bool the carrier of G
(G,(G,(G,x,y)),{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,x,y)) & b₂ in {H} ) } is set
(G,y,H) is Element of bool the carrier of G
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
(G,(G,y),{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) & b₂ in {H} ) } is set
(G,x,H) /\ (G,y,H) is Element of bool the carrier of G
y is set
B is Element of the carrier of G
B * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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 trivial finite 1 -element set
{{B,H},{B}} is non empty finite V57() set
the multF of G . [B,H] is set
y is set
B is Element of the carrier of G
B * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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 trivial finite 1 -element set
{{B,H},{B}} is non empty finite V57() set
the multF of G . [B,H] is set
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 trivial finite 1 -element set
{{B,H},{B}} is non empty finite V57() set
the multF of G . [B,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
H " is Element of the carrier of G
x is non empty unital Group-like associative (G)
(G,x,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
the carrier of x is non empty set
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,x)) 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 (G,x) ) } is set
(G,(H "),(G,x,H)) is Element of bool the carrier of G
{(H ")} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,x,H),{(H ")}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x,H) & b₂ in {(H ")} ) } is set
the Element of (G,x,H) is Element of (G,x,H)
B 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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{a,(H ")},{a}} is non empty finite V57() set
the multF of G . [a,(H ")] is set
Y 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} is non empty trivial finite 1 -element set
{{H,Y},{H}} is non empty finite V57() set
the multF of G . [H,Y] is set
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 trivial finite 1 -element set
{{(Y "),(H ")},{(Y ")}} is non empty finite V57() set
the multF of G . [(Y "),(H ")] is set
(G,x,(H ")) is Element of bool the carrier of G
(G,(H "),(G,x)) is Element of bool the carrier of G
(G,(G,x),{(H ")}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {(H ")} ) } is set
B " is Element of the carrier of G
(H ") " is Element of the carrier of G
(H * Y) " is Element of the carrier of G
((H ") ") * ((H * Y) ") is Element of the carrier of G
the multF of G . (((H ") "),((H * Y) ")) is Element of the carrier of G
[((H ") "),((H * Y) ")] is set
{((H ") "),((H * Y) ")} is non empty finite set
{((H ") ")} is non empty trivial finite 1 -element set
{{((H ") "),((H * Y) ")},{((H ") ")}} is non empty finite V57() set
the multF of G . [((H ") "),((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 ") * (H "))},{H}} is non empty finite V57() set
the multF of G . [H,((Y ") * (H "))] is set
(G,H,(G,x,(H "))) is Element of bool the carrier of G
(G,{H},(G,x,(H "))) 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 (G,x,(H ")) ) } is set
B is Element of the carrier of G
a 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 trivial finite 1 -element set
{{Y,(H ")},{Y}} is non empty finite V57() set
the multF of G . [Y,(H ")] is set
y 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 V57() set
the multF of G . [H,y] is set
Z is Element of the carrier of G
H * Z is Element of the carrier of G
the multF of G . (H,Z) is Element of the carrier of G
[H,Z] is set
{H,Z} is non empty finite set
{{H,Z},{H}} is non empty finite V57() set
the multF of G . [H,Z] is set
c is Element of the carrier of G
c * (H ") is Element of the carrier of G
the multF of G . (c,(H ")) is Element of the carrier of G
[c,(H ")] is set
{c,(H ")} is non empty finite set
{c} is non empty trivial finite 1 -element set
{{c,(H ")},{c}} is non empty finite V57() set
the multF of G . [c,(H ")] is set
y * c is Element of the carrier of G
the multF of G . (y,c) is Element of the carrier of G
[y,c] is set
{y,c} is non empty finite set
{y} is non empty trivial finite 1 -element set
{{y,c},{y}} is non empty finite V57() set
the multF of G . [y,c] is set
H * (y * c) is Element of the carrier of G
the multF of G . (H,(y * c)) is Element of the carrier of G
[H,(y * c)] is set
{H,(y * c)} is non empty finite set
{{H,(y * c)},{H}} is non empty finite V57() set
the multF of G . [H,(y * c)] is set
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 trivial finite 1 -element set
{{B,a},{B}} is non empty finite V57() set
the multF of G . [B,a] 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 * (H "))},{H}} is non empty finite V57() set
the multF of G . [H,(c * (H "))] is set
(H ") * (H * (c * (H "))) is Element of the carrier of G
the multF of G . ((H "),(H * (c * (H ")))) is Element of the carrier of G
[(H "),(H * (c * (H ")))] is set
{(H "),(H * (c * (H ")))} is non empty finite set
{(H ")} is non empty trivial finite 1 -element set
{{(H "),(H * (c * (H ")))},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),(H * (c * (H ")))] is set
(H * y) * ((H ") * (H * (c * (H ")))) is Element of the carrier of G
the multF of G . ((H * y),((H ") * (H * (c * (H "))))) is Element of the carrier of G
[(H * y),((H ") * (H * (c * (H "))))] is set
{(H * y),((H ") * (H * (c * (H "))))} is non empty finite set
{(H * y)} is non empty trivial finite 1 -element set
{{(H * y),((H ") * (H * (c * (H "))))},{(H * y)}} is non empty finite V57() set
the multF of G . [(H * y),((H ") * (H * (c * (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 V57() set
the multF of G . [(H "),H] is set
((H ") * H) * (c * (H ")) is Element of the carrier of G
the multF of G . (((H ") * H),(c * (H "))) is Element of the carrier of G
[((H ") * H),(c * (H "))] is set
{((H ") * H),(c * (H "))} is non empty finite set
{((H ") * H)} is non empty trivial finite 1 -element set
{{((H ") * H),(c * (H "))},{((H ") * H)}} is non empty finite V57() set
the multF of G . [((H ") * H),(c * (H "))] is set
(H * y) * (((H ") * H) * (c * (H "))) is Element of the carrier of G
the multF of G . ((H * y),(((H ") * H) * (c * (H ")))) is Element of the carrier of G
[(H * y),(((H ") * H) * (c * (H ")))] is set
{(H * y),(((H ") * H) * (c * (H ")))} is non empty finite set
{{(H * y),(((H ") * H) * (c * (H ")))},{(H * y)}} is non empty finite V57() set
the multF of G . [(H * y),(((H ") * H) * (c * (H ")))] is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * (c * (H ")) is Element of the carrier of G
the multF of G . ((1_ G),(c * (H "))) is Element of the carrier of G
[(1_ G),(c * (H "))] is set
{(1_ G),(c * (H "))} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),(c * (H "))},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),(c * (H "))] is set
(H * y) * ((1_ G) * (c * (H "))) is Element of the carrier of G
the multF of G . ((H * y),((1_ G) * (c * (H ")))) is Element of the carrier of G
[(H * y),((1_ G) * (c * (H ")))] is set
{(H * y),((1_ G) * (c * (H ")))} is non empty finite set
{{(H * y),((1_ G) * (c * (H ")))},{(H * y)}} is non empty finite V57() set
the multF of G . [(H * y),((1_ G) * (c * (H ")))] is set
(H * y) * (c * (H ")) is Element of the carrier of G
the multF of G . ((H * y),(c * (H "))) is Element of the carrier of G
[(H * y),(c * (H "))] is set
{(H * y),(c * (H "))} is non empty finite set
{{(H * y),(c * (H "))},{(H * y)}} is non empty finite V57() set
the multF of G . [(H * y),(c * (H "))] is set
(H * y) * c is Element of the carrier of G
the multF of G . ((H * y),c) is Element of the carrier of G
[(H * y),c] is set
{(H * y),c} is non empty finite set
{{(H * y),c},{(H * y)}} is non empty finite V57() set
the multF of G . [(H * y),c] is set
((H * y) * c) * (H ") is Element of the carrier of G
the multF of G . (((H * y) * c),(H ")) is Element of the carrier of G
[((H * y) * c),(H ")] is set
{((H * y) * c),(H ")} is non empty finite set
{((H * y) * c)} is non empty trivial finite 1 -element set
{{((H * y) * c),(H ")},{((H * y) * c)}} is non empty finite V57() set
the multF of G . [((H * y) * c),(H ")] is set
(H * (y * c)) * (H ") is Element of the carrier of G
the multF of G . ((H * (y * c)),(H ")) is Element of the carrier of G
[(H * (y * c)),(H ")] is set
{(H * (y * c)),(H ")} is non empty finite set
{(H * (y * c))} is non empty trivial finite 1 -element set
{{(H * (y * c)),(H ")},{(H * (y * c))}} is non empty finite V57() set
the multF of G . [(H * (y * c)),(H ")] is set
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 trivial finite 1 -element set
{{B,(H ")},{B}} is non empty finite V57() set
the multF of G . [B,(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
x is Element of the carrier of G
y is non empty unital Group-like associative (G)
(G,y,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,y)) 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 (G,y) ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{x},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in (G,y) ) } is set
H " is Element of the carrier of G
x * (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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (x,(H ")) is Element of the carrier of G
[x,(H ")] is set
{x,(H ")} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,(H ")},{x}} is non empty finite V57() set
the multF of G . [x,(H ")] is set
y is set
B is Element of the carrier of G
(x * (H ")) * B is Element of the carrier of G
the multF of G . ((x * (H ")),B) is Element of the carrier of G
[(x * (H ")),B] is set
{(x * (H ")),B} is non empty finite set
{(x * (H "))} is non empty trivial finite 1 -element set
{{(x * (H ")),B},{(x * (H "))}} is non empty finite V57() set
the multF of G . [(x * (H ")),B] is set
B is set
y is Relation-like Function-like set
dom y is set
rng y is set
B is set
B is set
y . B is set
a is Element of the carrier of G
(x * (H ")) * a is Element of the carrier of G
the multF of G . ((x * (H ")),a) is Element of the carrier of G
[(x * (H ")),a] is set
{(x * (H ")),a} is non empty finite set
{(x * (H "))} is non empty trivial finite 1 -element set
{{(x * (H ")),a},{(x * (H "))}} is non empty finite V57() set
the multF of G . [(x * (H ")),a] is set
Y 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} is non empty trivial finite 1 -element set
{{H,Y},{H}} is non empty finite V57() set
the multF of G . [H,Y] is set
(x * (H ")) * H is Element of the carrier of G
the multF of G . ((x * (H ")),H) is Element of the carrier of G
[(x * (H ")),H] is set
{(x * (H ")),H} is non empty finite set
{{(x * (H ")),H},{(x * (H "))}} is non empty finite V57() set
the multF of G . [(x * (H ")),H] is set
((x * (H ")) * H) * Y is Element of the carrier of G
the multF of G . (((x * (H ")) * H),Y) is Element of the carrier of G
[((x * (H ")) * H),Y] is set
{((x * (H ")) * H),Y} is non empty finite set
{((x * (H ")) * H)} is non empty trivial finite 1 -element set
{{((x * (H ")) * H),Y},{((x * (H ")) * H)}} is non empty finite V57() set
the multF of G . [((x * (H ")) * H),Y] 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 trivial finite 1 -element set
{{(H "),H},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),H] is set
x * ((H ") * H) is Element of the carrier of G
the multF of G . (x,((H ") * H)) is Element of the carrier of G
[x,((H ") * H)] is set
{x,((H ") * H)} is non empty finite set
{{x,((H ") * H)},{x}} is non empty finite V57() set
the multF of G . [x,((H ") * H)] is set
(x * ((H ") * H)) * Y is Element of the carrier of G
the multF of G . ((x * ((H ") * H)),Y) is Element of the carrier of G
[(x * ((H ") * H)),Y] is set
{(x * ((H ") * H)),Y} is non empty finite set
{(x * ((H ") * H))} is non empty trivial finite 1 -element set
{{(x * ((H ") * H)),Y},{(x * ((H ") * H))}} is non empty finite V57() set
the multF of G . [(x * ((H ") * H)),Y] is set
1_ G is non being_of_order_0 Element of the carrier of G
x * (1_ G) is Element of the carrier of G
the multF of G . (x,(1_ G)) is Element of the carrier of G
[x,(1_ G)] is set
{x,(1_ G)} is non empty finite set
{{x,(1_ G)},{x}} is non empty finite V57() set
the multF of G . [x,(1_ G)] is set
(x * (1_ G)) * Y is Element of the carrier of G
the multF of G . ((x * (1_ G)),Y) is Element of the carrier of G
[(x * (1_ G)),Y] is set
{(x * (1_ G)),Y} is non empty finite set
{(x * (1_ G))} is non empty trivial finite 1 -element set
{{(x * (1_ G)),Y},{(x * (1_ G))}} is non empty finite V57() set
the multF of G . [(x * (1_ G)),Y] is set
x * Y is Element of the carrier of G
the multF of G . (x,Y) is Element of the carrier of G
[x,Y] is set
{x,Y} is non empty finite set
{{x,Y},{x}} is non empty finite V57() set
the multF of G . [x,Y] is set
B is set
B is Element of the carrier of G
x * B is Element of the carrier of G
the multF of G . (x,B) is Element of the carrier of G
[x,B] is set
{x,B} is non empty finite set
{{x,B},{x}} is non empty finite V57() set
the multF of G . [x,B] is set
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} is non empty trivial finite 1 -element set
{{H,B},{H}} is non empty finite V57() set
the multF of G . [H,B] is set
y . (H * B) is set
(x * (H ")) * H is Element of the carrier of G
the multF of G . ((x * (H ")),H) is Element of the carrier of G
[(x * (H ")),H] is set
{(x * (H ")),H} is non empty finite set
{(x * (H "))} is non empty trivial finite 1 -element set
{{(x * (H ")),H},{(x * (H "))}} is non empty finite V57() set
the multF of G . [(x * (H ")),H] is set
((x * (H ")) * H) * B is Element of the carrier of G
the multF of G . (((x * (H ")) * H),B) is Element of the carrier of G
[((x * (H ")) * H),B] is set
{((x * (H ")) * H),B} is non empty finite set
{((x * (H ")) * H)} is non empty trivial finite 1 -element set
{{((x * (H ")) * H),B},{((x * (H ")) * H)}} is non empty finite V57() set
the multF of G . [((x * (H ")) * H),B] 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 trivial finite 1 -element set
{{(H "),H},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),H] is set
x * ((H ") * H) is Element of the carrier of G
the multF of G . (x,((H ") * H)) is Element of the carrier of G
[x,((H ") * H)] is set
{x,((H ") * H)} is non empty finite set
{{x,((H ") * H)},{x}} is non empty finite V57() set
the multF of G . [x,((H ") * H)] is set
(x * ((H ") * H)) * B is Element of the carrier of G
the multF of G . ((x * ((H ") * H)),B) is Element of the carrier of G
[(x * ((H ") * H)),B] is set
{(x * ((H ") * H)),B} is non empty finite set
{(x * ((H ") * H))} is non empty trivial finite 1 -element set
{{(x * ((H ") * H)),B},{(x * ((H ") * H))}} is non empty finite V57() set
the multF of G . [(x * ((H ") * H)),B] is set
1_ G is non being_of_order_0 Element of the carrier of G
x * (1_ G) is Element of the carrier of G
the multF of G . (x,(1_ G)) is Element of the carrier of G
[x,(1_ G)] is set
{x,(1_ G)} is non empty finite set
{{x,(1_ G)},{x}} is non empty finite V57() set
the multF of G . [x,(1_ G)] is set
(x * (1_ G)) * B is Element of the carrier of G
the multF of G . ((x * (1_ G)),B) is Element of the carrier of G
[(x * (1_ G)),B] is set
{(x * (1_ G)),B} is non empty finite set
{(x * (1_ G))} is non empty trivial finite 1 -element set
{{(x * (1_ G)),B},{(x * (1_ G))}} is non empty finite V57() set
the multF of G . [(x * (1_ G)),B] is set
a is Element of the carrier of G
(x * (H ")) * a is Element of the carrier of G
the multF of G . ((x * (H ")),a) is Element of the carrier of G
[(x * (H ")),a] is set
{(x * (H ")),a} is non empty finite set
{{(x * (H ")),a},{(x * (H "))}} is non empty finite V57() set
the multF of G . [(x * (H ")),a] is set
B is set
y . B is set
B is set
y . B is set
a is Element of the carrier of G
(x * (H ")) * a is Element of the carrier of G
the multF of G . ((x * (H ")),a) is Element of the carrier of G
[(x * (H ")),a] is set
{(x * (H ")),a} is non empty finite set
{(x * (H "))} is non empty trivial finite 1 -element set
{{(x * (H ")),a},{(x * (H "))}} is non empty finite V57() set
the multF of G . [(x * (H ")),a] is set
Y is Element of the carrier of G
(x * (H ")) * Y is Element of the carrier of G
the multF of G . ((x * (H ")),Y) is Element of the carrier of G
[(x * (H ")),Y] is set
{(x * (H ")),Y} is non empty finite set
{{(x * (H ")),Y},{(x * (H "))}} is non empty finite V57() set
the multF of G . [(x * (H ")),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
x is Element of the carrier of G
y is non empty unital Group-like associative (G)
(G,y,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,y)) 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 (G,y) ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{x}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & b₂ in {x} ) } is set
H " is Element of the carrier of G
y is set
B is Element of the carrier of G
(H ") * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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 ")} is non empty trivial finite 1 -element set
{{(H "),B},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),B] is set
((H ") * B) * x is Element of the carrier of G
the multF of G . (((H ") * B),x) is Element of the carrier of G
[((H ") * B),x] is set
{((H ") * B),x} is non empty finite set
{((H ") * B)} is non empty trivial finite 1 -element set
{{((H ") * B),x},{((H ") * B)}} is non empty finite V57() set
the multF of G . [((H ") * B),x] is set
B is set
y is Relation-like Function-like set
dom y is set
rng y is set
B is set
B is set
y . B is set
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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{(H "),a},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),a] is set
((H ") * a) * x is Element of the carrier of G
the multF of G . (((H ") * a),x) is Element of the carrier of G
[((H ") * a),x] is set
{((H ") * a),x} is non empty finite set
{((H ") * a)} is non empty trivial finite 1 -element set
{{((H ") * a),x},{((H ") * a)}} is non empty finite V57() set
the multF of G . [((H ") * a),x] is set
Y 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} is non empty trivial finite 1 -element set
{{H,Y},{H}} is non empty finite V57() set
the multF of G . [H,Y] 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 V57() set
the multF of G . [(H "),H] is set
((H ") * H) * Y is Element of the carrier of G
the multF of G . (((H ") * H),Y) is Element of the carrier of G
[((H ") * H),Y] is set
{((H ") * H),Y} is non empty finite set
{((H ") * H)} is non empty trivial finite 1 -element set
{{((H ") * H),Y},{((H ") * H)}} is non empty finite V57() set
the multF of G . [((H ") * H),Y] is set
(((H ") * H) * Y) * x is Element of the carrier of G
the multF of G . ((((H ") * H) * Y),x) is Element of the carrier of G
[(((H ") * H) * Y),x] is set
{(((H ") * H) * Y),x} is non empty finite set
{(((H ") * H) * Y)} is non empty trivial finite 1 -element set
{{(((H ") * H) * Y),x},{(((H ") * H) * Y)}} is non empty finite V57() set
the multF of G . [(((H ") * H) * Y),x] is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * Y is Element of the carrier of G
the multF of G . ((1_ G),Y) is Element of the carrier of G
[(1_ G),Y] is set
{(1_ G),Y} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),Y},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),Y] is set
((1_ G) * Y) * x is Element of the carrier of G
the multF of G . (((1_ G) * Y),x) is Element of the carrier of G
[((1_ G) * Y),x] is set
{((1_ G) * Y),x} is non empty finite set
{((1_ G) * Y)} is non empty trivial finite 1 -element set
{{((1_ G) * Y),x},{((1_ G) * Y)}} is non empty finite V57() set
the multF of G . [((1_ G) * Y),x] is set
Y * x is Element of the carrier of G
the multF of G . (Y,x) is Element of the carrier of G
[Y,x] is set
{Y,x} is non empty finite set
{Y} is non empty trivial finite 1 -element set
{{Y,x},{Y}} is non empty finite V57() set
the multF of G . [Y,x] is set
B is set
B is Element of the carrier of G
B * x is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (B,x) is Element of the carrier of G
[B,x] is set
{B,x} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,x},{B}} is non empty finite V57() set
the multF of G . [B,x] is set
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} is non empty trivial finite 1 -element set
{{H,B},{H}} is non empty finite V57() set
the multF of G . [H,B] is set
y . (H * B) 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 trivial finite 1 -element set
{{(H "),H},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),H] is set
((H ") * H) * B is Element of the carrier of G
the multF of G . (((H ") * H),B) is Element of the carrier of G
[((H ") * H),B] is set
{((H ") * H),B} is non empty finite set
{((H ") * H)} is non empty trivial finite 1 -element set
{{((H ") * H),B},{((H ") * H)}} is non empty finite V57() set
the multF of G . [((H ") * H),B] is set
(((H ") * H) * B) * x is Element of the carrier of G
the multF of G . ((((H ") * H) * B),x) is Element of the carrier of G
[(((H ") * H) * B),x] is set
{(((H ") * H) * B),x} is non empty finite set
{(((H ") * H) * B)} is non empty trivial finite 1 -element set
{{(((H ") * H) * B),x},{(((H ") * H) * B)}} is non empty finite V57() set
the multF of G . [(((H ") * H) * B),x] is set
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * B is Element of the carrier of G
the multF of G . ((1_ G),B) is Element of the carrier of G
[(1_ G),B] is set
{(1_ G),B} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),B},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),B] is set
((1_ G) * B) * x is Element of the carrier of G
the multF of G . (((1_ G) * B),x) is Element of the carrier of G
[((1_ G) * B),x] is set
{((1_ G) * B),x} is non empty finite set
{((1_ G) * B)} is non empty trivial finite 1 -element set
{{((1_ G) * B),x},{((1_ G) * B)}} is non empty finite V57() set
the multF of G . [((1_ G) * B),x] is set
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 "),a},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),a] is set
((H ") * a) * x is Element of the carrier of G
the multF of G . (((H ") * a),x) is Element of the carrier of G
[((H ") * a),x] is set
{((H ") * a),x} is non empty finite set
{((H ") * a)} is non empty trivial finite 1 -element set
{{((H ") * a),x},{((H ") * a)}} is non empty finite V57() set
the multF of G . [((H ") * a),x] is set
B is set
y . B is set
B is set
y . B is set
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 having_a_unity 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 cup-closed diff-closed preBoolean 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 trivial finite 1 -element set
{{(H "),a},{(H ")}} is non empty finite V57() set
the multF of G . [(H "),a] is set
((H ") * a) * x is Element of the carrier of G
the multF of G . (((H ") * a),x) is Element of the carrier of G
[((H ") * a),x] is set
{((H ") * a),x} is non empty finite set
{((H ") * a)} is non empty trivial finite 1 -element set
{{((H ") * a),x},{((H ") * a)}} is non empty finite V57() set
the multF of G . [((H ") * a),x] is set
Y 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 V57() set
the multF of G . [(H "),Y] is set
((H ") * Y) * x is Element of the carrier of G
the multF of G . (((H ") * Y),x) is Element of the carrier of G
[((H ") * Y),x] is set
{((H ") * Y),x} is non empty finite set
{((H ") * Y)} is non empty trivial finite 1 -element set
{{((H ") * Y),x},{((H ") * Y)}} is non empty finite V57() set
the multF of G . [((H ") * Y),x] 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
x is Element of the carrier of G
y is non empty unital Group-like associative (G)
(G,y,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,y) is Element of bool the carrier of G
the carrier of y is non empty set
(G,H,(G,y)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{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) & b₂ in {H} ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
{x} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,y),{x}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,y) & b₂ in {x} ) } is set
(G,y,x) is Element of bool the carrier of G
(G,x,(G,y)) is Element of bool the carrier of G
(G,{x},(G,y)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {x} & b₂ in (G,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
x is non empty unital Group-like associative (G)
(G,x) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
the carrier of x is non empty set
(G,x,H) is Element of bool the carrier of G
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,x)) 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 (G,x) ) } is set
(G,x,H) is Element of bool the carrier of G
(G,H,(G,x)) is Element of bool the carrier of G
(G,(G,x),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {H} ) } is set
1_ G is non being_of_order_0 Element of the carrier of G
(G,x,(1_ G)) is Element of bool the carrier of G
(G,(1_ G),(G,x)) is Element of bool the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(1_ G)},(G,x)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(1_ G)} & b₂ in (G,x) ) } is set
(G,x,(1_ G)) is Element of bool the carrier of G
(G,(1_ G),(G,x)) is Element of bool the carrier of G
(G,(G,x),{(1_ G)}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & 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
x is non empty unital Group-like associative (G)
card x is V20() cardinal set
the carrier of x is non empty set
card the carrier of x is non empty V20() cardinal set
(G,x,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,x)) 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 (G,x) ) } is set
card (G,x,H) is V20() cardinal set
(G,x,H) is Element of bool the carrier of G
(G,H,(G,x)) is Element of bool the carrier of G
(G,(G,x),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {H} ) } is set
card (G,x,H) is V20() 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
x is non empty finite unital Group-like associative (G)
(G,x,H) is Element of bool the carrier of G
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
(G,x) is Element of bool the carrier of G
the carrier of x is non empty finite set
(G,H,(G,x)) is Element of bool the carrier of G
{H} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{H},(G,x)) 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 (G,x) ) } is set
(G,x,H) is Element of bool the carrier of G
(G,H,(G,x)) is Element of bool the carrier of G
(G,(G,x),{H}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,x) & b₂ in {H} ) } is set
card x is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
card the carrier of x is non empty V20() V24() finite cardinal set
y is finite set
card y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
y is finite set
card y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
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 cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
H is non empty unital Group-like associative (G)
x is Element of bool (bool the carrier of G)
x is Element of bool (bool the carrier of G)
y is Element of bool (bool the carrier of G)
x is Element of bool (bool the carrier of G)
x is Element of bool (bool the carrier of G)
y is Element of bool (bool the carrier of G)
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G,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 cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
(G,H) is Element of bool (bool the carrier of G)
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G,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 cup-closed diff-closed preBoolean set
the carrier of H is non empty set
(G,H) is Element of bool (bool the carrier of G)
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
(G,H) is Element of bool (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
(G,(1_ G),(G,H)) is Element of bool the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(1_ G)},(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 (G,H) ) } is set
(G,H,(1_ G)) is Element of bool the carrier of G
(G,(1_ G),(G,H)) is Element of bool the carrier of G
(G,(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 (G,H) & b₂ in {(1_ G)} ) } is set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G,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 cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
(G,H) is Element of bool (bool the carrier of G)
x is set
y is Element of the carrier of G
(G,H,y) is Element of bool the carrier of G
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,y,(G,H)) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{y},(G,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 (G,H) ) } is set
y " is Element of the carrier of G
(G,H,(y ")) is Element of bool the carrier of G
(G,(y "),(G,H)) is Element of bool the carrier of G
{(y ")} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(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 (G,H) & b₂ in {(y ")} ) } is set
y is set
x is Relation-like Function-like set
dom x is set
rng x is set
y is set
y is set
x . y is set
B is Element of the carrier of G
(G,H,B) is Element of bool the carrier of G
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,B,(G,H)) is Element of bool the carrier of G
{B} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{B},(G,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 (G,H) ) } is set
B " is Element of the carrier of G
(G,H,(B ")) is Element of bool the carrier of G
(G,(B "),(G,H)) is Element of bool the carrier of G
{(B ")} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(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 (G,H) & b₂ in {(B ")} ) } is set
y is set
y is Element of bool the carrier of G
B is Element of the carrier of G
(G,H,B) is Element of bool the carrier of G
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,B,(G,H)) is Element of bool the carrier of G
{B} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(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 (G,H) & b₂ in {B} ) } is set
B " is Element of the carrier of G
(G,H,(B ")) is Element of bool the carrier of G
(G,(B "),(G,H)) is Element of bool the carrier of G
{(B ")} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(B ")},(G,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 (G,H) ) } is set
x . (G,H,(B ")) is set
B is Element of the carrier of G
(G,H,B) is Element of bool the carrier of G
(G,B,(G,H)) is Element of bool the carrier of G
{B} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{B},(G,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 (G,H) ) } is set
B " is Element of the carrier of G
(G,H,(B ")) is Element of bool the carrier of G
(G,(B "),(G,H)) is Element of bool the carrier of G
{(B ")} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(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 (G,H) & b₂ in {(B ")} ) } is set
(B ") * (B ") is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . ((B "),(B ")) is Element of the carrier of G
[(B "),(B ")] is set
{(B "),(B ")} is non empty finite set
{(B ")} is non empty trivial finite 1 -element set
{{(B "),(B ")},{(B ")}} is non empty finite V57() set
the multF of G . [(B "),(B ")] is set
y is set
x . y is set
y is set
x . y is set
B is Element of the carrier of G
(G,H,B) is Element of bool the carrier of G
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,B,(G,H)) is Element of bool the carrier of G
{B} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{B},(G,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 (G,H) ) } is set
B " is Element of the carrier of G
(G,H,(B ")) is Element of bool the carrier of G
(G,(B "),(G,H)) is Element of bool the carrier of G
{(B ")} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(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 (G,H) & b₂ in {(B ")} ) } is set
B is Element of the carrier of G
(G,H,B) is Element of bool the carrier of G
(G,B,(G,H)) is Element of bool the carrier of G
{B} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{B},(G,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 (G,H) ) } is set
B " is Element of the carrier of G
(G,H,(B ")) is Element of bool the carrier of G
(G,(B "),(G,H)) is Element of bool the carrier of G
{(B ")} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(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 (G,H) & b₂ in {(B ")} ) } is set
(B ") " is Element of the carrier of G
(B ") * ((B ") ") is Element of the carrier of G
the multF of G is Relation-like [: the carrier of G, the carrier of G:] -defined the carrier of G -valued Function-like V28([: the carrier of G, the carrier of G:], the carrier of G) associative having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . ((B "),((B ") ")) is Element of the carrier of G
[(B "),((B ") ")] is set
{(B "),((B ") ")} is non empty finite set
{(B ")} is non empty trivial finite 1 -element set
{{(B "),((B ") ")},{(B ")}} is non empty finite V57() set
the multF of G . [(B "),((B ") ")] 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 (G)
(G,H) is Element of bool (bool the carrier of G)
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
union (G,H) is Element of bool the carrier of G
(G,H) is Element of bool (bool the carrier of G)
union (G,H) is Element of bool the carrier of G
the carrier of H is non empty set
the Element of the carrier of H is Element of the carrier of H
y is set
B is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
B * (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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (B,(1_ G)) is Element of the carrier of G
[B,(1_ G)] is set
{B,(1_ G)} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,(1_ G)},{B}} is non empty finite V57() set
the multF of G . [B,(1_ G)] is set
y is Element of the carrier of G
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 ")} is non empty trivial finite 1 -element set
{{(y "),y},{(y ")}} is non empty finite V57() set
the multF of G . [(y "),y] is set
B * ((y ") * y) is Element of the carrier of G
the multF of G . (B,((y ") * y)) is Element of the carrier of G
[B,((y ") * y)] is set
{B,((y ") * y)} is non empty finite set
{{B,((y ") * y)},{B}} is non empty finite V57() set
the multF of G . [B,((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,(y ")},{B}} is non empty finite V57() set
the multF of G . [B,(y ")] is set
(B * (y ")) * y is Element of the carrier of G
the multF of G . ((B * (y ")),y) is Element of the carrier of G
[(B * (y ")),y] is set
{(B * (y ")),y} is non empty finite set
{(B * (y "))} is non empty trivial finite 1 -element set
{{(B * (y ")),y},{(B * (y "))}} is non empty finite V57() set
the multF of G . [(B * (y ")),y] is set
(G,H,(B * (y "))) is Element of bool the carrier of G
(G,H) is Element of bool the carrier of G
(G,(B * (y ")),(G,H)) is Element of bool the carrier of G
{(B * (y "))} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{(B * (y "))},(G,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {(B * (y "))} & b₂ in (G,H) ) } is set
the carrier of H is non empty set
the Element of the carrier of H is Element of the carrier of H
y is set
B is Element of the carrier of G
1_ G is non being_of_order_0 Element of the carrier of G
(1_ G) * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . ((1_ G),B) is Element of the carrier of G
[(1_ G),B] is set
{(1_ G),B} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G),B},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G),B] is set
y is Element of the carrier of G
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} is non empty trivial finite 1 -element set
{{y,(y ")},{y}} is non empty finite V57() set
the multF of G . [y,(y ")] is set
(y * (y ")) * B is Element of the carrier of G
the multF of G . ((y * (y ")),B) is Element of the carrier of G
[(y * (y ")),B] is set
{(y * (y ")),B} is non empty finite set
{(y * (y "))} is non empty trivial finite 1 -element set
{{(y * (y ")),B},{(y * (y "))}} is non empty finite V57() set
the multF of G . [(y * (y ")),B] is set
(y ") * B is Element of the carrier of G
the multF of G . ((y "),B) is Element of the carrier of G
[(y "),B] is set
{(y "),B} is non empty finite set
{(y ")} is non empty trivial finite 1 -element set
{{(y "),B},{(y ")}} is non empty finite V57() set
the multF of G . [(y "),B] is set
y * ((y ") * B) is Element of the carrier of G
the multF of G . (y,((y ") * B)) is Element of the carrier of G
[y,((y ") * B)] is set
{y,((y ") * B)} is non empty finite set
{{y,((y ") * B)},{y}} is non empty finite V57() set
the multF of G . [y,((y ") * B)] is set
(G,H,((y ") * B)) is Element of bool the carrier of G
(G,H) is Element of bool the carrier of G
(G,((y ") * B),(G,H)) is Element of bool the carrier of G
{((y ") * B)} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,H),{((y ") * B)}) 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 {((y ") * B)} ) } is set
G is non empty unital Group-like associative multMagma
(G) is non empty finite strict unital Group-like associative (G)
(G,(G)) 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 cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
{ {b₁} where b₁ is Element of the carrier of G : verum } is set
x is set
y is Element of bool the carrier of G
y is Element of the carrier of G
(G,(G),y) is Element of bool the carrier of G
(G,(G)) is Element of bool the carrier of G
the carrier of (G) is non empty finite set
(G,y,(G,(G))) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{y},(G,(G))) 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 (G,(G)) ) } is set
x is set
y is Element of the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G),y) is Element of bool the carrier of G
(G,(G)) is Element of bool the carrier of G
the carrier of (G) is non empty finite set
(G,y,(G,(G))) is Element of bool the carrier of G
(G,{y},(G,(G))) 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 (G,(G)) ) } is set
G is non empty unital Group-like associative multMagma
(G) is non empty finite strict unital Group-like associative (G)
(G,(G)) 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 cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
{ {b₁} where b₁ is Element of the carrier of G : verum } is set
x is set
y is Element of bool the carrier of G
y is Element of the carrier of G
(G,(G),y) is Element of bool the carrier of G
(G,(G)) is Element of bool the carrier of G
the carrier of (G) is non empty finite set
(G,y,(G,(G))) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,(G)),{y}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,(G)) & b₂ in {y} ) } is set
x is set
y is Element of the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G),y) is Element of bool the carrier of G
(G,(G)) is Element of bool the carrier of G
the carrier of (G) is non empty finite set
(G,y,(G,(G))) is Element of bool the carrier of G
(G,(G,(G)),{y}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,(G)) & b₂ in {y} ) } is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
{ {b₁} where b₁ is Element of the carrier of G : verum } is set
(G) is non empty finite strict unital Group-like associative (G)
H is non empty strict unital Group-like associative (G)
(G,H) is Element of bool (bool the carrier of G)
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
the carrier of H is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
the Element of the carrier of G is Element of the carrier of G
y is set
y is Element of the carrier of H
(G,H, the Element of the carrier of G) is Element of bool the carrier of G
(G,H) is Element of bool the carrier of G
(G, the Element of the carrier of G,(G,H)) is Element of bool the carrier of G
{ the Element of the carrier of G} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{ the Element of the carrier of G},(G,H)) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in { the Element of the carrier of G} & b₂ in (G,H) ) } is set
B is Element of the carrier of G
{B} is non empty trivial finite 1 -element Element of bool the carrier of G
B is Element of the carrier of G
the Element of the carrier of G * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . ( the Element of the carrier of G,B) is Element of the carrier of G
[ the Element of the carrier of G,B] is set
{ the Element of the carrier of G,B} is non empty finite set
{ the Element of the carrier of G} is non empty trivial finite 1 -element set
{{ the Element of the carrier of G,B},{ the Element of the carrier of G}} is non empty finite V57() set
the multF of G . [ the Element of the carrier of G,B] is set
the Element of the carrier of G * (1_ G) is Element of the carrier of G
the multF of G . ( the Element of the carrier of G,(1_ G)) is Element of the carrier of G
[ the Element of the carrier of G,(1_ G)] is set
{ the Element of the carrier of G,(1_ G)} is non empty finite set
{{ the Element of the carrier of G,(1_ G)},{ the Element of the carrier of G}} is non empty finite V57() set
the multF of G . [ the Element of the carrier of G,(1_ G)] is set
G is non empty unital Group-like associative multMagma
the carrier of G is non empty set
{ {b₁} where b₁ is Element of the carrier of G : verum } is set
(G) is non empty finite strict unital Group-like associative (G)
H is non empty strict unital Group-like associative (G)
(G,H) is Element of bool (bool the carrier of G)
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
the carrier of H is non empty set
1_ G is non being_of_order_0 Element of the carrier of G
{(1_ G)} is non empty trivial finite 1 -element Element of bool the carrier of G
the Element of the carrier of G is Element of the carrier of G
y is set
y is Element of the carrier of H
(G,H, the Element of the carrier of G) is Element of bool the carrier of G
(G,H) is Element of bool the carrier of G
(G, the Element of the carrier of G,(G,H)) is Element of bool the carrier of G
{ the Element of the carrier of G} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,H),{ the Element of the carrier of G}) 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 { the Element of the carrier of G} ) } is set
B is Element of the carrier of G
{B} is non empty trivial finite 1 -element Element of bool the carrier of G
B is Element of the carrier of G
B * the Element of the carrier of 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 having_a_unity 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 cup-closed diff-closed preBoolean set
the multF of G . (B, the Element of the carrier of G) is Element of the carrier of G
[B, the Element of the carrier of G] is set
{B, the Element of the carrier of G} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B, the Element of the carrier of G},{B}} is non empty finite V57() set
the multF of G . [B, the Element of the carrier of G] is set
(1_ G) * the Element of the carrier of G is Element of the carrier of G
the multF of G . ((1_ G), the Element of the carrier of G) is Element of the carrier of G
[(1_ G), the Element of the carrier of G] is set
{(1_ G), the Element of the carrier of G} is non empty finite set
{(1_ G)} is non empty trivial finite 1 -element set
{{(1_ G), the Element of the carrier of G},{(1_ G)}} is non empty finite V57() set
the multF of G . [(1_ G), the Element of the carrier of G] is set
G is non empty unital Group-like associative multMagma
(G) is non empty strict unital Group-like associative (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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
(G,(G)) is Element of bool (bool the carrier of G)
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
{ the carrier of G} is non empty trivial finite 1 -element set
(G,(G)) is Element of bool (bool the carrier of G)
the Element of the carrier of G is Element of the carrier of G
x is set
y is Element of bool the carrier of G
y is Element of the carrier of G
(G,(G),y) is Element of bool the carrier of G
(G,(G)) is Element of bool the carrier of G
the carrier of (G) is non empty set
(G,y,(G,(G))) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{y},(G,(G))) 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 (G,(G)) ) } is set
x is set
y is Element of bool the carrier of G
y is Element of the carrier of G
(G,(G),y) is Element of bool the carrier of G
(G,(G)) is Element of bool the carrier of G
the carrier of (G) is non empty set
(G,y,(G,(G))) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,(G)),{y}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,(G)) & b₂ in {y} ) } is set
(G,(G), the Element of the carrier of G) is Element of bool the carrier of G
(G,(G)) is Element of bool the carrier of G
the carrier of (G) is non empty set
(G, the Element of the carrier of G,(G,(G))) is Element of bool the carrier of G
{ the Element of the carrier of G} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(G,(G)),{ the Element of the carrier of G}) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in (G,(G)) & b₂ in { the Element of the carrier of G} ) } is set
(G,(G), the Element of the carrier of G) is Element of bool the carrier of G
(G, the Element of the carrier of G,(G,(G))) is Element of bool the carrier of G
(G,{ the Element of the carrier of G},(G,(G))) is Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in { the Element of the carrier of G} & b₂ in (G,(G)) ) } is set
G is non empty strict unital Group-like associative multMagma
the carrier of G is non empty set
{ the carrier of G} is non empty trivial finite 1 -element set
H is non empty strict unital Group-like associative (G)
(G,H) is Element of bool (bool the carrier of G)
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
x 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
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,y,(G,H)) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{y},(G,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 (G,H) ) } is set
y is Element of the carrier of G
y * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of G . [y,y] is set
B is Element of the carrier of G
y * B is Element of the carrier of G
the multF of G . (y,B) is Element of the carrier of G
[y,B] is set
{y,B} is non empty finite set
{{y,B},{y}} is non empty finite V57() set
the multF of G . [y,B] is set
G is non empty strict unital Group-like associative multMagma
the carrier of G is non empty set
{ the carrier of G} is non empty trivial finite 1 -element set
H is non empty strict unital Group-like associative (G)
(G,H) is Element of bool (bool the carrier of G)
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
x 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
(G,H) is Element of bool the carrier of G
the carrier of H is non empty set
(G,y,(G,H)) is Element of bool the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,(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 (G,H) & b₂ in {y} ) } is set
y is Element of the carrier of G
y * 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 having_a_unity 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 cup-closed diff-closed preBoolean set
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} is non empty trivial finite 1 -element set
{{y,y},{y}} is non empty finite V57() set
the multF of G . [y,y] is set
B is Element of the carrier of G
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 trivial finite 1 -element set
{{B,y},{B}} is non empty finite V57() set
the multF of G . [B,y] is set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G,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 cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
card (G,H) is V20() cardinal set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G,H) is V20() cardinal set
(G,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 cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
card (G,H) is V20() cardinal set
(G,H) is Element of bool (bool the carrier of G)
card (G,H) is V20() cardinal set
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G,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 cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
x is finite set
card x is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
y is finite set
x is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
card y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
B is finite set
y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
card B is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G is non empty unital Group-like associative multMagma
H is non empty unital Group-like associative (G)
(G,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 cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
(G,H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
(G,H) is Element of bool (bool the carrier of G)
x is finite set
y is finite set
card y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
y is finite set
card y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
y is finite set
card y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
card x is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G is V20() V24() finite cardinal set
H is finite set
union H is set
card H is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G * (card H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
x is finite set
card x is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
x is non empty set
Fin x is non empty cup-closed diff-closed preBoolean set
y is finite Element of Fin x
union y is set
card y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G * (card y) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
y is Element of x
{.y.} is non empty trivial finite 1 -element Element of Fin x
y \/ {.y.} is non empty finite Element of Fin x
union (y \/ {.y.}) is set
card (y \/ {.y.}) is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G * (card (y \/ {.y.})) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
{y} is non empty trivial finite 1 -element Element of bool x
bool x is non empty cup-closed diff-closed preBoolean set
y \/ {y} is non empty finite set
card (y \/ {y}) is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
(card y) + 1 is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
B is set
B is finite set
card B is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
B is set
Y is finite set
card Y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
y is finite set
card y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
Z is set
a is finite set
card a is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
union {y} is set
union (y \/ {y}) is set
B is finite set
B is finite set
B \/ B is finite set
a is finite set
card a is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
Y is set
y is set
Z is set
c is finite set
card c is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G * 1 is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
(G * (card y)) + (G * 1) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G * (card (y \/ {y})) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
Y is set
y is finite set
card y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
Z is finite set
card Z is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
{}. x is empty V20() V24() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element Element of Fin x
union ({}. x) is finite set
card ({}. x) is empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() V52() finite V57() cardinal {} -element Element of NAT
G * (card ({}. x)) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G is non empty finite unital Group-like associative multMagma
card G is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of G is non empty finite set
card the carrier of G is non empty V20() V24() finite cardinal set
H is non empty finite unital Group-like associative (G)
card H is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of H is non empty finite set
card the carrier of H is non empty V20() V24() finite cardinal set
(G,H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
(card H) * (G,H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
(G,H) is finite V57() Element of bool (bool the carrier of G)
bool the carrier of G is non empty cup-closed diff-closed preBoolean finite V57() set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean finite V57() set
y is set
x is finite set
y is finite Element of bool the carrier of G
B is finite set
B is finite set
a is Element of the carrier of G
(G,H,a) is finite Element of bool the carrier of G
(G,H) is finite Element of bool the carrier of G
(G,a,(G,H)) is finite Element of bool the carrier of G
{a} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{a},(G,H)) is finite 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
(G,H,a) is finite Element of bool the carrier of G
(G,a,(G,H)) is finite Element of bool the carrier of G
(G,(G,H),{a}) is finite 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
card B is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
Y is finite set
y is finite set
card Y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
card y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
Y is set
y is finite Element of bool the carrier of G
Z is Element of the carrier of G
(G,H,Z) is finite Element of bool the carrier of G
(G,Z,(G,H)) is finite Element of bool the carrier of G
{Z} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{Z},(G,H)) is finite Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {Z} & b₂ in (G,H) ) } is set
Z is Element of the carrier of G
(G,H,Z) is finite Element of bool the carrier of G
(G,Z,(G,H)) is finite Element of bool the carrier of G
{Z} is non empty trivial finite 1 -element Element of bool the carrier of G
(G,{Z},(G,H)) is finite Element of bool the carrier of G
{ (b₁ * b₂) where b₁, b₂ is Element of the carrier of G : ( b₁ in {Z} & b₂ in (G,H) ) } is set
union x is set
card x is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
(card H) * (card x) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
union (G,H) is finite Element of bool the carrier of G
y is finite set
card y is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G is non empty finite unital Group-like associative multMagma
card G is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of G is non empty finite set
card the carrier of G is non empty V20() V24() finite cardinal set
H is non empty finite unital Group-like associative (G)
card H is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of H is non empty finite set
card the carrier of H is non empty V20() V24() finite cardinal set
(G,H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
(card H) * (G,H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G is non empty finite unital Group-like associative multMagma
x is non empty finite unital Group-like associative (G)
H is non empty finite unital Group-like associative (G)
(G,H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
(G,x) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
y is non empty finite unital Group-like associative (x)
(x,y) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
(x,y) * (G,x) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
card G is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of G is non empty finite set
card the carrier of G is non empty V20() V24() finite cardinal set
card x is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of x is non empty finite set
card the carrier of x is non empty V20() V24() finite cardinal set
(card x) * (G,x) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
card y is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of y is non empty finite set
card the carrier of y is non empty V20() V24() finite cardinal set
(card y) * (x,y) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
card H is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of H is non empty finite set
card the carrier of H is non empty V20() V24() finite cardinal set
(card H) * ((x,y) * (G,x)) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
(card H) * (G,H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G is non empty unital Group-like associative multMagma
(G) is non empty strict unital Group-like associative (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 having_a_unity 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 cup-closed diff-closed preBoolean set
multMagma(# the carrier of G, the multF of G #) is non empty strict multMagma
(G,(G)) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
(G,(G)) is Element of bool (bool the carrier of G)
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
{ the carrier of G} is non empty trivial finite 1 -element set
card { the carrier of G} is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G is non empty strict unital Group-like associative multMagma
H is non empty strict unital Group-like associative (G)
(G,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 cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
(G,H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
x is finite set
card x is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
y is set
{y} is non empty trivial finite 1 -element set
union {y} is set
union (G,H) is Element of bool the carrier of G
G is non empty unital Group-like associative multMagma
(G) is non empty finite strict unital Group-like associative (G)
(G,(G)) is V20() cardinal set
(G,(G)) 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 cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
card (G,(G)) is V20() cardinal set
card G is V20() cardinal set
card the carrier of G is non empty V20() cardinal set
H is Relation-like Function-like set
dom H is set
rng H is set
x is set
y is set
H . y is set
y is Element of the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
{ {b₁} where b₁ is Element of the carrier of G : verum } is set
x is set
{ {b₁} where b₁ is Element of the carrier of G : verum } is set
y is Element of the carrier of G
{y} is non empty trivial finite 1 -element Element of bool the carrier of G
H . y is set
x is set
H . x is set
y is set
H . y is set
{y} is non empty trivial finite 1 -element set
{x} is non empty trivial finite 1 -element set
G is non empty finite unital Group-like associative multMagma
(G) is non empty finite strict unital Group-like associative (G)
(G,(G)) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
card G is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of G is non empty finite set
card the carrier of G is non empty V20() V24() finite cardinal set
card (G) is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of (G) is non empty finite set
card the carrier of (G) is non empty V20() V24() finite cardinal set
(card (G)) * (G,(G)) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
1 * (G,(G)) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G is non empty finite unital Group-like associative multMagma
card G is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of G is non empty finite set
card the carrier of G is non empty V20() V24() finite cardinal set
(G) is non empty finite strict unital Group-like associative (G)
H is non empty finite strict unital Group-like associative (G)
(G,H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
1 * (card G) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
card H is non empty V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
the carrier of H is non empty finite set
card the carrier of H is non empty V20() V24() finite cardinal set
(card H) * (card G) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G is non empty unital Group-like associative multMagma
card G is V20() cardinal set
the carrier of G is non empty set
card the carrier of G is non empty V20() cardinal set
(G) is non empty finite strict unital Group-like associative (G)
H is non empty strict unital Group-like associative (G)
(G,H) is Element of bool (bool the carrier of G)
bool the carrier of G is non empty cup-closed diff-closed preBoolean set
bool (bool the carrier of G) is non empty cup-closed diff-closed preBoolean set
(G,H) is V20() cardinal set
card (G,H) is V20() cardinal set
(G,H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
x is finite set
card x is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
H is finite set
G is V20() V24() finite cardinal set
union H is set
card H is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT
G * (card H) is V20() V24() ext-real V42() V43() V44() V46() V47() V48() V49() V50() V51() finite cardinal Element of NAT