REAL is set
NAT is epsilon-transitive epsilon-connected ordinal non empty non trivial non finite cardinal limit_cardinal Element of bool REAL
bool REAL is non empty cup-closed diff-closed preBoolean set
{} is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
the Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
COMPLEX is set
NAT is epsilon-transitive epsilon-connected ordinal non empty non trivial non finite cardinal limit_cardinal set
bool NAT is non empty non trivial cup-closed diff-closed preBoolean non finite set
bool NAT is non empty non trivial cup-closed diff-closed preBoolean non finite set
Fin NAT is non empty cup-closed diff-closed preBoolean set
INT is set
1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
[:1,1:] is Relation-like non empty finite set
bool [:1,1:] is non empty cup-closed diff-closed preBoolean finite V52() set
[:[:1,1:],1:] is Relation-like non empty finite set
bool [:[:1,1:],1:] is non empty cup-closed diff-closed preBoolean finite V52() set
3 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
0 is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of NAT
Seg 1 is non empty trivial finite 1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= 1 ) } is set
{1} is non empty trivial finite V52() 1 -element set
Seg 2 is non empty finite 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= 2 ) } is set
{1,2} is non empty finite V52() set
Permutations 1 is non empty permutational set
idseq 1 is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
id (Seg 1) is Relation-like Seg 1 -defined Seg 1 -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg 1),(Seg 1):]
[:(Seg 1),(Seg 1):] is Relation-like non empty finite set
bool [:(Seg 1),(Seg 1):] is non empty cup-closed diff-closed preBoolean finite V52() set
{(idseq 1)} is functional non empty trivial finite V52() 1 -element set
dom {} is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
rng {} is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty trivial ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V120() set
id {} is Relation-like non-empty empty-yielding {} -defined {} -valued V6() V8() V9() V13() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty total ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
len {} is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
{{}} is functional non empty trivial finite V52() 1 -element set
n is set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg K is finite K -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= K ) } is set
TWOELEMENTSETS (Seg K) is set
A is set
B is set
{A,B} is non empty finite set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{A,B} is non empty finite V52() set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{A,B} is non empty finite V52() set
Seg {} is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= {} ) } is set
TWOELEMENTSETS (Seg {}) is set
TWOELEMENTSETS (Seg 1) is set
n is set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{K,A} is non empty finite V52() set
n is set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{K,A} is non empty finite V52() set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
TWOELEMENTSETS (Seg n) is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is set
{} + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
A is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
K is set
A . K is set
dom A is finite set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
rng A is finite set
dom A is finite set
dom A is finite set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of n is non empty non trivial set
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
A is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
P is set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{KK,mm} is non empty finite V52() set
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
A . KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
{aa,AB} is non empty finite V52() set
A . AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
{aa,AB} is non empty finite V52() set
A . AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
aa is set
AB is set
P is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
A . mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{KK,mm} is non empty finite V52() set
P . {KK,mm} is set
P is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
KK is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
mm is set
P . mm is set
KK . mm is set
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{aa,AB} is non empty finite V52() set
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
A . aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P . {aa,AB} is set
P . {aa,AB} is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
1_ K is Element of the carrier of K
the carrier of K is non empty non trivial set
K254(K) is V70(K) Element of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
A is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,A) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
B is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
the multF of K $$ (B,(n,K,A)) is Element of the carrier of K
[:(Fin (TWOELEMENTSETS (Seg (n + 2)))), the carrier of K:] is Relation-like non empty set
bool [:(Fin (TWOELEMENTSETS (Seg (n + 2)))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
AB is Relation-like Fin (TWOELEMENTSETS (Seg (n + 2))) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (TWOELEMENTSETS (Seg (n + 2)))), the carrier of K:]
AB . B is Element of the carrier of K
AB . {} is set
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
SUM1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
F is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
card F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
AB . F is Element of the carrier of K
Ga is set
{Ga} is non empty trivial finite 1 -element set
Gs is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,A) . Gs is Element of the carrier of K
Ga is set
Gs is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,A) . Gs is Element of the carrier of K
{Gs} is non empty trivial finite 1 -element set
F \ {Gs} is finite Element of bool F
bool F is non empty cup-closed diff-closed preBoolean finite V52() set
B9 is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
B \ B9 is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
{Gs} \/ B9 is non empty finite set
card B9 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(card B9) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
AB . B9 is Element of the carrier of K
(1_ K) * (1_ K) is Element of the carrier of K
the multF of K . ((1_ K),(1_ K)) is Element of the carrier of K
[(1_ K),(1_ K)] is set
{(1_ K),(1_ K)} is non empty finite set
{(1_ K)} is non empty trivial finite 1 -element set
{{(1_ K),(1_ K)},{(1_ K)}} is non empty finite V52() set
the multF of K . [(1_ K),(1_ K)] is set
SUM1 is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
card SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
AB . SUM1 is Element of the carrier of K
card B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
SUM1 is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
card SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
AB . SUM1 is Element of the carrier of K
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
A is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,A) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
B is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,A) . B is Element of the carrier of K
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{P,KK} is non empty finite V52() set
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
A . P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
A is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,A) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the carrier of K is non empty non trivial set
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
B is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,B) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{KK,mm} is non empty finite V52() set
(n,K,A) . {KK,mm} is set
(n,K,B) . {KK,mm} is set
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
aa is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
aa . KK is set
aa . mm is set
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
A is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
B is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,B) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ (A,(n,K,B)) is Element of the carrier of K
P is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,P) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
{ b1 where b1 is Element of TWOELEMENTSETS (Seg (n + 2)) : ( b1 in A & not (n,K,B) . b1 = (n,K,P) . b1 ) } is set
the multF of K $$ (A,(n,K,P)) is Element of the carrier of K
- ( the multF of K $$ (A,(n,K,P))) is Element of the carrier of K
KK is finite set
card KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(card KK) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
A \ KK is finite Element of bool A
bool A is non empty cup-closed diff-closed preBoolean finite V52() set
Path is set
F is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,B) . F is Element of the carrier of K
(n,K,P) . F is Element of the carrier of K
SUM1 is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
KK \/ SUM1 is finite set
[:(Fin (TWOELEMENTSETS (Seg (n + 2)))), the carrier of K:] is Relation-like non empty set
bool [:(Fin (TWOELEMENTSETS (Seg (n + 2)))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
Path is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
the multF of K $$ (Path,(n,K,B)) is Element of the carrier of K
Gs is Relation-like Fin (TWOELEMENTSETS (Seg (n + 2))) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (TWOELEMENTSETS (Seg (n + 2)))), the carrier of K:]
Gs . KK is set
Gs . {} is set
the multF of K $$ (Path,(n,K,P)) is Element of the carrier of K
B9 is Relation-like Fin (TWOELEMENTSETS (Seg (n + 2))) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (TWOELEMENTSETS (Seg (n + 2)))), the carrier of K:]
B9 . KK is set
B9 . {} is set
b is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,B) . b is Element of the carrier of K
(n,K,P) . b is Element of the carrier of K
- ((n,K,P) . b) is Element of the carrier of K
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
((n,K,B) . b) + ((n,K,P) . b) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K . (((n,K,B) . b),((n,K,P) . b)) is Element of the carrier of K
[((n,K,B) . b),((n,K,P) . b)] is set
{((n,K,B) . b),((n,K,P) . b)} is non empty finite set
{((n,K,B) . b)} is non empty trivial finite 1 -element set
{{((n,K,B) . b),((n,K,P) . b)},{((n,K,B) . b)}} is non empty finite V52() set
the addF of K . [((n,K,B) . b),((n,K,P) . b)] is set
0. K is V70(K) Element of the carrier of K
mA is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,B) . mA is Element of the carrier of K
(n,K,P) . mA is Element of the carrier of K
b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
b + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Bb is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
card Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(card Bb) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Gs . Bb is Element of the carrier of K
B9 . Bb is Element of the carrier of K
- (B9 . Bb) is Element of the carrier of K
PM is set
{PM} is non empty trivial finite 1 -element set
i is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,P) . i is Element of the carrier of K
(n,K,B) . i is Element of the carrier of K
PM is set
i is Element of TWOELEMENTSETS (Seg (n + 2))
{i} is non empty trivial finite 1 -element set
Bb \ {i} is finite Element of bool Bb
bool Bb is non empty cup-closed diff-closed preBoolean finite V52() set
Pi is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
KK \ Pi is finite Element of bool KK
bool KK is non empty cup-closed diff-closed preBoolean finite V52() set
{i} \/ Pi is non empty finite set
card Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(card Pi) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
B9 . Pi is Element of the carrier of K
(n,K,P) . i is Element of the carrier of K
the multF of K . ((B9 . Pi),((n,K,P) . i)) is Element of the carrier of K
[(B9 . Pi),((n,K,P) . i)] is set
{(B9 . Pi),((n,K,P) . i)} is non empty finite set
{(B9 . Pi)} is non empty trivial finite 1 -element set
{{(B9 . Pi),((n,K,P) . i)},{(B9 . Pi)}} is non empty finite V52() set
the multF of K . [(B9 . Pi),((n,K,P) . i)] is set
Gs . Pi is Element of the carrier of K
(n,K,B) . i is Element of the carrier of K
the multF of K . ((Gs . Pi),((n,K,B) . i)) is Element of the carrier of K
[(Gs . Pi),((n,K,B) . i)] is set
{(Gs . Pi),((n,K,B) . i)} is non empty finite set
{(Gs . Pi)} is non empty trivial finite 1 -element set
{{(Gs . Pi),((n,K,B) . i)},{(Gs . Pi)}} is non empty finite V52() set
the multF of K . [(Gs . Pi),((n,K,B) . i)] is set
b mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
2 - 1 is ext-real V44() V45() set
(b + 1) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
{} + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
- ((n,K,P) . i) is Element of the carrier of K
(Gs . Pi) * (- ((n,K,P) . i)) is Element of the carrier of K
the multF of K . ((Gs . Pi),(- ((n,K,P) . i))) is Element of the carrier of K
[(Gs . Pi),(- ((n,K,P) . i))] is set
{(Gs . Pi),(- ((n,K,P) . i))} is non empty finite set
{{(Gs . Pi),(- ((n,K,P) . i))},{(Gs . Pi)}} is non empty finite V52() set
the multF of K . [(Gs . Pi),(- ((n,K,P) . i))] is set
(Gs . Pi) * ((n,K,P) . i) is Element of the carrier of K
the multF of K . ((Gs . Pi),((n,K,P) . i)) is Element of the carrier of K
[(Gs . Pi),((n,K,P) . i)] is set
{(Gs . Pi),((n,K,P) . i)} is non empty finite set
{{(Gs . Pi),((n,K,P) . i)},{(Gs . Pi)}} is non empty finite V52() set
the multF of K . [(Gs . Pi),((n,K,P) . i)] is set
b mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
- ((n,K,P) . i) is Element of the carrier of K
- (B9 . Pi) is Element of the carrier of K
(- (B9 . Pi)) * (- ((n,K,P) . i)) is Element of the carrier of K
the multF of K . ((- (B9 . Pi)),(- ((n,K,P) . i))) is Element of the carrier of K
[(- (B9 . Pi)),(- ((n,K,P) . i))] is set
{(- (B9 . Pi)),(- ((n,K,P) . i))} is non empty finite set
{(- (B9 . Pi))} is non empty trivial finite 1 -element set
{{(- (B9 . Pi)),(- ((n,K,P) . i))},{(- (B9 . Pi))}} is non empty finite V52() set
the multF of K . [(- (B9 . Pi)),(- ((n,K,P) . i))] is set
2 - 1 is ext-real V44() V45() set
(B9 . Pi) * ((n,K,P) . i) is Element of the carrier of K
b mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
b mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
b is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
card b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(card b) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Gs . b is Element of the carrier of K
B9 . b is Element of the carrier of K
- (B9 . b) is Element of the carrier of K
{} mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
the multF of K $$ (SUM1,(n,K,B)) is Element of the carrier of K
the multF of K . (( the multF of K $$ (SUM1,(n,K,B))),( the multF of K $$ (Path,(n,K,B)))) is Element of the carrier of K
[( the multF of K $$ (SUM1,(n,K,B))),( the multF of K $$ (Path,(n,K,B)))] is set
{( the multF of K $$ (SUM1,(n,K,B))),( the multF of K $$ (Path,(n,K,B)))} is non empty finite set
{( the multF of K $$ (SUM1,(n,K,B)))} is non empty trivial finite 1 -element set
{{( the multF of K $$ (SUM1,(n,K,B))),( the multF of K $$ (Path,(n,K,B)))},{( the multF of K $$ (SUM1,(n,K,B)))}} is non empty finite V52() set
the multF of K . [( the multF of K $$ (SUM1,(n,K,B))),( the multF of K $$ (Path,(n,K,B)))] is set
the multF of K $$ (SUM1,(n,K,P)) is Element of the carrier of K
the multF of K . (( the multF of K $$ (SUM1,(n,K,P))),( the multF of K $$ (Path,(n,K,P)))) is Element of the carrier of K
[( the multF of K $$ (SUM1,(n,K,P))),( the multF of K $$ (Path,(n,K,P)))] is set
{( the multF of K $$ (SUM1,(n,K,P))),( the multF of K $$ (Path,(n,K,P)))} is non empty finite set
{( the multF of K $$ (SUM1,(n,K,P)))} is non empty trivial finite 1 -element set
{{( the multF of K $$ (SUM1,(n,K,P))),( the multF of K $$ (Path,(n,K,P)))},{( the multF of K $$ (SUM1,(n,K,P)))}} is non empty finite V52() set
the multF of K . [( the multF of K $$ (SUM1,(n,K,P))),( the multF of K $$ (Path,(n,K,P)))] is set
dom (n,K,B) is non empty finite set
(n,K,B) | SUM1 is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined SUM1 -defined TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
dom ((n,K,B) | SUM1) is finite set
dom (n,K,P) is non empty finite set
(n,K,P) | SUM1 is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined SUM1 -defined TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
dom ((n,K,P) | SUM1) is finite set
b is set
((n,K,B) | SUM1) . b is set
((n,K,P) | SUM1) . b is set
mA is Element of TWOELEMENTSETS (Seg (n + 2))
((n,K,B) | SUM1) . mA is set
(n,K,B) . mA is Element of the carrier of K
((n,K,P) | SUM1) . mA is set
(n,K,P) . mA is Element of the carrier of K
( the multF of K $$ (SUM1,(n,K,B))) * ( the multF of K $$ (Path,(n,K,P))) is Element of the carrier of K
the multF of K . (( the multF of K $$ (SUM1,(n,K,B))),( the multF of K $$ (Path,(n,K,P)))) is Element of the carrier of K
[( the multF of K $$ (SUM1,(n,K,B))),( the multF of K $$ (Path,(n,K,P)))] is set
{( the multF of K $$ (SUM1,(n,K,B))),( the multF of K $$ (Path,(n,K,P)))} is non empty finite set
{{( the multF of K $$ (SUM1,(n,K,B))),( the multF of K $$ (Path,(n,K,P)))},{( the multF of K $$ (SUM1,(n,K,B)))}} is non empty finite V52() set
the multF of K . [( the multF of K $$ (SUM1,(n,K,B))),( the multF of K $$ (Path,(n,K,P)))] is set
- ( the multF of K $$ (Path,(n,K,P))) is Element of the carrier of K
( the multF of K $$ (SUM1,(n,K,B))) * (- ( the multF of K $$ (Path,(n,K,P)))) is Element of the carrier of K
the multF of K . (( the multF of K $$ (SUM1,(n,K,B))),(- ( the multF of K $$ (Path,(n,K,P))))) is Element of the carrier of K
[( the multF of K $$ (SUM1,(n,K,B))),(- ( the multF of K $$ (Path,(n,K,P))))] is set
{( the multF of K $$ (SUM1,(n,K,B))),(- ( the multF of K $$ (Path,(n,K,P))))} is non empty finite set
{{( the multF of K $$ (SUM1,(n,K,B))),(- ( the multF of K $$ (Path,(n,K,P))))},{( the multF of K $$ (SUM1,(n,K,B)))}} is non empty finite V52() set
the multF of K . [( the multF of K $$ (SUM1,(n,K,B))),(- ( the multF of K $$ (Path,(n,K,P))))] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
dom K is finite set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K . A is set
K . B is set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K . P is set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K . KK is set
rng K is finite set
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K . mm is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
KK is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
P is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
B is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
B * P is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):]
[:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
(n,K,B) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
(n,K,KK) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P . mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
Path is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,B) . Path is Element of the carrier of K
(n,K,KK) . Path is Element of the carrier of K
AB is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
dom AB is non empty finite set
SUM1 is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
dom SUM1 is non empty finite set
F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{F,Ga} is non empty finite V52() set
P . Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK . Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P . F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK . F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
1_ K is Element of the carrier of K
the carrier of K is non empty non trivial set
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
A is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,A) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{P,KK} is non empty finite V52() set
(n,K,A) . {P,KK} is set
A . KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
mm is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
mm . P is set
mm . KK is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
P is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
B is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
A is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
A * B is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):]
[:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{KK,mm} is non empty finite V52() set
aa is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
1_ aa is Element of the carrier of aa
the carrier of aa is non empty non trivial set
K254(aa) is V70(aa) Element of the carrier of aa
- (1_ aa) is Element of the carrier of aa
(n,aa,A) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of aa -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of aa:]
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of aa:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of aa:] is non empty cup-closed diff-closed preBoolean set
(n,aa,A) . {KK,mm} is set
(n,aa,P) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of aa -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of aa:]
(n,aa,P) . {KK,mm} is set
dom B is finite set
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
F is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
dom F is non empty finite set
Ga is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
dom Ga is non empty finite set
P . KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P . mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Path is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
dom Path is non empty finite set
Path . KK is set
Path . mm is set
Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{KK,Gs} is non empty finite V52() set
(n,aa,A) . {KK,Gs} is set
(n,aa,P) . {KK,Gs} is set
{mm,Gs} is non empty finite V52() set
(n,aa,A) . {mm,Gs} is set
(n,aa,P) . {mm,Gs} is set
B . Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P . Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . (B . Gs) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{Gs,KK} is non empty finite V52() set
(n,aa,A) . {Gs,KK} is set
A . Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{Gs,mm} is non empty finite V52() set
(n,aa,A) . {Gs,mm} is set
A . Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
A is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,A) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,A)) is Element of the carrier of K
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
1_ K is Element of the carrier of K
the carrier of K is non empty non trivial set
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
A is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,A) is Element of the carrier of K
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,A) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,A)) is Element of the carrier of K
[:(Fin (TWOELEMENTSETS (Seg (n + 2)))), the carrier of K:] is Relation-like non empty set
bool [:(Fin (TWOELEMENTSETS (Seg (n + 2)))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
AB is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
the multF of K $$ (AB,(n,K,A)) is Element of the carrier of K
SUM1 is Relation-like Fin (TWOELEMENTSETS (Seg (n + 2))) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (TWOELEMENTSETS (Seg (n + 2)))), the carrier of K:]
SUM1 . AB is Element of the carrier of K
SUM1 . {} is set
Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Path + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Ga is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
card Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
SUM1 . Ga is Element of the carrier of K
Gs is set
{Gs} is non empty trivial finite 1 -element set
B9 is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,A) . B9 is Element of the carrier of K
Gs is set
B9 is Element of TWOELEMENTSETS (Seg (n + 2))
{B9} is non empty trivial finite 1 -element set
Ga \ {B9} is finite Element of bool Ga
bool Ga is non empty cup-closed diff-closed preBoolean finite V52() set
b is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
{B9} \/ b is non empty finite set
card b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(card b) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
SUM1 . b is Element of the carrier of K
(TWOELEMENTSETS (Seg (n + 2))) \ b is finite Element of bool (TWOELEMENTSETS (Seg (n + 2)))
bool (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean finite V52() set
(n,K,A) . B9 is Element of the carrier of K
the multF of K . ((SUM1 . b),((n,K,A) . B9)) is Element of the carrier of K
[(SUM1 . b),((n,K,A) . B9)] is set
{(SUM1 . b),((n,K,A) . B9)} is non empty finite set
{(SUM1 . b)} is non empty trivial finite 1 -element set
{{(SUM1 . b),((n,K,A) . B9)},{(SUM1 . b)}} is non empty finite V52() set
the multF of K . [(SUM1 . b),((n,K,A) . B9)] is set
(1_ K) * (1_ K) is Element of the carrier of K
the multF of K . ((1_ K),(1_ K)) is Element of the carrier of K
[(1_ K),(1_ K)] is set
{(1_ K),(1_ K)} is non empty finite set
{(1_ K)} is non empty trivial finite 1 -element set
{{(1_ K),(1_ K)},{(1_ K)}} is non empty finite V52() set
the multF of K . [(1_ K),(1_ K)] is set
(1_ K) * (- (1_ K)) is Element of the carrier of K
the multF of K . ((1_ K),(- (1_ K))) is Element of the carrier of K
[(1_ K),(- (1_ K))] is set
{(1_ K),(- (1_ K))} is non empty finite set
{{(1_ K),(- (1_ K))},{(1_ K)}} is non empty finite V52() set
the multF of K . [(1_ K),(- (1_ K))] is set
(- (1_ K)) * (1_ K) is Element of the carrier of K
the multF of K . ((- (1_ K)),(1_ K)) is Element of the carrier of K
[(- (1_ K)),(1_ K)] is set
{(- (1_ K)),(1_ K)} is non empty finite set
{(- (1_ K))} is non empty trivial finite 1 -element set
{{(- (1_ K)),(1_ K)},{(- (1_ K))}} is non empty finite V52() set
the multF of K . [(- (1_ K)),(1_ K)] is set
(- (1_ K)) * (- (1_ K)) is Element of the carrier of K
the multF of K . ((- (1_ K)),(- (1_ K))) is Element of the carrier of K
[(- (1_ K)),(- (1_ K))] is set
{(- (1_ K)),(- (1_ K))} is non empty finite set
{{(- (1_ K)),(- (1_ K))},{(- (1_ K))}} is non empty finite V52() set
the multF of K . [(- (1_ K)),(- (1_ K))] is set
Path is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
card Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
SUM1 . Path is Element of the carrier of K
card AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Path is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
card Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
SUM1 . Path is Element of the carrier of K
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
idseq (n + 2) is Relation-like NAT -defined Function-like finite n + 2 -element FinSequence-like FinSubsequence-like set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
id (Seg (n + 2)) is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
1_ K is Element of the carrier of K
the carrier of K is non empty non trivial set
K254(K) is V70(K) Element of the carrier of K
B is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,B) is Element of the carrier of K
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,B) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,B)) is Element of the carrier of K
aa is set
mm is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{AB,SUM1} is non empty finite V52() set
B . SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(n,K,B) . aa is set
n is set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg K is finite K -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= K ) } is set
TWOELEMENTSETS (Seg K) is set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{B,P} is non empty finite V52() set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{A,KK} is non empty finite V52() set
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{A,mm} is non empty finite V52() set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
mm is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
KK is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
P is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
P * KK is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):]
[:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
(n,K,mm) is Element of the carrier of K
the carrier of K is non empty non trivial set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,mm) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,mm)) is Element of the carrier of K
(n,K,P) is Element of the carrier of K
(n,K,P) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,P)) is Element of the carrier of K
- (n,K,P) is Element of the carrier of K
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK . AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
{AB,SUM1} is non empty finite V52() set
(n,K,mm) . {AB,SUM1} is set
(n,K,P) . {AB,SUM1} is set
aa is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
{ b1 where b1 is Element of TWOELEMENTSETS (Seg (n + 2)) : ( b1 in aa & not (n,K,mm) . b1 = (n,K,P) . b1 ) } is set
Gs is set
B9 is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,mm) . B9 is Element of the carrier of K
(n,K,P) . B9 is Element of the carrier of K
{ b1 where b1 is Element of TWOELEMENTSETS (Seg (n + 2)) : ( b1 in aa & not (n,K,mm) . b1 = (n,K,P) . b1 & AB in b1 ) } is set
{ b1 where b1 is Element of TWOELEMENTSETS (Seg (n + 2)) : ( b1 in aa & not (n,K,mm) . b1 = (n,K,P) . b1 & SUM1 in b1 ) } is set
Gs is finite set
mA is set
Bb is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,mm) . Bb is Element of the carrier of K
(n,K,P) . Bb is Element of the carrier of K
mA is set
Bb is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,mm) . Bb is Element of the carrier of K
(n,K,P) . Bb is Element of the carrier of K
dom KK is finite set
mA is finite set
Bb is finite set
mA \/ Bb is finite set
PM is set
i is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,mm) . i is Element of the carrier of K
(n,K,P) . i is Element of the carrier of K
mA /\ Bb is finite set
{{AB,SUM1}} is non empty trivial finite V52() 1 -element set
PM is set
i is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,mm) . i is Element of the carrier of K
(n,K,P) . i is Element of the carrier of K
i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{i,Pi} is non empty finite V52() set
H is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,mm) . H is Element of the carrier of K
(n,K,P) . H is Element of the carrier of K
H is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,mm) . H is Element of the carrier of K
(n,K,P) . H is Element of the carrier of K
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
card mA is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(card mA) - 1 is ext-real V44() V45() set
i is set
Pi is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,mm) . Pi is Element of the carrier of K
(n,K,P) . Pi is Element of the carrier of K
H is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{AB,H} is non empty finite V52() set
SF is non empty finite V52() set
QQ is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{SUM1,QQ} is non empty finite V52() set
{SUM1,H} is non empty finite V52() set
SF is non empty finite V52() set
(n,K,mm) . SF is set
(n,K,P) . SF is set
QQ is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{SUM1,QQ} is non empty finite V52() set
SF is non empty finite V52() set
[:mA,Bb:] is Relation-like finite set
bool [:mA,Bb:] is non empty cup-closed diff-closed preBoolean finite V52() set
i is Relation-like mA -defined Bb -valued Function-like quasi_total finite Element of bool [:mA,Bb:]
dom i is finite set
Pi is set
H is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,mm) . H is Element of the carrier of K
(n,K,P) . H is Element of the carrier of K
SF is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{SUM1,SF} is non empty finite V52() set
i . H is set
{AB,SF} is non empty finite V52() set
(n,K,mm) . {AB,SF} is set
(n,K,P) . {AB,SF} is set
i . {AB,SF} is set
QQ is set
i . QQ is set
rng i is finite set
Pi is set
H is set
i . Pi is set
i . H is set
SF is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,mm) . SF is Element of the carrier of K
(n,K,P) . SF is Element of the carrier of K
QQ is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{AB,QQ} is non empty finite V52() set
h is Element of TWOELEMENTSETS (Seg (n + 2))
(n,K,mm) . h is Element of the carrier of K
(n,K,P) . h is Element of the carrier of K
Mh is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{AB,Mh} is non empty finite V52() set
{SUM1,QQ} is non empty finite V52() set
{SUM1,Mh} is non empty finite V52() set
{SUM1,Mh} is non empty finite V52() set
{SUM1,QQ} is non empty finite V52() set
card Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
card Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(card mA) + (card mA) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
card {{AB,SUM1}} is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
((card mA) + (card mA)) - (card {{AB,SUM1}}) is ext-real V44() V45() set
PM is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
PM + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
(PM + 1) + (PM + 1) is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
((PM + 1) + (PM + 1)) - 1 is ext-real V44() V45() set
2 * PM is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(2 * PM) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
(card Gs) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
1 mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
dom KK is finite set
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK . AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK . SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
1_ K is Element of the carrier of K
the carrier of K is non empty non trivial set
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
P is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,P) is Element of the carrier of K
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,P) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,P)) is Element of the carrier of K
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
idseq (n + 2) is Relation-like NAT -defined Function-like finite n + 2 -element FinSequence-like FinSubsequence-like set
id (Seg (n + 2)) is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
KK is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
(id (Seg (n + 2))) * KK is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
rng KK is non empty finite set
aa is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
mm is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,mm) is Element of the carrier of K
(n,K,mm) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,mm)) is Element of the carrier of K
- (n,K,mm) is Element of the carrier of K
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Group_of_Perm (n + 2) is non empty strict unital Group-like associative multMagma
the carrier of (Group_of_Perm (n + 2)) is non empty set
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
1_ K is Element of the carrier of K
the carrier of K is non empty non trivial set
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
P + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
mm is Relation-like NAT -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (n + 2))
Product mm is Element of the carrier of (Group_of_Perm (n + 2))
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom mm is finite Element of bool NAT
(len mm) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
aa is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,aa) is Element of the carrier of K
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,aa) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,aa)) is Element of the carrier of K
Seg (len mm) is finite len mm -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len mm ) } is set
AB is set
<*AB*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,AB] is set
{1,AB} is non empty finite set
{{1,AB},{1}} is non empty finite V52() set
{[1,AB]} is Relation-like Function-like constant non empty trivial finite 1 -element set
SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
<*AB*> ^ SUM1 is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
len SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len SUM1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
F is Relation-like NAT -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (n + 2))
dom F is finite Element of bool NAT
Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
F . Ga is set
Path is Relation-like NAT -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (n + 2))
len Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len Path) + Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
mm . ((len Path) + Ga) is set
1 + {} is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Seg (P + 1) is non empty finite P + 1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= P + 1 ) } is set
mm . 1 is set
Ga is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
Product F is Element of the carrier of (Group_of_Perm (n + 2))
B9 is Element of the carrier of (Group_of_Perm (n + 2))
B9 * (Product F) is Element of the carrier of (Group_of_Perm (n + 2))
the multF of (Group_of_Perm (n + 2)) is Relation-like [: the carrier of (Group_of_Perm (n + 2)), the carrier of (Group_of_Perm (n + 2)):] -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like non empty total quasi_total having_a_unity V155( the carrier of (Group_of_Perm (n + 2))) associative Element of bool [:[: the carrier of (Group_of_Perm (n + 2)), the carrier of (Group_of_Perm (n + 2)):], the carrier of (Group_of_Perm (n + 2)):]
[: the carrier of (Group_of_Perm (n + 2)), the carrier of (Group_of_Perm (n + 2)):] is Relation-like non empty set
[:[: the carrier of (Group_of_Perm (n + 2)), the carrier of (Group_of_Perm (n + 2)):], the carrier of (Group_of_Perm (n + 2)):] is Relation-like non empty set
bool [:[: the carrier of (Group_of_Perm (n + 2)), the carrier of (Group_of_Perm (n + 2)):], the carrier of (Group_of_Perm (n + 2)):] is non empty cup-closed diff-closed preBoolean set
the multF of (Group_of_Perm (n + 2)) . (B9,(Product F)) is Element of the carrier of (Group_of_Perm (n + 2))
[B9,(Product F)] is set
{B9,(Product F)} is non empty finite set
{B9} is non empty trivial finite 1 -element set
{{B9,(Product F)},{B9}} is non empty finite V52() set
the multF of (Group_of_Perm (n + 2)) . [B9,(Product F)] is set
Gs is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
Gs * Ga is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):]
[:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
(n,K,Gs) is Element of the carrier of K
(n,K,Gs) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,Gs)) is Element of the carrier of K
- (n,K,Gs) is Element of the carrier of K
len F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len F) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
2 - 1 is ext-real V44() V45() set
{} + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
len F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len F) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
2 - 1 is ext-real V44() V45() set
len F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len F) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
P is Relation-like NAT -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (n + 2))
Product P is Element of the carrier of (Group_of_Perm (n + 2))
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom P is finite Element of bool NAT
(len P) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
KK is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,KK) is Element of the carrier of K
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,KK) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,KK)) is Element of the carrier of K
<*> the carrier of (Group_of_Perm (n + 2)) is Relation-like non-empty empty-yielding NAT -defined the carrier of (Group_of_Perm (n + 2)) -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered FinSequence of the carrier of (Group_of_Perm (n + 2))
[:NAT, the carrier of (Group_of_Perm (n + 2)):] is Relation-like non empty non trivial non finite set
1_ (Group_of_Perm (n + 2)) is non being_of_order_0 Element of the carrier of (Group_of_Perm (n + 2))
idseq (n + 2) is Relation-like NAT -defined Function-like finite n + 2 -element FinSequence-like FinSubsequence-like set
id (Seg (n + 2)) is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
KK is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
P is Relation-like NAT -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (n + 2))
Product P is Element of the carrier of (Group_of_Perm (n + 2))
dom P is finite Element of bool NAT
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len P) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(n,K,KK) is Element of the carrier of K
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,KK) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,KK)) is Element of the carrier of K
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg A is finite A -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= A ) } is set
Permutations A is non empty permutational set
len (Permutations A) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations A)) is finite len (Permutations A) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations A) ) } is set
B is set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is set
[:(Seg A),(Seg A):] is Relation-like finite set
bool [:(Seg A),(Seg A):] is non empty cup-closed diff-closed preBoolean finite V52() set
B is Relation-like Seg A -defined Seg A -valued Function-like total quasi_total finite Element of bool [:(Seg A),(Seg A):]
P is set
KK is set
B . P is set
B . KK is set
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P is set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . K is set
B . n is set
B . KK is set
rng B is finite set
P is Relation-like Seg (len (Permutations A)) -defined Seg (len (Permutations A)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations A
P . K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
dom P is finite set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P . KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P . n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 1) is non empty permutational set
len (Permutations (n + 1)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 1))) is finite len (Permutations (n + 1)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 1)) ) } is set
A is Relation-like Seg (len (Permutations (n + 1))) -defined Seg (len (Permutations (n + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 1)
A . (n + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg (n + 1) is non empty finite n + 1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 1 ) } is set
[:(Seg (n + 1)),(Seg (n + 1)):] is Relation-like non empty finite set
bool [:(Seg (n + 1)),(Seg (n + 1)):] is non empty cup-closed diff-closed preBoolean finite V52() set
B is Relation-like Seg (n + 1) -defined Seg (n + 1) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 1)),(Seg (n + 1)):]
dom B is non empty finite set
rng B is non empty finite set
P is Relation-like Seg (len (Permutations (n + 1))) -defined Seg (len (Permutations (n + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 1)
P . (A . (n + 1)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is Relation-like Seg (n + 1) -defined Seg (n + 1) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 1)),(Seg (n + 1)):]
dom KK is non empty finite set
P * A is Relation-like Seg (len (Permutations (n + 1))) -defined Seg (len (Permutations (n + 1))) -defined Seg (len (Permutations (n + 1))) -valued Seg (len (Permutations (n + 1))) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations (n + 1)))),(Seg (len (Permutations (n + 1)))):]
[:(Seg (len (Permutations (n + 1)))),(Seg (len (Permutations (n + 1)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (n + 1)))),(Seg (len (Permutations (n + 1)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
dom (P * A) is finite set
(P * A) . (n + 1) is set
n is set
[:n,n:] is Relation-like set
bool [:n,n:] is non empty cup-closed diff-closed preBoolean set
K is set
{K} is non empty trivial finite 1 -element set
n \/ {K} is non empty set
[:(n \/ {K}),(n \/ {K}):] is Relation-like non empty set
bool [:(n \/ {K}),(n \/ {K}):] is non empty cup-closed diff-closed preBoolean set
A is Relation-like n \/ {K} -defined n \/ {K} -valued Function-like one-to-one non empty total quasi_total onto bijective Element of bool [:(n \/ {K}),(n \/ {K}):]
A . K is set
A | n is Relation-like n -defined n \/ {K} -defined n \/ {K} -valued Function-like Element of bool [:(n \/ {K}),(n \/ {K}):]
dom A is non empty set
dom (A | n) is set
rng A is non empty set
rng (A | n) is set
P is set
KK is set
(A | n) . KK is set
A . KK is set
P is set
KK is set
A . KK is set
(A | n) . KK is set
P is Relation-like n -defined n -valued Function-like total quasi_total Element of bool [:n,n:]
n is set
[:n,n:] is Relation-like set
bool [:n,n:] is non empty cup-closed diff-closed preBoolean set
K is set
{K} is non empty trivial finite 1 -element set
n \/ {K} is non empty set
[:(n \/ {K}),(n \/ {K}):] is Relation-like non empty set
bool [:(n \/ {K}),(n \/ {K}):] is non empty cup-closed diff-closed preBoolean set
A is Relation-like n -defined n -valued Function-like one-to-one total quasi_total onto bijective Element of bool [:n,n:]
B is Relation-like n -defined n -valued Function-like one-to-one total quasi_total onto bijective Element of bool [:n,n:]
A * B is Relation-like n -defined n -valued Function-like one-to-one total quasi_total onto bijective Element of bool [:n,n:]
P is Relation-like n \/ {K} -defined n \/ {K} -valued Function-like one-to-one non empty total quasi_total onto bijective Element of bool [:(n \/ {K}),(n \/ {K}):]
P | n is Relation-like n -defined n \/ {K} -defined n \/ {K} -valued Function-like Element of bool [:(n \/ {K}),(n \/ {K}):]
KK is Relation-like n \/ {K} -defined n \/ {K} -valued Function-like one-to-one non empty total quasi_total onto bijective Element of bool [:(n \/ {K}),(n \/ {K}):]
KK | n is Relation-like n -defined n \/ {K} -defined n \/ {K} -valued Function-like Element of bool [:(n \/ {K}),(n \/ {K}):]
P . K is set
KK . K is set
P * KK is Relation-like n \/ {K} -defined n \/ {K} -valued Function-like one-to-one non empty total quasi_total onto bijective Element of bool [:(n \/ {K}),(n \/ {K}):]
(P * KK) | n is Relation-like n -defined n \/ {K} -defined n \/ {K} -valued Function-like Element of bool [:(n \/ {K}),(n \/ {K}):]
(P * KK) . K is set
rng B is set
dom B is set
dom (P * KK) is non empty set
dom ((P * KK) | n) is set
dom (A * B) is set
dom A is set
SUM1 is set
(A * B) . SUM1 is set
((P * KK) | n) . SUM1 is set
B . SUM1 is set
A . (B . SUM1) is set
(P * KK) . SUM1 is set
KK . SUM1 is set
P . (KK . SUM1) is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
id (Seg n) is Relation-like Seg n -defined Seg n -valued V6() V8() V9() V13() Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
A is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
A * A is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
A " is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
dom A is finite set
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
KK * KK is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
dom (KK * KK) is finite set
dom KK is finite set
aa is set
(KK * KK) . aa is set
(idseq n) . aa is set
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(KK * KK) . AB is set
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A . AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(KK * KK) . AB is set
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(KK * KK) . AB is set
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(KK * KK) . AB is set
dom (idseq n) is finite n -element Element of bool NAT
rng KK is finite set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
Group_of_Perm n is non empty strict unital Group-like associative multMagma
the carrier of (Group_of_Perm n) is non empty set
K is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Permutations A is non empty permutational set
len (Permutations A) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations A)) is finite len (Permutations A) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations A) ) } is set
Group_of_Perm A is non empty strict unital Group-like associative multMagma
the carrier of (Group_of_Perm A) is non empty set
A + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (A + 1) is non empty permutational set
len (Permutations (A + 1)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (A + 1))) is finite len (Permutations (A + 1)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (A + 1)) ) } is set
Group_of_Perm (A + 1) is non empty strict unital Group-like associative multMagma
the carrier of (Group_of_Perm (A + 1)) is non empty set
P is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
Seg (A + 1) is non empty finite A + 1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= A + 1 ) } is set
[:(Seg (A + 1)),(Seg (A + 1)):] is Relation-like non empty finite set
bool [:(Seg (A + 1)),(Seg (A + 1)):] is non empty cup-closed diff-closed preBoolean finite V52() set
idseq A is Relation-like NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like set
Seg A is finite A -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= A ) } is set
id (Seg A) is Relation-like Seg A -defined Seg A -valued V6() V8() V9() V13() Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg A),(Seg A):]
[:(Seg A),(Seg A):] is Relation-like finite set
bool [:(Seg A),(Seg A):] is non empty cup-closed diff-closed preBoolean finite V52() set
idseq (A + 1) is Relation-like NAT -defined Function-like finite A + 1 -element FinSequence-like FinSubsequence-like set
id (Seg (A + 1)) is Relation-like Seg (A + 1) -defined Seg (A + 1) -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
F is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
F . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
the multF of (Group_of_Perm (A + 1)) is Relation-like [: the carrier of (Group_of_Perm (A + 1)), the carrier of (Group_of_Perm (A + 1)):] -defined the carrier of (Group_of_Perm (A + 1)) -valued Function-like non empty total quasi_total having_a_unity V155( the carrier of (Group_of_Perm (A + 1))) associative Element of bool [:[: the carrier of (Group_of_Perm (A + 1)), the carrier of (Group_of_Perm (A + 1)):], the carrier of (Group_of_Perm (A + 1)):]
[: the carrier of (Group_of_Perm (A + 1)), the carrier of (Group_of_Perm (A + 1)):] is Relation-like non empty set
[:[: the carrier of (Group_of_Perm (A + 1)), the carrier of (Group_of_Perm (A + 1)):], the carrier of (Group_of_Perm (A + 1)):] is Relation-like non empty set
bool [:[: the carrier of (Group_of_Perm (A + 1)), the carrier of (Group_of_Perm (A + 1)):], the carrier of (Group_of_Perm (A + 1)):] is non empty cup-closed diff-closed preBoolean set
the multF of (Group_of_Perm A) is Relation-like [: the carrier of (Group_of_Perm A), the carrier of (Group_of_Perm A):] -defined the carrier of (Group_of_Perm A) -valued Function-like non empty total quasi_total having_a_unity V155( the carrier of (Group_of_Perm A)) associative Element of bool [:[: the carrier of (Group_of_Perm A), the carrier of (Group_of_Perm A):], the carrier of (Group_of_Perm A):]
[: the carrier of (Group_of_Perm A), the carrier of (Group_of_Perm A):] is Relation-like non empty set
[:[: the carrier of (Group_of_Perm A), the carrier of (Group_of_Perm A):], the carrier of (Group_of_Perm A):] is Relation-like non empty set
bool [:[: the carrier of (Group_of_Perm A), the carrier of (Group_of_Perm A):], the carrier of (Group_of_Perm A):] is non empty cup-closed diff-closed preBoolean set
{(A + 1)} is non empty trivial finite V52() 1 -element set
(Seg A) \/ {(A + 1)} is non empty finite set
B9 is Relation-like Seg (A + 1) -defined Seg (A + 1) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
B9 | (Seg A) is Relation-like Seg (A + 1) -defined Seg A -defined Seg (A + 1) -defined Seg (A + 1) -valued Function-like finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
b is Relation-like Seg A -defined Seg A -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg A),(Seg A):]
mA is Relation-like Seg (len (Permutations A)) -defined Seg (len (Permutations A)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations A
Bb is Relation-like NAT -defined the carrier of (Group_of_Perm A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm A)
Product Bb is Element of the carrier of (Group_of_Perm A)
dom Bb is finite Element of bool NAT
the multF of (Group_of_Perm A) $$ Bb is Element of the carrier of (Group_of_Perm A)
len Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len Bb) is finite len Bb -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len Bb ) } is set
PM is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Bb . PM is set
i is Relation-like Seg (len (Permutations A)) -defined Seg (len (Permutations A)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations A
dom i is finite set
Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
H is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
i . Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
i . H is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
SF is Relation-like Seg A -defined Seg A -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg A),(Seg A):]
QQ is Relation-like Seg (A + 1) -defined Seg (A + 1) -valued Function-like non empty total quasi_total finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
QQ | (Seg A) is Relation-like Seg (A + 1) -defined Seg A -defined Seg (A + 1) -defined Seg (A + 1) -valued Function-like finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
QQ . (A + 1) is set
QQ . H is set
h is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
Mh is Element of the carrier of (Group_of_Perm (A + 1))
Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Bb . Path is set
QQ is Relation-like Seg (len (Permutations A)) -defined Seg (len (Permutations A)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations A
h . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
h | (Seg A) is Relation-like Seg (len (Permutations (A + 1))) -defined Seg A -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
[:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
dom QQ is non empty finite set
dom SF is finite set
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
QQ . SUM1 is set
i . SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
QQ . Pi is set
PM is Relation-like NAT -defined the carrier of (Group_of_Perm (A + 1)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (A + 1))
dom PM is finite Element of bool NAT
Product PM is Element of the carrier of (Group_of_Perm (A + 1))
the multF of (Group_of_Perm (A + 1)) $$ PM is Element of the carrier of (Group_of_Perm (A + 1))
len PM is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len PM) is finite len PM -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len PM ) } is set
the_unity_wrt the multF of (Group_of_Perm (A + 1)) is Element of the carrier of (Group_of_Perm (A + 1))
1_ (Group_of_Perm (A + 1)) is non being_of_order_0 Element of the carrier of (Group_of_Perm (A + 1))
the_unity_wrt the multF of (Group_of_Perm A) is Element of the carrier of (Group_of_Perm A)
dom B9 is non empty finite set
1_ (Group_of_Perm A) is non being_of_order_0 Element of the carrier of (Group_of_Perm A)
dom F is finite set
i is set
F . i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(idseq (A + 1)) . i is set
Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(idseq (A + 1)) . Pi is set
dom b is finite set
b . i is set
(idseq A) . Pi is set
dom (idseq (A + 1)) is finite A + 1 -element Element of bool NAT
i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
PM . i is set
len PM is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
[:NAT, the carrier of (Group_of_Perm (A + 1)):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Group_of_Perm (A + 1)):] is non empty non trivial cup-closed diff-closed preBoolean non finite set
PM . 1 is set
i is Relation-like NAT -defined the carrier of (Group_of_Perm (A + 1)) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (Group_of_Perm (A + 1)):]
i . 1 is Element of the carrier of (Group_of_Perm (A + 1))
i . (len PM) is Element of the carrier of (Group_of_Perm (A + 1))
[:NAT, the carrier of (Group_of_Perm A):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Group_of_Perm A):] is non empty non trivial cup-closed diff-closed preBoolean non finite set
Bb . 1 is set
Pi is Relation-like NAT -defined the carrier of (Group_of_Perm A) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (Group_of_Perm A):]
Pi . 1 is Element of the carrier of (Group_of_Perm A)
Pi . (len Bb) is Element of the carrier of (Group_of_Perm A)
H is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
i . H is Element of the carrier of (Group_of_Perm (A + 1))
Pi . H is Element of the carrier of (Group_of_Perm A)
H + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
i . (H + 1) is Element of the carrier of (Group_of_Perm (A + 1))
Pi . (H + 1) is Element of the carrier of (Group_of_Perm A)
(H + 1) + {} is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Bb . (H + 1) is set
QQ is Relation-like Seg (len (Permutations A)) -defined Seg (len (Permutations A)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations A
PM . (H + 1) is set
h is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
h . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
h | (Seg A) is Relation-like Seg (len (Permutations (A + 1))) -defined Seg A -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
[:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
H + {} is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Mh is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
Path is Relation-like Seg (len (Permutations A)) -defined Seg (len (Permutations A)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations A
Mh | (Seg A) is Relation-like Seg (len (Permutations (A + 1))) -defined Seg A -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
Mh . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Mh is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
Path is Relation-like Seg (len (Permutations A)) -defined Seg (len (Permutations A)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations A
Mh | (Seg A) is Relation-like Seg (len (Permutations (A + 1))) -defined Seg A -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
Mh . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
QQ is Relation-like Seg A -defined Seg A -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg A),(Seg A):]
SUM1 is Relation-like Seg A -defined Seg A -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg A),(Seg A):]
SUM1 * QQ is Relation-like Seg A -defined Seg A -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg A),(Seg A):]
the multF of (Group_of_Perm A) . (QQ,SUM1) is set
[QQ,SUM1] is set
{QQ,SUM1} is functional non empty finite V52() set
{QQ} is functional non empty trivial finite V52() 1 -element set
{{QQ,SUM1},{QQ}} is non empty finite V52() set
the multF of (Group_of_Perm A) . [QQ,SUM1] is set
PA is Relation-like Seg (len (Permutations A)) -defined Seg (len (Permutations A)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations A
PS is Relation-like Seg (A + 1) -defined Seg (A + 1) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
PSaa is Relation-like Seg (A + 1) -defined Seg (A + 1) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
PSaa * PS is Relation-like Seg (A + 1) -defined Seg (A + 1) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
Y9 is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
Y9 | (Seg A) is Relation-like Seg (len (Permutations (A + 1))) -defined Seg A -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
the multF of (Group_of_Perm (A + 1)) . (PS,PSaa) is set
[PS,PSaa] is set
{PS,PSaa} is functional non empty finite V52() set
{PS} is functional non empty trivial finite V52() 1 -element set
{{PS,PSaa},{PS}} is non empty finite V52() set
the multF of (Group_of_Perm (A + 1)) . [PS,PSaa] is set
Y9 . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Mh is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
Path is Relation-like Seg (len (Permutations A)) -defined Seg (len (Permutations A)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations A
Mh | (Seg A) is Relation-like Seg (len (Permutations (A + 1))) -defined Seg A -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
Mh . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
i . {} is Element of the carrier of (Group_of_Perm (A + 1))
Pi . {} is Element of the carrier of (Group_of_Perm A)
i . (len Bb) is Element of the carrier of (Group_of_Perm (A + 1))
H is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
SF is Relation-like Seg (len (Permutations A)) -defined Seg (len (Permutations A)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations A
H | (Seg A) is Relation-like Seg (len (Permutations (A + 1))) -defined Seg A -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
[:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
H . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
dom B9 is non empty finite set
dom F is finite set
QQ is Relation-like Seg (A + 1) -defined Seg (A + 1) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
h is set
F . h is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
QQ . h is set
dom b is finite set
SF . h is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
QQ . (A + 1) is set
dom QQ is non empty finite set
h is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
PM . h is set
Bb . h is set
Mh is Relation-like Seg (len (Permutations A)) -defined Seg (len (Permutations A)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations A
Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
PM . Path is set
QQ is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
QQ . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
QQ | (Seg A) is Relation-like Seg (len (Permutations (A + 1))) -defined Seg A -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
len PM is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
PM . i is set
P . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is Relation-like Seg (A + 1) -defined Seg (A + 1) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
rng KK is non empty finite set
aa is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
aa . (P . (A + 1)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
aa * P is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
[:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
(aa * P) . (A + 1) is set
AB is Relation-like Seg (A + 1) -defined Seg (A + 1) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
AB * KK is Relation-like Seg (A + 1) -defined Seg (A + 1) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
SUM1 is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
Path is Relation-like NAT -defined the carrier of (Group_of_Perm (A + 1)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (A + 1))
Product Path is Element of the carrier of (Group_of_Perm (A + 1))
dom Path is finite Element of bool NAT
Ga is Element of the carrier of (Group_of_Perm (A + 1))
<*Ga*> is Relation-like NAT -defined the carrier of (Group_of_Perm (A + 1)) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of (Group_of_Perm (A + 1))
1 -tuples_on the carrier of (Group_of_Perm (A + 1)) is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Group_of_Perm (A + 1))
the carrier of (Group_of_Perm (A + 1)) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Group_of_Perm (A + 1))
{ b1 where b1 is Relation-like NAT -defined the carrier of (Group_of_Perm (A + 1)) -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of (Group_of_Perm (A + 1)) * : len b1 = 1 } is set
[1,Ga] is set
{1,Ga} is non empty finite set
{{1,Ga},{1}} is non empty finite V52() set
{[1,Ga]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Path ^ <*Ga*> is Relation-like NAT -defined the carrier of (Group_of_Perm (A + 1)) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (A + 1))
Gs is Relation-like NAT -defined the carrier of (Group_of_Perm (A + 1)) -valued Function-like non empty finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (A + 1))
Product Gs is Element of the carrier of (Group_of_Perm (A + 1))
F is Element of the carrier of (Group_of_Perm (A + 1))
F * Ga is Element of the carrier of (Group_of_Perm (A + 1))
the multF of (Group_of_Perm (A + 1)) is Relation-like [: the carrier of (Group_of_Perm (A + 1)), the carrier of (Group_of_Perm (A + 1)):] -defined the carrier of (Group_of_Perm (A + 1)) -valued Function-like non empty total quasi_total having_a_unity V155( the carrier of (Group_of_Perm (A + 1))) associative Element of bool [:[: the carrier of (Group_of_Perm (A + 1)), the carrier of (Group_of_Perm (A + 1)):], the carrier of (Group_of_Perm (A + 1)):]
[: the carrier of (Group_of_Perm (A + 1)), the carrier of (Group_of_Perm (A + 1)):] is Relation-like non empty set
[:[: the carrier of (Group_of_Perm (A + 1)), the carrier of (Group_of_Perm (A + 1)):], the carrier of (Group_of_Perm (A + 1)):] is Relation-like non empty set
bool [:[: the carrier of (Group_of_Perm (A + 1)), the carrier of (Group_of_Perm (A + 1)):], the carrier of (Group_of_Perm (A + 1)):] is non empty cup-closed diff-closed preBoolean set
the multF of (Group_of_Perm (A + 1)) . (F,Ga) is Element of the carrier of (Group_of_Perm (A + 1))
[F,Ga] is set
{F,Ga} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,Ga},{F}} is non empty finite V52() set
the multF of (Group_of_Perm (A + 1)) . [F,Ga] is set
aa * (aa * P) is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total total quasi_total quasi_total quasi_total onto onto bijective bijective finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
aa * aa is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
(aa * aa) * P is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total total quasi_total quasi_total quasi_total onto onto bijective bijective finite Element of bool [:(Seg (len (Permutations (A + 1)))),(Seg (len (Permutations (A + 1)))):]
idseq (A + 1) is Relation-like NAT -defined Function-like finite A + 1 -element FinSequence-like FinSubsequence-like set
id (Seg (A + 1)) is Relation-like Seg (A + 1) -defined Seg (A + 1) -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (A + 1)),(Seg (A + 1)):]
P * (idseq (A + 1)) is Relation-like Seg (len (Permutations (A + 1))) -defined Function-like finite set
dom Gs is non empty finite Element of bool NAT
len Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len Path) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Seg ((len Path) + 1) is non empty finite (len Path) + 1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= (len Path) + 1 ) } is set
Seg (len Path) is finite len Path -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len Path ) } is set
{((len Path) + 1)} is non empty trivial finite V52() 1 -element set
(Seg (len Path)) \/ {((len Path) + 1)} is non empty finite set
len Gs is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
mA is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Gs . mA is set
Path . mA is set
Bb is Relation-like Seg (len (Permutations (A + 1))) -defined Seg (len (Permutations (A + 1))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (A + 1)
P . (A + 1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
aa is Relation-like NAT -defined the carrier of (Group_of_Perm (A + 1)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (A + 1))
Product aa is Element of the carrier of (Group_of_Perm (A + 1))
dom aa is finite Element of bool NAT
Permutations {} is non empty permutational set
len (Permutations {}) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations {})) is finite len (Permutations {}) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations {}) ) } is set
Group_of_Perm {} is non empty strict unital Group-like associative multMagma
the carrier of (Group_of_Perm {}) is non empty set
A is Relation-like Seg (len (Permutations {})) -defined Seg (len (Permutations {})) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations {}
<*> the carrier of (Group_of_Perm {}) is Relation-like non-empty empty-yielding NAT -defined the carrier of (Group_of_Perm {}) -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered FinSequence of the carrier of (Group_of_Perm {})
[:NAT, the carrier of (Group_of_Perm {}):] is Relation-like non empty non trivial non finite set
Product (<*> the carrier of (Group_of_Perm {})) is Element of the carrier of (Group_of_Perm {})
dom (<*> the carrier of (Group_of_Perm {})) is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of bool NAT
[:(Seg {}),(Seg {}):] is Relation-like finite set
bool [:(Seg {}),(Seg {}):] is non empty cup-closed diff-closed preBoolean finite V52() set
idseq {} is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element 0 -element FinSequence-like FinSubsequence-like FinSequence-membered set
id (Seg {}) is Relation-like non-empty empty-yielding Seg {} -defined Seg {} -valued V6() V8() V9() V13() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty total quasi_total onto bijective ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of bool [:(Seg {}),(Seg {}):]
1_ (Group_of_Perm {}) is non being_of_order_0 Element of the carrier of (Group_of_Perm {})
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(<*> the carrier of (Group_of_Perm {})) . B is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
1_ n is Element of the carrier of n
the carrier of n is non empty non trivial set
K254(n) is V70(n) Element of the carrier of n
- (1_ n) is Element of the carrier of n
(1_ n) + (1_ n) is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
the addF of n . ((1_ n),(1_ n)) is Element of the carrier of n
[(1_ n),(1_ n)] is set
{(1_ n),(1_ n)} is non empty finite set
{(1_ n)} is non empty trivial finite 1 -element set
{{(1_ n),(1_ n)},{(1_ n)}} is non empty finite V52() set
the addF of n . [(1_ n),(1_ n)] is set
0. n is V70(n) Element of the carrier of n
0. n is V70(n) Element of the carrier of n
K is Element of the carrier of n
K + K is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
the addF of n . (K,K) is Element of the carrier of n
[K,K] is set
{K,K} is non empty finite set
{K} is non empty trivial finite 1 -element set
{{K,K},{K}} is non empty finite V52() set
the addF of n . [K,K] is set
K * (1_ n) is Element of the carrier of n
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
the multF of n . (K,(1_ n)) is Element of the carrier of n
[K,(1_ n)] is set
{K,(1_ n)} is non empty finite set
{{K,(1_ n)},{K}} is non empty finite V52() set
the multF of n . [K,(1_ n)] is set
(1_ n) + (1_ n) is Element of the carrier of n
the addF of n . ((1_ n),(1_ n)) is Element of the carrier of n
[(1_ n),(1_ n)] is set
{(1_ n),(1_ n)} is non empty finite set
{(1_ n)} is non empty trivial finite 1 -element set
{{(1_ n),(1_ n)},{(1_ n)}} is non empty finite V52() set
the addF of n . [(1_ n),(1_ n)] is set
K * ((1_ n) + (1_ n)) is Element of the carrier of n
the multF of n . (K,((1_ n) + (1_ n))) is Element of the carrier of n
[K,((1_ n) + (1_ n))] is set
{K,((1_ n) + (1_ n))} is non empty finite set
{{K,((1_ n) + (1_ n))},{K}} is non empty finite V52() set
the multF of n . [K,((1_ n) + (1_ n))] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
K is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
B is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
(n,B,K) is Element of the carrier of B
the carrier of B is non empty non trivial set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of B is Relation-like [: the carrier of B, the carrier of B:] -defined the carrier of B -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of B, the carrier of B:], the carrier of B:]
[: the carrier of B, the carrier of B:] is Relation-like non empty set
[:[: the carrier of B, the carrier of B:], the carrier of B:] is Relation-like 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
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,B,K) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of B -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of B:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of B:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of B:] is non empty cup-closed diff-closed preBoolean set
the multF of B $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,B,K)) is Element of the carrier of B
1_ B is Element of the carrier of B
K254(B) is V70(B) Element of the carrier of B
- (1_ B) is Element of the carrier of B
Group_of_Perm (n + 2) is non empty strict unital Group-like associative multMagma
the carrier of (Group_of_Perm (n + 2)) is non empty set
P is Relation-like NAT -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (n + 2))
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len P) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Product P is Element of the carrier of (Group_of_Perm (n + 2))
dom P is finite Element of bool NAT
Group_of_Perm (n + 2) is non empty strict unital Group-like associative multMagma
the carrier of (Group_of_Perm (n + 2)) is non empty set
P is Relation-like NAT -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (n + 2))
Product P is Element of the carrier of (Group_of_Perm (n + 2))
dom P is finite Element of bool NAT
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len P) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
KK is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
P is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
B is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
B * P is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):]
[:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
(n,K,KK) is Element of the carrier of K
the carrier of K is non empty non trivial set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,KK) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,KK)) is Element of the carrier of K
(n,K,B) is Element of the carrier of K
(n,K,B) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,B)) is Element of the carrier of K
(n,K,P) is Element of the carrier of K
(n,K,P) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,P)) is Element of the carrier of K
(n,K,B) * (n,K,P) is Element of the carrier of K
the multF of K . ((n,K,B),(n,K,P)) is Element of the carrier of K
[(n,K,B),(n,K,P)] is set
{(n,K,B),(n,K,P)} is non empty finite set
{(n,K,B)} is non empty trivial finite 1 -element set
{{(n,K,B),(n,K,P)},{(n,K,B)}} is non empty finite V52() set
the multF of K . [(n,K,B),(n,K,P)] is set
Group_of_Perm (n + 2) is non empty strict unital Group-like associative multMagma
the carrier of (Group_of_Perm (n + 2)) is non empty set
mm is Relation-like NAT -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (n + 2))
Product mm is Element of the carrier of (Group_of_Perm (n + 2))
dom mm is finite Element of bool NAT
aa is Relation-like NAT -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (n + 2))
Product aa is Element of the carrier of (Group_of_Perm (n + 2))
dom aa is finite Element of bool NAT
mm ^ aa is Relation-like NAT -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Group_of_Perm (n + 2))
dom (mm ^ aa) is finite Element of bool NAT
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(mm ^ aa) . SUM1 is set
mm . SUM1 is set
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(len mm) + Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(len mm) + Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
aa . Path is set
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Path is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
Product (mm ^ aa) is Element of the carrier of (Group_of_Perm (n + 2))
(Product mm) * (Product aa) is Element of the carrier of (Group_of_Perm (n + 2))
the multF of (Group_of_Perm (n + 2)) is Relation-like [: the carrier of (Group_of_Perm (n + 2)), the carrier of (Group_of_Perm (n + 2)):] -defined the carrier of (Group_of_Perm (n + 2)) -valued Function-like non empty total quasi_total having_a_unity V155( the carrier of (Group_of_Perm (n + 2))) associative Element of bool [:[: the carrier of (Group_of_Perm (n + 2)), the carrier of (Group_of_Perm (n + 2)):], the carrier of (Group_of_Perm (n + 2)):]
[: the carrier of (Group_of_Perm (n + 2)), the carrier of (Group_of_Perm (n + 2)):] is Relation-like non empty set
[:[: the carrier of (Group_of_Perm (n + 2)), the carrier of (Group_of_Perm (n + 2)):], the carrier of (Group_of_Perm (n + 2)):] is Relation-like non empty set
bool [:[: the carrier of (Group_of_Perm (n + 2)), the carrier of (Group_of_Perm (n + 2)):], the carrier of (Group_of_Perm (n + 2)):] is non empty cup-closed diff-closed preBoolean set
the multF of (Group_of_Perm (n + 2)) . ((Product mm),(Product aa)) is Element of the carrier of (Group_of_Perm (n + 2))
[(Product mm),(Product aa)] is set
{(Product mm),(Product aa)} is non empty finite set
{(Product mm)} is non empty trivial finite 1 -element set
{{(Product mm),(Product aa)},{(Product mm)}} is non empty finite V52() set
the multF of (Group_of_Perm (n + 2)) . [(Product mm),(Product aa)] is set
len aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len aa) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len mm) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len (mm ^ aa) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len (mm ^ aa)) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len mm) + (len aa) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
((len mm) + (len aa)) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
{} + (len aa) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
({} + (len aa)) + {} is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(({} + (len aa)) + {}) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
len aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len aa) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len mm) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len (mm ^ aa) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len (mm ^ aa)) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len mm) + (len aa) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
((len mm) + (len aa)) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
{} + (len aa) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
({} + (len aa)) + {} is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(({} + (len aa)) + {}) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
len aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len aa) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len mm) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len (mm ^ aa) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len (mm ^ aa)) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len mm) + (len aa) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
((len mm) + (len aa)) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
1 + (len aa) is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
(1 + (len aa)) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
1 + {} is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
(1 + {}) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
len aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len aa) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len mm) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len (mm ^ aa) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len (mm ^ aa)) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len mm) + (len aa) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
((len mm) + (len aa)) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
1 + (len aa) is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
(1 + (len aa)) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
(1 + 1) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
(1_ K) * (1_ K) is Element of the carrier of K
the multF of K . ((1_ K),(1_ K)) is Element of the carrier of K
[(1_ K),(1_ K)] is set
{(1_ K),(1_ K)} is non empty finite set
{(1_ K)} is non empty trivial finite 1 -element set
{{(1_ K),(1_ K)},{(1_ K)}} is non empty finite V52() set
the multF of K . [(1_ K),(1_ K)] is set
- (1_ K) is Element of the carrier of K
len aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len aa) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len mm) mod 2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
id (Seg n) is Relation-like Seg n -defined Seg n -valued V6() V8() V9() V13() Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
A is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
A is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
K is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
A is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
K * A is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
B is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
n - 2 is ext-real V44() V45() set
the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (P + 2) is non empty permutational set
len (Permutations (P + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (P + 2))) is finite len (Permutations (P + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (P + 2)) ) } is set
B is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
mm is Relation-like Seg (len (Permutations (P + 2))) -defined Seg (len (Permutations (P + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (P + 2)
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr is non empty non trivial set
Seg (P + 2) is non empty finite P + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= P + 2 ) } is set
TWOELEMENTSETS (Seg (P + 2)) is non empty finite set
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr is Relation-like [: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :]
[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
[:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
bool [:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (P + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (P + 2)))
Fin (TWOELEMENTSETS (Seg (P + 2))) is non empty cup-closed diff-closed preBoolean set
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm) is Relation-like TWOELEMENTSETS (Seg (P + 2)) -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (P + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :]
[:(TWOELEMENTSETS (Seg (P + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (P + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is non empty cup-closed diff-closed preBoolean set
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr $$ ((FinOmega (TWOELEMENTSETS (Seg (P + 2)))),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm)) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
K254( the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) is V70( the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
aa is Relation-like Seg (len (Permutations (P + 2))) -defined Seg (len (Permutations (P + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (P + 2)
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa) is Relation-like TWOELEMENTSETS (Seg (P + 2)) -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (P + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :]
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr $$ ((FinOmega (TWOELEMENTSETS (Seg (P + 2)))),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
- (1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm) * (P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr . ((P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
[(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)] is set
{(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)} is non empty finite set
{(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm)} is non empty trivial finite 1 -element set
{{(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)},{(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm)}} is non empty finite V52() set
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr . [(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)] is set
(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) * (1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr . ((1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ),(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr )) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
[(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ),(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr )] is set
{(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ),(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr )} is non empty finite set
{(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr )} is non empty trivial finite 1 -element set
{{(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ),(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr )},{(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr )}} is non empty finite V52() set
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr . [(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ),(1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr )] is set
AB is Relation-like Seg (len (Permutations (P + 2))) -defined Seg (len (Permutations (P + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (P + 2)
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,AB) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,AB) is Relation-like TWOELEMENTSETS (Seg (P + 2)) -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (P + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :]
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr $$ ((FinOmega (TWOELEMENTSETS (Seg (P + 2)))),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,AB)) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
AB is Relation-like Seg (len (Permutations (P + 2))) -defined Seg (len (Permutations (P + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (P + 2)
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,AB) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr is non empty non trivial set
Seg (P + 2) is non empty finite P + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= P + 2 ) } is set
TWOELEMENTSETS (Seg (P + 2)) is non empty finite set
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr is Relation-like [: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :]
[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
[:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
bool [:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (P + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (P + 2)))
Fin (TWOELEMENTSETS (Seg (P + 2))) is non empty cup-closed diff-closed preBoolean set
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,AB) is Relation-like TWOELEMENTSETS (Seg (P + 2)) -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (P + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :]
[:(TWOELEMENTSETS (Seg (P + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (P + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is non empty cup-closed diff-closed preBoolean set
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr $$ ((FinOmega (TWOELEMENTSETS (Seg (P + 2)))),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,AB)) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
K254( the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) is V70( the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
mm is Relation-like Seg (len (Permutations (P + 2))) -defined Seg (len (Permutations (P + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (P + 2)
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm) is Relation-like TWOELEMENTSETS (Seg (P + 2)) -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (P + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :]
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr $$ ((FinOmega (TWOELEMENTSETS (Seg (P + 2)))),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm)) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
aa is Relation-like Seg (len (Permutations (P + 2))) -defined Seg (len (Permutations (P + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (P + 2)
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa) is Relation-like TWOELEMENTSETS (Seg (P + 2)) -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (P + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :]
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr $$ ((FinOmega (TWOELEMENTSETS (Seg (P + 2)))),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm) * (P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr . ((P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
[(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)] is set
{(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)} is non empty finite set
{(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm)} is non empty trivial finite 1 -element set
{{(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)},{(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm)}} is non empty finite V52() set
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr . [(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,mm),(P, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,aa)] is set
- (1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
- (1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
A is Element of the carrier of K
B is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
- (A,B) is Element of the carrier of K
(n,K,B) is Element of the carrier of K
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,B) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,B)) is Element of the carrier of K
(n,K,B) * A is Element of the carrier of K
the multF of K . ((n,K,B),A) is Element of the carrier of K
[(n,K,B),A] is set
{(n,K,B),A} is non empty finite set
{(n,K,B)} is non empty trivial finite 1 -element set
{{(n,K,B),A},{(n,K,B)}} is non empty finite V52() set
the multF of K . [(n,K,B),A] is set
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- A is Element of the carrier of K
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
(- (1_ K)) * A is Element of the carrier of K
the multF of K . ((- (1_ K)),A) is Element of the carrier of K
[(- (1_ K)),A] is set
{(- (1_ K)),A} is non empty finite set
{(- (1_ K))} is non empty trivial finite 1 -element set
{{(- (1_ K)),A},{(- (1_ K))}} is non empty finite V52() set
the multF of K . [(- (1_ K)),A] is set
(1_ K) * A is Element of the carrier of K
the multF of K . ((1_ K),A) is Element of the carrier of K
[(1_ K),A] is set
{(1_ K),A} is non empty finite set
{(1_ K)} is non empty trivial finite 1 -element set
{{(1_ K),A},{(1_ K)}} is non empty finite V52() set
the multF of K . [(1_ K),A] is set
- ((1_ K) * A) is Element of the carrier of K
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
- A is Element of the carrier of K
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
(- (1_ K)) * A is Element of the carrier of K
the multF of K . ((- (1_ K)),A) is Element of the carrier of K
[(- (1_ K)),A] is set
{(- (1_ K)),A} is non empty finite set
{(- (1_ K))} is non empty trivial finite 1 -element set
{{(- (1_ K)),A},{(- (1_ K))}} is non empty finite V52() set
the multF of K . [(- (1_ K)),A] is set
(1_ K) * A is Element of the carrier of K
the multF of K . ((1_ K),A) is Element of the carrier of K
[(1_ K),A] is set
{(1_ K),A} is non empty finite set
{(1_ K)} is non empty trivial finite 1 -element set
{{(1_ K),A},{(1_ K)}} is non empty finite V52() set
the multF of K . [(1_ K),A] is set
- ((1_ K) * A) is Element of the carrier of K
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
A is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,A) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr is non empty non trivial set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr is Relation-like [: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :]
[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
[:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
bool [:[: the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr , the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :], the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,A) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr :] is non empty cup-closed diff-closed preBoolean set
the multF of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n, the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ,A)) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
K254( the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) is V70( the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
- (1_ the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr ) is Element of the carrier of the non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital Fanoian doubleLoopStr
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
2 + {} is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{A,B} is non empty finite V52() set
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
P is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
P . A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is non empty set
B * is functional non empty FinSequence-membered FinSequenceSet of B
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of B
len KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
P is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,A,B
width P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Indices P is set
dom P is finite Element of bool NAT
Seg (width P) is finite width P -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg aa is finite aa -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= aa ) } is set
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg AB is finite AB -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= AB ) } is set
[:(Seg aa),(Seg AB):] is Relation-like finite set
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[SUM1,Path] is set
{SUM1,Path} is non empty finite V52() set
{SUM1} is non empty trivial finite V52() 1 -element set
{{SUM1,Path},{SUM1}} is non empty finite V52() set
rng KK is finite set
Seg (len KK) is finite len KK -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len KK ) } is set
dom KK is finite Element of bool NAT
KK . Path is set
F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P * (F,Ga) is Element of B
KK . Ga is set
P * (SUM1,Path) is Element of B
F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P * (F,Ga) is Element of B
KK . Ga is set
F is Element of B
Ga is Element of B
SUM1 is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of aa,AB,B
Indices SUM1 is set
dom SUM1 is finite Element of bool NAT
width SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (width SUM1) is finite width SUM1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width SUM1 ) } is set
[:(dom SUM1),(Seg (width SUM1)):] is Relation-like finite set
Path is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,A,B
len Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
width Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
mm is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of len P, width P,B
Indices mm is set
dom mm is finite Element of bool NAT
width mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (width mm) is finite width mm -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width mm ) } is set
[:(dom mm),(Seg (width mm)):] is Relation-like finite set
Indices Path is set
dom Path is finite Element of bool NAT
Seg (width Path) is finite width Path -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width Path ) } is set
[:(dom Path),(Seg (width Path)):] is Relation-like finite set
F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[F,Ga] is set
{F,Ga} is non empty finite V52() set
{F} is non empty trivial finite V52() 1 -element set
{{F,Ga},{F}} is non empty finite V52() set
Path * (F,Ga) is Element of B
P * (F,Ga) is Element of B
Path * (n,Ga) is Element of B
KK . Ga is set
mm is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,A,B
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
width mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
aa is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,A,B
len aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
width aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Indices mm is set
dom mm is finite Element of bool NAT
Seg (width mm) is finite width mm -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width mm ) } is set
[:(dom mm),(Seg (width mm)):] is Relation-like finite set
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[AB,SUM1] is set
{AB,SUM1} is non empty finite V52() set
{AB} is non empty trivial finite V52() 1 -element set
{{AB,SUM1},{AB}} is non empty finite V52() set
mm * (AB,SUM1) is Element of B
aa * (AB,SUM1) is Element of B
mm * (n,SUM1) is Element of B
KK . SUM1 is set
P * (AB,SUM1) is Element of B
B is non empty set
B * is functional non empty FinSequence-membered FinSequenceSet of B
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is non empty set
A * is functional non empty FinSequence-membered FinSequenceSet of A
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,n,A
Indices P is set
dom P is finite Element of bool NAT
width P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (width P) is finite width P -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
KK is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of A
(B,K,n,A,P,KK) is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,n,A
Indices (B,K,n,A,P,KK) is set
dom (B,K,n,A,P,KK) is finite Element of bool NAT
width (B,K,n,A,P,KK) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (width (B,K,n,A,P,KK)) is finite width (B,K,n,A,P,KK) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width (B,K,n,A,P,KK) ) } is set
[:(dom (B,K,n,A,P,KK)),(Seg (width (B,K,n,A,P,KK))):] is Relation-like finite set
len (B,K,n,A,P,KK) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg K is finite K -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= K ) } is set
A is non empty set
A * is functional non empty FinSequence-membered FinSequenceSet of A
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,n,A
width P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
KK is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of A
len KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(B,K,n,A,P,KK) is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,n,A
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Line ((B,K,n,A,P,KK),mm) is Relation-like NAT -defined A -valued Function-like finite width (B,K,n,A,P,KK) -element FinSequence-like FinSubsequence-like Element of (width (B,K,n,A,P,KK)) -tuples_on A
width (B,K,n,A,P,KK) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width (B,K,n,A,P,KK)) -tuples_on A is functional non empty FinSequence-membered FinSequenceSet of A
{ b1 where b1 is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like Element of A * : len b1 = width (B,K,n,A,P,KK) } is set
Line (P,mm) is Relation-like NAT -defined A -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on A
(width P) -tuples_on A is functional non empty FinSequence-membered FinSequenceSet of A
{ b1 where b1 is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like Element of A * : len b1 = width P } is set
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg (width (B,K,n,A,P,KK)) is finite width (B,K,n,A,P,KK) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width (B,K,n,A,P,KK) ) } is set
len (B,K,n,A,P,KK) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom (B,K,n,A,P,KK) is finite Element of bool NAT
[mm,SUM1] is set
{mm,SUM1} is non empty finite V52() set
{mm} is non empty trivial finite V52() 1 -element set
{{mm,SUM1},{mm}} is non empty finite V52() set
Indices (B,K,n,A,P,KK) is set
[:(dom (B,K,n,A,P,KK)),(Seg (width (B,K,n,A,P,KK))):] is Relation-like finite set
Indices P is set
dom P is finite Element of bool NAT
Seg (width P) is finite width P -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
(Line ((B,K,n,A,P,KK),mm)) . SUM1 is set
(B,K,n,A,P,KK) * (mm,SUM1) is Element of A
KK . SUM1 is set
len (Line ((B,K,n,A,P,KK),mm)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len (Line (P,mm)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg (len (Line (P,mm))) is finite len (Line (P,mm)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Line (P,mm)) ) } is set
(Line (P,mm)) . Path is set
P * (mm,Path) is Element of A
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len P) is finite len P -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len P ) } is set
dom P is finite Element of bool NAT
[mm,Path] is set
{mm,Path} is non empty finite V52() set
{mm} is non empty trivial finite V52() 1 -element set
{{mm,Path},{mm}} is non empty finite V52() set
Indices P is set
Seg (width P) is finite width P -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
(Line ((B,K,n,A,P,KK),mm)) . Path is set
(B,K,n,A,P,KK) * (mm,Path) is Element of A
len (Line ((B,K,n,A,P,KK),mm)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is non empty set
B * is functional non empty FinSequence-membered FinSequenceSet of B
P is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,n,B
width P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
KK is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of B
len KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(A,K,n,B,P,KK) is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,n,B
aa is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B *
Replace (P,A,aa) is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of B *
len (Replace (P,A,aa)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg (len (Replace (P,A,aa))) is finite len (Replace (P,A,aa)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Replace (P,A,aa)) ) } is set
dom (Replace (P,A,aa)) is finite Element of bool NAT
Seg K is finite K -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= K ) } is set
dom P is finite Element of bool NAT
Line ((A,K,n,B,P,KK),SUM1) is Relation-like NAT -defined B -valued Function-like finite width (A,K,n,B,P,KK) -element FinSequence-like FinSubsequence-like Element of (width (A,K,n,B,P,KK)) -tuples_on B
width (A,K,n,B,P,KK) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width (A,K,n,B,P,KK)) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b1 where b1 is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b1 = width (A,K,n,B,P,KK) } is set
(Replace (P,A,aa)) /. SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of B *
(Replace (P,A,aa)) . SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(A,K,n,B,P,KK) . SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (P,SUM1) is Relation-like NAT -defined B -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on B
(width P) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b1 where b1 is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b1 = width P } is set
Line ((A,K,n,B,P,KK),SUM1) is Relation-like NAT -defined B -valued Function-like finite width (A,K,n,B,P,KK) -element FinSequence-like FinSubsequence-like Element of (width (A,K,n,B,P,KK)) -tuples_on B
width (A,K,n,B,P,KK) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width (A,K,n,B,P,KK)) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b1 where b1 is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b1 = width (A,K,n,B,P,KK) } is set
(Replace (P,A,aa)) . SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Replace (P,A,aa)) /. SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of B *
P . SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
P /. SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of B *
(A,K,n,B,P,KK) . SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Replace (P,A,aa)) . SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(A,K,n,B,P,KK) . SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Replace (P,A,aa)) . SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(A,K,n,B,P,KK) . SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (A,K,n,B,P,KK) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is non empty set
B * is functional non empty FinSequence-membered FinSequenceSet of B
P is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,B
Line (P,A) is Relation-like NAT -defined B -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on B
width P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width P) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b1 where b1 is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b1 = width P } is set
(A,n,K,B,P,(Line (P,A))) is Relation-like NAT -defined B * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,B
len (Line (P,A)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len P) is finite len P -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len P ) } is set
(A,n,K,B,P,(Line (P,A))) . aa is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line ((A,n,K,B,P,(Line (P,A))),aa) is Relation-like NAT -defined B -valued Function-like finite width (A,n,K,B,P,(Line (P,A))) -element FinSequence-like FinSubsequence-like Element of (width (A,n,K,B,P,(Line (P,A)))) -tuples_on B
width (A,n,K,B,P,(Line (P,A))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width (A,n,K,B,P,(Line (P,A)))) -tuples_on B is functional non empty FinSequence-membered FinSequenceSet of B
{ b1 where b1 is Relation-like NAT -defined B -valued Function-like finite FinSequence-like FinSubsequence-like Element of B * : len b1 = width (A,n,K,B,P,(Line (P,A))) } is set
Line (P,aa) is Relation-like NAT -defined B -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on B
P . aa is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (A,n,K,B,P,(Line (P,A))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of n is non empty non trivial set
K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
A is Element of the carrier of n
A * K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
A multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
bool [: the carrier of n, the carrier of n:] is non empty cup-closed diff-closed preBoolean set
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of n, the carrier of n:]
K224( the carrier of n, the carrier of n, the multF of n,A,(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
(A multfield) * K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (A * K) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = len K } is set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of n is non empty non trivial set
K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
K + A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
the addF of n .: (K,A) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (K + A) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = len K } is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
A is Element of the carrier of K
B is Element of the carrier of K
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
A * KK is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,A,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(A multfield) * KK is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
mm is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
B * mm is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
B multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,B,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(B multfield) * mm is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(A * KK) + (B * mm) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K .: ((A * KK),(B * mm)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
aa is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
Path is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) is Element of the carrier of K
(P,n,n, the carrier of K,Path,KK) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Path_matrix (aa,(P,n,n, the carrier of K,Path,KK)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) is Element of the carrier of K
A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK)))) is Element of the carrier of K
the multF of K . (A,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))))) is Element of the carrier of K
[A,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))))] is set
{A,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))))} is non empty finite set
{A} is non empty trivial finite 1 -element set
{{A,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))))},{A}} is non empty finite V52() set
the multF of K . [A,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))))] is set
(P,n,n, the carrier of K,Path,mm) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Path_matrix (aa,(P,n,n, the carrier of K,Path,mm)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) is Element of the carrier of K
B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm)))) is Element of the carrier of K
the multF of K . (B,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))))) is Element of the carrier of K
[B,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))))] is set
{B,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))))} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))))},{B}} is non empty finite V52() set
the multF of K . [B,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))))] is set
(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))))) + (B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))))) is Element of the carrier of K
the addF of K . ((A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))))),(B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm)))))) is Element of the carrier of K
[(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))))),(B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm)))))] is set
{(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))))),(B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm)))))} is non empty finite set
{(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK)))))} is non empty trivial finite 1 -element set
{{(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))))),(B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm)))))},{(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK)))))}} is non empty finite V52() set
the addF of K . [(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))))),(B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm)))))] is set
len (Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
[:NAT, the carrier of K:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of K:] is non empty non trivial cup-closed diff-closed preBoolean non finite set
(Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) . 1 is set
Gs is Relation-like NAT -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of K:]
Gs . 1 is Element of the carrier of K
Gs . (len (Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm)))))) is Element of the carrier of K
len (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) . 1 is set
mA is Relation-like NAT -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of K:]
mA . 1 is Element of the carrier of K
mA . (len (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm)))) is Element of the carrier of K
len (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) . 1 is set
i is Relation-like NAT -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of K:]
i . 1 is Element of the carrier of K
i . (len (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK)))) is Element of the carrier of K
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) . Pi is set
(Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) . Pi is set
(Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) . Pi is set
aa . Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
dom (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) is finite Element of bool NAT
dom (Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) is finite Element of bool NAT
H is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
[Pi,H] is set
{Pi,H} is non empty finite V52() set
{Pi} is non empty trivial finite V52() 1 -element set
{{Pi,H},{Pi}} is non empty finite V52() set
[:(Seg n),(Seg n):] is Relation-like finite set
Indices Path is set
dom Path is finite Element of bool NAT
width Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (width Path) is finite width Path -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width Path ) } is set
[:(dom Path),(Seg (width Path)):] is Relation-like finite set
dom mm is finite Element of bool NAT
mm /. H is Element of the carrier of K
mm . H is set
dom KK is finite Element of bool NAT
KK /. H is Element of the carrier of K
KK . H is set
len (B * mm) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom (B * mm) is finite Element of bool NAT
(B * mm) . H is set
QQ is Element of the carrier of K
B * QQ is Element of the carrier of K
the multF of K . (B,QQ) is Element of the carrier of K
[B,QQ] is set
{B,QQ} is non empty finite set
{{B,QQ},{B}} is non empty finite V52() set
the multF of K . [B,QQ] is set
len (A * KK) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len ((A * KK) + (B * mm)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom ((A * KK) + (B * mm)) is finite Element of bool NAT
dom (A * KK) is finite Element of bool NAT
(A * KK) . H is set
SF is Element of the carrier of K
A * SF is Element of the carrier of K
the multF of K . (A,SF) is Element of the carrier of K
[A,SF] is set
{A,SF} is non empty finite set
{{A,SF},{A}} is non empty finite V52() set
the multF of K . [A,SF] is set
dom (Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) is finite Element of bool NAT
(P,n,n, the carrier of K,Path,mm) * (Pi,H) is Element of the carrier of K
Path * (Pi,H) is Element of the carrier of K
(P,n,n, the carrier of K,Path,KK) * (Pi,H) is Element of the carrier of K
(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))) * (Pi,H) is Element of the carrier of K
(P,n,n, the carrier of K,Path,KK) * (Pi,H) is Element of the carrier of K
(P,n,n, the carrier of K,Path,mm) * (Pi,H) is Element of the carrier of K
(A * SF) + (B * QQ) is Element of the carrier of K
the addF of K . ((A * SF),(B * QQ)) is Element of the carrier of K
[(A * SF),(B * QQ)] is set
{(A * SF),(B * QQ)} is non empty finite set
{(A * SF)} is non empty trivial finite 1 -element set
{{(A * SF),(B * QQ)},{(A * SF)}} is non empty finite V52() set
the addF of K . [(A * SF),(B * QQ)] is set
(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))) * (Pi,H) is Element of the carrier of K
((A * KK) + (B * mm)) . H is set
Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
i . Pi is Element of the carrier of K
mA . Pi is Element of the carrier of K
Gs . Pi is Element of the carrier of K
Pi + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
i . (Pi + 1) is Element of the carrier of K
mA . (Pi + 1) is Element of the carrier of K
Gs . (Pi + 1) is Element of the carrier of K
(Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) . (Pi + 1) is set
the multF of K . ((i . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) . (Pi + 1))) is set
[(i . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) . (Pi + 1))] is set
{(i . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) . (Pi + 1))} is non empty finite set
{(i . Pi)} is non empty trivial finite 1 -element set
{{(i . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) . (Pi + 1))},{(i . Pi)}} is non empty finite V52() set
the multF of K . [(i . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) . (Pi + 1))] is set
Pi + {} is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) . (Pi + 1) is set
the multF of K . ((mA . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) . (Pi + 1))) is set
[(mA . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) . (Pi + 1))] is set
{(mA . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) . (Pi + 1))} is non empty finite set
{(mA . Pi)} is non empty trivial finite 1 -element set
{{(mA . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) . (Pi + 1))},{(mA . Pi)}} is non empty finite V52() set
the multF of K . [(mA . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) . (Pi + 1))] is set
(Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) . (Pi + 1) is set
the multF of K . ((Gs . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) . (Pi + 1))) is set
[(Gs . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) . (Pi + 1))] is set
{(Gs . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) . (Pi + 1))} is non empty finite set
{(Gs . Pi)} is non empty trivial finite 1 -element set
{{(Gs . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) . (Pi + 1))},{(Gs . Pi)}} is non empty finite V52() set
the multF of K . [(Gs . Pi),((Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) . (Pi + 1))] is set
i . {} is Element of the carrier of K
mA . {} is Element of the carrier of K
Gs . {} is Element of the carrier of K
Gs . AB is Element of the carrier of K
i . AB is Element of the carrier of K
A * (i . AB) is Element of the carrier of K
the multF of K . (A,(i . AB)) is Element of the carrier of K
[A,(i . AB)] is set
{A,(i . AB)} is non empty finite set
{{A,(i . AB)},{A}} is non empty finite V52() set
the multF of K . [A,(i . AB)] is set
mA . AB is Element of the carrier of K
B * (mA . AB) is Element of the carrier of K
the multF of K . (B,(mA . AB)) is Element of the carrier of K
[B,(mA . AB)] is set
{B,(mA . AB)} is non empty finite set
{{B,(mA . AB)},{B}} is non empty finite V52() set
the multF of K . [B,(mA . AB)] is set
(A * (i . AB)) + (B * (mA . AB)) is Element of the carrier of K
the addF of K . ((A * (i . AB)),(B * (mA . AB))) is Element of the carrier of K
[(A * (i . AB)),(B * (mA . AB))] is set
{(A * (i . AB)),(B * (mA . AB))} is non empty finite set
{(A * (i . AB))} is non empty trivial finite 1 -element set
{{(A * (i . AB)),(B * (mA . AB))},{(A * (i . AB))}} is non empty finite V52() set
the addF of K . [(A * (i . AB)),(B * (mA . AB))] is set
(Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) . AB is set
(Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) . AB is set
(Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) . AB is set
Pi is Element of the carrier of K
H is Element of the carrier of K
A * Pi is Element of the carrier of K
the multF of K . (A,Pi) is Element of the carrier of K
[A,Pi] is set
{A,Pi} is non empty finite set
{{A,Pi},{A}} is non empty finite V52() set
the multF of K . [A,Pi] is set
B * H is Element of the carrier of K
the multF of K . (B,H) is Element of the carrier of K
[B,H] is set
{B,H} is non empty finite set
{{B,H},{B}} is non empty finite V52() set
the multF of K . [B,H] is set
(A * Pi) + (B * H) is Element of the carrier of K
the addF of K . ((A * Pi),(B * H)) is Element of the carrier of K
[(A * Pi),(B * H)] is set
{(A * Pi),(B * H)} is non empty finite set
{(A * Pi)} is non empty trivial finite 1 -element set
{{(A * Pi),(B * H)},{(A * Pi)}} is non empty finite V52() set
the addF of K . [(A * Pi),(B * H)] is set
AB - 1 is ext-real V44() V45() set
SF is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
SF + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
1 + {} is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
i . SF is Element of the carrier of K
(i . SF) * Pi is Element of the carrier of K
the multF of K . ((i . SF),Pi) is Element of the carrier of K
[(i . SF),Pi] is set
{(i . SF),Pi} is non empty finite set
{(i . SF)} is non empty trivial finite 1 -element set
{{(i . SF),Pi},{(i . SF)}} is non empty finite V52() set
the multF of K . [(i . SF),Pi] is set
(i . SF) * (A * Pi) is Element of the carrier of K
the multF of K . ((i . SF),(A * Pi)) is Element of the carrier of K
[(i . SF),(A * Pi)] is set
{(i . SF),(A * Pi)} is non empty finite set
{{(i . SF),(A * Pi)},{(i . SF)}} is non empty finite V52() set
the multF of K . [(i . SF),(A * Pi)] is set
mA . SF is Element of the carrier of K
(i . SF) * H is Element of the carrier of K
the multF of K . ((i . SF),H) is Element of the carrier of K
[(i . SF),H] is set
{(i . SF),H} is non empty finite set
{{(i . SF),H},{(i . SF)}} is non empty finite V52() set
the multF of K . [(i . SF),H] is set
(i . SF) * (B * H) is Element of the carrier of K
the multF of K . ((i . SF),(B * H)) is Element of the carrier of K
[(i . SF),(B * H)] is set
{(i . SF),(B * H)} is non empty finite set
{{(i . SF),(B * H)},{(i . SF)}} is non empty finite V52() set
the multF of K . [(i . SF),(B * H)] is set
Gs . SF is Element of the carrier of K
(i . SF) * ((A * Pi) + (B * H)) is Element of the carrier of K
the multF of K . ((i . SF),((A * Pi) + (B * H))) is Element of the carrier of K
[(i . SF),((A * Pi) + (B * H))] is set
{(i . SF),((A * Pi) + (B * H))} is non empty finite set
{{(i . SF),((A * Pi) + (B * H))},{(i . SF)}} is non empty finite V52() set
the multF of K . [(i . SF),((A * Pi) + (B * H))] is set
Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Pi + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
SF is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Gs . SF is Element of the carrier of K
i . SF is Element of the carrier of K
A * (i . SF) is Element of the carrier of K
the multF of K . (A,(i . SF)) is Element of the carrier of K
[A,(i . SF)] is set
{A,(i . SF)} is non empty finite set
{{A,(i . SF)},{A}} is non empty finite V52() set
the multF of K . [A,(i . SF)] is set
mA . SF is Element of the carrier of K
B * (mA . SF) is Element of the carrier of K
the multF of K . (B,(mA . SF)) is Element of the carrier of K
[B,(mA . SF)] is set
{B,(mA . SF)} is non empty finite set
{{B,(mA . SF)},{B}} is non empty finite V52() set
the multF of K . [B,(mA . SF)] is set
(A * (i . SF)) + (B * (mA . SF)) is Element of the carrier of K
the addF of K . ((A * (i . SF)),(B * (mA . SF))) is Element of the carrier of K
[(A * (i . SF)),(B * (mA . SF))] is set
{(A * (i . SF)),(B * (mA . SF))} is non empty finite set
{(A * (i . SF))} is non empty trivial finite 1 -element set
{{(A * (i . SF)),(B * (mA . SF))},{(A * (i . SF))}} is non empty finite V52() set
the addF of K . [(A * (i . SF)),(B * (mA . SF))] is set
(Path_matrix (aa,(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) . (Pi + 1) is set
(Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) . (Pi + 1) is set
dom (Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) is finite Element of bool NAT
(Path_matrix (aa,(P,n,n, the carrier of K,Path,KK))) /. (Pi + 1) is Element of the carrier of K
mA . Pi is Element of the carrier of K
B * (mA . Pi) is Element of the carrier of K
the multF of K . (B,(mA . Pi)) is Element of the carrier of K
[B,(mA . Pi)] is set
{B,(mA . Pi)} is non empty finite set
{{B,(mA . Pi)},{B}} is non empty finite V52() set
the multF of K . [B,(mA . Pi)] is set
QQ is Element of the carrier of K
(B * (mA . Pi)) * QQ is Element of the carrier of K
the multF of K . ((B * (mA . Pi)),QQ) is Element of the carrier of K
[(B * (mA . Pi)),QQ] is set
{(B * (mA . Pi)),QQ} is non empty finite set
{(B * (mA . Pi))} is non empty trivial finite 1 -element set
{{(B * (mA . Pi)),QQ},{(B * (mA . Pi))}} is non empty finite V52() set
the multF of K . [(B * (mA . Pi)),QQ] is set
(mA . Pi) * QQ is Element of the carrier of K
the multF of K . ((mA . Pi),QQ) is Element of the carrier of K
[(mA . Pi),QQ] is set
{(mA . Pi),QQ} is non empty finite set
{(mA . Pi)} is non empty trivial finite 1 -element set
{{(mA . Pi),QQ},{(mA . Pi)}} is non empty finite V52() set
the multF of K . [(mA . Pi),QQ] is set
B * ((mA . Pi) * QQ) is Element of the carrier of K
the multF of K . (B,((mA . Pi) * QQ)) is Element of the carrier of K
[B,((mA . Pi) * QQ)] is set
{B,((mA . Pi) * QQ)} is non empty finite set
{{B,((mA . Pi) * QQ)},{B}} is non empty finite V52() set
the multF of K . [B,((mA . Pi) * QQ)] is set
i . Pi is Element of the carrier of K
A * (i . Pi) is Element of the carrier of K
the multF of K . (A,(i . Pi)) is Element of the carrier of K
[A,(i . Pi)] is set
{A,(i . Pi)} is non empty finite set
{{A,(i . Pi)},{A}} is non empty finite V52() set
the multF of K . [A,(i . Pi)] is set
(A * (i . Pi)) * QQ is Element of the carrier of K
the multF of K . ((A * (i . Pi)),QQ) is Element of the carrier of K
[(A * (i . Pi)),QQ] is set
{(A * (i . Pi)),QQ} is non empty finite set
{(A * (i . Pi))} is non empty trivial finite 1 -element set
{{(A * (i . Pi)),QQ},{(A * (i . Pi))}} is non empty finite V52() set
the multF of K . [(A * (i . Pi)),QQ] is set
(i . Pi) * QQ is Element of the carrier of K
the multF of K . ((i . Pi),QQ) is Element of the carrier of K
[(i . Pi),QQ] is set
{(i . Pi),QQ} is non empty finite set
{(i . Pi)} is non empty trivial finite 1 -element set
{{(i . Pi),QQ},{(i . Pi)}} is non empty finite V52() set
the multF of K . [(i . Pi),QQ] is set
A * ((i . Pi) * QQ) is Element of the carrier of K
the multF of K . (A,((i . Pi) * QQ)) is Element of the carrier of K
[A,((i . Pi) * QQ)] is set
{A,((i . Pi) * QQ)} is non empty finite set
{{A,((i . Pi) * QQ)},{A}} is non empty finite V52() set
the multF of K . [A,((i . Pi) * QQ)] is set
Gs . Pi is Element of the carrier of K
(A * (i . Pi)) + (B * (mA . Pi)) is Element of the carrier of K
the addF of K . ((A * (i . Pi)),(B * (mA . Pi))) is Element of the carrier of K
[(A * (i . Pi)),(B * (mA . Pi))] is set
{(A * (i . Pi)),(B * (mA . Pi))} is non empty finite set
{{(A * (i . Pi)),(B * (mA . Pi))},{(A * (i . Pi))}} is non empty finite V52() set
the addF of K . [(A * (i . Pi)),(B * (mA . Pi))] is set
Gs . (Pi + 1) is Element of the carrier of K
((A * (i . Pi)) + (B * (mA . Pi))) * QQ is Element of the carrier of K
the multF of K . (((A * (i . Pi)) + (B * (mA . Pi))),QQ) is Element of the carrier of K
[((A * (i . Pi)) + (B * (mA . Pi))),QQ] is set
{((A * (i . Pi)) + (B * (mA . Pi))),QQ} is non empty finite set
{((A * (i . Pi)) + (B * (mA . Pi)))} is non empty trivial finite 1 -element set
{{((A * (i . Pi)) + (B * (mA . Pi))),QQ},{((A * (i . Pi)) + (B * (mA . Pi)))}} is non empty finite V52() set
the multF of K . [((A * (i . Pi)) + (B * (mA . Pi))),QQ] is set
(Path_matrix (aa,(P,n,n, the carrier of K,Path,mm))) . (Pi + 1) is set
mA . (Pi + 1) is Element of the carrier of K
i . (Pi + 1) is Element of the carrier of K
Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Gs . Pi is Element of the carrier of K
i . Pi is Element of the carrier of K
A * (i . Pi) is Element of the carrier of K
the multF of K . (A,(i . Pi)) is Element of the carrier of K
[A,(i . Pi)] is set
{A,(i . Pi)} is non empty finite set
{{A,(i . Pi)},{A}} is non empty finite V52() set
the multF of K . [A,(i . Pi)] is set
mA . Pi is Element of the carrier of K
B * (mA . Pi) is Element of the carrier of K
the multF of K . (B,(mA . Pi)) is Element of the carrier of K
[B,(mA . Pi)] is set
{B,(mA . Pi)} is non empty finite set
{{B,(mA . Pi)},{B}} is non empty finite V52() set
the multF of K . [B,(mA . Pi)] is set
(A * (i . Pi)) + (B * (mA . Pi)) is Element of the carrier of K
the addF of K . ((A * (i . Pi)),(B * (mA . Pi))) is Element of the carrier of K
[(A * (i . Pi)),(B * (mA . Pi))] is set
{(A * (i . Pi)),(B * (mA . Pi))} is non empty finite set
{(A * (i . Pi))} is non empty trivial finite 1 -element set
{{(A * (i . Pi)),(B * (mA . Pi))},{(A * (i . Pi))}} is non empty finite V52() set
the addF of K . [(A * (i . Pi)),(B * (mA . Pi))] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is Element of the carrier of K
B is Element of the carrier of K
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
A * KK is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,A,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(A multfield) * KK is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
mm is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
B * mm is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
B multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,B,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(B multfield) * mm is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(A * KK) + (B * mm) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K .: ((A * KK),(B * mm)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
aa is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
SUM1 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
(P,n,n, the carrier of K,SUM1,((A * KK) + (B * mm))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Path_product (P,n,n, the carrier of K,SUM1,((A * KK) + (B * mm))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
(Path_product (P,n,n, the carrier of K,SUM1,((A * KK) + (B * mm)))) . aa is Element of the carrier of K
(P,n,n, the carrier of K,SUM1,KK) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Path_product (P,n,n, the carrier of K,SUM1,KK) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
(Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa is Element of the carrier of K
A * ((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa) is Element of the carrier of K
the multF of K . (A,((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa)) is Element of the carrier of K
[A,((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa)] is set
{A,((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa)} is non empty finite set
{A} is non empty trivial finite 1 -element set
{{A,((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa)},{A}} is non empty finite V52() set
the multF of K . [A,((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa)] is set
(P,n,n, the carrier of K,SUM1,mm) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Path_product (P,n,n, the carrier of K,SUM1,mm) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
(Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa is Element of the carrier of K
B * ((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa) is Element of the carrier of K
the multF of K . (B,((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa)) is Element of the carrier of K
[B,((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa)] is set
{B,((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa)} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa)},{B}} is non empty finite V52() set
the multF of K . [B,((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa)] is set
(A * ((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa)) + (B * ((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa)) is Element of the carrier of K
the addF of K . ((A * ((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa)),(B * ((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa))) is Element of the carrier of K
[(A * ((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa)),(B * ((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa))] is set
{(A * ((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa)),(B * ((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa))} is non empty finite set
{(A * ((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa))} is non empty trivial finite 1 -element set
{{(A * ((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa)),(B * ((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa))},{(A * ((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa))}} is non empty finite V52() set
the addF of K . [(A * ((Path_product (P,n,n, the carrier of K,SUM1,KK)) . aa)),(B * ((Path_product (P,n,n, the carrier of K,SUM1,mm)) . aa))] is set
Path_matrix (aa,(P,n,n, the carrier of K,SUM1,((A * KK) + (B * mm)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,((A * KK) + (B * mm))))) is Element of the carrier of K
- (( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,((A * KK) + (B * mm)))))),aa) is Element of the carrier of K
the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))) is Element of the carrier of K
- (( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)))),aa) is Element of the carrier of K
the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))) is Element of the carrier of K
- (( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))),aa) is Element of the carrier of K
the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,((A * KK) + (B * mm))))) is Element of the carrier of K
- ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,((A * KK) + (B * mm)))))) is Element of the carrier of K
- (( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,((A * KK) + (B * mm)))))),aa) is Element of the carrier of K
the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))) is Element of the carrier of K
- ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))) is Element of the carrier of K
- (( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))),aa) is Element of the carrier of K
A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))) is Element of the carrier of K
the multF of K . (A,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))) is Element of the carrier of K
[A,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))] is set
{A,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))} is non empty finite set
{{A,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))},{A}} is non empty finite V52() set
the multF of K . [A,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))] is set
- (A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))) is Element of the carrier of K
A * (- ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))) is Element of the carrier of K
the multF of K . (A,(- ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))) is Element of the carrier of K
[A,(- ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))] is set
{A,(- ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))} is non empty finite set
{{A,(- ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))},{A}} is non empty finite V52() set
the multF of K . [A,(- ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))] is set
the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))) is Element of the carrier of K
- ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)))) is Element of the carrier of K
- (( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)))),aa) is Element of the carrier of K
B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)))) is Element of the carrier of K
the multF of K . (B,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))) is Element of the carrier of K
[B,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))] is set
{B,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))} is non empty finite set
{{B,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))},{B}} is non empty finite V52() set
the multF of K . [B,( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))] is set
(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))) + (B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))) is Element of the carrier of K
the addF of K . ((A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))),(B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)))))) is Element of the carrier of K
[(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))),(B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)))))] is set
{(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))),(B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)))))} is non empty finite set
{(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))} is non empty trivial finite 1 -element set
{{(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))),(B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)))))},{(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))}} is non empty finite V52() set
the addF of K . [(A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))),(B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)))))] is set
- ((A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))) + (B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)))))) is Element of the carrier of K
(- (A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))) - (B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))) is Element of the carrier of K
- (B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))) is Element of the carrier of K
(- (A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))) + (- (B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm)))))) is Element of the carrier of K
the addF of K . ((- (A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))),(- (B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))))) is Element of the carrier of K
[(- (A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))),(- (B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))))] is set
{(- (A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))),(- (B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))))} is non empty finite set
{(- (A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))))} is non empty trivial finite 1 -element set
{{(- (A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))),(- (B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))))},{(- (A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK))))))}} is non empty finite V52() set
the addF of K . [(- (A * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,KK)))))),(- (B * ( the multF of K $$ (Path_matrix (aa,(P,n,n, the carrier of K,SUM1,mm))))))] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is Element of the carrier of K
B is Element of the carrier of K
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
A * KK is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
A multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,A,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(A multfield) * KK is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
mm is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
B * mm is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
B multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,B,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(B multfield) * mm is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(A * KK) + (B * mm) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the addF of K .: ((A * KK),(B * mm)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Permutations n is non empty permutational set
Path is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
(P,n,n, the carrier of K,Path,((A * KK) + (B * mm))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (P,n,n, the carrier of K,Path,((A * KK) + (B * mm))) is Element of the carrier of K
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product (P,n,n, the carrier of K,Path,((A * KK) + (B * mm))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (P,n,n, the carrier of K,Path,((A * KK) + (B * mm))))) is Element of the carrier of K
(P,n,n, the carrier of K,Path,KK) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (P,n,n, the carrier of K,Path,KK) is Element of the carrier of K
Path_product (P,n,n, the carrier of K,Path,KK) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (P,n,n, the carrier of K,Path,KK))) is Element of the carrier of K
A * (Det (P,n,n, the carrier of K,Path,KK)) is Element of the carrier of K
the multF of K . (A,(Det (P,n,n, the carrier of K,Path,KK))) is Element of the carrier of K
[A,(Det (P,n,n, the carrier of K,Path,KK))] is set
{A,(Det (P,n,n, the carrier of K,Path,KK))} is non empty finite set
{A} is non empty trivial finite 1 -element set
{{A,(Det (P,n,n, the carrier of K,Path,KK))},{A}} is non empty finite V52() set
the multF of K . [A,(Det (P,n,n, the carrier of K,Path,KK))] is set
(P,n,n, the carrier of K,Path,mm) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (P,n,n, the carrier of K,Path,mm) is Element of the carrier of K
Path_product (P,n,n, the carrier of K,Path,mm) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (P,n,n, the carrier of K,Path,mm))) is Element of the carrier of K
B * (Det (P,n,n, the carrier of K,Path,mm)) is Element of the carrier of K
the multF of K . (B,(Det (P,n,n, the carrier of K,Path,mm))) is Element of the carrier of K
[B,(Det (P,n,n, the carrier of K,Path,mm))] is set
{B,(Det (P,n,n, the carrier of K,Path,mm))} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,(Det (P,n,n, the carrier of K,Path,mm))},{B}} is non empty finite V52() set
the multF of K . [B,(Det (P,n,n, the carrier of K,Path,mm))] is set
(A * (Det (P,n,n, the carrier of K,Path,KK))) + (B * (Det (P,n,n, the carrier of K,Path,mm))) is Element of the carrier of K
the addF of K . ((A * (Det (P,n,n, the carrier of K,Path,KK))),(B * (Det (P,n,n, the carrier of K,Path,mm)))) is Element of the carrier of K
[(A * (Det (P,n,n, the carrier of K,Path,KK))),(B * (Det (P,n,n, the carrier of K,Path,mm)))] is set
{(A * (Det (P,n,n, the carrier of K,Path,KK))),(B * (Det (P,n,n, the carrier of K,Path,mm)))} is non empty finite set
{(A * (Det (P,n,n, the carrier of K,Path,KK)))} is non empty trivial finite 1 -element set
{{(A * (Det (P,n,n, the carrier of K,Path,KK))),(B * (Det (P,n,n, the carrier of K,Path,mm)))},{(A * (Det (P,n,n, the carrier of K,Path,KK)))}} is non empty finite V52() set
the addF of K . [(A * (Det (P,n,n, the carrier of K,Path,KK))),(B * (Det (P,n,n, the carrier of K,Path,mm)))] is set
[:(Fin (Permutations n)), the carrier of K:] is Relation-like non empty set
bool [:(Fin (Permutations n)), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
PM is Relation-like Fin (Permutations n) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (Permutations n)), the carrier of K:]
PM . (FinOmega (Permutations n)) is Element of the carrier of K
PM . {} is set
i is Relation-like Fin (Permutations n) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (Permutations n)), the carrier of K:]
i . (FinOmega (Permutations n)) is Element of the carrier of K
i . {} is set
Pi is Relation-like Fin (Permutations n) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (Permutations n)), the carrier of K:]
Pi . (FinOmega (Permutations n)) is Element of the carrier of K
Pi . {} is set
H is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
H + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
SF is finite Element of Fin (Permutations n)
card SF is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
PM . SF is Element of the carrier of K
Pi . SF is Element of the carrier of K
A * (Pi . SF) is Element of the carrier of K
the multF of K . (A,(Pi . SF)) is Element of the carrier of K
[A,(Pi . SF)] is set
{A,(Pi . SF)} is non empty finite set
{{A,(Pi . SF)},{A}} is non empty finite V52() set
the multF of K . [A,(Pi . SF)] is set
i . SF is Element of the carrier of K
B * (i . SF) is Element of the carrier of K
the multF of K . (B,(i . SF)) is Element of the carrier of K
[B,(i . SF)] is set
{B,(i . SF)} is non empty finite set
{{B,(i . SF)},{B}} is non empty finite V52() set
the multF of K . [B,(i . SF)] is set
(A * (Pi . SF)) + (B * (i . SF)) is Element of the carrier of K
the addF of K . ((A * (Pi . SF)),(B * (i . SF))) is Element of the carrier of K
[(A * (Pi . SF)),(B * (i . SF))] is set
{(A * (Pi . SF)),(B * (i . SF))} is non empty finite set
{(A * (Pi . SF))} is non empty trivial finite 1 -element set
{{(A * (Pi . SF)),(B * (i . SF))},{(A * (Pi . SF))}} is non empty finite V52() set
the addF of K . [(A * (Pi . SF)),(B * (i . SF))] is set
QQ is set
{QQ} is non empty trivial finite 1 -element set
h is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
(Path_product (P,n,n, the carrier of K,Path,KK)) . h is Element of the carrier of K
(Path_product (P,n,n, the carrier of K,Path,mm)) . h is Element of the carrier of K
(Path_product (P,n,n, the carrier of K,Path,((A * KK) + (B * mm)))) . h is Element of the carrier of K
QQ is set
h is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
{h} is functional non empty trivial finite V52() 1 -element set
SF \ {h} is finite Element of bool SF
bool SF is non empty cup-closed diff-closed preBoolean finite V52() set
Mh is finite Element of Fin (Permutations n)
(FinOmega (Permutations n)) \ Mh is finite Element of Fin (Permutations n)
{h} \/ Mh is non empty finite set
card Mh is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(card Mh) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
PM . Mh is Element of the carrier of K
Pi . Mh is Element of the carrier of K
A * (Pi . Mh) is Element of the carrier of K
the multF of K . (A,(Pi . Mh)) is Element of the carrier of K
[A,(Pi . Mh)] is set
{A,(Pi . Mh)} is non empty finite set
{{A,(Pi . Mh)},{A}} is non empty finite V52() set
the multF of K . [A,(Pi . Mh)] is set
i . Mh is Element of the carrier of K
B * (i . Mh) is Element of the carrier of K
the multF of K . (B,(i . Mh)) is Element of the carrier of K
[B,(i . Mh)] is set
{B,(i . Mh)} is non empty finite set
{{B,(i . Mh)},{B}} is non empty finite V52() set
the multF of K . [B,(i . Mh)] is set
(A * (Pi . Mh)) + (B * (i . Mh)) is Element of the carrier of K
the addF of K . ((A * (Pi . Mh)),(B * (i . Mh))) is Element of the carrier of K
[(A * (Pi . Mh)),(B * (i . Mh))] is set
{(A * (Pi . Mh)),(B * (i . Mh))} is non empty finite set
{(A * (Pi . Mh))} is non empty trivial finite 1 -element set
{{(A * (Pi . Mh)),(B * (i . Mh))},{(A * (Pi . Mh))}} is non empty finite V52() set
the addF of K . [(A * (Pi . Mh)),(B * (i . Mh))] is set
(Path_product (P,n,n, the carrier of K,Path,((A * KK) + (B * mm)))) . h is Element of the carrier of K
the addF of K . ((PM . Mh),((Path_product (P,n,n, the carrier of K,Path,((A * KK) + (B * mm)))) . h)) is Element of the carrier of K
[(PM . Mh),((Path_product (P,n,n, the carrier of K,Path,((A * KK) + (B * mm)))) . h)] is set
{(PM . Mh),((Path_product (P,n,n, the carrier of K,Path,((A * KK) + (B * mm)))) . h)} is non empty finite set
{(PM . Mh)} is non empty trivial finite 1 -element set
{{(PM . Mh),((Path_product (P,n,n, the carrier of K,Path,((A * KK) + (B * mm)))) . h)},{(PM . Mh)}} is non empty finite V52() set
the addF of K . [(PM . Mh),((Path_product (P,n,n, the carrier of K,Path,((A * KK) + (B * mm)))) . h)] is set
(Path_product (P,n,n, the carrier of K,Path,KK)) . h is Element of the carrier of K
A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h) is Element of the carrier of K
the multF of K . (A,((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) is Element of the carrier of K
[A,((Path_product (P,n,n, the carrier of K,Path,KK)) . h)] is set
{A,((Path_product (P,n,n, the carrier of K,Path,KK)) . h)} is non empty finite set
{{A,((Path_product (P,n,n, the carrier of K,Path,KK)) . h)},{A}} is non empty finite V52() set
the multF of K . [A,((Path_product (P,n,n, the carrier of K,Path,KK)) . h)] is set
(Path_product (P,n,n, the carrier of K,Path,mm)) . h is Element of the carrier of K
B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h) is Element of the carrier of K
the multF of K . (B,((Path_product (P,n,n, the carrier of K,Path,mm)) . h)) is Element of the carrier of K
[B,((Path_product (P,n,n, the carrier of K,Path,mm)) . h)] is set
{B,((Path_product (P,n,n, the carrier of K,Path,mm)) . h)} is non empty finite set
{{B,((Path_product (P,n,n, the carrier of K,Path,mm)) . h)},{B}} is non empty finite V52() set
the multF of K . [B,((Path_product (P,n,n, the carrier of K,Path,mm)) . h)] is set
(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)) is Element of the carrier of K
the addF of K . ((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)),(B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))) is Element of the carrier of K
[(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)),(B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))] is set
{(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)),(B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))} is non empty finite set
{(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))} is non empty trivial finite 1 -element set
{{(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)),(B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))},{(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))}} is non empty finite V52() set
the addF of K . [(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)),(B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))] is set
((A * (Pi . Mh)) + (B * (i . Mh))) + ((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))) is Element of the carrier of K
the addF of K . (((A * (Pi . Mh)) + (B * (i . Mh))),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))) is Element of the carrier of K
[((A * (Pi . Mh)) + (B * (i . Mh))),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
{((A * (Pi . Mh)) + (B * (i . Mh))),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))} is non empty finite set
{((A * (Pi . Mh)) + (B * (i . Mh)))} is non empty trivial finite 1 -element set
{{((A * (Pi . Mh)) + (B * (i . Mh))),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))},{((A * (Pi . Mh)) + (B * (i . Mh)))}} is non empty finite V52() set
the addF of K . [((A * (Pi . Mh)) + (B * (i . Mh))),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
(B * (i . Mh)) + ((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))) is Element of the carrier of K
the addF of K . ((B * (i . Mh)),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))) is Element of the carrier of K
[(B * (i . Mh)),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
{(B * (i . Mh)),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))} is non empty finite set
{(B * (i . Mh))} is non empty trivial finite 1 -element set
{{(B * (i . Mh)),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))},{(B * (i . Mh))}} is non empty finite V52() set
the addF of K . [(B * (i . Mh)),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
(A * (Pi . Mh)) + ((B * (i . Mh)) + ((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))) is Element of the carrier of K
the addF of K . ((A * (Pi . Mh)),((B * (i . Mh)) + ((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))) is Element of the carrier of K
[(A * (Pi . Mh)),((B * (i . Mh)) + ((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))] is set
{(A * (Pi . Mh)),((B * (i . Mh)) + ((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))} is non empty finite set
{{(A * (Pi . Mh)),((B * (i . Mh)) + ((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))},{(A * (Pi . Mh))}} is non empty finite V52() set
the addF of K . [(A * (Pi . Mh)),((B * (i . Mh)) + ((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))] is set
(B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)) is Element of the carrier of K
the addF of K . ((B * (i . Mh)),(B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))) is Element of the carrier of K
[(B * (i . Mh)),(B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))] is set
{(B * (i . Mh)),(B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))} is non empty finite set
{{(B * (i . Mh)),(B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))},{(B * (i . Mh))}} is non empty finite V52() set
the addF of K . [(B * (i . Mh)),(B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))] is set
(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + ((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))) is Element of the carrier of K
the addF of K . ((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))) is Element of the carrier of K
[(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
{(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))} is non empty finite set
{{(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))},{(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))}} is non empty finite V52() set
the addF of K . [(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
(A * (Pi . Mh)) + ((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + ((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))) is Element of the carrier of K
the addF of K . ((A * (Pi . Mh)),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + ((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))) is Element of the carrier of K
[(A * (Pi . Mh)),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + ((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))] is set
{(A * (Pi . Mh)),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + ((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))} is non empty finite set
{{(A * (Pi . Mh)),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + ((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))},{(A * (Pi . Mh))}} is non empty finite V52() set
the addF of K . [(A * (Pi . Mh)),((A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) + ((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))] is set
(A * (Pi . Mh)) + (A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) is Element of the carrier of K
the addF of K . ((A * (Pi . Mh)),(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))) is Element of the carrier of K
[(A * (Pi . Mh)),(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))] is set
{(A * (Pi . Mh)),(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))} is non empty finite set
{{(A * (Pi . Mh)),(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))},{(A * (Pi . Mh))}} is non empty finite V52() set
the addF of K . [(A * (Pi . Mh)),(A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))] is set
((A * (Pi . Mh)) + (A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))) + ((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))) is Element of the carrier of K
the addF of K . (((A * (Pi . Mh)) + (A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))) is Element of the carrier of K
[((A * (Pi . Mh)) + (A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
{((A * (Pi . Mh)) + (A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))} is non empty finite set
{((A * (Pi . Mh)) + (A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))} is non empty trivial finite 1 -element set
{{((A * (Pi . Mh)) + (A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))},{((A * (Pi . Mh)) + (A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))}} is non empty finite V52() set
the addF of K . [((A * (Pi . Mh)) + (A * ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
(Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h) is Element of the carrier of K
the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) is Element of the carrier of K
[(Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)] is set
{(Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)} is non empty finite set
{(Pi . Mh)} is non empty trivial finite 1 -element set
{{(Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)},{(Pi . Mh)}} is non empty finite V52() set
the addF of K . [(Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)] is set
A * ((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)) is Element of the carrier of K
the multF of K . (A,((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))) is Element of the carrier of K
[A,((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))] is set
{A,((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))} is non empty finite set
{{A,((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))},{A}} is non empty finite V52() set
the multF of K . [A,((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))] is set
(A * ((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))) + ((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))) is Element of the carrier of K
the addF of K . ((A * ((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))) is Element of the carrier of K
[(A * ((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
{(A * ((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))} is non empty finite set
{(A * ((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))} is non empty trivial finite 1 -element set
{{(A * ((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))},{(A * ((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))}} is non empty finite V52() set
the addF of K . [(A * ((Pi . Mh) + ((Path_product (P,n,n, the carrier of K,Path,KK)) . h))),((B * (i . Mh)) + (B * ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h))) is Element of the carrier of K
the multF of K . (A,( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))) is Element of the carrier of K
[A,( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))] is set
{A,( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))} is non empty finite set
{{A,( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))},{A}} is non empty finite V52() set
the multF of K . [A,( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))] is set
(i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h) is Element of the carrier of K
the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h)) is Element of the carrier of K
[(i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h)] is set
{(i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h)} is non empty finite set
{(i . Mh)} is non empty trivial finite 1 -element set
{{(i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h)},{(i . Mh)}} is non empty finite V52() set
the addF of K . [(i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h)] is set
B * ((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)) is Element of the carrier of K
the multF of K . (B,((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))) is Element of the carrier of K
[B,((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))] is set
{B,((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))} is non empty finite set
{{B,((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))},{B}} is non empty finite V52() set
the multF of K . [B,((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))] is set
(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))) + (B * ((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h))) is Element of the carrier of K
the addF of K . ((A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))),(B * ((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))) is Element of the carrier of K
[(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))),(B * ((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
{(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))),(B * ((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))} is non empty finite set
{(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h))))} is non empty trivial finite 1 -element set
{{(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))),(B * ((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))},{(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h))))}} is non empty finite V52() set
the addF of K . [(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))),(B * ((i . Mh) + ((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
B * ( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h))) is Element of the carrier of K
the multF of K . (B,( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))) is Element of the carrier of K
[B,( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
{B,( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))} is non empty finite set
{{B,( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))},{B}} is non empty finite V52() set
the multF of K . [B,( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))] is set
(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))) + (B * ( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h)))) is Element of the carrier of K
the addF of K . ((A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))),(B * ( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))) is Element of the carrier of K
[(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))),(B * ( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))] is set
{(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))),(B * ( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))} is non empty finite set
{{(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))),(B * ( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))},{(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h))))}} is non empty finite V52() set
the addF of K . [(A * ( the addF of K . ((Pi . Mh),((Path_product (P,n,n, the carrier of K,Path,KK)) . h)))),(B * ( the addF of K . ((i . Mh),((Path_product (P,n,n, the carrier of K,Path,mm)) . h))))] is set
0. K is V70(K) Element of the carrier of K
B * (0. K) is Element of the carrier of K
the multF of K . (B,(0. K)) is Element of the carrier of K
[B,(0. K)] is set
{B,(0. K)} is non empty finite set
{{B,(0. K)},{B}} is non empty finite V52() set
the multF of K . [B,(0. K)] is set
A * (0. K) is Element of the carrier of K
the multF of K . (A,(0. K)) is Element of the carrier of K
[A,(0. K)] is set
{A,(0. K)} is non empty finite set
{{A,(0. K)},{A}} is non empty finite V52() set
the multF of K . [A,(0. K)] is set
H is finite Element of Fin (Permutations n)
card H is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
PM . H is Element of the carrier of K
Pi . H is Element of the carrier of K
A * (Pi . H) is Element of the carrier of K
the multF of K . (A,(Pi . H)) is Element of the carrier of K
[A,(Pi . H)] is set
{A,(Pi . H)} is non empty finite set
{{A,(Pi . H)},{A}} is non empty finite V52() set
the multF of K . [A,(Pi . H)] is set
i . H is Element of the carrier of K
B * (i . H) is Element of the carrier of K
the multF of K . (B,(i . H)) is Element of the carrier of K
[B,(i . H)] is set
{B,(i . H)} is non empty finite set
{{B,(i . H)},{B}} is non empty finite V52() set
the multF of K . [B,(i . H)] is set
(A * (Pi . H)) + (B * (i . H)) is Element of the carrier of K
the addF of K . ((A * (Pi . H)),(B * (i . H))) is Element of the carrier of K
[(A * (Pi . H)),(B * (i . H))] is set
{(A * (Pi . H)),(B * (i . H))} is non empty finite set
{(A * (Pi . H))} is non empty trivial finite 1 -element set
{{(A * (Pi . H)),(B * (i . H))},{(A * (Pi . H))}} is non empty finite V52() set
the addF of K . [(A * (Pi . H)),(B * (i . H))] is set
card (FinOmega (Permutations n)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
H is finite Element of Fin (Permutations n)
card H is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
PM . H is Element of the carrier of K
Pi . H is Element of the carrier of K
A * (Pi . H) is Element of the carrier of K
the multF of K . (A,(Pi . H)) is Element of the carrier of K
[A,(Pi . H)] is set
{A,(Pi . H)} is non empty finite set
{{A,(Pi . H)},{A}} is non empty finite V52() set
the multF of K . [A,(Pi . H)] is set
i . H is Element of the carrier of K
B * (i . H) is Element of the carrier of K
the multF of K . (B,(i . H)) is Element of the carrier of K
[B,(i . H)] is set
{B,(i . H)} is non empty finite set
{{B,(i . H)},{B}} is non empty finite V52() set
the multF of K . [B,(i . H)] is set
(A * (Pi . H)) + (B * (i . H)) is Element of the carrier of K
the addF of K . ((A * (Pi . H)),(B * (i . H))) is Element of the carrier of K
[(A * (Pi . H)),(B * (i . H))] is set
{(A * (Pi . H)),(B * (i . H))} is non empty finite set
{(A * (Pi . H))} is non empty trivial finite 1 -element set
{{(A * (Pi . H)),(B * (i . H))},{(A * (Pi . H))}} is non empty finite V52() set
the addF of K . [(A * (Pi . H)),(B * (i . H))] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg K is finite K -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= K ) } is set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of A is non empty non trivial set
the carrier of A * is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
B is Element of the carrier of A
P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
B * P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
B multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
[: the carrier of A, the carrier of A:] is Relation-like non empty set
bool [: the carrier of A, the carrier of A:] is non empty cup-closed diff-closed preBoolean set
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like non empty set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is non empty cup-closed diff-closed preBoolean set
id the carrier of A is Relation-like the carrier of A -defined the carrier of A -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of A, the carrier of A:]
K224( the carrier of A, the carrier of A, the multF of A,B,(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
(B multfield) * P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
KK is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of A
(n,K,K, the carrier of A,KK,(B * P)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of A
Det (n,K,K, the carrier of A,KK,(B * P)) is Element of the carrier of A
Permutations K is non empty permutational set
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
FinOmega (Permutations K) is finite Element of Fin (Permutations K)
Fin (Permutations K) is non empty cup-closed diff-closed preBoolean set
Path_product (n,K,K, the carrier of A,KK,(B * P)) is Relation-like Permutations K -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [:(Permutations K), the carrier of A:]
[:(Permutations K), the carrier of A:] is Relation-like non empty set
bool [:(Permutations K), the carrier of A:] is non empty cup-closed diff-closed preBoolean set
the addF of A $$ ((FinOmega (Permutations K)),(Path_product (n,K,K, the carrier of A,KK,(B * P)))) is Element of the carrier of A
(n,K,K, the carrier of A,KK,P) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of A
Det (n,K,K, the carrier of A,KK,P) is Element of the carrier of A
Path_product (n,K,K, the carrier of A,KK,P) is Relation-like Permutations K -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [:(Permutations K), the carrier of A:]
the addF of A $$ ((FinOmega (Permutations K)),(Path_product (n,K,K, the carrier of A,KK,P))) is Element of the carrier of A
B * (Det (n,K,K, the carrier of A,KK,P)) is Element of the carrier of A
the multF of A . (B,(Det (n,K,K, the carrier of A,KK,P))) is Element of the carrier of A
[B,(Det (n,K,K, the carrier of A,KK,P))] is set
{B,(Det (n,K,K, the carrier of A,KK,P))} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,(Det (n,K,K, the carrier of A,KK,P))},{B}} is non empty finite V52() set
the multF of A . [B,(Det (n,K,K, the carrier of A,KK,P))] is set
(len P) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b1 where b1 is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b1 = len P } is set
0. A is V70(A) Element of the carrier of A
(0. A) * P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(0. A) multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
K224( the carrier of A, the carrier of A, the multF of A,(0. A),(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
((0. A) multfield) * P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(B * P) + ((0. A) * P) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
the addF of A .: ((B * P),((0. A) * P)) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
B + (0. A) is Element of the carrier of A
the addF of A . (B,(0. A)) is Element of the carrier of A
[B,(0. A)] is set
{B,(0. A)} is non empty finite set
{{B,(0. A)},{B}} is non empty finite V52() set
the addF of A . [B,(0. A)] is set
(B + (0. A)) * P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(B + (0. A)) multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
K224( the carrier of A, the carrier of A, the multF of A,(B + (0. A)),(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
((B + (0. A)) multfield) * P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(0. A) * (Det (n,K,K, the carrier of A,KK,P)) is Element of the carrier of A
the multF of A . ((0. A),(Det (n,K,K, the carrier of A,KK,P))) is Element of the carrier of A
[(0. A),(Det (n,K,K, the carrier of A,KK,P))] is set
{(0. A),(Det (n,K,K, the carrier of A,KK,P))} is non empty finite set
{(0. A)} is non empty trivial finite 1 -element set
{{(0. A),(Det (n,K,K, the carrier of A,KK,P))},{(0. A)}} is non empty finite V52() set
the multF of A . [(0. A),(Det (n,K,K, the carrier of A,KK,P))] is set
(B * (Det (n,K,K, the carrier of A,KK,P))) + ((0. A) * (Det (n,K,K, the carrier of A,KK,P))) is Element of the carrier of A
the addF of A . ((B * (Det (n,K,K, the carrier of A,KK,P))),((0. A) * (Det (n,K,K, the carrier of A,KK,P)))) is Element of the carrier of A
[(B * (Det (n,K,K, the carrier of A,KK,P))),((0. A) * (Det (n,K,K, the carrier of A,KK,P)))] is set
{(B * (Det (n,K,K, the carrier of A,KK,P))),((0. A) * (Det (n,K,K, the carrier of A,KK,P)))} is non empty finite set
{(B * (Det (n,K,K, the carrier of A,KK,P)))} is non empty trivial finite 1 -element set
{{(B * (Det (n,K,K, the carrier of A,KK,P))),((0. A) * (Det (n,K,K, the carrier of A,KK,P)))},{(B * (Det (n,K,K, the carrier of A,KK,P)))}} is non empty finite V52() set
the addF of A . [(B * (Det (n,K,K, the carrier of A,KK,P))),((0. A) * (Det (n,K,K, the carrier of A,KK,P)))] is set
(B * (Det (n,K,K, the carrier of A,KK,P))) + (0. A) is Element of the carrier of A
the addF of A . ((B * (Det (n,K,K, the carrier of A,KK,P))),(0. A)) is Element of the carrier of A
[(B * (Det (n,K,K, the carrier of A,KK,P))),(0. A)] is set
{(B * (Det (n,K,K, the carrier of A,KK,P))),(0. A)} is non empty finite set
{{(B * (Det (n,K,K, the carrier of A,KK,P))),(0. A)},{(B * (Det (n,K,K, the carrier of A,KK,P)))}} is non empty finite V52() set
the addF of A . [(B * (Det (n,K,K, the carrier of A,KK,P))),(0. A)] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg K is finite K -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= K ) } is set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of A is non empty non trivial set
the carrier of A * is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
B is Element of the carrier of A
P is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of A
width P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Line (P,n) is Relation-like NAT -defined the carrier of A -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of A
(width P) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b1 where b1 is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b1 = width P } is set
B * (Line (P,n)) is Relation-like NAT -defined the carrier of A -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on the carrier of A
B multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
[: the carrier of A, the carrier of A:] is Relation-like non empty set
bool [: the carrier of A, the carrier of A:] is non empty cup-closed diff-closed preBoolean set
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like non empty set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is non empty cup-closed diff-closed preBoolean set
id the carrier of A is Relation-like the carrier of A -defined the carrier of A -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of A, the carrier of A:]
K224( the carrier of A, the carrier of A, the multF of A,B,(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
(B multfield) * (Line (P,n)) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(n,K,K, the carrier of A,P,(B * (Line (P,n)))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of A
Det (n,K,K, the carrier of A,P,(B * (Line (P,n)))) is Element of the carrier of A
Permutations K is non empty permutational set
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
FinOmega (Permutations K) is finite Element of Fin (Permutations K)
Fin (Permutations K) is non empty cup-closed diff-closed preBoolean set
Path_product (n,K,K, the carrier of A,P,(B * (Line (P,n)))) is Relation-like Permutations K -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [:(Permutations K), the carrier of A:]
[:(Permutations K), the carrier of A:] is Relation-like non empty set
bool [:(Permutations K), the carrier of A:] is non empty cup-closed diff-closed preBoolean set
the addF of A $$ ((FinOmega (Permutations K)),(Path_product (n,K,K, the carrier of A,P,(B * (Line (P,n)))))) is Element of the carrier of A
Det P is Element of the carrier of A
Path_product P is Relation-like Permutations K -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [:(Permutations K), the carrier of A:]
the addF of A $$ ((FinOmega (Permutations K)),(Path_product P)) is Element of the carrier of A
B * (Det P) is Element of the carrier of A
the multF of A . (B,(Det P)) is Element of the carrier of A
[B,(Det P)] is set
{B,(Det P)} is non empty finite set
{B} is non empty trivial finite 1 -element set
{{B,(Det P)},{B}} is non empty finite V52() set
the multF of A . [B,(Det P)] is set
len (Line (P,n)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(n,K,K, the carrier of A,P,(Line (P,n))) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of A
Det (n,K,K, the carrier of A,P,(Line (P,n))) is Element of the carrier of A
Path_product (n,K,K, the carrier of A,P,(Line (P,n))) is Relation-like Permutations K -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [:(Permutations K), the carrier of A:]
the addF of A $$ ((FinOmega (Permutations K)),(Path_product (n,K,K, the carrier of A,P,(Line (P,n))))) is Element of the carrier of A
B * (Det (n,K,K, the carrier of A,P,(Line (P,n)))) is Element of the carrier of A
the multF of A . (B,(Det (n,K,K, the carrier of A,P,(Line (P,n))))) is Element of the carrier of A
[B,(Det (n,K,K, the carrier of A,P,(Line (P,n))))] is set
{B,(Det (n,K,K, the carrier of A,P,(Line (P,n))))} is non empty finite set
{{B,(Det (n,K,K, the carrier of A,P,(Line (P,n))))},{B}} is non empty finite V52() set
the multF of A . [B,(Det (n,K,K, the carrier of A,P,(Line (P,n))))] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg K is finite K -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= K ) } is set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of A is non empty non trivial set
the carrier of A * is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
B is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
len B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
B + P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[: the carrier of A, the carrier of A:] is Relation-like non empty set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like non empty set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is non empty cup-closed diff-closed preBoolean set
the addF of A .: (B,P) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
KK is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of A
(n,K,K, the carrier of A,KK,(B + P)) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of A
Det (n,K,K, the carrier of A,KK,(B + P)) is Element of the carrier of A
Permutations K is non empty permutational set
FinOmega (Permutations K) is finite Element of Fin (Permutations K)
Fin (Permutations K) is non empty cup-closed diff-closed preBoolean set
Path_product (n,K,K, the carrier of A,KK,(B + P)) is Relation-like Permutations K -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [:(Permutations K), the carrier of A:]
[:(Permutations K), the carrier of A:] is Relation-like non empty set
bool [:(Permutations K), the carrier of A:] is non empty cup-closed diff-closed preBoolean set
the addF of A $$ ((FinOmega (Permutations K)),(Path_product (n,K,K, the carrier of A,KK,(B + P)))) is Element of the carrier of A
(n,K,K, the carrier of A,KK,B) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of A
Det (n,K,K, the carrier of A,KK,B) is Element of the carrier of A
Path_product (n,K,K, the carrier of A,KK,B) is Relation-like Permutations K -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [:(Permutations K), the carrier of A:]
the addF of A $$ ((FinOmega (Permutations K)),(Path_product (n,K,K, the carrier of A,KK,B))) is Element of the carrier of A
(n,K,K, the carrier of A,KK,P) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of A
Det (n,K,K, the carrier of A,KK,P) is Element of the carrier of A
Path_product (n,K,K, the carrier of A,KK,P) is Relation-like Permutations K -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [:(Permutations K), the carrier of A:]
the addF of A $$ ((FinOmega (Permutations K)),(Path_product (n,K,K, the carrier of A,KK,P))) is Element of the carrier of A
(Det (n,K,K, the carrier of A,KK,B)) + (Det (n,K,K, the carrier of A,KK,P)) is Element of the carrier of A
the addF of A . ((Det (n,K,K, the carrier of A,KK,B)),(Det (n,K,K, the carrier of A,KK,P))) is Element of the carrier of A
[(Det (n,K,K, the carrier of A,KK,B)),(Det (n,K,K, the carrier of A,KK,P))] is set
{(Det (n,K,K, the carrier of A,KK,B)),(Det (n,K,K, the carrier of A,KK,P))} is non empty finite set
{(Det (n,K,K, the carrier of A,KK,B))} is non empty trivial finite 1 -element set
{{(Det (n,K,K, the carrier of A,KK,B)),(Det (n,K,K, the carrier of A,KK,P))},{(Det (n,K,K, the carrier of A,KK,B))}} is non empty finite V52() set
the addF of A . [(Det (n,K,K, the carrier of A,KK,B)),(Det (n,K,K, the carrier of A,KK,P))] is set
(len B) -tuples_on the carrier of A is functional non empty FinSequence-membered FinSequenceSet of the carrier of A
{ b1 where b1 is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : len b1 = len B } is set
1_ A is Element of the carrier of A
K254(A) is V70(A) Element of the carrier of A
(1_ A) * B is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(1_ A) multfield is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
bool [: the carrier of A, the carrier of A:] is non empty cup-closed diff-closed preBoolean set
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
id the carrier of A is Relation-like the carrier of A -defined the carrier of A -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of A, the carrier of A:]
K224( the carrier of A, the carrier of A, the multF of A,(1_ A),(id the carrier of A)) is Relation-like the carrier of A -defined the carrier of A -valued Function-like non empty total quasi_total Element of bool [: the carrier of A, the carrier of A:]
((1_ A) multfield) * B is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(1_ A) * P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
((1_ A) multfield) * P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(1_ A) * (Det (n,K,K, the carrier of A,KK,B)) is Element of the carrier of A
the multF of A . ((1_ A),(Det (n,K,K, the carrier of A,KK,B))) is Element of the carrier of A
[(1_ A),(Det (n,K,K, the carrier of A,KK,B))] is set
{(1_ A),(Det (n,K,K, the carrier of A,KK,B))} is non empty finite set
{(1_ A)} is non empty trivial finite 1 -element set
{{(1_ A),(Det (n,K,K, the carrier of A,KK,B))},{(1_ A)}} is non empty finite V52() set
the multF of A . [(1_ A),(Det (n,K,K, the carrier of A,KK,B))] is set
(1_ A) * (Det (n,K,K, the carrier of A,KK,P)) is Element of the carrier of A
the multF of A . ((1_ A),(Det (n,K,K, the carrier of A,KK,P))) is Element of the carrier of A
[(1_ A),(Det (n,K,K, the carrier of A,KK,P))] is set
{(1_ A),(Det (n,K,K, the carrier of A,KK,P))} is non empty finite set
{{(1_ A),(Det (n,K,K, the carrier of A,KK,P))},{(1_ A)}} is non empty finite V52() set
the multF of A . [(1_ A),(Det (n,K,K, the carrier of A,KK,P))] is set
((1_ A) * (Det (n,K,K, the carrier of A,KK,B))) + ((1_ A) * (Det (n,K,K, the carrier of A,KK,P))) is Element of the carrier of A
the addF of A . (((1_ A) * (Det (n,K,K, the carrier of A,KK,B))),((1_ A) * (Det (n,K,K, the carrier of A,KK,P)))) is Element of the carrier of A
[((1_ A) * (Det (n,K,K, the carrier of A,KK,B))),((1_ A) * (Det (n,K,K, the carrier of A,KK,P)))] is set
{((1_ A) * (Det (n,K,K, the carrier of A,KK,B))),((1_ A) * (Det (n,K,K, the carrier of A,KK,P)))} is non empty finite set
{((1_ A) * (Det (n,K,K, the carrier of A,KK,B)))} is non empty trivial finite 1 -element set
{{((1_ A) * (Det (n,K,K, the carrier of A,KK,B))),((1_ A) * (Det (n,K,K, the carrier of A,KK,P)))},{((1_ A) * (Det (n,K,K, the carrier of A,KK,B)))}} is non empty finite V52() set
the addF of A . [((1_ A) * (Det (n,K,K, the carrier of A,KK,B))),((1_ A) * (Det (n,K,K, the carrier of A,KK,P)))] is set
(Det (n,K,K, the carrier of A,KK,B)) + ((1_ A) * (Det (n,K,K, the carrier of A,KK,P))) is Element of the carrier of A
the addF of A . ((Det (n,K,K, the carrier of A,KK,B)),((1_ A) * (Det (n,K,K, the carrier of A,KK,P)))) is Element of the carrier of A
[(Det (n,K,K, the carrier of A,KK,B)),((1_ A) * (Det (n,K,K, the carrier of A,KK,P)))] is set
{(Det (n,K,K, the carrier of A,KK,B)),((1_ A) * (Det (n,K,K, the carrier of A,KK,P)))} is non empty finite set
{{(Det (n,K,K, the carrier of A,KK,B)),((1_ A) * (Det (n,K,K, the carrier of A,KK,P)))},{(Det (n,K,K, the carrier of A,KK,B))}} is non empty finite V52() set
the addF of A . [(Det (n,K,K, the carrier of A,KK,B)),((1_ A) * (Det (n,K,K, the carrier of A,KK,P)))] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is non empty set
A * is functional non empty FinSequence-membered FinSequenceSet of A
B is Relation-like Seg n -defined Seg n -valued Function-like total quasi_total finite Element of bool [:(Seg n),(Seg n):]
P is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
P * B is Relation-like Seg n -defined A * -valued Function-like finite Element of bool [:(Seg n),(A *):]
[:(Seg n),(A *):] is Relation-like set
bool [:(Seg n),(A *):] is non empty cup-closed diff-closed preBoolean set
rng B is finite set
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom P is finite Element of bool NAT
dom B is finite set
dom (P * B) is finite set
KK is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng KK is finite set
rng P is finite set
mm is set
aa is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of A
len aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
mm is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of A *
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
rng mm is finite set
aa is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of A
len aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
AB is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of A
len AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
A is non empty set
A * is functional non empty FinSequence-membered FinSequenceSet of A
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is Relation-like Seg n -defined Seg n -valued Function-like total quasi_total finite Element of bool [:(Seg n),(Seg n):]
P is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
B * P is Relation-like Seg n -defined A * -valued Function-like finite set
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
width P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Indices P is set
dom P is finite Element of bool NAT
Seg (width P) is finite width P -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
P * B is Relation-like Seg n -defined A * -valued Function-like finite Element of bool [:(Seg n),(A *):]
[:(Seg n),(A *):] is Relation-like set
bool [:(Seg n),(A *):] is non empty cup-closed diff-closed preBoolean set
mm is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
width mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
rng B is finite set
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . aa is set
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[aa,AB] is set
{aa,AB} is non empty finite V52() set
{aa} is non empty trivial finite V52() 1 -element set
{{aa,AB},{aa}} is non empty finite V52() set
mm * (aa,AB) is Element of A
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P * (SUM1,AB) is Element of A
[:(Seg n),(Seg (width P)):] is Relation-like finite set
Line (mm,aa) is Relation-like NAT -defined A -valued Function-like finite width mm -element FinSequence-like FinSubsequence-like Element of (width mm) -tuples_on A
(width mm) -tuples_on A is functional non empty FinSequence-membered FinSequenceSet of A
{ b1 where b1 is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like Element of A * : len b1 = width mm } is set
mm . aa is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom B is finite set
dom mm is finite Element of bool NAT
P . SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (P,SUM1) is Relation-like NAT -defined A -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on A
(width P) -tuples_on A is functional non empty FinSequence-membered FinSequenceSet of A
{ b1 where b1 is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like Element of A * : len b1 = width P } is set
(Line (P,SUM1)) . AB is set
Indices mm is set
dom mm is finite Element of bool NAT
Seg (width mm) is finite width mm -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width mm ) } is set
[:(dom mm),(Seg (width mm)):] is Relation-like finite set
KK is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[aa,AB] is set
{aa,AB} is non empty finite V52() set
{aa} is non empty trivial finite V52() 1 -element set
{{aa,AB},{aa}} is non empty finite V52() set
mm * (aa,AB) is Element of A
KK * (aa,AB) is Element of A
[:(Seg n),(Seg (width P)):] is Relation-like finite set
Line (KK,aa) is Relation-like NAT -defined A -valued Function-like finite width KK -element FinSequence-like FinSubsequence-like Element of (width KK) -tuples_on A
width KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width KK) -tuples_on A is functional non empty FinSequence-membered FinSequenceSet of A
{ b1 where b1 is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like Element of A * : len b1 = width KK } is set
KK . aa is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng B is finite set
dom B is finite set
B . aa is set
len KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom KK is finite Element of bool NAT
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
P . SUM1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (P,SUM1) is Relation-like NAT -defined A -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on A
(width P) -tuples_on A is functional non empty FinSequence-membered FinSequenceSet of A
{ b1 where b1 is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like Element of A * : len b1 = width P } is set
len (Line (P,SUM1)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len (Line (KK,aa)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(Line (P,SUM1)) . AB is set
P * (SUM1,AB) is Element of A
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is non empty set
A * is functional non empty FinSequence-membered FinSequenceSet of A
B is Relation-like Seg n -defined Seg n -valued Function-like total quasi_total finite Element of bool [:(Seg n),(Seg n):]
P is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
Indices P is set
dom P is finite Element of bool NAT
width P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (width P) is finite width P -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width P ) } is set
[:(dom P),(Seg (width P)):] is Relation-like finite set
(n,K,A,B,P) is Relation-like NAT -defined Seg n -defined A * -valued A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
Indices (n,K,A,B,P) is set
dom (n,K,A,B,P) is finite Element of bool NAT
width (n,K,A,B,P) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (width (n,K,A,B,P)) is finite width (n,K,A,B,P) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width (n,K,A,B,P) ) } is set
[:(dom (n,K,A,B,P)),(Seg (width (n,K,A,B,P))):] is Relation-like finite set
dom B is finite set
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len (n,K,A,B,P) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . mm is set
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[mm,aa] is set
{mm,aa} is non empty finite V52() set
{mm} is non empty trivial finite V52() 1 -element set
{{mm,aa},{mm}} is non empty finite V52() set
(n,K,A,B,P) * (mm,aa) is Element of A
[:(Seg n),(Seg (width P)):] is Relation-like finite set
rng B is finite set
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
[AB,aa] is set
{AB,aa} is non empty finite V52() set
{AB} is non empty trivial finite V52() 1 -element set
{{AB,aa},{AB}} is non empty finite V52() set
P * (AB,aa) is Element of A
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is non empty set
A * is functional non empty FinSequence-membered FinSequenceSet of A
B is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
P is Relation-like Seg n -defined Seg n -valued Function-like total quasi_total finite Element of bool [:(Seg n),(Seg n):]
(n,K,A,P,B) is Relation-like NAT -defined Seg n -defined A * -valued A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Line ((n,K,A,P,B),KK) is Relation-like NAT -defined A -valued Function-like finite width (n,K,A,P,B) -element FinSequence-like FinSubsequence-like Element of (width (n,K,A,P,B)) -tuples_on A
width (n,K,A,P,B) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width (n,K,A,P,B)) -tuples_on A is functional non empty FinSequence-membered FinSequenceSet of A
{ b1 where b1 is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like Element of A * : len b1 = width (n,K,A,P,B) } is set
P . KK is set
B . (P . KK) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (n,K,A,P,B) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom (n,K,A,P,B) is finite Element of bool NAT
(n,K,A,P,B) . KK is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
idseq K is Relation-like NAT -defined Function-like finite K -element FinSequence-like FinSubsequence-like set
Seg K is finite K -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= K ) } is set
id (Seg K) is Relation-like Seg K -defined Seg K -valued V6() V8() V9() V13() Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg K),(Seg K):]
[:(Seg K),(Seg K):] is Relation-like finite set
bool [:(Seg K),(Seg K):] is non empty cup-closed diff-closed preBoolean finite V52() set
A is non empty set
A * is functional non empty FinSequence-membered FinSequenceSet of A
B is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,n,A
(idseq K) * B is Relation-like NAT -defined A * -valued Function-like finite set
P is Relation-like Seg K -defined Seg K -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg K),(Seg K):]
(K,n,A,P,B) is Relation-like NAT -defined Seg K -defined A * -valued A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,n,A
width (K,n,A,P,B) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
width B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Indices B is set
dom B is finite Element of bool NAT
Seg (width B) is finite width B -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width B ) } is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[KK,mm] is set
{KK,mm} is non empty finite V52() set
{KK} is non empty trivial finite V52() 1 -element set
{{KK,mm},{KK}} is non empty finite V52() set
B * (KK,mm) is Element of A
(K,n,A,P,B) * (KK,mm) is Element of A
[:(Seg K),(Seg (width B)):] is Relation-like finite set
P . KK is set
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[aa,mm] is set
{aa,mm} is non empty finite V52() set
{aa} is non empty trivial finite V52() 1 -element set
{{aa,mm},{aa}} is non empty finite V52() set
B * (aa,mm) is Element of A
len (K,n,A,P,B) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
B is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
P is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
P " is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
B * (P ") is Relation-like Seg n -defined Seg (len (Permutations n)) -valued Function-like one-to-one finite Element of bool [:(Seg n),(Seg (len (Permutations n))):]
[:(Seg n),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg n),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
(n,n, the carrier of K,P,A) is Relation-like NAT -defined Seg n -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Path_matrix (B,(n,n, the carrier of K,P,A)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
KK is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
Path_matrix (KK,A) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(Path_matrix (KK,A)) * P is Relation-like Seg n -defined the carrier of K -valued Function-like finite Element of bool [:(Seg n), the carrier of K:]
[:(Seg n), the carrier of K:] is Relation-like set
bool [:(Seg n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
mm is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
dom mm is finite set
dom B is finite set
rng B is finite set
dom KK is finite set
len (Path_matrix (KK,A)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom (Path_matrix (KK,A)) is finite Element of bool NAT
rng mm is finite set
(Path_matrix (KK,A)) * mm is Relation-like Seg (len (Permutations n)) -defined the carrier of K -valued Function-like finite Element of bool [:(Seg (len (Permutations n))), the carrier of K:]
[:(Seg (len (Permutations n))), the carrier of K:] is Relation-like set
bool [:(Seg (len (Permutations n))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
dom ((Path_matrix (KK,A)) * mm) is finite set
len (Path_matrix (B,(n,n, the carrier of K,P,A))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
rng KK is finite set
F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
mm " is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
mm . F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(mm ") . (mm . F) is set
Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
[F,Ga] is set
{F,Ga} is non empty finite V52() set
{F} is non empty trivial finite V52() 1 -element set
{{F,Ga},{F}} is non empty finite V52() set
Indices A is set
dom A is finite Element of bool NAT
width A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (width A) is finite width A -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width A ) } is set
[:(dom A),(Seg (width A)):] is Relation-like finite set
(n,n, the carrier of K,P,A) * (F,Ga) is Element of the carrier of K
Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[Gs,Ga] is set
{Gs,Ga} is non empty finite V52() set
{Gs} is non empty trivial finite V52() 1 -element set
{{Gs,Ga},{Gs}} is non empty finite V52() set
A * (Gs,Ga) is Element of the carrier of K
Path is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom Path is finite Element of bool NAT
Path . F is set
(Path_matrix (KK,A)) . Gs is set
KK . Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B9 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
A * (Gs,B9) is Element of the carrier of K
dom (Path_matrix (B,(n,n, the carrier of K,P,A))) is finite Element of bool NAT
B . ((mm ") . (mm . F)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(Path_matrix (B,(n,n, the carrier of K,P,A))) . F is set
len Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
B is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
P is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
P " is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
B * (P ") is Relation-like Seg n -defined Seg (len (Permutations n)) -valued Function-like one-to-one finite Element of bool [:(Seg n),(Seg (len (Permutations n))):]
[:(Seg n),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg n),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
(n,n, the carrier of K,P,A) is Relation-like NAT -defined Seg n -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Path_matrix (B,(n,n, the carrier of K,P,A)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K $$ (Path_matrix (B,(n,n, the carrier of K,P,A))) is Element of the carrier of K
KK is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
Path_matrix (KK,A) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K $$ (Path_matrix (KK,A)) is Element of the carrier of K
len (Path_matrix (B,(n,n, the carrier of K,P,A))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
the_unity_wrt the multF of K is Element of the carrier of K
len (Path_matrix (KK,A)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
n + {} is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
len (Path_matrix (KK,A)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(Path_matrix (KK,A)) * P is Relation-like Seg n -defined the carrier of K -valued Function-like finite Element of bool [:(Seg n), the carrier of K:]
[:(Seg n), the carrier of K:] is Relation-like set
bool [:(Seg n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
n + {} is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + {} is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
{} + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
n + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
(n + 1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
A is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
A " is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):]
[:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
(n,K,A) is Element of the carrier of K
the carrier of K is non empty non trivial set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,A) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,A)) is Element of the carrier of K
B is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,B) is Element of the carrier of K
(n,K,B) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,B)) is Element of the carrier of K
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- ((1_ K),A) is Element of the carrier of K
- ((1_ K),B) is Element of the carrier of K
(n,K,B) * (1_ K) is Element of the carrier of K
the multF of K . ((n,K,B),(1_ K)) is Element of the carrier of K
[(n,K,B),(1_ K)] is set
{(n,K,B),(1_ K)} is non empty finite set
{(n,K,B)} is non empty trivial finite 1 -element set
{{(n,K,B),(1_ K)},{(n,K,B)}} is non empty finite V52() set
the multF of K . [(n,K,B),(1_ K)] is set
(n,K,A) * (1_ K) is Element of the carrier of K
the multF of K . ((n,K,A),(1_ K)) is Element of the carrier of K
[(n,K,A),(1_ K)] is set
{(n,K,A),(1_ K)} is non empty finite set
{(n,K,A)} is non empty trivial finite 1 -element set
{{(n,K,A),(1_ K)},{(n,K,A)}} is non empty finite V52() set
the multF of K . [(n,K,A),(1_ K)] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n + 2,n + 2, the carrier of K
Path_product A is Relation-like Permutations (n + 2) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (n + 2)), the carrier of K:]
[:(Permutations (n + 2)), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (n + 2)), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
B is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,B) is Element of the carrier of K
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,B) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,B)) is Element of the carrier of K
P is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
P " is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
((n + 2),(n + 2), the carrier of K,P,A) is Relation-like NAT -defined Seg (n + 2) -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n + 2,n + 2, the carrier of K
Path_product ((n + 2),(n + 2), the carrier of K,P,A) is Relation-like Permutations (n + 2) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (n + 2)), the carrier of K:]
aa is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
aa * (P ") is Relation-like Seg (n + 2) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one finite Element of bool [:(Seg (n + 2)),(Seg (len (Permutations (n + 2)))):]
[:(Seg (n + 2)),(Seg (len (Permutations (n + 2)))):] is Relation-like finite set
bool [:(Seg (n + 2)),(Seg (len (Permutations (n + 2)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
(Path_product ((n + 2),(n + 2), the carrier of K,P,A)) . aa is Element of the carrier of K
(n,K,B) * ((Path_product ((n + 2),(n + 2), the carrier of K,P,A)) . aa) is Element of the carrier of K
the multF of K . ((n,K,B),((Path_product ((n + 2),(n + 2), the carrier of K,P,A)) . aa)) is Element of the carrier of K
[(n,K,B),((Path_product ((n + 2),(n + 2), the carrier of K,P,A)) . aa)] is set
{(n,K,B),((Path_product ((n + 2),(n + 2), the carrier of K,P,A)) . aa)} is non empty finite set
{(n,K,B)} is non empty trivial finite 1 -element set
{{(n,K,B),((Path_product ((n + 2),(n + 2), the carrier of K,P,A)) . aa)},{(n,K,B)}} is non empty finite V52() set
the multF of K . [(n,K,B),((Path_product ((n + 2),(n + 2), the carrier of K,P,A)) . aa)] is set
AB is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(Path_product A) . AB is Element of the carrier of K
B " is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):]
[:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
Path_matrix (AB,A) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K $$ (Path_matrix (AB,A)) is Element of the carrier of K
Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A))) is Element of the carrier of K
(n,K,AB) is Element of the carrier of K
(n,K,AB) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,AB)) is Element of the carrier of K
(n,K,aa) is Element of the carrier of K
(n,K,aa) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,aa)) is Element of the carrier of K
SUM1 is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,SUM1) is Element of the carrier of K
(n,K,SUM1) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,SUM1)) is Element of the carrier of K
(n,K,aa) * (n,K,SUM1) is Element of the carrier of K
the multF of K . ((n,K,aa),(n,K,SUM1)) is Element of the carrier of K
[(n,K,aa),(n,K,SUM1)] is set
{(n,K,aa),(n,K,SUM1)} is non empty finite set
{(n,K,aa)} is non empty trivial finite 1 -element set
{{(n,K,aa),(n,K,SUM1)},{(n,K,aa)}} is non empty finite V52() set
the multF of K . [(n,K,aa),(n,K,SUM1)] is set
(n,K,aa) * (n,K,B) is Element of the carrier of K
the multF of K . ((n,K,aa),(n,K,B)) is Element of the carrier of K
[(n,K,aa),(n,K,B)] is set
{(n,K,aa),(n,K,B)} is non empty finite set
{{(n,K,aa),(n,K,B)},{(n,K,aa)}} is non empty finite V52() set
the multF of K . [(n,K,aa),(n,K,B)] is set
- (( the multF of K $$ (Path_matrix (AB,A))),AB) is Element of the carrier of K
(n,K,B) * (n,K,aa) is Element of the carrier of K
the multF of K . ((n,K,B),(n,K,aa)) is Element of the carrier of K
[(n,K,B),(n,K,aa)] is set
{(n,K,B),(n,K,aa)} is non empty finite set
{{(n,K,B),(n,K,aa)},{(n,K,B)}} is non empty finite V52() set
the multF of K . [(n,K,B),(n,K,aa)] is set
((n,K,B) * (n,K,aa)) * ( the multF of K $$ (Path_matrix (AB,A))) is Element of the carrier of K
the multF of K . (((n,K,B) * (n,K,aa)),( the multF of K $$ (Path_matrix (AB,A)))) is Element of the carrier of K
[((n,K,B) * (n,K,aa)),( the multF of K $$ (Path_matrix (AB,A)))] is set
{((n,K,B) * (n,K,aa)),( the multF of K $$ (Path_matrix (AB,A)))} is non empty finite set
{((n,K,B) * (n,K,aa))} is non empty trivial finite 1 -element set
{{((n,K,B) * (n,K,aa)),( the multF of K $$ (Path_matrix (AB,A)))},{((n,K,B) * (n,K,aa))}} is non empty finite V52() set
the multF of K . [((n,K,B) * (n,K,aa)),( the multF of K $$ (Path_matrix (AB,A)))] is set
(n,K,aa) * ( the multF of K $$ (Path_matrix (AB,A))) is Element of the carrier of K
the multF of K . ((n,K,aa),( the multF of K $$ (Path_matrix (AB,A)))) is Element of the carrier of K
[(n,K,aa),( the multF of K $$ (Path_matrix (AB,A)))] is set
{(n,K,aa),( the multF of K $$ (Path_matrix (AB,A)))} is non empty finite set
{{(n,K,aa),( the multF of K $$ (Path_matrix (AB,A)))},{(n,K,aa)}} is non empty finite V52() set
the multF of K . [(n,K,aa),( the multF of K $$ (Path_matrix (AB,A)))] is set
(n,K,B) * ((n,K,aa) * ( the multF of K $$ (Path_matrix (AB,A)))) is Element of the carrier of K
the multF of K . ((n,K,B),((n,K,aa) * ( the multF of K $$ (Path_matrix (AB,A))))) is Element of the carrier of K
[(n,K,B),((n,K,aa) * ( the multF of K $$ (Path_matrix (AB,A))))] is set
{(n,K,B),((n,K,aa) * ( the multF of K $$ (Path_matrix (AB,A))))} is non empty finite set
{{(n,K,B),((n,K,aa) * ( the multF of K $$ (Path_matrix (AB,A))))},{(n,K,B)}} is non empty finite V52() set
the multF of K . [(n,K,B),((n,K,aa) * ( the multF of K $$ (Path_matrix (AB,A))))] is set
(n,K,aa) * ( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))) is Element of the carrier of K
the multF of K . ((n,K,aa),( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A))))) is Element of the carrier of K
[(n,K,aa),( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A))))] is set
{(n,K,aa),( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A))))} is non empty finite set
{{(n,K,aa),( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A))))},{(n,K,aa)}} is non empty finite V52() set
the multF of K . [(n,K,aa),( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A))))] is set
(n,K,B) * ((n,K,aa) * ( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A))))) is Element of the carrier of K
the multF of K . ((n,K,B),((n,K,aa) * ( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))))) is Element of the carrier of K
[(n,K,B),((n,K,aa) * ( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))))] is set
{(n,K,B),((n,K,aa) * ( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))))} is non empty finite set
{{(n,K,B),((n,K,aa) * ( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))))},{(n,K,B)}} is non empty finite V52() set
the multF of K . [(n,K,B),((n,K,aa) * ( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))))] is set
- (( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))),aa) is Element of the carrier of K
(n,K,B) * (- (( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))),aa)) is Element of the carrier of K
the multF of K . ((n,K,B),(- (( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))),aa))) is Element of the carrier of K
[(n,K,B),(- (( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))),aa))] is set
{(n,K,B),(- (( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))),aa))} is non empty finite set
{{(n,K,B),(- (( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))),aa))},{(n,K,B)}} is non empty finite V52() set
the multF of K . [(n,K,B),(- (( the multF of K $$ (Path_matrix (aa,((n + 2),(n + 2), the carrier of K,P,A)))),aa))] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
[:(Permutations n),(Permutations n):] is Relation-like non empty set
bool [:(Permutations n),(Permutations n):] is non empty cup-closed diff-closed preBoolean set
K is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
len (Permutations n) is epsilon-transitive epsilon-connected ordinal non empty cardinal set
B is set
P is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
P * K is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
KK is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
mm is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
mm * K is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
B is Relation-like Permutations n -defined Permutations n -valued Function-like non empty total quasi_total Element of bool [:(Permutations n),(Permutations n):]
P is set
KK is set
B . P is set
B . KK is set
aa is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
dom aa is finite set
B . aa is Element of Permutations n
aa * K is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
mm is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
B . mm is Element of Permutations n
mm * K is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
rng K is finite set
dom mm is finite set
id (rng K) is Relation-like rng K -defined rng K -valued V6() V8() V9() V13() Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(rng K),(rng K):]
[:(rng K),(rng K):] is Relation-like finite set
bool [:(rng K),(rng K):] is non empty cup-closed diff-closed preBoolean finite V52() set
mm * (id (rng K)) is Relation-like rng K -defined Seg (len (Permutations n)) -valued Function-like one-to-one finite Element of bool [:(rng K),(Seg (len (Permutations n))):]
[:(rng K),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(rng K),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
K " is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
K * (K ") is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total total quasi_total quasi_total quasi_total onto onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
mm * (K * (K ")) is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total total quasi_total quasi_total quasi_total onto onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
(aa * K) * (K ") is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
aa * (K * (K ")) is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total total quasi_total quasi_total quasi_total onto onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
aa * (id (rng K)) is Relation-like rng K -defined Seg (len (Permutations n)) -valued Function-like one-to-one finite Element of bool [:(rng K),(Seg (len (Permutations n))):]
P is Relation-like Permutations n -defined Permutations n -valued Function-like one-to-one non empty total quasi_total onto bijective Element of bool [:(Permutations n),(Permutations n):]
KK is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
P . KK is Element of Permutations n
KK * K is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
n + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (n + 2)) ) } is set
Seg (n + 2) is non empty finite n + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n + 2 ) } is set
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like non empty finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n + 2,n + 2, the carrier of K
Det B is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (Permutations (n + 2)) is finite Element of Fin (Permutations (n + 2))
Fin (Permutations (n + 2)) is non empty cup-closed diff-closed preBoolean set
Path_product B is Relation-like Permutations (n + 2) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (n + 2)), the carrier of K:]
[:(Permutations (n + 2)), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (n + 2)), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations (n + 2))),(Path_product B)) is Element of the carrier of K
P is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
(n,K,P) is Element of the carrier of K
TWOELEMENTSETS (Seg (n + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
FinOmega (TWOELEMENTSETS (Seg (n + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (n + 2)))
Fin (TWOELEMENTSETS (Seg (n + 2))) is non empty cup-closed diff-closed preBoolean set
(n,K,P) is Relation-like TWOELEMENTSETS (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (n + 2)))),(n,K,P)) is Element of the carrier of K
(n,K,P) * (Det B) is Element of the carrier of K
the multF of K . ((n,K,P),(Det B)) is Element of the carrier of K
[(n,K,P),(Det B)] is set
{(n,K,P),(Det B)} is non empty finite set
{(n,K,P)} is non empty trivial finite 1 -element set
{{(n,K,P),(Det B)},{(n,K,P)}} is non empty finite V52() set
the multF of K . [(n,K,P),(Det B)] is set
KK is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
((n + 2),(n + 2), the carrier of K,KK,B) is Relation-like NAT -defined Seg (n + 2) -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n + 2,n + 2, the carrier of K
Det ((n + 2),(n + 2), the carrier of K,KK,B) is Element of the carrier of K
Path_product ((n + 2),(n + 2), the carrier of K,KK,B) is Relation-like Permutations (n + 2) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (n + 2)), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations (n + 2))),(Path_product ((n + 2),(n + 2), the carrier of K,KK,B))) is Element of the carrier of K
P " is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):]
[:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is Relation-like finite set
bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):] is non empty cup-closed diff-closed preBoolean finite V52() set
[:(Fin (Permutations (n + 2))), the carrier of K:] is Relation-like non empty set
bool [:(Fin (Permutations (n + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
B9 is Relation-like Fin (Permutations (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (Permutations (n + 2))), the carrier of K:]
B9 . (FinOmega (Permutations (n + 2))) is Element of the carrier of K
B9 . {} is set
[:(Permutations (n + 2)),(Permutations (n + 2)):] is Relation-like non empty set
bool [:(Permutations (n + 2)),(Permutations (n + 2)):] is non empty cup-closed diff-closed preBoolean set
Gs is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
b is Relation-like Permutations (n + 2) -defined Permutations (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective Element of bool [:(Permutations (n + 2)),(Permutations (n + 2)):]
mA is Relation-like Fin (Permutations (n + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (Permutations (n + 2))), the carrier of K:]
mA . (FinOmega (Permutations (n + 2))) is Element of the carrier of K
mA . {} is set
Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Bb + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
i is finite Element of Fin (Permutations (n + 2))
card i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
mA . i is Element of the carrier of K
(n,K,P) * (mA . i) is Element of the carrier of K
the multF of K . ((n,K,P),(mA . i)) is Element of the carrier of K
[(n,K,P),(mA . i)] is set
{(n,K,P),(mA . i)} is non empty finite set
{{(n,K,P),(mA . i)},{(n,K,P)}} is non empty finite V52() set
the multF of K . [(n,K,P),(mA . i)] is set
b .: i is finite Element of Fin (Permutations (n + 2))
B9 . (b .: i) is Element of the carrier of K
Pi is set
{Pi} is non empty trivial finite 1 -element set
H is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
b . H is Element of Permutations (n + 2)
{(b . H)} is non empty trivial finite 1 -element set
B9 . {(b . H)} is set
(Path_product B) . (b . H) is Element of the carrier of K
H * Gs is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):]
dom b is non empty set
{H} is functional non empty trivial finite V52() 1 -element set
mA . {H} is set
(Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H is Element of the carrier of K
Im (b,H) is set
b .: {H} is finite set
B9 . (Im (b,H)) is set
Pi is set
H is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
{H} is functional non empty trivial finite V52() 1 -element set
i \ {H} is finite Element of bool i
bool i is non empty cup-closed diff-closed preBoolean finite V52() set
b .: (i \ {H}) is finite set
rng b is non empty set
b . H is Element of Permutations (n + 2)
SF is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (n + 2)
{SF} is functional non empty trivial finite V52() 1 -element set
dom b is non empty set
Im (b,H) is set
b .: {H} is finite set
QQ is finite Element of Fin (Permutations (n + 2))
{H} \/ QQ is non empty finite set
h is finite Element of Fin (Permutations (n + 2))
(Im (b,H)) \/ h is set
H * Gs is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Seg (len (Permutations (n + 2))) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations (n + 2)))),(Seg (len (Permutations (n + 2)))):]
(Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H is Element of the carrier of K
(n,K,P) * ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H) is Element of the carrier of K
the multF of K . ((n,K,P),((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)) is Element of the carrier of K
[(n,K,P),((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)] is set
{(n,K,P),((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)} is non empty finite set
{{(n,K,P),((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)},{(n,K,P)}} is non empty finite V52() set
the multF of K . [(n,K,P),((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)] is set
(Path_product B) . SF is Element of the carrier of K
QQ /\ {H} is finite set
b .: {} is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
{SF} /\ h is finite set
(FinOmega (Permutations (n + 2))) \ h is finite Element of Fin (Permutations (n + 2))
(FinOmega (Permutations (n + 2))) \ QQ is finite Element of Fin (Permutations (n + 2))
card QQ is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(card QQ) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Mh is finite Element of Fin (Permutations (n + 2))
B9 . Mh is Element of the carrier of K
B9 . h is Element of the carrier of K
the addF of K . ((B9 . h),((Path_product B) . SF)) is Element of the carrier of K
[(B9 . h),((Path_product B) . SF)] is set
{(B9 . h),((Path_product B) . SF)} is non empty finite set
{(B9 . h)} is non empty trivial finite 1 -element set
{{(B9 . h),((Path_product B) . SF)},{(B9 . h)}} is non empty finite V52() set
the addF of K . [(B9 . h),((Path_product B) . SF)] is set
Path is set
mA . QQ is Element of the carrier of K
(n,K,P) * (mA . QQ) is Element of the carrier of K
the multF of K . ((n,K,P),(mA . QQ)) is Element of the carrier of K
[(n,K,P),(mA . QQ)] is set
{(n,K,P),(mA . QQ)} is non empty finite set
{{(n,K,P),(mA . QQ)},{(n,K,P)}} is non empty finite V52() set
the multF of K . [(n,K,P),(mA . QQ)] is set
((n,K,P) * (mA . QQ)) + ((n,K,P) * ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)) is Element of the carrier of K
the addF of K . (((n,K,P) * (mA . QQ)),((n,K,P) * ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H))) is Element of the carrier of K
[((n,K,P) * (mA . QQ)),((n,K,P) * ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H))] is set
{((n,K,P) * (mA . QQ)),((n,K,P) * ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H))} is non empty finite set
{((n,K,P) * (mA . QQ))} is non empty trivial finite 1 -element set
{{((n,K,P) * (mA . QQ)),((n,K,P) * ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H))},{((n,K,P) * (mA . QQ))}} is non empty finite V52() set
the addF of K . [((n,K,P) * (mA . QQ)),((n,K,P) * ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H))] is set
(mA . QQ) + ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H) is Element of the carrier of K
the addF of K . ((mA . QQ),((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)) is Element of the carrier of K
[(mA . QQ),((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)] is set
{(mA . QQ),((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)} is non empty finite set
{(mA . QQ)} is non empty trivial finite 1 -element set
{{(mA . QQ),((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)},{(mA . QQ)}} is non empty finite V52() set
the addF of K . [(mA . QQ),((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)] is set
(n,K,P) * ((mA . QQ) + ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H)) is Element of the carrier of K
the multF of K . ((n,K,P),((mA . QQ) + ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H))) is Element of the carrier of K
[(n,K,P),((mA . QQ) + ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H))] is set
{(n,K,P),((mA . QQ) + ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H))} is non empty finite set
{{(n,K,P),((mA . QQ) + ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H))},{(n,K,P)}} is non empty finite V52() set
the multF of K . [(n,K,P),((mA . QQ) + ((Path_product ((n + 2),(n + 2), the carrier of K,KK,B)) . H))] is set
Bb is finite Element of Fin (Permutations (n + 2))
card Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
mA . Bb is Element of the carrier of K
(n,K,P) * (mA . Bb) is Element of the carrier of K
the multF of K . ((n,K,P),(mA . Bb)) is Element of the carrier of K
[(n,K,P),(mA . Bb)] is set
{(n,K,P),(mA . Bb)} is non empty finite set
{{(n,K,P),(mA . Bb)},{(n,K,P)}} is non empty finite V52() set
the multF of K . [(n,K,P),(mA . Bb)] is set
b .: Bb is finite Element of Fin (Permutations (n + 2))
B9 . (b .: Bb) is Element of the carrier of K
rng b is non empty set
dom b is non empty set
b .: (dom b) is set
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
(1_ K) * (1_ K) is Element of the carrier of K
the multF of K . ((1_ K),(1_ K)) is Element of the carrier of K
[(1_ K),(1_ K)] is set
{(1_ K),(1_ K)} is non empty finite set
{(1_ K)} is non empty trivial finite 1 -element set
{{(1_ K),(1_ K)},{(1_ K)}} is non empty finite V52() set
the multF of K . [(1_ K),(1_ K)] is set
- (1_ K) is Element of the carrier of K
(- (1_ K)) * (- (1_ K)) is Element of the carrier of K
the multF of K . ((- (1_ K)),(- (1_ K))) is Element of the carrier of K
[(- (1_ K)),(- (1_ K))] is set
{(- (1_ K)),(- (1_ K))} is non empty finite set
{(- (1_ K))} is non empty trivial finite 1 -element set
{{(- (1_ K)),(- (1_ K))},{(- (1_ K))}} is non empty finite V52() set
the multF of K . [(- (1_ K)),(- (1_ K))] is set
card (FinOmega (Permutations (n + 2))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(n,K,P) * (Det ((n + 2),(n + 2), the carrier of K,KK,B)) is Element of the carrier of K
the multF of K . ((n,K,P),(Det ((n + 2),(n + 2), the carrier of K,KK,B))) is Element of the carrier of K
[(n,K,P),(Det ((n + 2),(n + 2), the carrier of K,KK,B))] is set
{(n,K,P),(Det ((n + 2),(n + 2), the carrier of K,KK,B))} is non empty finite set
{{(n,K,P),(Det ((n + 2),(n + 2), the carrier of K,KK,B))},{(n,K,P)}} is non empty finite V52() set
the multF of K . [(n,K,P),(Det ((n + 2),(n + 2), the carrier of K,KK,B))] is set
(1_ K) * (Det ((n + 2),(n + 2), the carrier of K,KK,B)) is Element of the carrier of K
the multF of K . ((1_ K),(Det ((n + 2),(n + 2), the carrier of K,KK,B))) is Element of the carrier of K
[(1_ K),(Det ((n + 2),(n + 2), the carrier of K,KK,B))] is set
{(1_ K),(Det ((n + 2),(n + 2), the carrier of K,KK,B))} is non empty finite set
{{(1_ K),(Det ((n + 2),(n + 2), the carrier of K,KK,B))},{(1_ K)}} is non empty finite V52() set
the multF of K . [(1_ K),(Det ((n + 2),(n + 2), the carrier of K,KK,B))] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det A is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product A is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product A)) is Element of the carrier of K
B is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
- ((Det A),B) is Element of the carrier of K
P is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
(n,n, the carrier of K,P,A) is Relation-like NAT -defined Seg n -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (n,n, the carrier of K,P,A) is Element of the carrier of K
Path_product (n,n, the carrier of K,P,A) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (n,n, the carrier of K,P,A))) is Element of the carrier of K
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
id (Seg n) is Relation-like Seg n -defined Seg n -valued V6() V8() V9() V13() Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
A * B is Relation-like Seg (len (Permutations n)) -defined the carrier of K * -valued Function-like finite Element of bool [:(Seg (len (Permutations n))),( the carrier of K *):]
[:(Seg (len (Permutations n))),( the carrier of K *):] is Relation-like set
bool [:(Seg (len (Permutations n))),( the carrier of K *):] is non empty cup-closed diff-closed preBoolean set
n - 2 is ext-real V44() V45() set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Seg (KK + 2) is non empty finite KK + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= KK + 2 ) } is set
[:(Seg (KK + 2)),(Seg (KK + 2)):] is Relation-like non empty finite set
bool [:(Seg (KK + 2)),(Seg (KK + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
Permutations (KK + 2) is non empty permutational set
len (Permutations (KK + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (KK + 2))) is finite len (Permutations (KK + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (KK + 2)) ) } is set
aa is Relation-like Seg (KK + 2) -defined Seg (KK + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (KK + 2)),(Seg (KK + 2)):]
mm is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of KK + 2,KK + 2, the carrier of K
((KK + 2),(KK + 2), the carrier of K,aa,mm) is Relation-like NAT -defined Seg (KK + 2) -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of KK + 2,KK + 2, the carrier of K
Det ((KK + 2),(KK + 2), the carrier of K,aa,mm) is Element of the carrier of K
FinOmega (Permutations (KK + 2)) is finite Element of Fin (Permutations (KK + 2))
Fin (Permutations (KK + 2)) is non empty cup-closed diff-closed preBoolean set
Path_product ((KK + 2),(KK + 2), the carrier of K,aa,mm) is Relation-like Permutations (KK + 2) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (KK + 2)), the carrier of K:]
[:(Permutations (KK + 2)), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (KK + 2)), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations (KK + 2))),(Path_product ((KK + 2),(KK + 2), the carrier of K,aa,mm))) is Element of the carrier of K
AB is Relation-like Seg (len (Permutations (KK + 2))) -defined Seg (len (Permutations (KK + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (KK + 2)
(KK,K,AB) is Element of the carrier of K
TWOELEMENTSETS (Seg (KK + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
FinOmega (TWOELEMENTSETS (Seg (KK + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (KK + 2)))
Fin (TWOELEMENTSETS (Seg (KK + 2))) is non empty cup-closed diff-closed preBoolean set
(KK,K,AB) is Relation-like TWOELEMENTSETS (Seg (KK + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (KK + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (KK + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (KK + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (KK + 2)))),(KK,K,AB)) is Element of the carrier of K
Det mm is Element of the carrier of K
Path_product mm is Relation-like Permutations (KK + 2) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (KK + 2)), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations (KK + 2))),(Path_product mm)) is Element of the carrier of K
(KK,K,AB) * (Det mm) is Element of the carrier of K
the multF of K . ((KK,K,AB),(Det mm)) is Element of the carrier of K
[(KK,K,AB),(Det mm)] is set
{(KK,K,AB),(Det mm)} is non empty finite set
{(KK,K,AB)} is non empty trivial finite 1 -element set
{{(KK,K,AB),(Det mm)},{(KK,K,AB)}} is non empty finite V52() set
the multF of K . [(KK,K,AB),(Det mm)] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
[:(Permutations n),(Permutations n):] is Relation-like non empty set
bool [:(Permutations n),(Permutations n):] is non empty cup-closed diff-closed preBoolean set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
{ b1 where b1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n : b1 is even } is set
{ b1 where b1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n : not b1 is even } is set
A is Relation-like Permutations n -defined Permutations n -valued Function-like one-to-one non empty total quasi_total onto bijective Element of bool [:(Permutations n),(Permutations n):]
A .: { b1 where b1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n : b1 is even } is set
B is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
dom A is non empty set
mm is set
B " is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
AB is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
id (Seg n) is Relation-like Seg n -defined Seg n -valued V6() V8() V9() V13() Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
(idseq n) * AB is Relation-like NAT -defined Seg (len (Permutations n)) -valued Function-like finite set
aa is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
AB * aa is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
SUM1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
A . SUM1 is set
aa * B is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
A . SUM1 is Element of Permutations n
SUM1 * B is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
aa is set
A . aa is set
aa is set
A . aa is set
AB is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
AB * B is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
SUM1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
{ b1 where b1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n : b1 is even } is set
{ b1 where b1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n : not b1 is even } is set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
P . K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{K,A} is non empty finite V52() set
TWOELEMENTSETS (Seg n) is set
n - 2 is ext-real V44() V45() set
AB is set
SUM1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
AB is set
SUM1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
AB is finite set
SUM1 is finite set
SUM1 /\ AB is finite set
SUM1 \/ AB is finite set
[:SUM1,AB:] is Relation-like finite set
bool [:SUM1,AB:] is non empty cup-closed diff-closed preBoolean finite V52() set
Path is set
F is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
Ga is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
Path is set
F is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
[:(Permutations n),(Permutations n):] is Relation-like non empty set
bool [:(Permutations n),(Permutations n):] is non empty cup-closed diff-closed preBoolean set
Path is Relation-like Permutations n -defined Permutations n -valued Function-like one-to-one non empty total quasi_total onto bijective Element of bool [:(Permutations n),(Permutations n):]
Path | SUM1 is Relation-like Permutations n -defined SUM1 -defined Permutations n -defined Permutations n -valued Function-like finite Element of bool [:(Permutations n),(Permutations n):]
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Permutations (KK + 2) is non empty permutational set
len (Permutations (KK + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (KK + 2))) is finite len (Permutations (KK + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (KK + 2)) ) } is set
Path .: SUM1 is finite set
rng (Path | SUM1) is finite set
dom Path is non empty set
dom (Path | SUM1) is finite set
Ga is Relation-like SUM1 -defined AB -valued Function-like quasi_total finite Element of bool [:SUM1,AB:]
dom Ga is finite set
Gs is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
Ga . Gs is set
Gs * P is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
Path . Gs is Element of Permutations n
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
{ b1 where b1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n : b1 is even } is set
{ b1 where b1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n : not b1 is even } is set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
A is finite set
K is finite set
A /\ K is finite set
A \/ K is finite set
[:A,K:] is Relation-like finite set
bool [:A,K:] is non empty cup-closed diff-closed preBoolean finite V52() set
P is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
P . 1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is Relation-like A -defined K -valued Function-like quasi_total finite Element of bool [:A,K:]
dom B is finite set
P is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
P . 1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is Relation-like A -defined K -valued Function-like quasi_total finite Element of bool [:A,K:]
dom B is finite set
rng B is finite set
card A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
card K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
{A,B} is non empty finite V52() set
TWOELEMENTSETS (Seg n) is set
n - 2 is ext-real V44() V45() set
KK is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Line (KK,A) is Relation-like NAT -defined the carrier of K -valued Function-like finite width KK -element FinSequence-like FinSubsequence-like Element of (width KK) -tuples_on the carrier of K
width KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width KK) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width KK } is set
Line (KK,B) is Relation-like NAT -defined the carrier of K -valued Function-like finite width KK -element FinSequence-like FinSubsequence-like Element of (width KK) -tuples_on the carrier of K
Path_product KK is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P + 2 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
AB is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
SUM1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
aa is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
aa * SUM1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
SUM1 . A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(Path_product KK) . AB is Element of the carrier of K
(Path_product KK) . aa is Element of the carrier of K
- ((Path_product KK) . aa) is Element of the carrier of K
Seg (P + 2) is non empty finite P + 2 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= P + 2 ) } is set
[:(Seg (P + 2)),(Seg (P + 2)):] is Relation-like non empty finite set
bool [:(Seg (P + 2)),(Seg (P + 2)):] is non empty cup-closed diff-closed preBoolean finite V52() set
Path is Relation-like Seg (P + 2) -defined Seg (P + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite Element of bool [:(Seg (P + 2)),(Seg (P + 2)):]
mm is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of P + 2,P + 2, the carrier of K
((P + 2),(P + 2), the carrier of K,Path,mm) is Relation-like NAT -defined Seg (P + 2) -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of P + 2,P + 2, the carrier of K
len mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
mm . Ga is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
((P + 2),(P + 2), the carrier of K,Path,mm) . Ga is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Seg (len mm) is finite len mm -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len mm ) } is set
KK . B is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom Path is non empty finite set
KK . A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (((P + 2),(P + 2), the carrier of K,Path,mm),Ga) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((P + 2),(P + 2), the carrier of K,Path,mm) -element FinSequence-like FinSubsequence-like Element of (width ((P + 2),(P + 2), the carrier of K,Path,mm)) -tuples_on the carrier of K
width ((P + 2),(P + 2), the carrier of K,Path,mm) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width ((P + 2),(P + 2), the carrier of K,Path,mm)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width ((P + 2),(P + 2), the carrier of K,Path,mm) } is set
SUM1 . Ga is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK . (SUM1 . Ga) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
KK . Ga is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
KK . Ga is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len ((P + 2),(P + 2), the carrier of K,Path,mm) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Permutations (P + 2) is non empty permutational set
len (Permutations (P + 2)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations (P + 2))) is finite len (Permutations (P + 2)) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations (P + 2)) ) } is set
Ga is Relation-like Seg (len (Permutations (P + 2))) -defined Seg (len (Permutations (P + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (P + 2)
(P,K,Ga) is Element of the carrier of K
TWOELEMENTSETS (Seg (P + 2)) is non empty finite set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (TWOELEMENTSETS (Seg (P + 2))) is finite Element of Fin (TWOELEMENTSETS (Seg (P + 2)))
Fin (TWOELEMENTSETS (Seg (P + 2))) is non empty cup-closed diff-closed preBoolean set
(P,K,Ga) is Relation-like TWOELEMENTSETS (Seg (P + 2)) -defined the carrier of K -valued Function-like non empty total quasi_total finite Element of bool [:(TWOELEMENTSETS (Seg (P + 2))), the carrier of K:]
[:(TWOELEMENTSETS (Seg (P + 2))), the carrier of K:] is Relation-like non empty set
bool [:(TWOELEMENTSETS (Seg (P + 2))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K $$ ((FinOmega (TWOELEMENTSETS (Seg (P + 2)))),(P,K,Ga)) is Element of the carrier of K
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
SUM1 " is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
Path_product mm is Relation-like Permutations (P + 2) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations (P + 2)), the carrier of K:]
[:(Permutations (P + 2)), the carrier of K:] is Relation-like non empty set
bool [:(Permutations (P + 2)), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
Gs is Relation-like Seg (len (Permutations (P + 2))) -defined Seg (len (Permutations (P + 2))) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations (P + 2)
(Path_product mm) . Gs is Element of the carrier of K
(- (1_ K)) * ((Path_product mm) . Gs) is Element of the carrier of K
the multF of K . ((- (1_ K)),((Path_product mm) . Gs)) is Element of the carrier of K
[(- (1_ K)),((Path_product mm) . Gs)] is set
{(- (1_ K)),((Path_product mm) . Gs)} is non empty finite set
{(- (1_ K))} is non empty trivial finite 1 -element set
{{(- (1_ K)),((Path_product mm) . Gs)},{(- (1_ K))}} is non empty finite V52() set
the multF of K . [(- (1_ K)),((Path_product mm) . Gs)] is set
(1_ K) * ((Path_product mm) . Gs) is Element of the carrier of K
the multF of K . ((1_ K),((Path_product mm) . Gs)) is Element of the carrier of K
[(1_ K),((Path_product mm) . Gs)] is set
{(1_ K),((Path_product mm) . Gs)} is non empty finite set
{(1_ K)} is non empty trivial finite 1 -element set
{{(1_ K),((Path_product mm) . Gs)},{(1_ K)}} is non empty finite V52() set
the multF of K . [(1_ K),((Path_product mm) . Gs)] is set
- ((1_ K) * ((Path_product mm) . Gs)) is Element of the carrier of K
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is V70(K) Element of the carrier of K
A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
{ b1 where b1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n : b1 is even } is set
{ b1 where b1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n : not b1 is even } is set
mm is finite set
KK is finite set
mm /\ KK is finite set
mm \/ KK is finite set
[:mm,KK:] is Relation-like finite set
bool [:mm,KK:] is non empty cup-closed diff-closed preBoolean finite V52() set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
SUM1 is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Line (SUM1,A) is Relation-like NAT -defined the carrier of K -valued Function-like finite width SUM1 -element FinSequence-like FinSubsequence-like Element of (width SUM1) -tuples_on the carrier of K
width SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width SUM1) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width SUM1 } is set
Line (SUM1,B) is Relation-like NAT -defined the carrier of K -valued Function-like finite width SUM1 -element FinSequence-like FinSubsequence-like Element of (width SUM1) -tuples_on the carrier of K
Det SUM1 is Element of the carrier of K
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product SUM1 is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product SUM1)) is Element of the carrier of K
Ga is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
Ga . A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
F is Relation-like mm -defined KK -valued Function-like quasi_total finite Element of bool [:mm,KK:]
dom F is finite set
Ga is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
Ga . A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
F is Relation-like mm -defined KK -valued Function-like quasi_total finite Element of bool [:mm,KK:]
dom F is finite set
[:(Fin (Permutations n)), the carrier of K:] is Relation-like non empty set
bool [:(Fin (Permutations n)), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
Gs is finite Element of Fin (Permutations n)
the addF of K $$ (Gs,(Path_product SUM1)) is Element of the carrier of K
b is Relation-like Fin (Permutations n) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (Permutations n)), the carrier of K:]
b . Gs is Element of the carrier of K
b . {} is set
B9 is finite Element of Fin (Permutations n)
the addF of K $$ (B9,(Path_product SUM1)) is Element of the carrier of K
mA is Relation-like Fin (Permutations n) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (Permutations n)), the carrier of K:]
mA . B9 is Element of the carrier of K
mA . {} is set
Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Bb + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
PM is finite Element of Fin (Permutations n)
card PM is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
F .: PM is finite set
i is finite Element of Fin (Permutations n)
b . PM is Element of the carrier of K
mA . i is Element of the carrier of K
(b . PM) + (mA . i) is Element of the carrier of K
the addF of K . ((b . PM),(mA . i)) is Element of the carrier of K
[(b . PM),(mA . i)] is set
{(b . PM),(mA . i)} is non empty finite set
{(b . PM)} is non empty trivial finite 1 -element set
{{(b . PM),(mA . i)},{(b . PM)}} is non empty finite V52() set
the addF of K . [(b . PM),(mA . i)] is set
Pi is set
{Pi} is non empty trivial finite 1 -element set
H is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
H * Ga is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
F . H is set
Im (F,H) is set
{H} is functional non empty trivial finite V52() 1 -element set
F .: {H} is finite set
SF is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
{SF} is functional non empty trivial finite V52() 1 -element set
b . {H} is set
(Path_product SUM1) . H is Element of the carrier of K
{(F . H)} is non empty trivial finite 1 -element set
mA . {(F . H)} is set
(Path_product SUM1) . SF is Element of the carrier of K
- (b . PM) is Element of the carrier of K
Pi is set
H is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
H * Ga is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective bijective finite Element of bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):]
[:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is Relation-like finite set
bool [:(Seg (len (Permutations n))),(Seg (len (Permutations n))):] is non empty cup-closed diff-closed preBoolean finite V52() set
F . H is set
Im (F,H) is set
{H} is functional non empty trivial finite V52() 1 -element set
F .: {H} is finite set
SF is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
{SF} is functional non empty trivial finite V52() 1 -element set
(Path_product SUM1) . H is Element of the carrier of K
- ((Path_product SUM1) . H) is Element of the carrier of K
(Path_product SUM1) . SF is Element of the carrier of K
PM \ {H} is finite Element of bool PM
bool PM is non empty cup-closed diff-closed preBoolean finite V52() set
rng F is finite set
F .: (PM \ {H}) is finite set
QQ is finite Element of Fin (Permutations n)
{H} \/ QQ is non empty finite set
h is finite Element of Fin (Permutations n)
(Im (F,H)) \/ h is set
QQ /\ {H} is finite set
F .: {} is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
{SF} /\ h is finite set
B9 \ h is finite Element of Fin (Permutations n)
Gs \ QQ is finite Element of Fin (Permutations n)
card QQ is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(card QQ) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Mh is set
F . Mh is set
mA . h is Element of the carrier of K
the addF of K . ((mA . h),((Path_product SUM1) . SF)) is Element of the carrier of K
[(mA . h),((Path_product SUM1) . SF)] is set
{(mA . h),((Path_product SUM1) . SF)} is non empty finite set
{(mA . h)} is non empty trivial finite 1 -element set
{{(mA . h),((Path_product SUM1) . SF)},{(mA . h)}} is non empty finite V52() set
the addF of K . [(mA . h),((Path_product SUM1) . SF)] is set
(mA . i) + (b . PM) is Element of the carrier of K
the addF of K . ((mA . i),(b . PM)) is Element of the carrier of K
[(mA . i),(b . PM)] is set
{(mA . i),(b . PM)} is non empty finite set
{(mA . i)} is non empty trivial finite 1 -element set
{{(mA . i),(b . PM)},{(mA . i)}} is non empty finite V52() set
the addF of K . [(mA . i),(b . PM)] is set
(mA . h) - ((Path_product SUM1) . H) is Element of the carrier of K
(mA . h) + (- ((Path_product SUM1) . H)) is Element of the carrier of K
the addF of K . ((mA . h),(- ((Path_product SUM1) . H))) is Element of the carrier of K
[(mA . h),(- ((Path_product SUM1) . H))] is set
{(mA . h),(- ((Path_product SUM1) . H))} is non empty finite set
{{(mA . h),(- ((Path_product SUM1) . H))},{(mA . h)}} is non empty finite V52() set
the addF of K . [(mA . h),(- ((Path_product SUM1) . H))] is set
b . QQ is Element of the carrier of K
(b . QQ) + ((Path_product SUM1) . H) is Element of the carrier of K
the addF of K . ((b . QQ),((Path_product SUM1) . H)) is Element of the carrier of K
[(b . QQ),((Path_product SUM1) . H)] is set
{(b . QQ),((Path_product SUM1) . H)} is non empty finite set
{(b . QQ)} is non empty trivial finite 1 -element set
{{(b . QQ),((Path_product SUM1) . H)},{(b . QQ)}} is non empty finite V52() set
the addF of K . [(b . QQ),((Path_product SUM1) . H)] is set
((mA . h) - ((Path_product SUM1) . H)) + ((b . QQ) + ((Path_product SUM1) . H)) is Element of the carrier of K
the addF of K . (((mA . h) - ((Path_product SUM1) . H)),((b . QQ) + ((Path_product SUM1) . H))) is Element of the carrier of K
[((mA . h) - ((Path_product SUM1) . H)),((b . QQ) + ((Path_product SUM1) . H))] is set
{((mA . h) - ((Path_product SUM1) . H)),((b . QQ) + ((Path_product SUM1) . H))} is non empty finite set
{((mA . h) - ((Path_product SUM1) . H))} is non empty trivial finite 1 -element set
{{((mA . h) - ((Path_product SUM1) . H)),((b . QQ) + ((Path_product SUM1) . H))},{((mA . h) - ((Path_product SUM1) . H))}} is non empty finite V52() set
the addF of K . [((mA . h) - ((Path_product SUM1) . H)),((b . QQ) + ((Path_product SUM1) . H))] is set
(- ((Path_product SUM1) . H)) + ((b . QQ) + ((Path_product SUM1) . H)) is Element of the carrier of K
the addF of K . ((- ((Path_product SUM1) . H)),((b . QQ) + ((Path_product SUM1) . H))) is Element of the carrier of K
[(- ((Path_product SUM1) . H)),((b . QQ) + ((Path_product SUM1) . H))] is set
{(- ((Path_product SUM1) . H)),((b . QQ) + ((Path_product SUM1) . H))} is non empty finite set
{(- ((Path_product SUM1) . H))} is non empty trivial finite 1 -element set
{{(- ((Path_product SUM1) . H)),((b . QQ) + ((Path_product SUM1) . H))},{(- ((Path_product SUM1) . H))}} is non empty finite V52() set
the addF of K . [(- ((Path_product SUM1) . H)),((b . QQ) + ((Path_product SUM1) . H))] is set
(mA . h) + ((- ((Path_product SUM1) . H)) + ((b . QQ) + ((Path_product SUM1) . H))) is Element of the carrier of K
the addF of K . ((mA . h),((- ((Path_product SUM1) . H)) + ((b . QQ) + ((Path_product SUM1) . H)))) is Element of the carrier of K
[(mA . h),((- ((Path_product SUM1) . H)) + ((b . QQ) + ((Path_product SUM1) . H)))] is set
{(mA . h),((- ((Path_product SUM1) . H)) + ((b . QQ) + ((Path_product SUM1) . H)))} is non empty finite set
{{(mA . h),((- ((Path_product SUM1) . H)) + ((b . QQ) + ((Path_product SUM1) . H)))},{(mA . h)}} is non empty finite V52() set
the addF of K . [(mA . h),((- ((Path_product SUM1) . H)) + ((b . QQ) + ((Path_product SUM1) . H)))] is set
((Path_product SUM1) . H) - ((Path_product SUM1) . H) is Element of the carrier of K
((Path_product SUM1) . H) + (- ((Path_product SUM1) . H)) is Element of the carrier of K
the addF of K . (((Path_product SUM1) . H),(- ((Path_product SUM1) . H))) is Element of the carrier of K
[((Path_product SUM1) . H),(- ((Path_product SUM1) . H))] is set
{((Path_product SUM1) . H),(- ((Path_product SUM1) . H))} is non empty finite set
{((Path_product SUM1) . H)} is non empty trivial finite 1 -element set
{{((Path_product SUM1) . H),(- ((Path_product SUM1) . H))},{((Path_product SUM1) . H)}} is non empty finite V52() set
the addF of K . [((Path_product SUM1) . H),(- ((Path_product SUM1) . H))] is set
(b . QQ) + (((Path_product SUM1) . H) - ((Path_product SUM1) . H)) is Element of the carrier of K
the addF of K . ((b . QQ),(((Path_product SUM1) . H) - ((Path_product SUM1) . H))) is Element of the carrier of K
[(b . QQ),(((Path_product SUM1) . H) - ((Path_product SUM1) . H))] is set
{(b . QQ),(((Path_product SUM1) . H) - ((Path_product SUM1) . H))} is non empty finite set
{{(b . QQ),(((Path_product SUM1) . H) - ((Path_product SUM1) . H))},{(b . QQ)}} is non empty finite V52() set
the addF of K . [(b . QQ),(((Path_product SUM1) . H) - ((Path_product SUM1) . H))] is set
(mA . h) + ((b . QQ) + (((Path_product SUM1) . H) - ((Path_product SUM1) . H))) is Element of the carrier of K
the addF of K . ((mA . h),((b . QQ) + (((Path_product SUM1) . H) - ((Path_product SUM1) . H)))) is Element of the carrier of K
[(mA . h),((b . QQ) + (((Path_product SUM1) . H) - ((Path_product SUM1) . H)))] is set
{(mA . h),((b . QQ) + (((Path_product SUM1) . H) - ((Path_product SUM1) . H)))} is non empty finite set
{{(mA . h),((b . QQ) + (((Path_product SUM1) . H) - ((Path_product SUM1) . H)))},{(mA . h)}} is non empty finite V52() set
the addF of K . [(mA . h),((b . QQ) + (((Path_product SUM1) . H) - ((Path_product SUM1) . H)))] is set
(b . QQ) + (0. K) is Element of the carrier of K
the addF of K . ((b . QQ),(0. K)) is Element of the carrier of K
[(b . QQ),(0. K)] is set
{(b . QQ),(0. K)} is non empty finite set
{{(b . QQ),(0. K)},{(b . QQ)}} is non empty finite V52() set
the addF of K . [(b . QQ),(0. K)] is set
(mA . h) + ((b . QQ) + (0. K)) is Element of the carrier of K
the addF of K . ((mA . h),((b . QQ) + (0. K))) is Element of the carrier of K
[(mA . h),((b . QQ) + (0. K))] is set
{(mA . h),((b . QQ) + (0. K))} is non empty finite set
{{(mA . h),((b . QQ) + (0. K))},{(mA . h)}} is non empty finite V52() set
the addF of K . [(mA . h),((b . QQ) + (0. K))] is set
(mA . h) + (b . QQ) is Element of the carrier of K
the addF of K . ((mA . h),(b . QQ)) is Element of the carrier of K
[(mA . h),(b . QQ)] is set
{(mA . h),(b . QQ)} is non empty finite set
{{(mA . h),(b . QQ)},{(mA . h)}} is non empty finite V52() set
the addF of K . [(mA . h),(b . QQ)] is set
((mA . h) + (b . QQ)) + (0. K) is Element of the carrier of K
the addF of K . (((mA . h) + (b . QQ)),(0. K)) is Element of the carrier of K
[((mA . h) + (b . QQ)),(0. K)] is set
{((mA . h) + (b . QQ)),(0. K)} is non empty finite set
{((mA . h) + (b . QQ))} is non empty trivial finite 1 -element set
{{((mA . h) + (b . QQ)),(0. K)},{((mA . h) + (b . QQ))}} is non empty finite V52() set
the addF of K . [((mA . h) + (b . QQ)),(0. K)] is set
(0. K) + (0. K) is Element of the carrier of K
the addF of K . ((0. K),(0. K)) is Element of the carrier of K
[(0. K),(0. K)] is set
{(0. K),(0. K)} is non empty finite set
{(0. K)} is non empty trivial finite 1 -element set
{{(0. K),(0. K)},{(0. K)}} is non empty finite V52() set
the addF of K . [(0. K),(0. K)] is set
(mA . i) + (b . PM) is Element of the carrier of K
the addF of K . ((mA . i),(b . PM)) is Element of the carrier of K
[(mA . i),(b . PM)] is set
{(mA . i),(b . PM)} is non empty finite set
{(mA . i)} is non empty trivial finite 1 -element set
{{(mA . i),(b . PM)},{(mA . i)}} is non empty finite V52() set
the addF of K . [(mA . i),(b . PM)] is set
(mA . i) + (b . PM) is Element of the carrier of K
the addF of K . ((mA . i),(b . PM)) is Element of the carrier of K
[(mA . i),(b . PM)] is set
{(mA . i),(b . PM)} is non empty finite set
{(mA . i)} is non empty trivial finite 1 -element set
{{(mA . i),(b . PM)},{(mA . i)}} is non empty finite V52() set
the addF of K . [(mA . i),(b . PM)] is set
rng F is finite set
F .: Gs is finite set
PM is finite Element of Fin (Permutations n)
card PM is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
F .: PM is finite set
i is finite Element of Fin (Permutations n)
b . PM is Element of the carrier of K
mA . i is Element of the carrier of K
(b . PM) + (mA . i) is Element of the carrier of K
the addF of K . ((b . PM),(mA . i)) is Element of the carrier of K
[(b . PM),(mA . i)] is set
{(b . PM),(mA . i)} is non empty finite set
{(b . PM)} is non empty trivial finite 1 -element set
{{(b . PM),(mA . i)},{(b . PM)}} is non empty finite V52() set
the addF of K . [(b . PM),(mA . i)] is set
F .: {} is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
card Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
PM is finite Element of Fin (Permutations n)
card PM is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
F .: PM is finite set
i is finite Element of Fin (Permutations n)
b . PM is Element of the carrier of K
mA . i is Element of the carrier of K
(b . PM) + (mA . i) is Element of the carrier of K
the addF of K . ((b . PM),(mA . i)) is Element of the carrier of K
[(b . PM),(mA . i)] is set
{(b . PM),(mA . i)} is non empty finite set
{(b . PM)} is non empty trivial finite 1 -element set
{{(b . PM),(mA . i)},{(b . PM)}} is non empty finite V52() set
the addF of K . [(b . PM),(mA . i)] is set
( the addF of K $$ (Gs,(Path_product SUM1))) + ( the addF of K $$ (B9,(Path_product SUM1))) is Element of the carrier of K
the addF of K . (( the addF of K $$ (Gs,(Path_product SUM1))),( the addF of K $$ (B9,(Path_product SUM1)))) is Element of the carrier of K
[( the addF of K $$ (Gs,(Path_product SUM1))),( the addF of K $$ (B9,(Path_product SUM1)))] is set
{( the addF of K $$ (Gs,(Path_product SUM1))),( the addF of K $$ (B9,(Path_product SUM1)))} is non empty finite set
{( the addF of K $$ (Gs,(Path_product SUM1)))} is non empty trivial finite 1 -element set
{{( the addF of K $$ (Gs,(Path_product SUM1))),( the addF of K $$ (B9,(Path_product SUM1)))},{( the addF of K $$ (Gs,(Path_product SUM1)))}} is non empty finite V52() set
the addF of K . [( the addF of K $$ (Gs,(Path_product SUM1))),( the addF of K $$ (B9,(Path_product SUM1)))] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is V70(K) Element of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Line (A,P) is Relation-like NAT -defined the carrier of K -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of K
width A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width A) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width A } is set
(B,n,n, the carrier of K,A,(Line (A,P))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (B,n,n, the carrier of K,A,(Line (A,P))) is Element of the carrier of K
Permutations n is non empty permutational set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product (B,n,n, the carrier of K,A,(Line (A,P))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (B,n,n, the carrier of K,A,(Line (A,P))))) is Element of the carrier of K
len (Line (A,P)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Line ((B,n,n, the carrier of K,A,(Line (A,P))),B) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (B,n,n, the carrier of K,A,(Line (A,P))) -element FinSequence-like FinSubsequence-like Element of (width (B,n,n, the carrier of K,A,(Line (A,P)))) -tuples_on the carrier of K
width (B,n,n, the carrier of K,A,(Line (A,P))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width (B,n,n, the carrier of K,A,(Line (A,P)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width (B,n,n, the carrier of K,A,(Line (A,P))) } is set
Line ((B,n,n, the carrier of K,A,(Line (A,P))),P) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (B,n,n, the carrier of K,A,(Line (A,P))) -element FinSequence-like FinSubsequence-like Element of (width (B,n,n, the carrier of K,A,(Line (A,P)))) -tuples_on the carrier of K
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is V70(K) Element of the carrier of K
A is Element of the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
width B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Line (B,KK) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
(width B) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width B } is set
A * (Line (B,KK)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
A multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
bool [: the carrier of K, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,A,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(A multfield) * (Line (B,KK)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(P,n,n, the carrier of K,B,(A * (Line (B,KK)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))) is Element of the carrier of K
Permutations n is non empty permutational set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product (P,n,n, the carrier of K,B,(A * (Line (B,KK)))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))) is Element of the carrier of K
len (Line (B,KK)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(P,n,n, the carrier of K,B,(Line (B,KK))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (P,n,n, the carrier of K,B,(Line (B,KK))) is Element of the carrier of K
Path_product (P,n,n, the carrier of K,B,(Line (B,KK))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (P,n,n, the carrier of K,B,(Line (B,KK))))) is Element of the carrier of K
A * (Det (P,n,n, the carrier of K,B,(Line (B,KK)))) is Element of the carrier of K
the multF of K . (A,(Det (P,n,n, the carrier of K,B,(Line (B,KK))))) is Element of the carrier of K
[A,(Det (P,n,n, the carrier of K,B,(Line (B,KK))))] is set
{A,(Det (P,n,n, the carrier of K,B,(Line (B,KK))))} is non empty finite set
{A} is non empty trivial finite 1 -element set
{{A,(Det (P,n,n, the carrier of K,B,(Line (B,KK))))},{A}} is non empty finite V52() set
the multF of K . [A,(Det (P,n,n, the carrier of K,B,(Line (B,KK))))] is set
A * (0. K) is Element of the carrier of K
the multF of K . (A,(0. K)) is Element of the carrier of K
[A,(0. K)] is set
{A,(0. K)} is non empty finite set
{{A,(0. K)},{A}} is non empty finite V52() set
the multF of K . [A,(0. K)] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is Element of the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det B is Element of the carrier of K
Permutations n is non empty permutational set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product B is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product B)) is Element of the carrier of K
width B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Line (B,P) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
(width B) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width B } is set
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Line (B,KK) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
A * (Line (B,KK)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
A multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,A,(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(A multfield) * (Line (B,KK)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(Line (B,P)) + (A * (Line (B,KK))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
the addF of K .: ((Line (B,P)),(A * (Line (B,KK)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(P,n,n, the carrier of K,B,((Line (B,P)) + (A * (Line (B,KK))))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (P,n,n, the carrier of K,B,((Line (B,P)) + (A * (Line (B,KK))))) is Element of the carrier of K
Path_product (P,n,n, the carrier of K,B,((Line (B,P)) + (A * (Line (B,KK))))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (P,n,n, the carrier of K,B,((Line (B,P)) + (A * (Line (B,KK))))))) is Element of the carrier of K
len (Line (B,KK)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len (A * (Line (B,KK))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len (Line (B,P)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(P,n,n, the carrier of K,B,(Line (B,P))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (P,n,n, the carrier of K,B,(Line (B,P))) is Element of the carrier of K
Path_product (P,n,n, the carrier of K,B,(Line (B,P))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (P,n,n, the carrier of K,B,(Line (B,P))))) is Element of the carrier of K
(P,n,n, the carrier of K,B,(A * (Line (B,KK)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))) is Element of the carrier of K
Path_product (P,n,n, the carrier of K,B,(A * (Line (B,KK)))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))) is Element of the carrier of K
(Det (P,n,n, the carrier of K,B,(Line (B,P)))) + (Det (P,n,n, the carrier of K,B,(A * (Line (B,KK))))) is Element of the carrier of K
the addF of K . ((Det (P,n,n, the carrier of K,B,(Line (B,P)))),(Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))) is Element of the carrier of K
[(Det (P,n,n, the carrier of K,B,(Line (B,P)))),(Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))] is set
{(Det (P,n,n, the carrier of K,B,(Line (B,P)))),(Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))} is non empty finite set
{(Det (P,n,n, the carrier of K,B,(Line (B,P))))} is non empty trivial finite 1 -element set
{{(Det (P,n,n, the carrier of K,B,(Line (B,P)))),(Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))},{(Det (P,n,n, the carrier of K,B,(Line (B,P))))}} is non empty finite V52() set
the addF of K . [(Det (P,n,n, the carrier of K,B,(Line (B,P)))),(Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))] is set
(Det B) + (Det (P,n,n, the carrier of K,B,(A * (Line (B,KK))))) is Element of the carrier of K
the addF of K . ((Det B),(Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))) is Element of the carrier of K
[(Det B),(Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))] is set
{(Det B),(Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))} is non empty finite set
{(Det B)} is non empty trivial finite 1 -element set
{{(Det B),(Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))},{(Det B)}} is non empty finite V52() set
the addF of K . [(Det B),(Det (P,n,n, the carrier of K,B,(A * (Line (B,KK)))))] is set
0. K is V70(K) Element of the carrier of K
(Det B) + (0. K) is Element of the carrier of K
the addF of K . ((Det B),(0. K)) is Element of the carrier of K
[(Det B),(0. K)] is set
{(Det B),(0. K)} is non empty finite set
{{(Det B),(0. K)},{(Det B)}} is non empty finite V52() set
the addF of K . [(Det B),(0. K)] is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
Permutations n is non empty permutational set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is V70(K) Element of the carrier of K
A is Relation-like Seg n -defined Seg n -valued Function-like total quasi_total finite Element of bool [:(Seg n),(Seg n):]
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
(n,n, the carrier of K,A,B) is Relation-like NAT -defined Seg n -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (n,n, the carrier of K,A,B) is Element of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product (n,n, the carrier of K,A,B) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (n,n, the carrier of K,A,B))) is Element of the carrier of K
card (Seg n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom A is finite set
P is set
KK is set
A . P is set
A . KK is set
mm is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Line ((n,n, the carrier of K,A,B),mm) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (n,n, the carrier of K,A,B) -element FinSequence-like FinSubsequence-like Element of (width (n,n, the carrier of K,A,B)) -tuples_on the carrier of K
width (n,n, the carrier of K,A,B) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width (n,n, the carrier of K,A,B)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width (n,n, the carrier of K,A,B) } is set
A . mm is set
B . (A . mm) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Line ((n,n, the carrier of K,A,B),aa) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (n,n, the carrier of K,A,B) -element FinSequence-like FinSubsequence-like Element of (width (n,n, the carrier of K,A,B)) -tuples_on the carrier of K
n is non empty addLoopStr
the carrier of n is non empty set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
[:( the carrier of n *),( the carrier of n *):] is Relation-like non empty set
[:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):] is Relation-like non empty set
bool [:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):] is non empty cup-closed diff-closed preBoolean set
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
P is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
KK is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
P + KK is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
the addF of n .: (P,KK) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
mm is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
aa is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
AB is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
aa + AB is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n .: (aa,AB) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
A is Relation-like [:( the carrier of n *),( the carrier of n *):] -defined the carrier of n * -valued Function-like non empty total quasi_total Element of bool [:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):]
B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
P is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
A . (B,P) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[B,P] is set
{B,P} is functional non empty finite V52() set
{B} is functional non empty trivial finite V52() 1 -element set
{{B,P},{B}} is non empty finite V52() set
A . [B,P] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B + P is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
the addF of n .: (B,P) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
A is Relation-like [:( the carrier of n *),( the carrier of n *):] -defined the carrier of n * -valued Function-like non empty total quasi_total Element of bool [:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):]
B is Relation-like [:( the carrier of n *),( the carrier of n *):] -defined the carrier of n * -valued Function-like non empty total quasi_total Element of bool [:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):]
P is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
KK is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
A . (P,KK) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[P,KK] is set
{P,KK} is functional non empty finite V52() set
{P} is functional non empty trivial finite V52() 1 -element set
{{P,KK},{P}} is non empty finite V52() set
A . [P,KK] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
P + KK is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
the addF of n .: (P,KK) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
B . (P,KK) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
B . [P,KK] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
n is non empty addLoopStr
the carrier of n is non empty set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
K is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
K + A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
the addF of n .: (K,A) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom (K + A) is finite Element of bool NAT
dom K is finite Element of bool NAT
dom A is finite Element of bool NAT
(dom K) /\ (dom A) is finite Element of bool NAT
rng A is finite set
rng K is finite set
[:(rng K),(rng A):] is Relation-like finite set
dom the addF of n is Relation-like non empty set
n is non empty Abelian addLoopStr
the carrier of n is non empty set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
(n) is Relation-like [:( the carrier of n *),( the carrier of n *):] -defined the carrier of n * -valued Function-like non empty total quasi_total Element of bool [:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):]
[:( the carrier of n *),( the carrier of n *):] is Relation-like non empty set
[:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):] is Relation-like non empty set
bool [:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):] is non empty cup-closed diff-closed preBoolean set
A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
(n) . (A,B) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[A,B] is set
{A,B} is functional non empty finite V52() set
{A} is functional non empty trivial finite V52() 1 -element set
{{A,B},{A}} is non empty finite V52() set
(n) . [A,B] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(n) . (B,A) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[B,A] is set
{B,A} is functional non empty finite V52() set
{B} is functional non empty trivial finite V52() 1 -element set
{{B,A},{B}} is non empty finite V52() set
(n) . [B,A] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng B is finite set
A + B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
the addF of n .: (A,B) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom (A + B) is finite Element of bool NAT
dom A is finite Element of bool NAT
dom B is finite Element of bool NAT
(dom A) /\ (dom B) is finite Element of bool NAT
B + A is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n .: (B,A) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom (B + A) is finite Element of bool NAT
(dom B) /\ (dom A) is finite Element of bool NAT
rng A is finite set
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B . P is set
A . P is set
(A + B) . P is set
KK is Element of the carrier of n
mm is Element of the carrier of n
KK + mm is Element of the carrier of n
the addF of n . (KK,mm) is Element of the carrier of n
[KK,mm] is set
{KK,mm} is non empty finite set
{KK} is non empty trivial finite 1 -element set
{{KK,mm},{KK}} is non empty finite V52() set
the addF of n . [KK,mm] is set
(B + A) . P is set
(n) . (A,B) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
n is non empty add-associative addLoopStr
the carrier of n is non empty set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
(n) is Relation-like [:( the carrier of n *),( the carrier of n *):] -defined the carrier of n * -valued Function-like non empty total quasi_total Element of bool [:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):]
[:( the carrier of n *),( the carrier of n *):] is Relation-like non empty set
[:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):] is Relation-like non empty set
bool [:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):] is non empty cup-closed diff-closed preBoolean set
B is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
P is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
KK is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
(n) . (P,KK) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[P,KK] is set
{P,KK} is functional non empty finite V52() set
{P} is functional non empty trivial finite V52() 1 -element set
{{P,KK},{P}} is non empty finite V52() set
(n) . [P,KK] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(n) . (B,((n) . (P,KK))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[B,((n) . (P,KK))] is set
{B,((n) . (P,KK))} is functional non empty finite V52() set
{B} is functional non empty trivial finite V52() 1 -element set
{{B,((n) . (P,KK))},{B}} is non empty finite V52() set
(n) . [B,((n) . (P,KK))] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(n) . (B,P) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[B,P] is set
{B,P} is functional non empty finite V52() set
{{B,P},{B}} is non empty finite V52() set
(n) . [B,P] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(n) . (((n) . (B,P)),KK) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[((n) . (B,P)),KK] is set
{((n) . (B,P)),KK} is functional non empty finite V52() set
{((n) . (B,P))} is functional non empty trivial finite V52() 1 -element set
{{((n) . (B,P)),KK},{((n) . (B,P))}} is non empty finite V52() set
(n) . [((n) . (B,P)),KK] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B + P is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
the addF of n .: (B,P) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
P + KK is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n .: (P,KK) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
rng B is finite set
rng P is finite set
mm is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
rng mm is finite set
rng KK is finite set
aa is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
rng aa is finite set
dom mm is finite Element of bool NAT
dom B is finite Element of bool NAT
dom P is finite Element of bool NAT
(dom B) /\ (dom P) is finite Element of bool NAT
dom aa is finite Element of bool NAT
dom KK is finite Element of bool NAT
(dom P) /\ (dom KK) is finite Element of bool NAT
mm + KK is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n .: (mm,KK) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom (mm + KK) is finite Element of bool NAT
(dom mm) /\ (dom KK) is finite Element of bool NAT
B + aa is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n .: (B,aa) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom (B + aa) is finite Element of bool NAT
(dom B) /\ (dom aa) is finite Element of bool NAT
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
mm . AB is set
B . AB is set
KK . AB is set
P . AB is set
aa . AB is set
SUM1 is Element of the carrier of n
F is Element of the carrier of n
SUM1 + F is Element of the carrier of n
the addF of n . (SUM1,F) is Element of the carrier of n
[SUM1,F] is set
{SUM1,F} is non empty finite set
{SUM1} is non empty trivial finite 1 -element set
{{SUM1,F},{SUM1}} is non empty finite V52() set
the addF of n . [SUM1,F] is set
(mm + KK) . AB is set
Path is Element of the carrier of n
Gs is Element of the carrier of n
Path + Gs is Element of the carrier of n
the addF of n . (Path,Gs) is Element of the carrier of n
[Path,Gs] is set
{Path,Gs} is non empty finite set
{Path} is non empty trivial finite 1 -element set
{{Path,Gs},{Path}} is non empty finite V52() set
the addF of n . [Path,Gs] is set
F + Gs is Element of the carrier of n
the addF of n . (F,Gs) is Element of the carrier of n
[F,Gs] is set
{F,Gs} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,Gs},{F}} is non empty finite V52() set
the addF of n . [F,Gs] is set
(B + aa) . AB is set
Ga is Element of the carrier of n
SUM1 + Ga is Element of the carrier of n
the addF of n . (SUM1,Ga) is Element of the carrier of n
[SUM1,Ga] is set
{SUM1,Ga} is non empty finite set
{{SUM1,Ga},{SUM1}} is non empty finite V52() set
the addF of n . [SUM1,Ga] is set
(n) . (P,KK) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
(n) . (B,((n) . (P,KK))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[B,((n) . (P,KK))] is set
{B,((n) . (P,KK))} is functional non empty finite V52() set
{{B,((n) . (P,KK))},{B}} is non empty finite V52() set
(n) . [B,((n) . (P,KK))] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(n) . (B,aa) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[B,aa] is set
{B,aa} is functional non empty finite V52() set
{{B,aa},{B}} is non empty finite V52() set
(n) . [B,aa] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(n) . (mm,KK) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[mm,KK] is set
{mm,KK} is functional non empty finite V52() set
{mm} is functional non empty trivial finite V52() 1 -element set
{{mm,KK},{mm}} is non empty finite V52() set
(n) . [mm,KK] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(n) . (B,P) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
(n) . (((n) . (B,P)),KK) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
[((n) . (B,P)),KK] is set
{((n) . (B,P)),KK} is functional non empty finite V52() set
{((n) . (B,P))} is functional non empty trivial finite V52() 1 -element set
{{((n) . (B,P)),KK},{((n) . (B,P))}} is non empty finite V52() set
(n) . [((n) . (B,P)),KK] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
n is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
(n) is Relation-like [:( the carrier of n *),( the carrier of n *):] -defined the carrier of n * -valued Function-like non empty total quasi_total commutative associative Element of bool [:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):]
[:( the carrier of n *),( the carrier of n *):] is Relation-like non empty set
[:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):] is Relation-like non empty set
bool [:[:( the carrier of n *),( the carrier of n *):],( the carrier of n *):] is non empty cup-closed diff-closed preBoolean set
K is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of n *
width K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of n *
len A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len K) is finite len K -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len K ) } is set
K * A is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of n *
Seg (len A) is finite len A -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len A ) } is set
width A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty cup-closed diff-closed preBoolean set
the multF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Line ((K * A),SUM1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (K * A) -element FinSequence-like FinSubsequence-like Element of (width (K * A)) -tuples_on the carrier of n
width (K * A) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width (K * A)) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = width (K * A) } is set
(width A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = width A } is set
Path is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom Path is finite Element of bool NAT
dom A is finite Element of bool NAT
rng Path is finite set
F is set
Ga is set
Path . Ga is set
Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Path . Gs is set
Line (A,Gs) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
K * (SUM1,Gs) is Element of the carrier of n
(K * (SUM1,Gs)) * (Line (A,Gs)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
(K * (SUM1,Gs)) multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
bool [: the carrier of n, the carrier of n:] is non empty cup-closed diff-closed preBoolean set
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of n, the carrier of n:]
K224( the carrier of n, the carrier of n, the multF of n,(K * (SUM1,Gs)),(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
((K * (SUM1,Gs)) multfield) * (Line (A,Gs)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
aa is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
[:NAT,( the carrier of n *):] is Relation-like non empty non trivial non finite set
bool [:NAT,( the carrier of n *):] is non empty non trivial cup-closed diff-closed preBoolean non finite set
F is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n *
F . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(n) $$ F is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
Ga is Relation-like NAT -defined the carrier of n * -valued Function-like non empty total quasi_total Element of bool [:NAT,( the carrier of n *):]
Ga . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
Ga . (len F) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
AB is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg AB is finite AB -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= AB ) } is set
Line (K,SUM1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width K -element FinSequence-like FinSubsequence-like Element of (width K) -tuples_on the carrier of n
(width K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = width K } is set
Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Ga . Gs is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
Seg Gs is finite Gs -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= Gs ) } is set
Gs + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Ga . (Gs + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
Seg (Gs + 1) is non empty finite Gs + 1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= Gs + 1 ) } is set
Seg aa is finite aa -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= aa ) } is set
b is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Line (A,1) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
K * (SUM1,1) is Element of the carrier of n
(K * (SUM1,1)) * (Line (A,1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
(K * (SUM1,1)) multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
bool [: the carrier of n, the carrier of n:] is non empty cup-closed diff-closed preBoolean set
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of n, the carrier of n:]
K224( the carrier of n, the carrier of n, the multF of n,(K * (SUM1,1)),(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
((K * (SUM1,1)) multfield) * (Line (A,1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (Line (A,1)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
mA is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Col (A,mA) is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
(len A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = len A } is set
mlt ((Line (K,SUM1)),(Col (A,mA))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n .: ((Line (K,SUM1)),(Col (A,mA))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(mlt ((Line (K,SUM1)),(Col (A,mA)))) | (Seg (Gs + 1)) is Relation-like NAT -defined Seg (Gs + 1) -defined NAT -defined the carrier of n -valued Function-like finite FinSubsequence-like Element of bool [:NAT, the carrier of n:]
[:NAT, the carrier of n:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of n:] is non empty non trivial cup-closed diff-closed preBoolean non finite set
b . mA is set
dom b is finite Element of bool NAT
(Line (A,1)) . mA is set
A * (1,mA) is Element of the carrier of n
(K * (SUM1,1)) * (A * (1,mA)) is Element of the carrier of n
the multF of n . ((K * (SUM1,1)),(A * (1,mA))) is Element of the carrier of n
[(K * (SUM1,1)),(A * (1,mA))] is set
{(K * (SUM1,1)),(A * (1,mA))} is non empty finite set
{(K * (SUM1,1))} is non empty trivial finite 1 -element set
{{(K * (SUM1,1)),(A * (1,mA))},{(K * (SUM1,1))}} is non empty finite V52() set
the multF of n . [(K * (SUM1,1)),(A * (1,mA))] is set
aa -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = aa } is set
Pi is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom Pi is finite Element of bool NAT
(Seg aa) /\ (Seg 1) is finite Element of bool NAT
i is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
dom i is finite Element of bool NAT
len i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
i . 1 is set
Pi . 1 is set
<*(Pi . 1)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(Pi . 1)] is set
{1,(Pi . 1)} is non empty finite set
{{1,(Pi . 1)},{1}} is non empty finite V52() set
{[1,(Pi . 1)]} is Relation-like Function-like constant non empty trivial finite 1 -element set
the addF of n $$ i is Element of the carrier of n
(Col (A,mA)) . 1 is set
(Line (K,SUM1)) . 1 is set
dom F is finite Element of bool NAT
F . (Gs + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng F is finite set
mA is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Bb is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(n) . (Bb,mA) is set
[Bb,mA] is set
{Bb,mA} is functional non empty finite V52() set
{Bb} is functional non empty trivial finite V52() 1 -element set
{{Bb,mA},{Bb}} is non empty finite V52() set
(n) . [Bb,mA] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Bb + mA is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n .: (Bb,mA) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Gs + {} is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Line (A,(Gs + 1)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
len (Line (A,(Gs + 1))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
K * (SUM1,(Gs + 1)) is Element of the carrier of n
(K * (SUM1,(Gs + 1))) * (Line (A,(Gs + 1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
(K * (SUM1,(Gs + 1))) multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
bool [: the carrier of n, the carrier of n:] is non empty cup-closed diff-closed preBoolean set
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of n, the carrier of n:]
K224( the carrier of n, the carrier of n, the multF of n,(K * (SUM1,(Gs + 1))),(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
((K * (SUM1,(Gs + 1))) multfield) * (Line (A,(Gs + 1))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len mA is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom mA is finite Element of bool NAT
len Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom Bb is finite Element of bool NAT
dom (Bb + mA) is finite Element of bool NAT
(Seg AB) /\ (Seg AB) is finite Element of bool NAT
rng Bb is finite set
i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Col (A,i) is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
(len A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = len A } is set
mlt ((Line (K,SUM1)),(Col (A,i))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n .: ((Line (K,SUM1)),(Col (A,i))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(mlt ((Line (K,SUM1)),(Col (A,i)))) | (Seg (Gs + 1)) is Relation-like NAT -defined Seg (Gs + 1) -defined NAT -defined the carrier of n -valued Function-like finite FinSubsequence-like Element of bool [:NAT, the carrier of n:]
[:NAT, the carrier of n:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of n:] is non empty non trivial cup-closed diff-closed preBoolean non finite set
b . i is set
(Line (A,(Gs + 1))) . i is set
A * ((Gs + 1),i) is Element of the carrier of n
mA . i is set
(K * (SUM1,(Gs + 1))) * (A * ((Gs + 1),i)) is Element of the carrier of n
the multF of n . ((K * (SUM1,(Gs + 1))),(A * ((Gs + 1),i))) is Element of the carrier of n
[(K * (SUM1,(Gs + 1))),(A * ((Gs + 1),i))] is set
{(K * (SUM1,(Gs + 1))),(A * ((Gs + 1),i))} is non empty finite set
{(K * (SUM1,(Gs + 1)))} is non empty trivial finite 1 -element set
{{(K * (SUM1,(Gs + 1))),(A * ((Gs + 1),i))},{(K * (SUM1,(Gs + 1)))}} is non empty finite V52() set
the multF of n . [(K * (SUM1,(Gs + 1))),(A * ((Gs + 1),i))] is set
(mlt ((Line (K,SUM1)),(Col (A,i)))) | (Seg Gs) is Relation-like NAT -defined Seg Gs -defined NAT -defined the carrier of n -valued Function-like finite FinSubsequence-like Element of bool [:NAT, the carrier of n:]
Bb . i is set
SF is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n $$ SF is Element of the carrier of n
QQ is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n $$ QQ is Element of the carrier of n
aa -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = aa } is set
h is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len h is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom h is finite Element of bool NAT
(Line (K,SUM1)) . (Gs + 1) is set
(Col (A,i)) . (Gs + 1) is set
h . (Gs + 1) is set
<*(h . (Gs + 1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(h . (Gs + 1))] is set
{1,(h . (Gs + 1))} is non empty finite set
{{1,(h . (Gs + 1))},{1}} is non empty finite V52() set
{[1,(h . (Gs + 1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
SF ^ <*(h . (Gs + 1))*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
the addF of n . ((Bb . i),((K * (SUM1,(Gs + 1))) * (A * ((Gs + 1),i)))) is set
[(Bb . i),((K * (SUM1,(Gs + 1))) * (A * ((Gs + 1),i)))] is set
{(Bb . i),((K * (SUM1,(Gs + 1))) * (A * ((Gs + 1),i)))} is non empty finite set
{(Bb . i)} is non empty trivial finite 1 -element set
{{(Bb . i),((K * (SUM1,(Gs + 1))) * (A * ((Gs + 1),i)))},{(Bb . i)}} is non empty finite V52() set
the addF of n . [(Bb . i),((K * (SUM1,(Gs + 1))) * (A * ((Gs + 1),i)))] is set
Mh is Element of the carrier of n
Pi is Element of the carrier of n
Mh + Pi is Element of the carrier of n
the addF of n . (Mh,Pi) is Element of the carrier of n
[Mh,Pi] is set
{Mh,Pi} is non empty finite set
{Mh} is non empty trivial finite 1 -element set
{{Mh,Pi},{Mh}} is non empty finite V52() set
the addF of n . [Mh,Pi] is set
len (Line ((K * A),SUM1)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Ga . aa is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
Ga . {} is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n *
b is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
mA is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Col (A,mA) is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
(len A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = len A } is set
mlt ((Line (K,SUM1)),(Col (A,mA))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n .: ((Line (K,SUM1)),(Col (A,mA))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(mlt ((Line (K,SUM1)),(Col (A,mA)))) | (Seg {}) is Relation-like non-empty empty-yielding NAT -defined Seg {} -defined NAT -defined the carrier of n -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of bool [:NAT, the carrier of n:]
[:NAT, the carrier of n:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of n:] is non empty non trivial cup-closed diff-closed preBoolean non finite set
b . mA is set
B9 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(Line ((K * A),SUM1)) . Bb is set
B9 . Bb is set
Col (A,Bb) is Relation-like NAT -defined the carrier of n -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
(len A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = len A } is set
the multF of n .: ((Line (K,SUM1)),(Col (A,Bb))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
aa -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = aa } is set
mlt ((Line (K,SUM1)),(Col (A,Bb))) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (mlt ((Line (K,SUM1)),(Col (A,Bb)))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom (mlt ((Line (K,SUM1)),(Col (A,Bb)))) is finite Element of bool NAT
Seg aa is finite aa -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= aa ) } is set
(mlt ((Line (K,SUM1)),(Col (A,Bb)))) | (Seg aa) is Relation-like NAT -defined Seg aa -defined NAT -defined the carrier of n -valued Function-like finite FinSubsequence-like Element of bool [:NAT, the carrier of n:]
[:NAT, the carrier of n:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of n:] is non empty non trivial cup-closed diff-closed preBoolean non finite set
(Line (K,SUM1)) "*" (Col (A,Bb)) is Element of the carrier of n
Sum (mlt ((Line (K,SUM1)),(Col (A,Bb)))) is Element of the carrier of n
the addF of n $$ (mlt ((Line (K,SUM1)),(Col (A,Bb)))) is Element of the carrier of n
i is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the addF of n $$ i is Element of the carrier of n
len (K * A) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Indices (K * A) is set
dom (K * A) is finite Element of bool NAT
Seg (width (K * A)) is finite width (K * A) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= width (K * A) ) } is set
[:(dom (K * A)),(Seg (width (K * A))):] is Relation-like finite set
[:(Seg (len K)),(Seg AB):] is Relation-like finite set
[SUM1,Bb] is set
{SUM1,Bb} is non empty finite V52() set
{SUM1} is non empty trivial finite V52() 1 -element set
{{SUM1,Bb},{SUM1}} is non empty finite V52() set
(K * A) * (SUM1,Bb) is Element of the carrier of n
len B9 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
F . b is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (A,b) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
K * (SUM1,b) is Element of the carrier of n
(K * (SUM1,b)) * (Line (A,b)) is Relation-like NAT -defined the carrier of n -valued Function-like finite width A -element FinSequence-like FinSubsequence-like Element of (width A) -tuples_on the carrier of n
(K * (SUM1,b)) multfield is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
bool [: the carrier of n, the carrier of n:] is non empty cup-closed diff-closed preBoolean set
id the carrier of n is Relation-like the carrier of n -defined the carrier of n -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of n, the carrier of n:]
K224( the carrier of n, the carrier of n, the multF of n,(K * (SUM1,b)),(id the carrier of n)) is Relation-like the carrier of n -defined the carrier of n -valued Function-like non empty total quasi_total Element of bool [: the carrier of n, the carrier of n:]
((K * (SUM1,b)) multfield) * (Line (A,b)) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
P is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
A * B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Line ((A * B),KK) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (A * B) -element FinSequence-like FinSubsequence-like Element of (width (A * B)) -tuples_on the carrier of K
width (A * B) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width (A * B)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width (A * B) } is set
(KK,n,n, the carrier of K,P,(Line ((A * B),KK))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (KK,n,n, the carrier of K,P,(Line ((A * B),KK))) is Element of the carrier of K
Permutations n is non empty permutational set
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product (KK,n,n, the carrier of K,P,(Line ((A * B),KK))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (KK,n,n, the carrier of K,P,(Line ((A * B),KK))))) is Element of the carrier of K
(K) is Relation-like [:( the carrier of K *),( the carrier of K *):] -defined the carrier of K * -valued Function-like non empty total quasi_total commutative associative Element of bool [:[:( the carrier of K *),( the carrier of K *):],( the carrier of K *):]
[:( the carrier of K *),( the carrier of K *):] is Relation-like non empty set
[:[:( the carrier of K *),( the carrier of K *):],( the carrier of K *):] is Relation-like non empty set
bool [:[:( the carrier of K *),( the carrier of K *):],( the carrier of K *):] is non empty cup-closed diff-closed preBoolean set
len B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
SUM1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom SUM1 is finite Element of bool NAT
[:NAT, the carrier of K:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of K:] is non empty non trivial cup-closed diff-closed preBoolean non finite set
SUM1 . 1 is set
the addF of K $$ SUM1 is Element of the carrier of K
Path is Relation-like NAT -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of K:]
Path . 1 is Element of the carrier of K
Path . n is Element of the carrier of K
Seg (len A) is finite len A -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len A ) } is set
width A is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len B) is finite len B -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len B ) } is set
width B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
F is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K *
len F is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(K) $$ F is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
[:NAT,( the carrier of K *):] is Relation-like non empty non trivial non finite set
bool [:NAT,( the carrier of K *):] is non empty non trivial cup-closed diff-closed preBoolean non finite set
F . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Ga is Relation-like NAT -defined the carrier of K * -valued Function-like non empty total quasi_total Element of bool [:NAT,( the carrier of K *):]
Ga . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
Ga . n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Ga . Gs is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
Path . Gs is Element of the carrier of K
Gs + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Ga . (Gs + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
Path . (Gs + 1) is Element of the carrier of K
A * (KK,(Gs + 1)) is Element of the carrier of K
Line (B,(Gs + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
(width B) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width B } is set
Bb is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(KK,n,n, the carrier of K,P,Bb) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (KK,n,n, the carrier of K,P,Bb) is Element of the carrier of K
Path_product (KK,n,n, the carrier of K,P,Bb) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (KK,n,n, the carrier of K,P,Bb))) is Element of the carrier of K
F . (Gs + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(A * (KK,(Gs + 1))) * (Line (B,(Gs + 1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
(A * (KK,(Gs + 1))) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,(A * (KK,(Gs + 1))),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
((A * (KK,(Gs + 1))) multfield) * (Line (B,(Gs + 1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len (Line (B,(Gs + 1))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
SUM1 . (Gs + 1) is set
A * (KK,1) is Element of the carrier of K
Line (B,1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
(KK,n,n, the carrier of K,P,(Line (B,1))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (KK,n,n, the carrier of K,P,(Line (B,1))) is Element of the carrier of K
Path_product (KK,n,n, the carrier of K,P,(Line (B,1))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (KK,n,n, the carrier of K,P,(Line (B,1))))) is Element of the carrier of K
(A * (KK,1)) * (Det (KK,n,n, the carrier of K,P,(Line (B,1)))) is Element of the carrier of K
the multF of K . ((A * (KK,1)),(Det (KK,n,n, the carrier of K,P,(Line (B,1))))) is Element of the carrier of K
[(A * (KK,1)),(Det (KK,n,n, the carrier of K,P,(Line (B,1))))] is set
{(A * (KK,1)),(Det (KK,n,n, the carrier of K,P,(Line (B,1))))} is non empty finite set
{(A * (KK,1))} is non empty trivial finite 1 -element set
{{(A * (KK,1)),(Det (KK,n,n, the carrier of K,P,(Line (B,1))))},{(A * (KK,1))}} is non empty finite V52() set
the multF of K . [(A * (KK,1)),(Det (KK,n,n, the carrier of K,P,(Line (B,1))))] is set
dom F is finite Element of bool NAT
F . (Gs + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng F is finite set
Gs + {} is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
i is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
(KK,n,n, the carrier of K,P,i) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (KK,n,n, the carrier of K,P,i) is Element of the carrier of K
Path_product (KK,n,n, the carrier of K,P,i) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (KK,n,n, the carrier of K,P,i))) is Element of the carrier of K
len i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
SUM1 . (Gs + 1) is set
the addF of K . ((Path . Gs),(SUM1 . (Gs + 1))) is set
[(Path . Gs),(SUM1 . (Gs + 1))] is set
{(Path . Gs),(SUM1 . (Gs + 1))} is non empty finite set
{(Path . Gs)} is non empty trivial finite 1 -element set
{{(Path . Gs),(SUM1 . (Gs + 1))},{(Path . Gs)}} is non empty finite V52() set
the addF of K . [(Path . Gs),(SUM1 . (Gs + 1))] is set
(A * (KK,(Gs + 1))) * (Line (B,(Gs + 1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
(A * (KK,(Gs + 1))) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,(A * (KK,(Gs + 1))),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
((A * (KK,(Gs + 1))) multfield) * (Line (B,(Gs + 1))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
PM is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
(K) . (i,PM) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
[i,PM] is set
{i,PM} is functional non empty finite V52() set
{i} is functional non empty trivial finite V52() 1 -element set
{{i,PM},{i}} is non empty finite V52() set
(K) . [i,PM] is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
i + ((A * (KK,(Gs + 1))) * (Line (B,(Gs + 1)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the addF of K .: (i,((A * (KK,(Gs + 1))) * (Line (B,(Gs + 1))))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len (Line (B,(Gs + 1))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len ((A * (KK,(Gs + 1))) * (Line (B,(Gs + 1)))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(KK,n,n, the carrier of K,P,((A * (KK,(Gs + 1))) * (Line (B,(Gs + 1))))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (KK,n,n, the carrier of K,P,((A * (KK,(Gs + 1))) * (Line (B,(Gs + 1))))) is Element of the carrier of K
Path_product (KK,n,n, the carrier of K,P,((A * (KK,(Gs + 1))) * (Line (B,(Gs + 1))))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (KK,n,n, the carrier of K,P,((A * (KK,(Gs + 1))) * (Line (B,(Gs + 1))))))) is Element of the carrier of K
(KK,n,n, the carrier of K,P,(Line (B,(Gs + 1)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1)))) is Element of the carrier of K
Path_product (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1)))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1)))))) is Element of the carrier of K
(A * (KK,(Gs + 1))) * (Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1))))) is Element of the carrier of K
the multF of K . ((A * (KK,(Gs + 1))),(Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1)))))) is Element of the carrier of K
[(A * (KK,(Gs + 1))),(Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1)))))] is set
{(A * (KK,(Gs + 1))),(Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1)))))} is non empty finite set
{(A * (KK,(Gs + 1)))} is non empty trivial finite 1 -element set
{{(A * (KK,(Gs + 1))),(Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1)))))},{(A * (KK,(Gs + 1)))}} is non empty finite V52() set
the multF of K . [(A * (KK,(Gs + 1))),(Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1)))))] is set
(Det (KK,n,n, the carrier of K,P,i)) + ((A * (KK,(Gs + 1))) * (Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1)))))) is Element of the carrier of K
the addF of K . ((Det (KK,n,n, the carrier of K,P,i)),((A * (KK,(Gs + 1))) * (Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1))))))) is Element of the carrier of K
[(Det (KK,n,n, the carrier of K,P,i)),((A * (KK,(Gs + 1))) * (Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1))))))] is set
{(Det (KK,n,n, the carrier of K,P,i)),((A * (KK,(Gs + 1))) * (Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1))))))} is non empty finite set
{(Det (KK,n,n, the carrier of K,P,i))} is non empty trivial finite 1 -element set
{{(Det (KK,n,n, the carrier of K,P,i)),((A * (KK,(Gs + 1))) * (Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1))))))},{(Det (KK,n,n, the carrier of K,P,i))}} is non empty finite V52() set
the addF of K . [(Det (KK,n,n, the carrier of K,P,i)),((A * (KK,(Gs + 1))) * (Det (KK,n,n, the carrier of K,P,(Line (B,(Gs + 1))))))] is set
Ga . {} is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
Path . {} is Element of the carrier of K
Gs is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(KK,n,n, the carrier of K,P,Gs) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (KK,n,n, the carrier of K,P,Gs) is Element of the carrier of K
Path_product (KK,n,n, the carrier of K,P,Gs) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (KK,n,n, the carrier of K,P,Gs))) is Element of the carrier of K
Ga . (len F) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K *
Path . (len F) is Element of the carrier of K
Gs is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(KK,n,n, the carrier of K,P,Gs) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (KK,n,n, the carrier of K,P,Gs) is Element of the carrier of K
Path_product (KK,n,n, the carrier of K,P,Gs) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (KK,n,n, the carrier of K,P,Gs))) is Element of the carrier of K
Gs is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
SUM1 . Gs is set
A * (KK,Gs) is Element of the carrier of K
Line (B,Gs) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
(width B) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width B } is set
(KK,n,n, the carrier of K,P,(Line (B,Gs))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (KK,n,n, the carrier of K,P,(Line (B,Gs))) is Element of the carrier of K
Path_product (KK,n,n, the carrier of K,P,(Line (B,Gs))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (KK,n,n, the carrier of K,P,(Line (B,Gs))))) is Element of the carrier of K
(A * (KK,Gs)) * (Det (KK,n,n, the carrier of K,P,(Line (B,Gs)))) is Element of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the multF of K . ((A * (KK,Gs)),(Det (KK,n,n, the carrier of K,P,(Line (B,Gs))))) is Element of the carrier of K
[(A * (KK,Gs)),(Det (KK,n,n, the carrier of K,P,(Line (B,Gs))))] is set
{(A * (KK,Gs)),(Det (KK,n,n, the carrier of K,P,(Line (B,Gs))))} is non empty finite set
{(A * (KK,Gs))} is non empty trivial finite 1 -element set
{{(A * (KK,Gs)),(Det (KK,n,n, the carrier of K,P,(Line (B,Gs))))},{(A * (KK,Gs))}} is non empty finite V52() set
the multF of K . [(A * (KK,Gs)),(Det (KK,n,n, the carrier of K,P,(Line (B,Gs))))] is set
n is set
K is non empty set
Funcs (n,K) is functional non empty FUNCTION_DOMAIN of n,K
[:(Funcs (n,K)),K:] is Relation-like non empty set
[:n,K:] is Relation-like set
bool [:n,K:] is non empty cup-closed diff-closed preBoolean set
A is set
{A} is non empty trivial finite 1 -element set
n \/ {A} is non empty set
Funcs ((n \/ {A}),K) is functional non empty FUNCTION_DOMAIN of n \/ {A},K
[:[:(Funcs (n,K)),K:],(Funcs ((n \/ {A}),K)):] is Relation-like non empty set
bool [:[:(Funcs (n,K)),K:],(Funcs ((n \/ {A}),K)):] is non empty cup-closed diff-closed preBoolean set
[:(n \/ {A}),K:] is Relation-like non empty set
bool [:(n \/ {A}),K:] is non empty cup-closed diff-closed preBoolean set
mm is Element of [:(Funcs (n,K)),K:]
aa is set
AB is set
[aa,AB] is set
{aa,AB} is non empty finite set
{aa} is non empty trivial finite 1 -element set
{{aa,AB},{aa}} is non empty finite V52() set
{AB} is non empty trivial finite 1 -element set
K \/ {AB} is non empty set
SUM1 is Relation-like n -defined K -valued Function-like total quasi_total Element of bool [:n,K:]
Path is Relation-like n \/ {A} -defined K -valued Function-like non empty total quasi_total Element of bool [:(n \/ {A}),K:]
Path | n is Relation-like n -defined n \/ {A} -defined K -valued Function-like Element of bool [:(n \/ {A}),K:]
Path . A is set
F is Relation-like n \/ {A} -defined K -valued Function-like total quasi_total Element of Funcs ((n \/ {A}),K)
Ga is Relation-like n -defined K -valued Function-like total quasi_total Element of bool [:n,K:]
Gs is Relation-like n \/ {A} -defined K -valued Function-like non empty total quasi_total Element of bool [:(n \/ {A}),K:]
Gs . A is set
Gs | n is Relation-like n -defined n \/ {A} -defined K -valued Function-like Element of bool [:(n \/ {A}),K:]
B9 is set
[Ga,B9] is set
{Ga,B9} is non empty finite set
{Ga} is functional non empty trivial finite 1 -element set
{{Ga,B9},{Ga}} is non empty finite V52() set
b is set
dom SUM1 is set
dom Ga is set
Gs . b is set
Ga . b is set
Path . b is set
SUM1 . b is set
mm is Relation-like [:(Funcs (n,K)),K:] -defined Funcs ((n \/ {A}),K) -valued Function-like non empty total quasi_total Element of bool [:[:(Funcs (n,K)),K:],(Funcs ((n \/ {A}),K)):]
aa is set
AB is set
mm . aa is Relation-like Function-like set
mm . AB is Relation-like Function-like set
SUM1 is set
Path is set
[SUM1,Path] is set
{SUM1,Path} is non empty finite set
{SUM1} is non empty trivial finite 1 -element set
{{SUM1,Path},{SUM1}} is non empty finite V52() set
F is set
Ga is set
[F,Ga] is set
{F,Ga} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,Ga},{F}} is non empty finite V52() set
{Path} is non empty trivial finite 1 -element set
K \/ {Path} is non empty set
B9 is Relation-like n -defined K -valued Function-like total quasi_total Element of bool [:n,K:]
b is Relation-like n \/ {A} -defined K -valued Function-like non empty total quasi_total Element of bool [:(n \/ {A}),K:]
b | n is Relation-like n -defined n \/ {A} -defined K -valued Function-like Element of bool [:(n \/ {A}),K:]
b . A is set
{Ga} is non empty trivial finite 1 -element set
K \/ {Ga} is non empty set
Gs is Relation-like n -defined K -valued Function-like total quasi_total Element of bool [:n,K:]
mA is Relation-like n \/ {A} -defined K -valued Function-like non empty total quasi_total Element of bool [:(n \/ {A}),K:]
mA | n is Relation-like n -defined n \/ {A} -defined K -valued Function-like Element of bool [:(n \/ {A}),K:]
mA . A is set
rng mm is non empty set
aa is set
AB is Relation-like n \/ {A} -defined K -valued Function-like non empty total quasi_total Element of bool [:(n \/ {A}),K:]
dom AB is non empty set
AB | n is Relation-like n -defined n \/ {A} -defined K -valued Function-like Element of bool [:(n \/ {A}),K:]
dom (AB | n) is set
rng (AB | n) is set
SUM1 is Relation-like n -defined K -valued Function-like total quasi_total Element of bool [:n,K:]
AB . A is set
rng AB is non empty set
[SUM1,(AB . A)] is set
{SUM1,(AB . A)} is non empty finite set
{SUM1} is functional non empty trivial finite 1 -element set
{{SUM1,(AB . A)},{SUM1}} is non empty finite V52() set
dom mm is Relation-like non empty set
mm . [SUM1,(AB . A)] is Relation-like Function-like set
aa is Relation-like n -defined K -valued Function-like total quasi_total Element of bool [:n,K:]
AB is Relation-like n \/ {A} -defined K -valued Function-like non empty total quasi_total Element of bool [:(n \/ {A}),K:]
AB | n is Relation-like n -defined n \/ {A} -defined K -valued Function-like Element of bool [:(n \/ {A}),K:]
AB . A is set
mm . (aa,(AB . A)) is set
[aa,(AB . A)] is set
{aa,(AB . A)} is non empty finite set
{aa} is functional non empty trivial finite 1 -element set
{{aa,(AB . A)},{aa}} is non empty finite V52() set
mm . [aa,(AB . A)] is Relation-like Function-like set
dom AB is non empty set
rng AB is non empty set
n is non empty set
[:n,n:] is Relation-like non empty set
[:[:n,n:],n:] is Relation-like non empty set
bool [:[:n,n:],n:] is non empty cup-closed diff-closed preBoolean set
K is finite set
A is non empty finite set
Funcs (K,A) is functional non empty FUNCTION_DOMAIN of K,A
[:(Funcs (K,A)),n:] is Relation-like non empty set
bool [:(Funcs (K,A)),n:] is non empty cup-closed diff-closed preBoolean set
[:K,A:] is Relation-like finite set
bool [:K,A:] is non empty cup-closed diff-closed preBoolean finite V52() set
FinOmega (Funcs (K,A)) is finite Element of Fin (Funcs (K,A))
Fin (Funcs (K,A)) is non empty cup-closed diff-closed preBoolean set
B is set
{B} is non empty trivial finite 1 -element set
K \/ {B} is non empty finite set
Funcs ((K \/ {B}),A) is functional non empty FUNCTION_DOMAIN of K \/ {B},A
[:(Funcs ((K \/ {B}),A)),n:] is Relation-like non empty set
bool [:(Funcs ((K \/ {B}),A)),n:] is non empty cup-closed diff-closed preBoolean set
Fin (Funcs ((K \/ {B}),A)) is non empty cup-closed diff-closed preBoolean set
[:(K \/ {B}),A:] is Relation-like non empty finite set
bool [:(K \/ {B}),A:] is non empty cup-closed diff-closed preBoolean finite V52() set
FinOmega (Funcs ((K \/ {B}),A)) is finite Element of Fin (Funcs ((K \/ {B}),A))
[:(Funcs (K,A)),A:] is Relation-like non empty set
[:[:(Funcs (K,A)),A:],(Funcs ((K \/ {B}),A)):] is Relation-like non empty set
bool [:[:(Funcs (K,A)),A:],(Funcs ((K \/ {B}),A)):] is non empty cup-closed diff-closed preBoolean set
aa is Relation-like [:(Funcs (K,A)),A:] -defined Funcs ((K \/ {B}),A) -valued Function-like non empty total quasi_total Element of bool [:[:(Funcs (K,A)),A:],(Funcs ((K \/ {B}),A)):]
dom aa is Relation-like non empty set
Fin [:(Funcs (K,A)),A:] is non empty cup-closed diff-closed preBoolean set
Fin A is non empty cup-closed diff-closed preBoolean set
F is Relation-like [:n,n:] -defined n -valued Function-like non empty total quasi_total Element of bool [:[:n,n:],n:]
Ga is Relation-like Funcs (K,A) -defined n -valued Function-like non empty total quasi_total Element of bool [:(Funcs (K,A)),n:]
F $$ ((FinOmega (Funcs (K,A))),Ga) is Element of n
Gs is Relation-like Funcs ((K \/ {B}),A) -defined n -valued Function-like non empty total quasi_total Element of bool [:(Funcs ((K \/ {B}),A)),n:]
F $$ ((FinOmega (Funcs ((K \/ {B}),A))),Gs) is Element of n
[:[:(Funcs (K,A)),A:],n:] is Relation-like non empty set
bool [:[:(Funcs (K,A)),A:],n:] is non empty cup-closed diff-closed preBoolean set
Gs * aa is Relation-like [:(Funcs (K,A)),A:] -defined n -valued Function-like non empty total quasi_total Element of bool [:[:(Funcs (K,A)),A:],n:]
Path is finite Element of Fin A
Funcs (A,n) is functional non empty FUNCTION_DOMAIN of A,n
B9 is Relation-like [:(Funcs (K,A)),A:] -defined n -valued Function-like non empty total quasi_total Element of bool [:[:(Funcs (K,A)),A:],n:]
curry B9 is Relation-like Funcs (K,A) -defined Funcs (A,n) -valued Function-like non empty total quasi_total Element of bool [:(Funcs (K,A)),(Funcs (A,n)):]
[:(Funcs (K,A)),(Funcs (A,n)):] is Relation-like non empty set
bool [:(Funcs (K,A)),(Funcs (A,n)):] is non empty cup-closed diff-closed preBoolean set
b is Relation-like K -defined A -valued Function-like total quasi_total Element of Funcs (K,A)
Ga . b is Element of n
(curry B9) . b is Relation-like A -defined n -valued Function-like total quasi_total Element of Funcs (A,n)
F $$ (Path,((curry B9) . b)) is Element of n
mA is Relation-like K -defined A -valued Function-like total quasi_total finite Element of bool [:K,A:]
{ b1 where b1 is Relation-like K \/ {B} -defined A -valued Function-like non empty total quasi_total finite Element of bool [:(K \/ {B}),A:] : b1 | K = mA } is set
PM is Relation-like Function-like set
dom PM is set
{b} is functional non empty trivial finite 1 -element set
{mA} is functional non empty trivial finite V52() 1 -element set
[:{mA},A:] is Relation-like non empty finite set
i is finite Element of Fin [:(Funcs (K,A)),A:]
PM .: A is finite set
Pi is set
H is set
SF is set
[H,SF] is set
{H,SF} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,SF},{H}} is non empty finite V52() set
PM . SF is set
[b,SF] is set
{b,SF} is non empty finite set
{{b,SF},{b}} is non empty finite V52() set
H is set
PM . H is set
[b,H] is set
{b,H} is non empty finite set
{{b,H},{b}} is non empty finite V52() set
rng PM is set
Pi is set
H is set
PM . Pi is set
PM . H is set
[b,H] is set
{b,H} is non empty finite set
{{b,H},{b}} is non empty finite V52() set
[b,Pi] is set
{b,Pi} is non empty finite set
{{b,Pi},{b}} is non empty finite V52() set
[:A,[:(Funcs (K,A)),A:]:] is Relation-like non empty set
bool [:A,[:(Funcs (K,A)),A:]:] is non empty cup-closed diff-closed preBoolean set
[:A,n:] is Relation-like non empty set
bool [:A,n:] is non empty cup-closed diff-closed preBoolean set
Pi is Relation-like A -defined [:(Funcs (K,A)),A:] -valued Function-like non empty total quasi_total finite Element of bool [:A,[:(Funcs (K,A)),A:]:]
B9 * Pi is Relation-like A -defined n -valued Function-like non empty total quasi_total finite Element of bool [:A,n:]
dom B9 is Relation-like non empty set
rng B9 is non empty set
SF is Relation-like Function-like set
dom SF is set
rng SF is set
h is set
Pi . h is set
[b,h] is set
{b,h} is non empty finite set
{{b,h},{b}} is non empty finite V52() set
QQ is Relation-like A -defined n -valued Function-like non empty total quasi_total finite Element of bool [:A,n:]
QQ . h is set
B9 . (b,h) is set
B9 . [b,h] is set
H is Relation-like A -defined n -valued Function-like non empty total quasi_total finite Element of bool [:A,n:]
H . h is set
aa .: i is finite Element of Fin (Funcs ((K \/ {B}),A))
h is set
Mh is set
aa . Mh is Relation-like Function-like set
Path is set
QQ is set
[Path,QQ] is set
{Path,QQ} is non empty finite set
{Path} is non empty trivial finite 1 -element set
{{Path,QQ},{Path}} is non empty finite V52() set
{QQ} is non empty trivial finite 1 -element set
A \/ {QQ} is non empty finite set
SUM1 is Relation-like K \/ {B} -defined A -valued Function-like non empty total quasi_total finite Element of bool [:(K \/ {B}),A:]
SUM1 | K is Relation-like K -defined K \/ {B} -defined A -valued Function-like finite Element of bool [:(K \/ {B}),A:]
SUM1 . B is set
aa . (Path,QQ) is set
aa . [Path,QQ] is Relation-like Function-like set
h is set
Mh is Relation-like K \/ {B} -defined A -valued Function-like non empty total quasi_total finite Element of bool [:(K \/ {B}),A:]
Mh | K is Relation-like K -defined K \/ {B} -defined A -valued Function-like finite Element of bool [:(K \/ {B}),A:]
dom Mh is non empty finite set
Mh . B is set
rng Mh is non empty finite set
[b,(Mh . B)] is set
{b,(Mh . B)} is non empty finite set
{{b,(Mh . B)},{b}} is non empty finite V52() set
aa . (b,(Mh . B)) is set
aa . [b,(Mh . B)] is Relation-like Function-like set
h is finite Element of Fin (Funcs ((K \/ {B}),A))
F $$ ((aa .: i),Gs) is Element of n
Ga . mA is set
F $$ (i,B9) is Element of n
SUM1 is finite Element of Fin (Funcs (K,A))
[:SUM1,Path:] is Relation-like finite Element of Fin [:(Funcs (K,A)),A:]
F $$ ([:SUM1,Path:],B9) is Element of n
F $$ (SUM1,Ga) is Element of n
AB is finite Element of Fin [:(Funcs (K,A)),A:]
F $$ (AB,B9) is Element of n
rng aa is non empty set
aa .: AB is finite Element of Fin (Funcs ((K \/ {B}),A))
F $$ ((aa .: AB),Gs) is Element of n
F $$ (AB,(Gs * aa)) is Element of n
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
id (Seg n) is Relation-like Seg n -defined Seg n -valued V6() V8() V9() V13() Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A is non empty set
A * is functional non empty FinSequence-membered FinSequenceSet of A
B is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
P is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
KK is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg KK is finite KK -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= KK ) } is set
[:(Seg KK),(Seg n):] is Relation-like finite set
bool [:(Seg KK),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
aa is Relation-like Seg KK -defined Seg n -valued Function-like quasi_total finite Element of bool [:(Seg KK),(Seg n):]
(idseq n) +* aa is Relation-like Function-like finite set
((idseq n) +* aa) * P is Relation-like A * -valued Function-like finite set
(((idseq n) +* aa) * P) | (Seg KK) is Relation-like A * -valued Function-like finite set
B +* ((((idseq n) +* aa) * P) | (Seg KK)) is Relation-like A * -valued Function-like finite set
dom (idseq n) is finite n -element Element of bool NAT
rng (idseq n) is finite set
rng aa is finite set
(rng aa) \/ (Seg n) is finite set
rng ((idseq n) +* aa) is finite set
(Seg KK) \/ (Seg n) is finite Element of bool NAT
dom aa is finite set
(dom aa) \/ (Seg n) is finite set
dom ((idseq n) +* aa) is finite set
SUM1 is Relation-like Seg n -defined Seg n -valued Function-like total quasi_total finite Element of bool [:(Seg n),(Seg n):]
(n,K,A,SUM1,P) is Relation-like NAT -defined Seg n -defined A * -valued A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
Path is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
Path | (Seg KK) is Relation-like NAT -defined Seg KK -defined NAT -defined A * -valued Function-like finite FinSubsequence-like Element of bool [:NAT,(A *):]
[:NAT,(A *):] is Relation-like non empty non trivial non finite set
bool [:NAT,(A *):] is non empty non trivial cup-closed diff-closed preBoolean non finite set
B +* (Path | (Seg KK)) is Relation-like NAT -defined A * -valued Function-like finite Element of bool [:NAT,(A *):]
len P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom Path is finite Element of bool NAT
dom (Path | (Seg KK)) is finite set
len B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom B is finite Element of bool NAT
dom (B +* (Path | (Seg KK))) is finite set
Gs is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B9 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Gs . B9 is set
aa . B9 is set
P . (aa . B9) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B . B9 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Path | (Seg KK)) . B9 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Path . B9 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
SUM1 . B9 is set
Line (Path,B9) is Relation-like NAT -defined A -valued Function-like finite width Path -element FinSequence-like FinSubsequence-like Element of (width Path) -tuples_on A
width Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width Path) -tuples_on A is functional non empty FinSequence-membered FinSequenceSet of A
{ b1 where b1 is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like Element of A * : len b1 = width Path } is set
rng Gs is finite set
B9 is set
dom Gs is finite Element of bool NAT
b is set
Gs . b is set
mA is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
aa . mA is set
width P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width P) -tuples_on A is functional non empty FinSequence-membered FinSequenceSet of A
{ b1 where b1 is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like Element of A * : len b1 = width P } is set
Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Line (P,Bb) is Relation-like NAT -defined A -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on A
PM is Relation-like NAT -defined A -valued Function-like finite width P -element FinSequence-like FinSubsequence-like Element of (width P) -tuples_on A
len PM is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
P . (aa . mA) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
mA is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
width B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width B) -tuples_on A is functional non empty FinSequence-membered FinSequenceSet of A
{ b1 where b1 is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like Element of A * : len b1 = width B } is set
Line (B,mA) is Relation-like NAT -defined A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on A
Bb is Relation-like NAT -defined A -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on A
len Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Gs . mA is set
B . mA is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
mA is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
B9 is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of A *
len B9 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
b is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of A
rng B9 is finite set
len b is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
mA is Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of A
len mA is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
b is Relation-like NAT -defined A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,A
mA is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
b . mA is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
aa . mA is set
P . (aa . mA) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
b . Bb is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B . Bb is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
id (Seg n) is Relation-like Seg n -defined Seg n -valued V6() V8() V9() V13() Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
A * B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
P is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg P is finite P -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= P ) } is set
Funcs ((Seg P),(Seg n)) is functional set
[:(Funcs ((Seg P),(Seg n))), the carrier of K:] is Relation-like set
bool [:(Funcs ((Seg P),(Seg n))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
[:(Seg P),(Seg n):] is Relation-like finite set
bool [:(Seg P),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
F is non empty set
b is Relation-like Function-like set
dom b is set
rng b is set
B9 is non empty set
b is Element of B9
[:(Seg P),F:] is Relation-like set
bool [:(Seg P),F:] is non empty cup-closed diff-closed preBoolean set
mA is Relation-like Seg P -defined F -valued Function-like total quasi_total finite Element of bool [:(Seg P),F:]
Bb is set
rng mA is finite set
dom mA is finite set
PM is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
mA . PM is set
i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A * (PM,i) is Element of the carrier of K
H is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
mA . H is set
A * (H,Pi) is Element of the carrier of K
[:(Seg P), the carrier of K:] is Relation-like set
bool [:(Seg P), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
Bb is Relation-like Seg P -defined the carrier of K -valued Function-like total quasi_total finite Element of bool [:(Seg P), the carrier of K:]
dom Bb is finite set
rng Bb is finite set
PM is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Gs is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
(idseq n) +* mA is Relation-like Function-like finite set
((idseq n) +* mA) * B is Relation-like the carrier of K * -valued Function-like finite set
(((idseq n) +* mA) * B) | (Seg P) is Relation-like the carrier of K * -valued Function-like finite set
Gs +* ((((idseq n) +* mA) * B) | (Seg P)) is Relation-like the carrier of K * -valued Function-like finite set
Pi is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
i is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K $$ i is Element of the carrier of K
Det Pi is Element of the carrier of K
Permutations n is non empty permutational set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product Pi is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product Pi)) is Element of the carrier of K
( the multF of K $$ i) * (Det Pi) is Element of the carrier of K
the multF of K . (( the multF of K $$ i),(Det Pi)) is Element of the carrier of K
[( the multF of K $$ i),(Det Pi)] is set
{( the multF of K $$ i),(Det Pi)} is non empty finite set
{( the multF of K $$ i)} is non empty trivial finite 1 -element set
{{( the multF of K $$ i),(Det Pi)},{( the multF of K $$ i)}} is non empty finite V52() set
the multF of K . [( the multF of K $$ i),(Det Pi)] is set
H is Relation-like Seg P -defined Seg n -valued Function-like quasi_total finite Element of bool [:(Seg P),(Seg n):]
(idseq n) +* H is Relation-like Function-like finite set
((idseq n) +* H) * B is Relation-like the carrier of K * -valued Function-like finite set
(((idseq n) +* H) * B) | (Seg P) is Relation-like the carrier of K * -valued Function-like finite set
(A * B) +* ((((idseq n) +* H) * B) | (Seg P)) is Relation-like the carrier of K * -valued Function-like finite set
SF is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det SF is Element of the carrier of K
Path_product SF is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product SF)) is Element of the carrier of K
len i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
( the multF of K $$ i) * (Det SF) is Element of the carrier of K
the multF of K . (( the multF of K $$ i),(Det SF)) is Element of the carrier of K
[( the multF of K $$ i),(Det SF)] is set
{( the multF of K $$ i),(Det SF)} is non empty finite set
{{( the multF of K $$ i),(Det SF)},{( the multF of K $$ i)}} is non empty finite V52() set
the multF of K . [( the multF of K $$ i),(Det SF)] is set
h is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
QQ is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
H . h is set
i . h is set
A * (h,QQ) is Element of the carrier of K
len SF is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
QQ is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
len Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Pi . QQ is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
mA . QQ is set
B . (mA . QQ) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
SF . QQ is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Pi . QQ is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Gs . QQ is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
SF . QQ is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
[:B9, the carrier of K:] is Relation-like non empty set
bool [:B9, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
b is Relation-like B9 -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:B9, the carrier of K:]
mA is Relation-like Funcs ((Seg P),(Seg n)) -defined the carrier of K -valued Function-like total quasi_total Element of bool [:(Funcs ((Seg P),(Seg n))), the carrier of K:]
Bb is Relation-like Seg P -defined Seg n -valued Function-like quasi_total finite Element of bool [:(Seg P),(Seg n):]
(idseq n) +* Bb is Relation-like Function-like finite set
((idseq n) +* Bb) * B is Relation-like the carrier of K * -valued Function-like finite set
(((idseq n) +* Bb) * B) | (Seg P) is Relation-like the carrier of K * -valued Function-like finite set
(A * B) +* ((((idseq n) +* Bb) * B) | (Seg P)) is Relation-like the carrier of K * -valued Function-like finite set
mA . Bb is set
[:(Seg P),F:] is Relation-like set
bool [:(Seg P),F:] is non empty cup-closed diff-closed preBoolean set
i is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det i is Element of the carrier of K
Permutations n is non empty permutational set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product i is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product i)) is Element of the carrier of K
PM is Relation-like Seg P -defined F -valued Function-like total quasi_total finite Element of bool [:(Seg P),F:]
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
Funcs ((Seg n),(Seg n)) is functional non empty set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
FinOmega (Funcs ((Seg n),(Seg n))) is finite Element of Fin (Funcs ((Seg n),(Seg n)))
Fin (Funcs ((Seg n),(Seg n))) is non empty cup-closed diff-closed preBoolean set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
[:(Funcs ((Seg n),(Seg n))), the carrier of K:] is Relation-like non empty set
bool [:(Funcs ((Seg n),(Seg n))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
A * B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (A * B) is Element of the carrier of K
Permutations n is non empty permutational set
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product (A * B) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (A * B))) is Element of the carrier of K
idseq n is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like set
id (Seg n) is Relation-like Seg n -defined Seg n -valued V6() V8() V9() V13() Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg Path is finite Path -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= Path ) } is set
Funcs ((Seg Path),(Seg n)) is functional set
[:(Seg Path),(Seg n):] is Relation-like finite set
bool [:(Seg Path),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
Path + 1 is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Seg (Path + 1) is non empty finite Path + 1 -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= Path + 1 ) } is set
Funcs ((Seg (Path + 1)),(Seg n)) is functional set
[:(Seg (Path + 1)),(Seg n):] is Relation-like finite set
bool [:(Seg (Path + 1)),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
{(Path + 1)} is non empty trivial finite V52() 1 -element set
(Seg Path) \/ {(Path + 1)} is non empty finite set
b is non empty set
[:b, the carrier of K:] is Relation-like non empty set
bool [:b, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega b is finite Element of Fin b
Fin b is non empty cup-closed diff-closed preBoolean set
mA is Relation-like b -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:b, the carrier of K:]
Funcs (((Seg Path) \/ {(Path + 1)}),(Seg n)) is functional set
Bb is non empty set
[:Bb, the carrier of K:] is Relation-like non empty set
bool [:Bb, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
Path + {} is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Ga is non empty set
[:Ga, the carrier of K:] is Relation-like non empty set
bool [:Ga, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega Ga is finite Element of Fin Ga
Fin Ga is non empty cup-closed diff-closed preBoolean set
i is Relation-like Ga -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:Ga, the carrier of K:]
the addF of K $$ ((FinOmega Ga),i) is Element of the carrier of K
Fin Bb is non empty cup-closed diff-closed preBoolean set
[:((Seg Path) \/ {(Path + 1)}),(Seg n):] is Relation-like finite set
bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
PM is Relation-like Bb -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:Bb, the carrier of K:]
H is Relation-like Seg Path -defined Seg n -valued Function-like quasi_total finite Element of bool [:(Seg Path),(Seg n):]
{ b1 where b1 is Relation-like (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like quasi_total finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):] : b1 | (Seg Path) = H } is set
i . H is set
SF is finite Element of Fin Bb
the addF of K $$ (SF,PM) is Element of the carrier of K
Pi is non empty set
QQ is set
{QQ} is non empty trivial finite 1 -element set
(Seg n) \/ {QQ} is non empty finite set
h is Relation-like (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like quasi_total finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
h | (Seg Path) is Relation-like Seg Path -defined (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
h . (Path + 1) is set
Mh is Relation-like (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like quasi_total finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
Mh | (Seg Path) is Relation-like Seg Path -defined (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
Mh . (Path + 1) is set
Path is set
dom H is finite set
h . Path is set
H . Path is set
Mh . Path is set
[:Pi,SF:] is Relation-like set
bool [:Pi,SF:] is non empty cup-closed diff-closed preBoolean set
QQ is Relation-like Pi -defined SF -valued Function-like quasi_total Element of bool [:Pi,SF:]
rng QQ is set
h is set
Mh is Relation-like (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like quasi_total finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
Mh | (Seg Path) is Relation-like Seg Path -defined (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
dom Mh is finite set
Mh . (Path + 1) is set
rng Mh is finite set
dom QQ is set
QQ . (Mh . (Path + 1)) is set
{n} is non empty trivial finite V52() 1 -element set
(Seg n) \/ {n} is non empty finite set
h is Relation-like (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like quasi_total finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
h | (Seg Path) is Relation-like Seg Path -defined (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
h . (Path + 1) is set
(idseq n) +* H is Relation-like Function-like finite set
((idseq n) +* H) * B is Relation-like the carrier of K * -valued Function-like finite set
(((idseq n) +* H) * B) | (Seg Path) is Relation-like the carrier of K * -valued Function-like finite set
(A * B) +* ((((idseq n) +* H) * B) | (Seg Path)) is Relation-like the carrier of K * -valued Function-like finite set
Mh is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det Mh is Element of the carrier of K
Path_product Mh is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product Mh)) is Element of the carrier of K
Path is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len Path is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
the multF of K $$ Path is Element of the carrier of K
( the multF of K $$ Path) * (Det Mh) is Element of the carrier of K
the multF of K . (( the multF of K $$ Path),(Det Mh)) is Element of the carrier of K
[( the multF of K $$ Path),(Det Mh)] is set
{( the multF of K $$ Path),(Det Mh)} is non empty finite set
{( the multF of K $$ Path)} is non empty trivial finite 1 -element set
{{( the multF of K $$ Path),(Det Mh)},{( the multF of K $$ Path)}} is non empty finite V52() set
the multF of K . [( the multF of K $$ Path),(Det Mh)] is set
Mh . (Path + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(A * B) . (Path + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
[:Pi,Bb:] is Relation-like non empty set
bool [:Pi,Bb:] is non empty cup-closed diff-closed preBoolean set
QQ is Relation-like Pi -defined Bb -valued Function-like non empty total quasi_total Element of bool [:Pi,Bb:]
PM * QQ is Relation-like Pi -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:Pi, the carrier of K:]
[:Pi, the carrier of K:] is Relation-like non empty set
bool [:Pi, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
dom (PM * QQ) is non empty set
dom QQ is non empty set
QQ .: (dom QQ) is set
1 + {} is epsilon-transitive epsilon-connected ordinal natural non empty ext-real positive non negative V44() V45() finite cardinal Element of NAT
Line ((A * B),(Path + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (A * B) -element FinSequence-like FinSubsequence-like Element of (width (A * B)) -tuples_on the carrier of K
width (A * B) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width (A * B)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width (A * B) } is set
Line (Mh,(Path + 1)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Mh -element FinSequence-like FinSubsequence-like Element of (width Mh) -tuples_on the carrier of K
width Mh is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width Mh) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width Mh } is set
((Path + 1),n,n, the carrier of K,Mh,(Line ((A * B),(Path + 1)))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
SUM1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len SUM1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
the addF of K $$ SUM1 is Element of the carrier of K
dom (id (Seg n)) is finite set
( the multF of K $$ Path) * SUM1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
( the multF of K $$ Path) multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,( the multF of K $$ Path),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [: the carrier of K, the carrier of K:]
(( the multF of K $$ Path) multfield) * SUM1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len (( the multF of K $$ Path) * SUM1) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom (( the multF of K $$ Path) * SUM1) is finite Element of bool NAT
the_unity_wrt the addF of K is Element of the carrier of K
[#] ((( the multF of K $$ Path) * SUM1),(the_unity_wrt the addF of K)) is Relation-like NAT -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of K:]
[:NAT, the carrier of K:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of K:] is non empty non trivial cup-closed diff-closed preBoolean non finite set
width B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len Mh is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Y9 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
(PM * QQ) . Y9 is set
A * ((Path + 1),Y9) is Element of the carrier of K
( the multF of K $$ Path) * (A * ((Path + 1),Y9)) is Element of the carrier of K
the multF of K . (( the multF of K $$ Path),(A * ((Path + 1),Y9))) is Element of the carrier of K
[( the multF of K $$ Path),(A * ((Path + 1),Y9))] is set
{( the multF of K $$ Path),(A * ((Path + 1),Y9))} is non empty finite set
{{( the multF of K $$ Path),(A * ((Path + 1),Y9))},{( the multF of K $$ Path)}} is non empty finite V52() set
the multF of K . [( the multF of K $$ Path),(A * ((Path + 1),Y9))] is set
Line (B,Y9) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
(width B) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width B } is set
((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) is Element of the carrier of K
Path_product ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))))) is Element of the carrier of K
(( the multF of K $$ Path) * (A * ((Path + 1),Y9))) * (Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9)))) is Element of the carrier of K
the multF of K . ((( the multF of K $$ Path) * (A * ((Path + 1),Y9))),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))))) is Element of the carrier of K
[(( the multF of K $$ Path) * (A * ((Path + 1),Y9))),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))))] is set
{(( the multF of K $$ Path) * (A * ((Path + 1),Y9))),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))))} is non empty finite set
{(( the multF of K $$ Path) * (A * ((Path + 1),Y9)))} is non empty trivial finite 1 -element set
{{(( the multF of K $$ Path) * (A * ((Path + 1),Y9))),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))))},{(( the multF of K $$ Path) * (A * ((Path + 1),Y9)))}} is non empty finite V52() set
the multF of K . [(( the multF of K $$ Path) * (A * ((Path + 1),Y9))),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))))] is set
{Y9} is non empty trivial finite V52() 1 -element set
(Seg n) \/ {Y9} is non empty finite set
x2 is Relation-like (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like quasi_total finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
x2 | (Seg Path) is Relation-like Seg Path -defined (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
x2 . (Path + 1) is set
hj9 is Relation-like Seg (Path + 1) -defined Seg n -valued Function-like quasi_total finite Element of bool [:(Seg (Path + 1)),(Seg n):]
(idseq n) +* hj9 is Relation-like Function-like finite set
((idseq n) +* hj9) * B is Relation-like the carrier of K * -valued Function-like finite set
(((idseq n) +* hj9) * B) | (Seg (Path + 1)) is Relation-like the carrier of K * -valued Function-like finite set
(A * B) +* ((((idseq n) +* hj9) * B) | (Seg (Path + 1))) is Relation-like the carrier of K * -valued Function-like finite set
Mhj is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
len Mhj is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len (Line (B,Y9)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Pathj is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Line (((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))),Pathj) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) -element FinSequence-like FinSubsequence-like Element of (width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9)))) -tuples_on the carrier of K
width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) } is set
((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) . Pathj is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Mh . Pathj is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
H . Pathj is set
B . (H . Pathj) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom H is finite set
x2 . Pathj is set
Line (Mh,Pathj) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Mh -element FinSequence-like FinSubsequence-like Element of (width Mh) -tuples_on the carrier of K
Mhj . Pathj is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B . Y9 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) . Pathj is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))),Pathj) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) -element FinSequence-like FinSubsequence-like Element of (width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9)))) -tuples_on the carrier of K
width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) } is set
Mhj . Pathj is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Mhj . Pathj is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(A * B) . Pathj is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))),Pathj) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) -element FinSequence-like FinSubsequence-like Element of (width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9)))) -tuples_on the carrier of K
width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9)))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) } is set
((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) . Pathj is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (Mh,Pathj) is Relation-like NAT -defined the carrier of K -valued Function-like finite width Mh -element FinSequence-like FinSubsequence-like Element of (width Mh) -tuples_on the carrier of K
Mh . Pathj is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
mA . x2 is set
Pathj is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len Pathj is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
the multF of K $$ Pathj is Element of the carrier of K
( the multF of K $$ Pathj) * (Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9)))) is Element of the carrier of K
the multF of K . (( the multF of K $$ Pathj),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))))) is Element of the carrier of K
[( the multF of K $$ Pathj),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))))] is set
{( the multF of K $$ Pathj),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))))} is non empty finite set
{( the multF of K $$ Pathj)} is non empty trivial finite 1 -element set
{{( the multF of K $$ Pathj),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))))},{( the multF of K $$ Pathj)}} is non empty finite V52() set
the multF of K . [( the multF of K $$ Pathj),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,Y9))))] is set
Pathj . (Path + 1) is set
rng H is finite set
dom H is finite set
i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Pathj | Path is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Pathj | (Seg Path) is Relation-like NAT -defined Seg Path -defined NAT -defined the carrier of K -valued Function-like finite FinSubsequence-like set
(Pathj | Path) . i is set
Pathj . i is set
H . i is set
Path . i is set
Hi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
A * (i,Hi) is Element of the carrier of K
x2 . i is set
Pathj | (Seg Path) is Relation-like NAT -defined Seg Path -defined NAT -defined the carrier of K -valued Function-like finite FinSubsequence-like Element of bool [:NAT, the carrier of K:]
(Pathj | (Seg Path)) . i is set
len (Pathj | Path) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
<*(Pathj . (Path + 1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(Pathj . (Path + 1))] is set
{1,(Pathj . (Path + 1))} is non empty finite set
{{1,(Pathj . (Path + 1))},{1}} is non empty finite V52() set
{[1,(Pathj . (Path + 1))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
Path ^ <*(Pathj . (Path + 1))*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
QQ . Y9 is set
Y9 is set
x2 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
SUM1 . x2 is set
A * ((Path + 1),x2) is Element of the carrier of K
Line (B,x2) is Relation-like NAT -defined the carrier of K -valued Function-like finite width B -element FinSequence-like FinSubsequence-like Element of (width B) -tuples_on the carrier of K
width B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
(width B) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b1 where b1 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b1 = width B } is set
((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))) is Element of the carrier of K
Path_product ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))) is Element of the carrier of K
(A * ((Path + 1),x2)) * (Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2)))) is Element of the carrier of K
the multF of K . ((A * ((Path + 1),x2)),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))) is Element of the carrier of K
[(A * ((Path + 1),x2)),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))] is set
{(A * ((Path + 1),x2)),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))} is non empty finite set
{(A * ((Path + 1),x2))} is non empty trivial finite 1 -element set
{{(A * ((Path + 1),x2)),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))},{(A * ((Path + 1),x2))}} is non empty finite V52() set
the multF of K . [(A * ((Path + 1),x2)),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))] is set
(( the multF of K $$ Path) * SUM1) . Y9 is set
( the multF of K $$ Path) * ((A * ((Path + 1),x2)) * (Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))) is Element of the carrier of K
the multF of K . (( the multF of K $$ Path),((A * ((Path + 1),x2)) * (Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2)))))) is Element of the carrier of K
[( the multF of K $$ Path),((A * ((Path + 1),x2)) * (Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2)))))] is set
{( the multF of K $$ Path),((A * ((Path + 1),x2)) * (Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2)))))} is non empty finite set
{{( the multF of K $$ Path),((A * ((Path + 1),x2)) * (Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2)))))},{( the multF of K $$ Path)}} is non empty finite V52() set
the multF of K . [( the multF of K $$ Path),((A * ((Path + 1),x2)) * (Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2)))))] is set
( the multF of K $$ Path) * (A * ((Path + 1),x2)) is Element of the carrier of K
the multF of K . (( the multF of K $$ Path),(A * ((Path + 1),x2))) is Element of the carrier of K
[( the multF of K $$ Path),(A * ((Path + 1),x2))] is set
{( the multF of K $$ Path),(A * ((Path + 1),x2))} is non empty finite set
{{( the multF of K $$ Path),(A * ((Path + 1),x2))},{( the multF of K $$ Path)}} is non empty finite V52() set
the multF of K . [( the multF of K $$ Path),(A * ((Path + 1),x2))] is set
(( the multF of K $$ Path) * (A * ((Path + 1),x2))) * (Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2)))) is Element of the carrier of K
the multF of K . ((( the multF of K $$ Path) * (A * ((Path + 1),x2))),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))) is Element of the carrier of K
[(( the multF of K $$ Path) * (A * ((Path + 1),x2))),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))] is set
{(( the multF of K $$ Path) * (A * ((Path + 1),x2))),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))} is non empty finite set
{(( the multF of K $$ Path) * (A * ((Path + 1),x2)))} is non empty trivial finite 1 -element set
{{(( the multF of K $$ Path) * (A * ((Path + 1),x2))),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))},{(( the multF of K $$ Path) * (A * ((Path + 1),x2)))}} is non empty finite V52() set
the multF of K . [(( the multF of K $$ Path) * (A * ((Path + 1),x2))),(Det ((Path + 1),n,n, the carrier of K,Mh,(Line (B,x2))))] is set
(PM * QQ) . Y9 is set
([#] ((( the multF of K $$ Path) * SUM1),(the_unity_wrt the addF of K))) | (dom (( the multF of K $$ Path) * SUM1)) is Relation-like NAT -defined dom (( the multF of K $$ Path) * SUM1) -defined NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
Y9 is set
x2 is set
QQ . Y9 is set
QQ . x2 is set
{x2} is non empty trivial finite 1 -element set
(Seg n) \/ {x2} is non empty finite set
{Y9} is non empty trivial finite 1 -element set
(Seg n) \/ {Y9} is non empty finite set
h1 is Relation-like (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like quasi_total finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
h1 | (Seg Path) is Relation-like Seg Path -defined (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
h1 . (Path + 1) is set
R is Relation-like (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like quasi_total finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
R | (Seg Path) is Relation-like Seg Path -defined (Seg Path) \/ {(Path + 1)} -defined Seg n -valued Function-like finite Element of bool [:((Seg Path) \/ {(Path + 1)}),(Seg n):]
R . (Path + 1) is set
Fin Pi is non empty cup-closed diff-closed preBoolean set
Y9 is finite Element of Fin Pi
rng (id (Seg n)) is finite set
(PM * QQ) * (id (Seg n)) is Relation-like Seg n -defined the carrier of K -valued Function-like finite Element of bool [:(Seg n), the carrier of K:]
[:(Seg n), the carrier of K:] is Relation-like set
bool [:(Seg n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ (Y9,(PM * QQ)) is Element of the carrier of K
findom (( the multF of K $$ Path) * SUM1) is finite Element of Fin NAT
the addF of K $$ ((findom (( the multF of K $$ Path) * SUM1)),([#] ((( the multF of K $$ Path) * SUM1),(the_unity_wrt the addF of K)))) is Element of the carrier of K
Sum (( the multF of K $$ Path) * SUM1) is Element of the carrier of K
the addF of K $$ (( the multF of K $$ Path) * SUM1) is Element of the carrier of K
Sum SUM1 is Element of the carrier of K
( the multF of K $$ Path) * (Sum SUM1) is Element of the carrier of K
the multF of K . (( the multF of K $$ Path),(Sum SUM1)) is Element of the carrier of K
[( the multF of K $$ Path),(Sum SUM1)] is set
{( the multF of K $$ Path),(Sum SUM1)} is non empty finite set
{{( the multF of K $$ Path),(Sum SUM1)},{( the multF of K $$ Path)}} is non empty finite V52() set
the multF of K . [( the multF of K $$ Path),(Sum SUM1)] is set
FinOmega Bb is finite Element of Fin Bb
the addF of K $$ ((FinOmega Bb),PM) is Element of the carrier of K
Funcs ((Seg {}),(Seg n)) is functional set
[:(Seg {}),(Seg n):] is Relation-like finite set
bool [:(Seg {}),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
the_unity_wrt the multF of K is Element of the carrier of K
1_ K is Element of the carrier of K
K254(K) is V70(K) Element of the carrier of K
F is non empty set
[:F, the carrier of K:] is Relation-like non empty set
bool [:F, the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega F is finite Element of Fin F
Fin F is non empty cup-closed diff-closed preBoolean set
Ga is Relation-like F -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:F, the carrier of K:]
Path is Relation-like non-empty empty-yielding Seg {} -defined Seg n -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty total quasi_total ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of bool [:(Seg {}),(Seg n):]
{Path} is functional non empty trivial finite V52() 1 -element set
the addF of K $$ ((FinOmega F),Ga) is Element of the carrier of K
Ga . Path is set
(idseq n) +* Path is Relation-like Function-like finite set
((idseq n) +* Path) * B is Relation-like the carrier of K * -valued Function-like finite set
(((idseq n) +* Path) * B) | (Seg {}) is Relation-like non-empty empty-yielding NAT -defined the carrier of K * -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered set
(A * B) +* ((((idseq n) +* Path) * B) | (Seg {})) is Relation-like NAT -defined the carrier of K * -valued Function-like finite set
Gs is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
(A * B) +* {} is Relation-like NAT -defined Function-like finite set
Det Gs is Element of the carrier of K
Path_product Gs is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product Gs)) is Element of the carrier of K
B9 is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len B9 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
the multF of K $$ B9 is Element of the carrier of K
( the multF of K $$ B9) * (Det Gs) is Element of the carrier of K
the multF of K . (( the multF of K $$ B9),(Det Gs)) is Element of the carrier of K
[( the multF of K $$ B9),(Det Gs)] is set
{( the multF of K $$ B9),(Det Gs)} is non empty finite set
{( the multF of K $$ B9)} is non empty trivial finite 1 -element set
{{( the multF of K $$ B9),(Det Gs)},{( the multF of K $$ B9)}} is non empty finite V52() set
the multF of K . [( the multF of K $$ B9),(Det Gs)] is set
<*> the carrier of K is Relation-like non-empty empty-yielding NAT -defined the carrier of K -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered FinSequence of the carrier of K
[:NAT, the carrier of K:] is Relation-like non empty non trivial non finite set
Path is Relation-like Funcs ((Seg n),(Seg n)) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Funcs ((Seg n),(Seg n))), the carrier of K:]
the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),Path) is Element of the carrier of K
len (A * B) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom (A * B) is finite Element of bool NAT
F is Relation-like Seg n -defined Seg n -valued Function-like total quasi_total finite Element of bool [:(Seg n),(Seg n):]
Path . F is set
(n,n, the carrier of K,F,B) is Relation-like NAT -defined Seg n -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (n,n, the carrier of K,F,B) is Element of the carrier of K
Path_product (n,n, the carrier of K,F,B) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (n,n, the carrier of K,F,B))) is Element of the carrier of K
dom (idseq n) is finite n -element Element of bool NAT
dom F is finite set
(idseq n) +* F is Relation-like Function-like finite set
len B is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
len (n,n, the carrier of K,F,B) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom (n,n, the carrier of K,F,B) is finite Element of bool NAT
((idseq n) +* F) * B is Relation-like the carrier of K * -valued Function-like finite set
(((idseq n) +* F) * B) | (Seg n) is Relation-like the carrier of K * -valued Function-like finite set
(A * B) +* ((((idseq n) +* F) * B) | (Seg n)) is Relation-like the carrier of K * -valued Function-like finite set
Ga is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
(n,n, the carrier of K,F,B) | n is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K *
(n,n, the carrier of K,F,B) | (Seg n) is Relation-like NAT -defined Seg n -defined NAT -defined the carrier of K * -valued Function-like finite FinSubsequence-like set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
Permutations n is non empty permutational set
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
A * B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (A * B) is Element of the carrier of K
Path_product (A * B) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (A * B))) is Element of the carrier of K
Det B is Element of the carrier of K
Path_product B is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product B)) is Element of the carrier of K
id (Permutations n) is Relation-like Permutations n -defined Permutations n -valued V6() V8() V9() V13() Function-like one-to-one non empty total quasi_total onto bijective Element of bool [:(Permutations n),(Permutations n):]
[:(Permutations n),(Permutations n):] is Relation-like non empty set
bool [:(Permutations n),(Permutations n):] is non empty cup-closed diff-closed preBoolean set
dom (id (Permutations n)) is non empty set
Seg n is finite n -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
Funcs ((Seg n),(Seg n)) is functional non empty set
[:(Funcs ((Seg n),(Seg n))), the carrier of K:] is Relation-like non empty set
bool [:(Funcs ((Seg n),(Seg n))), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is non empty cup-closed diff-closed preBoolean finite V52() set
FinOmega (Funcs ((Seg n),(Seg n))) is finite Element of Fin (Funcs ((Seg n),(Seg n)))
Fin (Funcs ((Seg n),(Seg n))) is non empty cup-closed diff-closed preBoolean set
F is Relation-like Funcs ((Seg n),(Seg n)) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Funcs ((Seg n),(Seg n))), the carrier of K:]
the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),F) is Element of the carrier of K
(Funcs ((Seg n),(Seg n))) \ (Permutations n) is functional Element of bool (Funcs ((Seg n),(Seg n)))
bool (Funcs ((Seg n),(Seg n))) is non empty cup-closed diff-closed preBoolean set
Gs is set
B9 is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
Ga is finite Element of Fin (Funcs ((Seg n),(Seg n)))
Gs is finite Element of Fin (Funcs ((Seg n),(Seg n)))
the addF of K $$ (Gs,F) is Element of the carrier of K
Ga is finite Element of Fin (Funcs ((Seg n),(Seg n)))
0. K is V70(K) Element of the carrier of K
the_unity_wrt the addF of K is Element of the carrier of K
F .: Ga is finite Element of Fin the carrier of K
Fin the carrier of K is non empty cup-closed diff-closed preBoolean set
{(0. K)} is non empty trivial finite 1 -element set
B9 is set
dom F is non empty set
b is set
F . b is set
mA is Relation-like Seg n -defined Seg n -valued Function-like total quasi_total finite Element of bool [:(Seg n),(Seg n):]
(n,n, the carrier of K,mA,B) is Relation-like NAT -defined Seg n -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (n,n, the carrier of K,mA,B) is Element of the carrier of K
Path_product (n,n, the carrier of K,mA,B) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (n,n, the carrier of K,mA,B))) is Element of the carrier of K
F . mA is set
Bb is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
the multF of K $$ Bb is Element of the carrier of K
( the multF of K $$ Bb) * (Det (n,n, the carrier of K,mA,B)) is Element of the carrier of K
the multF of K . (( the multF of K $$ Bb),(Det (n,n, the carrier of K,mA,B))) is Element of the carrier of K
[( the multF of K $$ Bb),(Det (n,n, the carrier of K,mA,B))] is set
{( the multF of K $$ Bb),(Det (n,n, the carrier of K,mA,B))} is non empty finite set
{( the multF of K $$ Bb)} is non empty trivial finite 1 -element set
{{( the multF of K $$ Bb),(Det (n,n, the carrier of K,mA,B))},{( the multF of K $$ Bb)}} is non empty finite V52() set
the multF of K . [( the multF of K $$ Bb),(Det (n,n, the carrier of K,mA,B))] is set
dom F is non empty set
the addF of K $$ (Ga,F) is Element of the carrier of K
Ga \/ (Permutations n) is non empty set
(Funcs ((Seg n),(Seg n))) \/ (Permutations n) is non empty set
Gs is finite Element of Fin (Funcs ((Seg n),(Seg n)))
the addF of K $$ (Gs,F) is Element of the carrier of K
( the addF of K $$ (Gs,F)) + (0. K) is Element of the carrier of K
the addF of K . (( the addF of K $$ (Gs,F)),(0. K)) is Element of the carrier of K
[( the addF of K $$ (Gs,F)),(0. K)] is set
{( the addF of K $$ (Gs,F)),(0. K)} is non empty finite set
{( the addF of K $$ (Gs,F))} is non empty trivial finite 1 -element set
{{( the addF of K $$ (Gs,F)),(0. K)},{( the addF of K $$ (Gs,F))}} is non empty finite V52() set
the addF of K . [( the addF of K $$ (Gs,F)),(0. K)] is set
Ga is finite Element of Fin (Funcs ((Seg n),(Seg n)))
Gs is finite Element of Fin (Funcs ((Seg n),(Seg n)))
the addF of K $$ (Gs,F) is Element of the carrier of K
Gs is finite Element of Fin (Funcs ((Seg n),(Seg n)))
the addF of K $$ (Gs,F) is Element of the carrier of K
dom F is non empty set
F | (Permutations n) is Relation-like Permutations n -defined Funcs ((Seg n),(Seg n)) -defined the carrier of K -valued Function-like Element of bool [:(Funcs ((Seg n),(Seg n))), the carrier of K:]
dom (F | (Permutations n)) is set
rng (F | (Permutations n)) is set
B9 is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),B9) is Element of the carrier of K
rng (id (Permutations n)) is non empty set
B9 * (id (Permutations n)) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
b is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
B9 . b is Element of the carrier of K
Path_matrix (b,A) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K $$ (Path_matrix (b,A)) is Element of the carrier of K
- ((Det B),b) is Element of the carrier of K
( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b)) is Element of the carrier of K
the multF of K . (( the multF of K $$ (Path_matrix (b,A))),(- ((Det B),b))) is Element of the carrier of K
[( the multF of K $$ (Path_matrix (b,A))),(- ((Det B),b))] is set
{( the multF of K $$ (Path_matrix (b,A))),(- ((Det B),b))} is non empty finite set
{( the multF of K $$ (Path_matrix (b,A)))} is non empty trivial finite 1 -element set
{{( the multF of K $$ (Path_matrix (b,A))),(- ((Det B),b))},{( the multF of K $$ (Path_matrix (b,A)))}} is non empty finite V52() set
the multF of K . [( the multF of K $$ (Path_matrix (b,A))),(- ((Det B),b))] is set
mA is Relation-like Seg n -defined Seg n -valued Function-like one-to-one total quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
F . mA is set
B9 . mA is set
(n,n, the carrier of K,mA,B) is Relation-like NAT -defined Seg n -defined the carrier of K * -valued the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (n,n, the carrier of K,mA,B) is Element of the carrier of K
Path_product (n,n, the carrier of K,mA,B) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (n,n, the carrier of K,mA,B))) is Element of the carrier of K
Bb is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len Bb is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
the multF of K $$ Bb is Element of the carrier of K
( the multF of K $$ Bb) * (Det (n,n, the carrier of K,mA,B)) is Element of the carrier of K
the multF of K . (( the multF of K $$ Bb),(Det (n,n, the carrier of K,mA,B))) is Element of the carrier of K
[( the multF of K $$ Bb),(Det (n,n, the carrier of K,mA,B))] is set
{( the multF of K $$ Bb),(Det (n,n, the carrier of K,mA,B))} is non empty finite set
{( the multF of K $$ Bb)} is non empty trivial finite 1 -element set
{{( the multF of K $$ Bb),(Det (n,n, the carrier of K,mA,B))},{( the multF of K $$ Bb)}} is non empty finite V52() set
the multF of K . [( the multF of K $$ Bb),(Det (n,n, the carrier of K,mA,B))] is set
len (Path_matrix (b,A)) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
dom mA is finite set
i is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
mA . i is set
rng mA is finite set
dom (Path_matrix (b,A)) is finite Element of bool NAT
(Path_matrix (b,A)) . i is set
Pi is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
A * (i,Pi) is Element of the carrier of K
Bb . i is set
n is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V113() unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital V171() left_unital doubleLoopStr
the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
A * B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (A * B) is Element of the carrier of K
Permutations n is non empty permutational set
the addF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty cup-closed diff-closed preBoolean set
FinOmega (Permutations n) is finite Element of Fin (Permutations n)
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
Path_product (A * B) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like non empty set
bool [:(Permutations n), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (A * B))) is Element of the carrier of K
Det A is Element of the carrier of K
Path_product A is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product A)) is Element of the carrier of K
Det B is Element of the carrier of K
Path_product B is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product B)) is Element of the carrier of K
(Det A) * (Det B) is Element of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like non empty total quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the multF of K . ((Det A),(Det B)) is Element of the carrier of K
[(Det A),(Det B)] is set
{(Det A),(Det B)} is non empty finite set
{(Det A)} is non empty trivial finite 1 -element set
{{(Det A),(Det B)},{(Det A)}} is non empty finite V52() set
the multF of K . [(Det A),(Det B)] is set
len (Permutations n) is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b1 where b1 is epsilon-transitive epsilon-connected ordinal natural ext-real non negative V44() V45() finite cardinal Element of NAT : ( 1 <= b1 & b1 <= len (Permutations n) ) } is set
SUM1 is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),SUM1) is Element of the carrier of K
[:(Fin (Permutations n)), the carrier of K:] is Relation-like non empty set
bool [:(Fin (Permutations n)), the carrier of K:] is non empty cup-closed diff-closed preBoolean set
Ga is Relation-like Fin (Permutations n) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (Permutations n)), the carrier of K:]
Ga . (FinOmega (Permutations n)) is Element of the carrier of K
Ga . {} is set
0. K is V70(K) Element of the carrier of K
Gs is Relation-like Fin (Permutations n) -defined the carrier of K -valued Function-like non empty total quasi_total Element of bool [:(Fin (Permutations n)), the carrier of K:]
Gs . (FinOmega (Permutations n)) is Element of the carrier of K
Gs . {} is set
B9 is finite Element of Fin (Permutations n)
b is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one total quasi_total onto bijective finite Element of Permutations n
{b} is functional non empty trivial finite V52() 1 -element set
B9 \/ {b} is non empty finite set
Path_matrix (b,A) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
the multF of K $$ (Path_matrix (b,A)) is Element of the carrier of K
Bb is finite Element of Fin (Permutations n)
Gs . Bb is Element of the carrier of K
Ga . Bb is Element of the carrier of K
(Ga . Bb) * (Det B) is Element of the carrier of K
the multF of K . ((Ga . Bb),(Det B)) is Element of the carrier of K
[(Ga . Bb),(Det B)] is set
{(Ga . Bb),(Det B)} is non empty finite set
{(Ga . Bb)} is non empty trivial finite 1 -element set
{{(Ga . Bb),(Det B)},{(Ga . Bb)}} is non empty finite V52() set
the multF of K . [(Ga . Bb),(Det B)] is set
- (( the multF of K $$ (Path_matrix (b,A))),b) is Element of the carrier of K
- ((Det B),b) is Element of the carrier of K
SUM1 . b is Element of the carrier of K
(- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B) is Element of the carrier of K
the multF of K . ((- (( the multF of K $$ (Path_matrix (b,A))),b)),(Det B)) is Element of the carrier of K
[(- (( the multF of K $$ (Path_matrix (b,A))),b)),(Det B)] is set
{(- (( the multF of K $$ (Path_matrix (b,A))),b)),(Det B)} is non empty finite set
{(- (( the multF of K $$ (Path_matrix (b,A))),b))} is non empty trivial finite 1 -element set
{{(- (( the multF of K $$ (Path_matrix (b,A))),b)),(Det B)},{(- (( the multF of K $$ (Path_matrix (b,A))),b))}} is non empty finite V52() set
the multF of K . [(- (( the multF of K $$ (Path_matrix (b,A))),b)),(Det B)] is set
(Path_product A) . b is Element of the carrier of K
((Path_product A) . b) * (Det B) is Element of the carrier of K
the multF of K . (((Path_product A) . b),(Det B)) is Element of the carrier of K
[((Path_product A) . b),(Det B)] is set
{((Path_product A) . b),(Det B)} is non empty finite set
{((Path_product A) . b)} is non empty trivial finite 1 -element set
{{((Path_product A) . b),(Det B)},{((Path_product A) . b)}} is non empty finite V52() set
the multF of K . [((Path_product A) . b),(Det B)] is set
- (( the multF of K $$ (Path_matrix (b,A))),b) is Element of the carrier of K
- ( the multF of K $$ (Path_matrix (b,A))) is Element of the carrier of K
- ((Det B),b) is Element of the carrier of K
- (Det B) is Element of the carrier of K
(- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B) is Element of the carrier of K
the multF of K . ((- (( the multF of K $$ (Path_matrix (b,A))),b)),(Det B)) is Element of the carrier of K
[(- (( the multF of K $$ (Path_matrix (b,A))),b)),(Det B)] is set
{(- (( the multF of K $$ (Path_matrix (b,A))),b)),(Det B)} is non empty finite set
{(- (( the multF of K $$ (Path_matrix (b,A))),b))} is non empty trivial finite 1 -element set
{{(- (( the multF of K $$ (Path_matrix (b,A))),b)),(Det B)},{(- (( the multF of K $$ (Path_matrix (b,A))),b))}} is non empty finite V52() set
the multF of K . [(- (( the multF of K $$ (Path_matrix (b,A))),b)),(Det B)] is set
- ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B)) is Element of the carrier of K
(- ( the multF of K $$ (Path_matrix (b,A)))) * (- ((Det B),b)) is Element of the carrier of K
the multF of K . ((- ( the multF of K $$ (Path_matrix (b,A)))),(- ((Det B),b))) is Element of the carrier of K
[(- ( the multF of K $$ (Path_matrix (b,A)))),(- ((Det B),b))] is set
{(- ( the multF of K $$ (Path_matrix (b,A)))),(- ((Det B),b))} is non empty finite set
{(- ( the multF of K $$ (Path_matrix (b,A))))} is non empty trivial finite 1 -element set
{{(- ( the multF of K $$ (Path_matrix (b,A)))),(- ((Det B),b))},{(- ( the multF of K $$ (Path_matrix (b,A))))}} is non empty finite V52() set
the multF of K . [(- ( the multF of K $$ (Path_matrix (b,A)))),(- ((Det B),b))] is set
( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b)) is Element of the carrier of K
the multF of K . (( the multF of K $$ (Path_matrix (b,A))),(- ((Det B),b))) is Element of the carrier of K
[( the multF of K $$ (Path_matrix (b,A))),(- ((Det B),b))] is set
{( the multF of K $$ (Path_matrix (b,A))),(- ((Det B),b))} is non empty finite set
{( the multF of K $$ (Path_matrix (b,A)))} is non empty trivial finite 1 -element set
{{( the multF of K $$ (Path_matrix (b,A))),(- ((Det B),b))},{( the multF of K $$ (Path_matrix (b,A)))}} is non empty finite V52() set
the multF of K . [( the multF of K $$ (Path_matrix (b,A))),(- ((Det B),b))] is set
- (( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))) is Element of the carrier of K
(( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))) - ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B)) is Element of the carrier of K
(( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))) + (- ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B))) is Element of the carrier of K
the addF of K . ((( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))),(- ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B)))) is Element of the carrier of K
[(( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))),(- ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B)))] is set
{(( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))),(- ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B)))} is non empty finite set
{(( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b)))} is non empty trivial finite 1 -element set
{{(( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))),(- ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B)))},{(( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b)))}} is non empty finite V52() set
the addF of K . [(( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))),(- ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B)))] is set
(Path_product A) . b is Element of the carrier of K
SUM1 . b is Element of the carrier of K
((Path_product A) . b) * (Det B) is Element of the carrier of K
the multF of K . (((Path_product A) . b),(Det B)) is Element of the carrier of K
[((Path_product A) . b),(Det B)] is set
{((Path_product A) . b),(Det B)} is non empty finite set
{((Path_product A) . b)} is non empty trivial finite 1 -element set
{{((Path_product A) . b),(Det B)},{((Path_product A) . b)}} is non empty finite V52() set
the multF of K . [((Path_product A) . b),(Det B)] is set
SUM1 . b is Element of the carrier of K
(Path_product A) . b is Element of the carrier of K
((Path_product A) . b) * (Det B) is Element of the carrier of K
the multF of K . (((Path_product A) . b),(Det B)) is Element of the carrier of K
[((Path_product A) . b),(Det B)] is set
{((Path_product A) . b),(Det B)} is non empty finite set
{((Path_product A) . b)} is non empty trivial finite 1 -element set
{{((Path_product A) . b),(Det B)},{((Path_product A) . b)}} is non empty finite V52() set
the multF of K . [((Path_product A) . b),(Det B)] is set
SUM1 . b is Element of the carrier of K
(Path_product A) . b is Element of the carrier of K
((Path_product A) . b) * (Det B) is Element of the carrier of K
the multF of K . (((Path_product A) . b),(Det B)) is Element of the carrier of K
[((Path_product A) . b),(Det B)] is set
{((Path_product A) . b),(Det B)} is non empty finite set
{((Path_product A) . b)} is non empty trivial finite 1 -element set
{{((Path_product A) . b),(Det B)},{((Path_product A) . b)}} is non empty finite V52() set
the multF of K . [((Path_product A) . b),(Det B)] is set
(FinOmega (Permutations n)) \ B9 is finite Element of Fin (Permutations n)
Gs . B9 is Element of the carrier of K
the addF of K . ((Gs . B9),(SUM1 . b)) is Element of the carrier of K
[(Gs . B9),(SUM1 . b)] is set
{(Gs . B9),(SUM1 . b)} is non empty finite set
{(Gs . B9)} is non empty trivial finite 1 -element set
{{(Gs . B9),(SUM1 . b)},{(Gs . B9)}} is non empty finite V52() set
the addF of K . [(Gs . B9),(SUM1 . b)] is set
Ga . B9 is Element of the carrier of K
(Ga . B9) * (Det B) is Element of the carrier of K
the multF of K . ((Ga . B9),(Det B)) is Element of the carrier of K
[(Ga . B9),(Det B)] is set
{(Ga . B9),(Det B)} is non empty finite set
{(Ga . B9)} is non empty trivial finite 1 -element set
{{(Ga . B9),(Det B)},{(Ga . B9)}} is non empty finite V52() set
the multF of K . [(Ga . B9),(Det B)] is set
((Ga . B9) * (Det B)) + (((Path_product A) . b) * (Det B)) is Element of the carrier of K
the addF of K . (((Ga . B9) * (Det B)),(((Path_product A) . b) * (Det B))) is Element of the carrier of K
[((Ga . B9) * (Det B)),(((Path_product A) . b) * (Det B))] is set
{((Ga . B9) * (Det B)),(((Path_product A) . b) * (Det B))} is non empty finite set
{((Ga . B9) * (Det B))} is non empty trivial finite 1 -element set
{{((Ga . B9) * (Det B)),(((Path_product A) . b) * (Det B))},{((Ga . B9) * (Det B))}} is non empty finite V52() set
the addF of K . [((Ga . B9) * (Det B)),(((Path_product A) . b) * (Det B))] is set
(Ga . B9) + ((Path_product A) . b) is Element of the carrier of K
the addF of K . ((Ga . B9),((Path_product A) . b)) is Element of the carrier of K
[(Ga . B9),((Path_product A) . b)] is set
{(Ga . B9),((Path_product A) . b)} is non empty finite set
{{(Ga . B9),((Path_product A) . b)},{(Ga . B9)}} is non empty finite V52() set
the addF of K . [(Ga . B9),((Path_product A) . b)] is set
{}. (Permutations n) is Relation-like non-empty empty-yielding NAT -defined epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty ext-real non positive non negative V44() V45() finite finite-yielding V52() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered Element of Fin (Permutations n)
B9 is finite Element of Fin (Permutations n)
Gs . B9 is Element of the carrier of K
Ga . B9 is Element of the carrier of K
(Ga . B9) * (Det B) is Element of the carrier of K
the multF of K . ((Ga . B9),(Det B)) is Element of the carrier of K
[(Ga . B9),(Det B)] is set
{(Ga . B9),(Det B)} is non empty finite set
{(Ga . B9)} is non empty trivial finite 1 -element set
{{(Ga . B9),(Det B)},{(Ga . B9)}} is non empty finite V52() set
the multF of K . [(Ga . B9),(Det B)] is set