:: LAPLACE semantic presentation

REAL is set
NAT is non empty non trivial V26() V27() V28() non finite cardinal limit_cardinal Element of bool REAL
bool REAL is cup-closed diff-closed preBoolean set
{} is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V108() V109() V110() ext-real non positive non negative set
NAT is non empty non trivial V26() V27() V28() non finite cardinal limit_cardinal set
bool NAT is non trivial cup-closed diff-closed preBoolean non finite set
bool NAT is non trivial cup-closed diff-closed preBoolean non finite set
Fin NAT is non empty cup-closed diff-closed preBoolean set
COMPLEX is set
1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
2 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
RAT is set
INT is set
[:COMPLEX,COMPLEX:] is Relation-like set
bool [:COMPLEX,COMPLEX:] is cup-closed diff-closed preBoolean set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is cup-closed diff-closed preBoolean set
[:REAL,REAL:] is Relation-like set
bool [:REAL,REAL:] is cup-closed diff-closed preBoolean set
[:[:REAL,REAL:],REAL:] is Relation-like set
bool [:[:REAL,REAL:],REAL:] is cup-closed diff-closed preBoolean set
[:RAT,RAT:] is Relation-like set
bool [:RAT,RAT:] is cup-closed diff-closed preBoolean set
[:[:RAT,RAT:],RAT:] is Relation-like set
bool [:[:RAT,RAT:],RAT:] is cup-closed diff-closed preBoolean set
[:INT,INT:] is Relation-like set
bool [:INT,INT:] is cup-closed diff-closed preBoolean set
[:[:INT,INT:],INT:] is Relation-like set
bool [:[:INT,INT:],INT:] is cup-closed diff-closed preBoolean set
[:NAT,NAT:] is Relation-like non trivial non finite set
[:[:NAT,NAT:],NAT:] is Relation-like non trivial non finite set
bool [:[:NAT,NAT:],NAT:] is non trivial cup-closed diff-closed preBoolean non finite set
0 is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V108() V109() V110() ext-real non positive non negative Element of NAT
3 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Seg 1 is non empty trivial finite 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
{1} is non empty trivial finite V37() 1 -element set
Seg 2 is non empty finite 2 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
{1,2} is non empty finite V37() set
idseq 2 is Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
id (Seg 2) is Relation-like Seg 2 -defined Seg 2 -valued Function-like one-to-one non empty V14( Seg 2) quasi_total onto bijective finite V112() V114() V115() V119() Element of bool [:(Seg 2),(Seg 2):]
[:(Seg 2),(Seg 2):] is Relation-like finite set
bool [:(Seg 2),(Seg 2):] is cup-closed diff-closed preBoolean finite V37() set
<*1,2*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*1*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,1] is set
{1,1} is non empty finite V37() set
{{1,1},{1}} is non empty finite V37() set
{[1,1]} is Relation-like Function-like constant non empty trivial finite 1 -element set
<*2*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,2] is set
{{1,2},{1}} is non empty finite V37() set
{[1,2]} is Relation-like Function-like constant non empty trivial finite 1 -element set
K139(<*1*>,<*2*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
1 + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
union {} is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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 Function-like one-to-one non empty V14( Seg 1) quasi_total onto bijective finite V112() V114() V115() V119() Element of bool [:(Seg 1),(Seg 1):]
[:(Seg 1),(Seg 1):] is Relation-like finite set
bool [:(Seg 1),(Seg 1):] is cup-closed diff-closed preBoolean finite V37() set
{(idseq 1)} is functional non empty trivial finite V37() 1 -element set
id {} is Relation-like non-empty empty-yielding {} -defined {} -valued Function-like one-to-one constant functional empty V14( {} ) V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V108() V109() V110() V112() V114() V115() V119() ext-real non positive non negative set
<*2,1*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
K139(<*2*>,<*1*>) is Relation-like NAT -defined Function-like non empty finite 1 + 1 -element FinSequence-like FinSubsequence-like set
K206(NAT,2,1) is Relation-like NAT -defined NAT -valued Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like FinSequence of NAT
0 ! is V108() Element of REAL
n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom n is finite Element of bool NAT
len n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len n) -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
DelLine (,n) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (DelLine (,n)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
b is Relation-like NAT -defined the carrier of n * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of x,x, the carrier of n
dom b is finite Element of bool NAT
Deleting (b,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 *
DelLine (b,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 *
DelCol ((DelLine (b,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 *
len (Deleting (b,K,A)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len (DelLine (b,K)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 FinSequence of the carrier of K *
width A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width A) is finite width A -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
DelCol (A,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
width (DelCol (A,n)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width A) -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len (DelCol (A,n)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
rng (DelCol (A,n)) is finite set
b is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom (DelCol (A,n)) is finite Element of bool NAT
X is set
(DelCol (A,n)) . X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Seg (len (DelCol (A,n))) is finite len (DelCol (A,n)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (DelCol (A,n)) ) } is set
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom A is finite Element of bool NAT
Line (A,B) 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) -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
DelLine (,(Line (A,B))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (Line (A,B)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom (Line (A,B)) is finite width A -element Element of bool NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
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 *
len K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
DelLine (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 *
width (DelLine (K,A)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom K is finite Element of bool NAT
DelLine (,K) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like tabular set
len (DelLine (,K)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
dom (DelLine (,K)) is finite Element of bool NAT
(DelLine (K,A)) . x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng (DelLine (,K)) is finite set
rng K is finite set
Line ((DelLine (K,A)),x) is Relation-like NAT -defined the carrier of n -valued Function-like finite width (DelLine (K,A)) -element FinSequence-like FinSubsequence-like Element of (width (DelLine (K,A))) -tuples_on the carrier of n
(width (DelLine (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 (DelLine (K,A)) } is set
len (Line ((DelLine (K,A)),x)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom K is finite Element of bool NAT
dom K is finite Element of bool NAT
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
x 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 x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width x) is finite width x -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width x ) } is set
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Deleting (x,b,n) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of A *
DelLine (x,b) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of A *
DelCol ((DelLine (x,b)),n) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of A *
width (Deleting (x,b,n)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom x is finite Element of bool NAT
len (Deleting (x,b,n)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width (DelLine (x,b)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom x is finite Element of bool NAT
len x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom x is finite Element of bool NAT
n is non empty multMagma
the carrier of n is non empty set
[: the carrier of n,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of n,NAT:], the carrier of n:] is Relation-like non trivial non finite set
bool [:[: the carrier of n,NAT:], the carrier of n:] is non trivial cup-closed diff-closed preBoolean non finite set
K is Relation-like [: the carrier of n,NAT:] -defined the carrier of n -valued Function-like V14([: the carrier of n,NAT:]) quasi_total Element of bool [:[: the carrier of n,NAT:], the carrier of n:]
A is Element of the carrier of n
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K . (A,x) is set
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
K . (A,b) is Element of the carrier of n
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Permutations n is non empty permutational set
len (Permutations n) is non empty V26() V27() V28() cardinal set
n ! is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
finSeg A is finite A -element Element of Fin NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= A ) } is set
[:(finSeg A),(finSeg A):] is Relation-like finite set
bool [:(finSeg A),(finSeg A):] is cup-closed diff-closed preBoolean finite V37() set
{ b₁ where b₁ is Relation-like finSeg A -defined finSeg A -valued Function-like V14( finSeg A) quasi_total finite Element of bool [:(finSeg A),(finSeg A):] : b₁ is Relation-like finSeg A -defined finSeg A -valued Function-like one-to-one V14( finSeg A) quasi_total onto bijective finite Element of bool [:(finSeg A),(finSeg A):] } is set
X is set
X is set
B is Relation-like finSeg A -defined finSeg A -valued Function-like V14( finSeg A) quasi_total finite Element of bool [:(finSeg A),(finSeg A):]
card (finSeg A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(card (finSeg A)) ! is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
n + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Seg (n + 1) is non empty finite n + 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n + 1 ) } is set
Permutations (n + 1) is non empty permutational set
len (Permutations (n + 1)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (n + 1))) is finite len (Permutations (n + 1)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (n + 1)) ) } is set
n ! is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{ b₁ where b₁ is Relation-like Seg (len (Permutations (n + 1))) -defined Seg (len (Permutations (n + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (n + 1)))) quasi_total onto bijective finite Element of Permutations (n + 1) : b₁ . K = A } is set
len { b₁ where b₁ is Relation-like Seg (len (Permutations (n + 1))) -defined Seg (len (Permutations (n + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (n + 1)))) quasi_total onto bijective finite Element of Permutations (n + 1) : b₁ . K = A } is V26() V27() V28() cardinal set
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
finSeg (x + 1) is non empty finite x + 1 -element Element of Fin NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x + 1 ) } is set
{A} is non empty trivial finite V37() 1 -element set
(finSeg (x + 1)) \ {A} is finite Element of bool (finSeg (x + 1))
bool (finSeg (x + 1)) is cup-closed diff-closed preBoolean finite V37() set
((finSeg (x + 1)) \ {A}) \/ {A} is non empty finite set
{K} is non empty trivial finite V37() 1 -element set
(finSeg (x + 1)) \ {K} is finite Element of bool (finSeg (x + 1))
Permutations (x + 1) is non empty permutational set
len (Permutations (x + 1)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (x + 1))) is finite len (Permutations (x + 1)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (x + 1)) ) } is set
{ b₁ where b₁ is Relation-like Seg (len (Permutations (x + 1))) -defined Seg (len (Permutations (x + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (x + 1)))) quasi_total onto bijective finite Element of Permutations (x + 1) : b₁ . K = A } is set
((finSeg (x + 1)) \ {K}) \/ {K} is non empty finite set
[:(((finSeg (x + 1)) \ {K}) \/ {K}),(((finSeg (x + 1)) \ {A}) \/ {A}):] is Relation-like finite set
bool [:(((finSeg (x + 1)) \ {K}) \/ {K}),(((finSeg (x + 1)) \ {A}) \/ {A}):] is cup-closed diff-closed preBoolean finite V37() set
{ b₁ where b₁ is Relation-like ((finSeg (x + 1)) \ {K}) \/ {K} -defined ((finSeg (x + 1)) \ {A}) \/ {A} -valued Function-like non empty V14(((finSeg (x + 1)) \ {K}) \/ {K}) quasi_total finite Element of bool [:(((finSeg (x + 1)) \ {K}) \/ {K}),(((finSeg (x + 1)) \ {A}) \/ {A}):] : ( b₁ is one-to-one & b₁ . K = A ) } is set
j is set
[:(finSeg (x + 1)),(finSeg (x + 1)):] is Relation-like finite set
bool [:(finSeg (x + 1)),(finSeg (x + 1)):] is cup-closed diff-closed preBoolean finite V37() set
COL is Relation-like finSeg (x + 1) -defined finSeg (x + 1) -valued Function-like non empty V14( finSeg (x + 1)) quasi_total finite Element of bool [:(finSeg (x + 1)),(finSeg (x + 1)):]
COL . K is set
card (finSeg (x + 1)) is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j is set
COL is Relation-like Seg (len (Permutations (x + 1))) -defined Seg (len (Permutations (x + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (x + 1)))) quasi_total onto bijective finite Element of Permutations (x + 1)
COL . K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[:(finSeg (x + 1)),(finSeg (x + 1)):] is Relation-like finite set
bool [:(finSeg (x + 1)),(finSeg (x + 1)):] is cup-closed diff-closed preBoolean finite V37() set
L is Relation-like finSeg (x + 1) -defined finSeg (x + 1) -valued Function-like one-to-one non empty V14( finSeg (x + 1)) quasi_total onto bijective finite Element of bool [:(finSeg (x + 1)),(finSeg (x + 1)):]
L . K is set
card (finSeg (x + 1)) is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
card ((finSeg (x + 1)) \ {A}) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(card ((finSeg (x + 1)) \ {A})) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
card ((finSeg (x + 1)) \ {K}) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(card ((finSeg (x + 1)) \ {K})) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
[:((finSeg (x + 1)) \ {K}),((finSeg (x + 1)) \ {A}):] is Relation-like finite set
bool [:((finSeg (x + 1)) \ {K}),((finSeg (x + 1)) \ {A}):] is cup-closed diff-closed preBoolean finite V37() set
{ b₁ where b₁ is Relation-like (finSeg (x + 1)) \ {K} -defined (finSeg (x + 1)) \ {A} -valued Function-like quasi_total finite Element of bool [:((finSeg (x + 1)) \ {K}),((finSeg (x + 1)) \ {A}):] : b₁ is one-to-one } is set
len { b₁ where b₁ is Relation-like (finSeg (x + 1)) \ {K} -defined (finSeg (x + 1)) \ {A} -valued Function-like quasi_total finite Element of bool [:((finSeg (x + 1)) \ {K}),((finSeg (x + 1)) \ {A}):] : b₁ is one-to-one } is V26() V27() V28() cardinal set
(card ((finSeg (x + 1)) \ {A})) ! is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(card ((finSeg (x + 1)) \ {A})) -' (card ((finSeg (x + 1)) \ {K})) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
((card ((finSeg (x + 1)) \ {A})) -' (card ((finSeg (x + 1)) \ {K}))) ! is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
((card ((finSeg (x + 1)) \ {A})) !) / (((card ((finSeg (x + 1)) \ {A})) -' (card ((finSeg (x + 1)) \ {K}))) !) is V108() set
((card ((finSeg (x + 1)) \ {A})) !) / 1 is V108() set
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
n + 2 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n + 2 ) } is set
2Set (Seg (n + 2)) is non empty finite set
Fin (2Set (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 V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() left_unital Fanoian doubleLoopStr
the carrier of K is non empty non trivial set
1_ K is Element of the carrier of K
1. K is V52(K) Element of the carrier of K
- (1_ K) 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is cup-closed diff-closed preBoolean set
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like V14([: the carrier of K,NAT:]) quasi_total Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial cup-closed diff-closed preBoolean non finite set
A is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one V14( Seg (len (Permutations (n + 2)))) quasi_total onto bijective finite Element of Permutations (n + 2)
Part_sgn (A,K) is Relation-like 2Set (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty V14( 2Set (Seg (n + 2))) quasi_total finite Element of bool [:(2Set (Seg (n + 2))), the carrier of K:]
[:(2Set (Seg (n + 2))), the carrier of K:] is Relation-like set
bool [:(2Set (Seg (n + 2))), the carrier of K:] is cup-closed diff-closed preBoolean set
X is finite Element of Fin (2Set (Seg (n + 2)))
b is finite Element of Fin (2Set (Seg (n + 2)))
{ b₁ where b₁ is Element of 2Set (Seg (n + 2)) : ( b₁ in b & (Part_sgn (A,K)) . b₁ = - (1_ K) ) } is set
the multF of K $$ (b,(Part_sgn (A,K))) is Element of the carrier of K
card X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,(power K),(- (1_ K)),(card X)) is Element of the carrier of K
id (Seg (n + 2)) is Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty V14( Seg (n + 2)) quasi_total onto bijective finite V112() V114() V115() V119() Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]
[:(Seg (n + 2)),(Seg (n + 2)):] is Relation-like finite set
bool [:(Seg (n + 2)),(Seg (n + 2)):] is cup-closed diff-closed preBoolean finite V37() set
B is Relation-like Seg (len (Permutations (n + 2))) -defined Seg (len (Permutations (n + 2))) -valued Function-like one-to-one V14( Seg (len (Permutations (n + 2)))) quasi_total onto bijective finite Element of Permutations (n + 2)
Part_sgn (B,K) is Relation-like 2Set (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty V14( 2Set (Seg (n + 2))) quasi_total finite Element of bool [:(2Set (Seg (n + 2))), the carrier of K:]
{ b₁ where b₁ is Element of 2Set (Seg (n + 2)) : ( b₁ in b & not (Part_sgn (A,K)) . b₁ = (Part_sgn (B,K)) . b₁ ) } is set
j is set
(Part_sgn (B,K)) . j is set
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{N,i} is non empty finite V37() set
B . i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
B . N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j is set
N is Element of 2Set (Seg (n + 2))
(Part_sgn (A,K)) . N is Element of the carrier of K
(Part_sgn (B,K)) . N is Element of the carrier of K
j is set
N is Element of 2Set (Seg (n + 2))
(Part_sgn (A,K)) . N is Element of the carrier of K
(Part_sgn (B,K)) . N is Element of the carrier of K
(card X) mod 2 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
2 * j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(2 * j) + {} is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
the multF of K $$ (b,(Part_sgn (B,K))) is Element of the carrier of K
(card X) mod 2 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
2 * j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(2 * j) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
the multF of K $$ (b,(Part_sgn (B,K))) is Element of the carrier of K
- ( the multF of K $$ (b,(Part_sgn (B,K)))) is Element of the carrier of K
(card X) mod 2 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
n + 2 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Permutations (n + 2) is non empty permutational set
len (Permutations (n + 2)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (n + 2))) is finite len (Permutations (n + 2)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n + 2 ) } is set
2Set (Seg (n + 2)) is non empty finite set
Fin (2Set (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 V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() left_unital Fanoian 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is cup-closed diff-closed preBoolean set
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like V14([: the carrier of K,NAT:]) quasi_total Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ K is Element of the carrier of K
1. K is V52(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 V14( Seg (len (Permutations (n + 2)))) quasi_total onto bijective finite Element of Permutations (n + 2)
Part_sgn (A,K) is Relation-like 2Set (Seg (n + 2)) -defined the carrier of K -valued Function-like non empty V14( 2Set (Seg (n + 2))) quasi_total finite Element of bool [:(2Set (Seg (n + 2))), the carrier of K:]
[:(2Set (Seg (n + 2))), the carrier of K:] is Relation-like set
bool [:(2Set (Seg (n + 2))), the carrier of K:] is cup-closed diff-closed preBoolean set
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{ {b₁,x} where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : {b₁,x} in 2Set (Seg (n + 2)) } is set
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x + b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,(power K),(- (1_ K)),(x + b)) is Element of the carrier of K
X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
X + 2 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
finSeg (X + 2) is non empty finite X + 2 -element Element of Fin NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= X + 2 ) } is set
[:(finSeg (X + 2)),(finSeg (X + 2)):] is Relation-like finite set
bool [:(finSeg (X + 2)),(finSeg (X + 2)):] is cup-closed diff-closed preBoolean finite V37() set
i is Relation-like finSeg (X + 2) -defined finSeg (X + 2) -valued Function-like one-to-one non empty V14( finSeg (X + 2)) quasi_total onto bijective finite Element of bool [:(finSeg (X + 2)),(finSeg (X + 2)):]
rng i is non empty finite set
Seg (X + 2) is non empty finite X + 2 -element Element of bool NAT
x - 1 is V108() V109() V110() set
Seg x is finite x -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
(finSeg (X + 2)) \ (Seg x) is finite Element of bool (finSeg (X + 2))
bool (finSeg (X + 2)) is cup-closed diff-closed preBoolean finite V37() set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
finSeg j is finite j -element Element of Fin NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= j ) } is set
L is set
2Set (Seg (X + 2)) is non empty finite set
i9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
{i9,x} is non empty finite V37() set
L is finite Element of Fin (2Set (Seg (n + 2)))
{ b₁ where b₁ is Element of 2Set (Seg (n + 2)) : ( b₁ in L & (Part_sgn (A,K)) . b₁ = - (1_ K) ) } is set
j9 is set
Laa is Element of 2Set (Seg (n + 2))
(Part_sgn (A,K)) . Laa is Element of the carrier of K
dom i is non empty finite Element of bool (finSeg (X + 2))
rng A is finite set
b - 1 is V108() V109() V110() set
(finSeg j) \/ ((finSeg (X + 2)) \ (Seg x)) is finite set
x is Relation-like Function-like set
dom x is set
Laa is finite Element of Fin (2Set (Seg (n + 2)))
x " Laa is set
{x} is non empty trivial finite V37() 1 -element set
(Seg (X + 2)) \ {x} is finite Element of bool NAT
z is set
ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
z is set
ProjD is set
x . z is set
x . ProjD is set
{ProjD,x} is non empty finite set
{z,x} is non empty finite set
{x,ProjD} is non empty finite set
y is finite set
x .: y is finite set
card (x .: y) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
card y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(finSeg (X + 2)) \ {x} is finite Element of bool (finSeg (X + 2))
z is set
ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
rng x is set
z is set
ProjD is set
x . ProjD is set
Q1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x . Q1 is set
{x,Q1} is non empty finite V37() set
(finSeg j) \ (x " Laa) is finite Element of bool (finSeg j)
bool (finSeg j) is cup-closed diff-closed preBoolean finite V37() set
((finSeg (X + 2)) \ (Seg x)) /\ (x " Laa) is finite Element of bool (finSeg (X + 2))
((finSeg j) \ (x " Laa)) \/ (((finSeg (X + 2)) \ (Seg x)) /\ (x " Laa)) is finite set
i .: (((finSeg j) \ (x " Laa)) \/ (((finSeg (X + 2)) \ (Seg x)) /\ (x " Laa))) is finite set
j9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg j9 is finite j9 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= j9 ) } is set
z is set
ProjD is set
i . ProjD is set
Q1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
x . Q1 is set
A . Q1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{Q1,x} is non empty finite V37() set
(Part_sgn (A,K)) . {Q1,x} is set
Q is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
{Q,x} is non empty finite V37() set
j9 + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
x . Q1 is set
Q is Element of 2Set (Seg (n + 2))
(Part_sgn (A,K)) . Q is Element of the carrier of K
{x,Q1} is non empty finite V37() set
i . x is set
A . Q1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j9 + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
the multF of K $$ (L,(Part_sgn (A,K))) is Element of the carrier of K
(finSeg j) /\ (x " Laa) is finite set
Q is set
Q19 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j9 + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Pf is set
i . Pf is set
x . Pf is set
{Pf,x} is non empty finite set
2Set (Seg (X + 2)) is non empty finite set
domf is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
{domf,x} is non empty finite V37() set
(Part_sgn (A,K)) . {domf,x} is set
k is Element of 2Set (Seg (n + 2))
(Part_sgn (A,K)) . k is Element of the carrier of K
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(Part_sgn (A,K)) . {domf,x} is set
(x " Laa) /\ (finSeg j) is finite set
(finSeg j) \ ((x " Laa) /\ (finSeg j)) is finite Element of bool (finSeg j)
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Q is set
Q19 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
{Q19,x} is non empty finite V37() set
x . Q19 is set
((finSeg j) /\ (x " Laa)) \/ (((finSeg (X + 2)) \ (Seg x)) /\ (x " Laa)) is finite set
(dom x) /\ (x " Laa) is set
finSeg j9 is finite j9 -element Element of Fin NAT
card (finSeg j9) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
card (((finSeg j) \ (x " Laa)) \/ (((finSeg (X + 2)) \ (Seg x)) /\ (x " Laa))) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
card ((finSeg j) \ ((x " Laa) /\ (finSeg j))) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
card (((finSeg (X + 2)) \ (Seg x)) /\ (x " Laa)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(card ((finSeg j) \ ((x " Laa) /\ (finSeg j)))) + (card (((finSeg (X + 2)) \ (Seg x)) /\ (x " Laa))) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
card (finSeg j) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
card ((x " Laa) /\ (finSeg j)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(card (finSeg j)) - (card ((x " Laa) /\ (finSeg j))) is V108() V109() V110() set
((card (finSeg j)) - (card ((x " Laa) /\ (finSeg j)))) + (card (((finSeg (X + 2)) \ (Seg x)) /\ (x " Laa))) is V108() V109() V110() set
b - x is V108() V109() V110() set
card Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(finSeg j) /\ y is finite set
card ((finSeg j) /\ y) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(card ((finSeg j) /\ y)) + (card (finSeg j9)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
((card ((finSeg j) /\ y)) + (card (finSeg j9))) - (card (finSeg j)) is V108() V109() V110() set
y /\ (finSeg j) is finite set
card (y /\ (finSeg j)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(((card ((finSeg j) /\ y)) + (card (finSeg j9))) - (card (finSeg j))) + (card (y /\ (finSeg j))) is V108() V109() V110() set
2 * (card ((finSeg j) /\ y)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(2 * (card ((finSeg j) /\ y))) + (card (finSeg j9)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
((2 * (card ((finSeg j) /\ y))) + (card (finSeg j9))) - (card (finSeg j)) is V108() V109() V110() set
(2 * (card ((finSeg j) /\ y))) + j9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
((2 * (card ((finSeg j) /\ y))) + j9) - (card (finSeg j)) is V108() V109() V110() set
((2 * (card ((finSeg j) /\ y))) + j9) - j is V108() V109() V110() set
Q is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(2 * (card ((finSeg j) /\ y))) + Q is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,(power K),(- (1_ K)),((2 * (card ((finSeg j) /\ y))) + Q)) is Element of the carrier of K
(K,(power K),(- (1_ K)),(2 * (card ((finSeg j) /\ y)))) is Element of the carrier of K
(K,(power K),(- (1_ K)),Q) is Element of the carrier of K
(K,(power K),(- (1_ K)),(2 * (card ((finSeg j) /\ y)))) * (K,(power K),(- (1_ K)),Q) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(2 * (card ((finSeg j) /\ y)))),(K,(power K),(- (1_ K)),Q)) is Element of the carrier of K
(1_ K) * (K,(power K),(- (1_ K)),Q) is Element of the carrier of K
the multF of K . ((1_ K),(K,(power K),(- (1_ K)),Q)) is Element of the carrier of K
z is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
2 * z is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,(power K),(- (1_ K)),(2 * z)) is Element of the carrier of K
(K,(power K),(- (1_ K)),(2 * z)) * (K,(power K),(- (1_ K)),Q) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(2 * z)),(K,(power K),(- (1_ K)),Q)) is Element of the carrier of K
2 * x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(2 * x) + Q is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,(power K),(- (1_ K)),((2 * x) + Q)) is Element of the carrier of K
x - b is V108() V109() V110() set
card Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
((finSeg (X + 2)) \ (Seg x)) /\ y is finite Element of bool (finSeg (X + 2))
card (((finSeg (X + 2)) \ (Seg x)) /\ y) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(card (finSeg j)) + (card (((finSeg (X + 2)) \ (Seg x)) /\ y)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
((card (finSeg j)) + (card (((finSeg (X + 2)) \ (Seg x)) /\ y))) - (card (finSeg j9)) is V108() V109() V110() set
(((card (finSeg j)) + (card (((finSeg (X + 2)) \ (Seg x)) /\ y))) - (card (finSeg j9))) + (card (((finSeg (X + 2)) \ (Seg x)) /\ y)) is V108() V109() V110() set
2 * (card (((finSeg (X + 2)) \ (Seg x)) /\ y)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(2 * (card (((finSeg (X + 2)) \ (Seg x)) /\ y))) - (card (finSeg j9)) is V108() V109() V110() set
((2 * (card (((finSeg (X + 2)) \ (Seg x)) /\ y))) - (card (finSeg j9))) + (card (finSeg j)) is V108() V109() V110() set
(2 * (card (((finSeg (X + 2)) \ (Seg x)) /\ y))) - j9 is V108() V109() V110() set
((2 * (card (((finSeg (X + 2)) \ (Seg x)) /\ y))) - j9) + (card (finSeg j)) is V108() V109() V110() set
((2 * (card (((finSeg (X + 2)) \ (Seg x)) /\ y))) - j9) + j is V108() V109() V110() set
Q is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(2 * (card (((finSeg (X + 2)) \ (Seg x)) /\ y))) + Q is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,(power K),(- (1_ K)),((2 * (card (((finSeg (X + 2)) \ (Seg x)) /\ y))) + Q)) is Element of the carrier of K
(K,(power K),(- (1_ K)),(2 * (card (((finSeg (X + 2)) \ (Seg x)) /\ y)))) is Element of the carrier of K
(K,(power K),(- (1_ K)),Q) is Element of the carrier of K
(K,(power K),(- (1_ K)),(2 * (card (((finSeg (X + 2)) \ (Seg x)) /\ y)))) * (K,(power K),(- (1_ K)),Q) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(2 * (card (((finSeg (X + 2)) \ (Seg x)) /\ y)))),(K,(power K),(- (1_ K)),Q)) is Element of the carrier of K
(1_ K) * (K,(power K),(- (1_ K)),Q) is Element of the carrier of K
the multF of K . ((1_ K),(K,(power K),(- (1_ K)),Q)) is Element of the carrier of K
ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
2 * ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,(power K),(- (1_ K)),(2 * ProjD)) is Element of the carrier of K
(K,(power K),(- (1_ K)),(2 * ProjD)) * (K,(power K),(- (1_ K)),Q) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(2 * ProjD)),(K,(power K),(- (1_ K)),Q)) is Element of the carrier of K
2 * b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(2 * b) + Q is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,(power K),(- (1_ K)),((2 * b) + Q)) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
n + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Seg (n + 1) is non empty finite n + 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n + 1 ) } is set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
2Set (Seg n) is set
2Set (Seg (n + 1)) is set
[:(2Set (Seg n)),(2Set (Seg (n + 1))):] is Relation-like set
bool [:(2Set (Seg n)),(2Set (Seg (n + 1))):] is cup-closed diff-closed preBoolean set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{ {b₁,K} where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : {b₁,K} in 2Set (Seg (n + 1)) } is set
(2Set (Seg (n + 1))) \ { {b₁,K} where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : {b₁,K} in 2Set (Seg (n + 1)) } is Element of bool (2Set (Seg (n + 1)))
bool (2Set (Seg (n + 1))) is cup-closed diff-closed preBoolean set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{i,j} is non empty finite V37() set
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
{N,K} is non empty finite V37() set
{K,N} is non empty finite V37() set
j is set
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{N,i} is non empty finite V37() set
i + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Seg (i + 1) is non empty finite i + 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i + 1 ) } is set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{COL,L} is non empty finite V37() set
L + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{COL,(L + 1)} is non empty finite V37() set
COL + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{(COL + 1),(L + 1)} is non empty finite V37() set
{N,(i + 1)} is non empty finite V37() set
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{COL,L} is non empty finite V37() set
L + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{COL,(L + 1)} is non empty finite V37() set
COL + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{(COL + 1),(L + 1)} is non empty finite V37() set
N + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{(N + 1),(i + 1)} is non empty finite V37() set
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{COL,L} is non empty finite V37() set
L + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{COL,(L + 1)} is non empty finite V37() set
COL + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{(COL + 1),(L + 1)} is non empty finite V37() set
[:(2Set (Seg n)),((2Set (Seg (n + 1))) \ { {b₁,K} where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : {b₁,K} in 2Set (Seg (n + 1)) } ):] is Relation-like set
bool [:(2Set (Seg n)),((2Set (Seg (n + 1))) \ { {b₁,K} where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : {b₁,K} in 2Set (Seg (n + 1)) } ):] is cup-closed diff-closed preBoolean set
i is Relation-like 2Set (Seg n) -defined (2Set (Seg (n + 1))) \ { {b₁,K} where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : {b₁,K} in 2Set (Seg (n + 1)) } -valued Function-like quasi_total Element of bool [:(2Set (Seg n)),((2Set (Seg (n + 1))) \ { {b₁,K} where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : {b₁,K} in 2Set (Seg (n + 1)) } ):]
N is set
j is non empty set
[:(2Set (Seg n)),j:] is Relation-like set
bool [:(2Set (Seg n)),j:] is cup-closed diff-closed preBoolean set
N is Relation-like 2Set (Seg n) -defined j -valued Function-like V14( 2Set (Seg n)) quasi_total Element of bool [:(2Set (Seg n)),j:]
rng N is set
i is set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{j,COL} is non empty finite V37() set
j - 1 is V108() V109() V110() set
COL - 1 is V108() V109() V110() set
dom N is Element of bool (2Set (Seg n))
bool (2Set (Seg n)) is cup-closed diff-closed preBoolean set
Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
{Laa,j9} is non empty finite V37() set
N . i is set
i9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i9 + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
dom N is Element of bool (2Set (Seg n))
bool (2Set (Seg n)) is cup-closed diff-closed preBoolean set
Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
{Laa,i9} is non empty finite V37() set
N . {Laa,i9} is set
{Laa,(i9 + 1)} is non empty finite V37() set
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
L + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
dom N is Element of bool (2Set (Seg n))
bool (2Set (Seg n)) is cup-closed diff-closed preBoolean set
i9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i9 + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{L,i9} is non empty finite V37() set
N . {L,i9} is set
{(L + 1),(i9 + 1)} is non empty finite V37() set
dom N is Element of bool (2Set (Seg n))
bool (2Set (Seg n)) is cup-closed diff-closed preBoolean set
i is Relation-like 2Set (Seg n) -defined 2Set (Seg (n + 1)) -valued Function-like quasi_total Element of bool [:(2Set (Seg n)),(2Set (Seg (n + 1))):]
rng i is set
j is set
COL is set
i . j is set
i . COL is set
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{L,i9} is non empty finite V37() set
j9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{j9,Laa} is non empty finite V37() set
z is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
z is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
y + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{ProjD,(y + 1)} is non empty finite V37() set
ProjD + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{(ProjD + 1),(y + 1)} is non empty finite V37() set
{z,x} is non empty finite V37() set
z is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
y + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{ProjD,(y + 1)} is non empty finite V37() set
x + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{z,(x + 1)} is non empty finite V37() set
z is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ProjD + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
y + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{(ProjD + 1),(y + 1)} is non empty finite V37() set
{ProjD,y} is non empty finite V37() set
x + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{z,(x + 1)} is non empty finite V37() set
z is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
z is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ProjD + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
y + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{(ProjD + 1),(y + 1)} is non empty finite V37() set
z + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
x + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{(z + 1),(x + 1)} is non empty finite V37() set
z is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
{ProjD,y} is non empty finite V37() set
y + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{ProjD,(y + 1)} is non empty finite V37() set
z + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
x + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{(z + 1),(x + 1)} is non empty finite V37() set
z is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ProjD is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{j,COL} is non empty finite V37() set
i . {j,COL} is set
COL + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{j,(COL + 1)} is non empty finite V37() set
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{(j + 1),(COL + 1)} is non empty finite V37() set
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Permutations n is non empty permutational set
len (Permutations n) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
id (Seg n) is Relation-like Seg n -defined Seg n -valued Function-like one-to-one V14( Seg n) quasi_total onto bijective finite V112() V114() V115() V119() Element of bool [:(Seg n),(Seg n):]
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is cup-closed diff-closed preBoolean finite V37() set
K is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n
A is Relation-like Seg n -defined Seg n -valued Function-like one-to-one V14( Seg n) 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 V14( Seg n) quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
n is non empty set
K is non empty set
Fin K is non empty cup-closed diff-closed preBoolean set
[:n,(Fin K):] is Relation-like set
bool [:n,(Fin K):] is cup-closed diff-closed preBoolean set
A is non empty set
[:(Fin K),A:] is Relation-like set
bool [:(Fin K),A:] is cup-closed diff-closed preBoolean set
[:A,A:] is Relation-like set
[:[:A,A:],A:] is Relation-like set
bool [:[:A,A:],A:] is cup-closed diff-closed preBoolean set
Fin n is non empty cup-closed diff-closed preBoolean set
x is Relation-like n -defined Fin K -valued Function-like non empty V14(n) quasi_total Element of bool [:n,(Fin K):]
b is Relation-like Fin K -defined A -valued Function-like non empty V14( Fin K) quasi_total Element of bool [:(Fin K),A:]
b . {} is set
b * x is Relation-like n -defined A -valued Function-like non empty V14(n) quasi_total Element of bool [:n,A:]
[:n,A:] is Relation-like set
bool [:n,A:] is cup-closed diff-closed preBoolean set
X is Relation-like [:A,A:] -defined A -valued Function-like V14([:A,A:]) quasi_total Element of bool [:[:A,A:],A:]
the_unity_wrt X is Element of A
B is finite Element of Fin n
i is Element of n
{i} is non empty trivial finite 1 -element set
B \/ {i} is non empty finite set
j is finite Element of Fin n
X $$ (j,(b * x)) is Element of A
x .: j is finite Element of Fin (Fin K)
Fin (Fin K) is non empty cup-closed diff-closed preBoolean set
X $$ ((x .: j),b) is Element of A
union (x .: j) is set
b . (union (x .: j)) is set
N is set
x . N is set
i is set
x . i is set
X $$ (B,(b * x)) is Element of A
x .: B is finite Element of Fin (Fin K)
X $$ ((x .: B),b) is Element of A
union (x .: B) is set
dom x is non empty Element of bool n
bool n is cup-closed diff-closed preBoolean set
Im (x,i) is set
x .: {i} is finite set
x . i is finite Element of Fin K
{(x . i)} is non empty trivial finite V37() 1 -element set
(x .: B) \/ {(x . i)} is non empty finite set
b . (union (x .: B)) is set
dom (b * x) is non empty Element of bool n
b . (x . i) is Element of A
(b * x) . i is Element of A
N is set
x . N is set
X . ((X $$ ((x .: B),b)),((b * x) . i)) is Element of A
(union (x .: B)) \/ (x . i) is set
N is set
i is set
x . i is set
union {(x . i)} is finite set
(union (x .: B)) \/ (union {(x . i)}) is set
X . ((X $$ ((x .: B),b)),(the_unity_wrt X)) is Element of A
{}. n is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V108() V109() V110() ext-real non positive non negative Element of Fin n
B is finite Element of Fin n
X $$ (B,(b * x)) is Element of A
x .: B is finite Element of Fin (Fin K)
Fin (Fin K) is non empty cup-closed diff-closed preBoolean set
X $$ ((x .: B),b) is Element of A
union (x .: B) is set
b . (union (x .: B)) is set
{}. (Fin K) is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V108() V109() V110() ext-real non positive non negative Element of Fin (Fin K)
B is finite Element of Fin n
X $$ (B,(b * x)) is Element of A
x .: B is finite Element of Fin (Fin K)
Fin (Fin K) is non empty cup-closed diff-closed preBoolean set
X $$ ((x .: B),b) is Element of A
union (x .: B) is set
b . (union (x .: B)) is set
x is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() left_unital doubleLoopStr
the carrier of x is non empty non trivial set
the carrier of x * is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg A is finite A -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= A ) } is set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of A,A, the carrier of x
Deleting (b,n,K) is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of x *
DelLine (b,n) is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of x *
DelCol ((DelLine (b,n)),K) is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of x *
A -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom b is finite Element of bool NAT
len (Deleting (b,n,K)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom (Deleting (b,n,K)) is finite Element of bool NAT
Seg {} is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element {} -element FinSequence-like FinSubsequence-like FinSequence-membered V108() V109() V110() ext-real non positive non negative Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= {} ) } is set
rng (Deleting (b,n,K)) is finite set
B is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
len B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width (Deleting (b,n,K)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (n -' 1) is finite n -' 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n -' 1 ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(x,b,n,K,A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Deleting (A,x,b) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
DelLine (A,x) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
DelCol ((DelLine (A,x)),b) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
n - 1 is V108() V109() V110() set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(x,b,n,K,A) * (j,N) is Element of the carrier of K
A * (j,N) is Element of the carrier of K
N + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A * (j,(N + 1)) is Element of the carrier of K
A * ((j + 1),N) is Element of the carrier of K
A * ((j + 1),(N + 1)) is Element of the carrier of K
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
B + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Seg (B + 1) is non empty finite B + 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= B + 1 ) } is set
Seg B is finite B -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= B ) } is set
len A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom A is finite Element of bool NAT
len (DelLine (A,x)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom (DelLine (A,x)) is finite Element of bool NAT
(Deleting (A,x,b)) . j is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line ((DelLine (A,x)),j) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (DelLine (A,x)) -element FinSequence-like FinSubsequence-like Element of (width (DelLine (A,x))) -tuples_on the carrier of K
width (DelLine (A,x)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width (DelLine (A,x))) -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 (DelLine (A,x)) } is set
DelLine (,(Line ((DelLine (A,x)),j))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len (x,b,n,K,A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom (x,b,n,K,A) is finite Element of bool NAT
(x,b,n,K,A) . j is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line ((x,b,n,K,A),j) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (x,b,n,K,A) -element FinSequence-like FinSubsequence-like Element of (width (x,b,n,K,A)) -tuples_on the carrier of K
width (x,b,n,K,A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width (x,b,n,K,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 (x,b,n,K,A) } is set
(Line ((x,b,n,K,A),j)) . N is set
(DelLine (A,x)) . j is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(x,b,n,K,A) * (COL,L) is Element of the carrier of K
COL + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A * ((COL + 1),L) is Element of the carrier of K
L + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A * ((COL + 1),(L + 1)) is Element of the carrier of K
(DelLine (A,x)) . COL is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
A . (COL + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (A,(COL + 1)) 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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
(Line (A,(COL + 1))) . L is set
Line ((DelLine (A,x)),COL) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (DelLine (A,x)) -element FinSequence-like FinSubsequence-like Element of (width (DelLine (A,x))) -tuples_on the carrier of K
len (Line ((DelLine (A,x)),COL)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom (Line ((DelLine (A,x)),COL)) is finite width (DelLine (A,x)) -element Element of bool NAT
(Line (A,(COL + 1))) . (L + 1) is set
A * (COL,L) is Element of the carrier of K
A * (COL,(L + 1)) is Element of the carrier of K
(DelLine (A,x)) . COL is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
A . COL is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Line (A,COL) 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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
(Line (A,COL)) . L is set
Line ((DelLine (A,x)),COL) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (DelLine (A,x)) -element FinSequence-like FinSubsequence-like Element of (width (DelLine (A,x))) -tuples_on the carrier of K
len (Line ((DelLine (A,x)),COL)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom (Line ((DelLine (A,x)),COL)) is finite width (DelLine (A,x)) -element Element of bool NAT
(Line (A,COL)) . (L + 1) is set
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
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
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(x,b,n,K,A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
(x,b,n,K,A) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
(b,x,n,K,(A @)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
n - 1 is V108() V109() V110() 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
(b,x,n,K,i) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Seg (n -' 1) is finite n -' 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n -' 1 ) } is set
X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
X + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(X + 1) -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[i9,j9] is set
{i9,j9} is non empty finite V37() set
{i9} is non empty trivial finite V37() 1 -element set
{{i9,j9},{i9}} is non empty finite V37() set
j is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Indices j is set
dom j is finite Element of bool NAT
width j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width j) is finite width j -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width j ) } is set
[:(dom j),(Seg (width j)):] is Relation-like finite set
[j9,i9] is set
{j9,i9} is non empty finite V37() set
{j9} is non empty trivial finite V37() 1 -element set
{{j9,i9},{j9}} is non empty finite V37() set
Indices (x,b,n,K,A) is set
dom (x,b,n,K,A) is finite Element of bool NAT
width (x,b,n,K,A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width (x,b,n,K,A)) is finite width (x,b,n,K,A) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (x,b,n,K,A) ) } is set
[:(dom (x,b,n,K,A)),(Seg (width (x,b,n,K,A))):] is Relation-like finite set
j * (i9,j9) is Element of the carrier of K
(x,b,n,K,A) * (j9,i9) is Element of the carrier of K
[:(Seg (n -' 1)),(Seg (n -' 1)):] is Relation-like finite set
i9 + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Indices A is set
dom A is finite Element of bool NAT
width A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width A) is finite width A -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
[:(dom A),(Seg (width A)):] is Relation-like finite set
[:(Seg n),(Seg n):] is Relation-like finite set
j9 + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(b,x,n,K,i) * (i9,j9) is Element of the carrier of K
i * (i9,j9) is Element of the carrier of K
A * (j9,i9) is Element of the carrier of K
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(b,x,n,K,i) * (i9,j9) is Element of the carrier of K
i * ((i9 + 1),j9) is Element of the carrier of K
[j9,(i9 + 1)] is set
{j9,(i9 + 1)} is non empty finite V37() set
{{j9,(i9 + 1)},{j9}} is non empty finite V37() set
A * (j9,(i9 + 1)) is Element of the carrier of K
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(b,x,n,K,i) * (i9,j9) is Element of the carrier of K
i * (i9,(j9 + 1)) is Element of the carrier of K
[(j9 + 1),i9] is set
{(j9 + 1),i9} is non empty finite V37() set
{(j9 + 1)} is non empty trivial finite V37() 1 -element set
{{(j9 + 1),i9},{(j9 + 1)}} is non empty finite V37() set
A * ((j9 + 1),i9) is Element of the carrier of K
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(b,x,n,K,i) * (i9,j9) is Element of the carrier of K
i * ((i9 + 1),(j9 + 1)) is Element of the carrier of K
[(j9 + 1),(i9 + 1)] is set
{(j9 + 1),(i9 + 1)} is non empty finite V37() set
{(j9 + 1)} is non empty trivial finite V37() 1 -element set
{{(j9 + 1),(i9 + 1)},{(j9 + 1)}} is non empty finite V37() set
A * ((j9 + 1),(i9 + 1)) is Element of the carrier of K
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
x is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
ReplaceLine (A,b,x) 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
X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(b,X,n,K,A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(b,X,n,K,(ReplaceLine (A,b,x))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Deleting (A,b,X) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
DelLine (A,b) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
DelCol ((DelLine (A,b)),X) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
Deleting ((ReplaceLine (A,b,x)),b,X) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
DelLine ((ReplaceLine (A,b,x)),b) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
DelCol ((DelLine ((ReplaceLine (A,b,x)),b)),X) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
len x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ReplaceLine (A,i,x) 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 Element of the carrier of K *
Replace (A,b,B) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K *
len x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is non empty set
x * is functional non empty FinSequence-membered FinSequenceSet of x
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
X is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of x
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
b is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,A,x
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width b ) } is set
[:(dom b),(Seg (width b)):] is Relation-like finite set
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg i is finite i -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i ) } is set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg j is finite j -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= j ) } is set
[:(Seg i),(Seg j):] is Relation-like finite set
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[N,i] is set
{N,i} is non empty finite V37() set
{N} is non empty trivial finite V37() 1 -element set
{{N,i},{N}} is non empty finite V37() set
rng X is finite set
Seg (len X) is finite len X -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
dom X is finite Element of bool NAT
X . N is set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b * (j,COL) is Element of x
X . j is set
b * (N,i) is Element of x
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b * (j,COL) is Element of x
X . j is set
j is Element of x
COL is Element of x
N is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of i,j,x
Indices N is set
dom N is finite Element of bool NAT
width N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width N) is finite width N -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width N ) } is set
[:(dom N),(Seg (width N)):] is Relation-like finite set
i is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,A,x
len i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
B is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of len b, width b,x
Indices B is set
dom B is finite Element of bool NAT
width B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width B) is finite width B -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
Indices i is set
dom i is finite Element of bool NAT
Seg (width i) is finite width i -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width i ) } is set
[:(dom i),(Seg (width i)):] is Relation-like finite set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[j,COL] is set
{j,COL} is non empty finite V37() set
{j} is non empty trivial finite V37() 1 -element set
{{j,COL},{j}} is non empty finite V37() set
i * (j,COL) is Element of x
b * (j,COL) is Element of x
i * (j,n) is Element of x
X . j is set
B is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,A,x
len B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,A,x
len i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[j,N] is set
{j,N} is non empty finite V37() set
{j} is non empty trivial finite V37() 1 -element set
{{j,N},{j}} is non empty finite V37() set
B * (j,N) is Element of x
i * (j,N) is Element of x
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
B * (i,n) is Element of x
X . i is set
i * (i,j) is Element of x
b * (i,j) is Element of x
B * (i,j) is Element of x
x is non empty set
x * is functional non empty FinSequence-membered FinSequenceSet of x
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x is non empty set
x * is functional non empty FinSequence-membered FinSequenceSet of x
b is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,x
width b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width b) is finite width b -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width b ) } is set
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
X is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of x
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(A,n,K,x,b,X) is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,x
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Col ((A,n,K,x,b,X),B) is Relation-like NAT -defined x -valued Function-like finite len (A,n,K,x,b,X) -element FinSequence-like FinSubsequence-like Element of (len (A,n,K,x,b,X)) -tuples_on x
len (A,n,K,x,b,X) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len (A,n,K,x,b,X)) -tuples_on x is functional non empty FinSequence-membered FinSequenceSet of x
{ b₁ where b₁ is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like Element of x * : len b₁ = len (A,n,K,x,b,X) } is set
Col (b,B) is Relation-like NAT -defined x -valued Function-like finite len b -element FinSequence-like FinSubsequence-like Element of (len b) -tuples_on x
(len b) -tuples_on x is functional non empty FinSequence-membered FinSequenceSet of x
{ b₁ where b₁ is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like Element of x * : len b₁ = len b } is set
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg (len X) is finite len X -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len X ) } is set
dom (A,n,K,x,b,X) is finite Element of bool NAT
width (A,n,K,x,b,X) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width (A,n,K,x,b,X)) is finite width (A,n,K,x,b,X) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (A,n,K,x,b,X) ) } is set
[N,A] is set
{N,A} is non empty finite V37() set
{N} is non empty trivial finite V37() 1 -element set
{{N,A},{N}} is non empty finite V37() set
Indices (A,n,K,x,b,X) is set
[:(dom (A,n,K,x,b,X)),(Seg (width (A,n,K,x,b,X))):] is Relation-like finite set
Indices b is set
dom b is finite Element of bool NAT
[:(dom b),(Seg (width b)):] is Relation-like finite set
(Col ((A,n,K,x,b,X),B)) . N is set
(A,n,K,x,b,X) * (N,A) is Element of x
X . N is set
len (Col ((A,n,K,x,b,X),B)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len (Col (b,B)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg (len b) is finite len b -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len b ) } is set
dom b is finite Element of bool NAT
(Col (b,B)) . i is set
b * (i,B) is Element of x
dom (A,n,K,x,b,X) is finite Element of bool NAT
(Col ((A,n,K,x,b,X),B)) . i is set
(A,n,K,x,b,X) * (i,B) is Element of x
[i,B] is set
{i,B} is non empty finite V37() set
{i} is non empty trivial finite V37() 1 -element set
{{i,B},{i}} is non empty finite V37() set
Indices b is set
[:(dom b),(Seg (width b)):] is Relation-like finite set
len (Col ((A,n,K,x,b,X),B)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x is non empty set
x * is functional non empty FinSequence-membered FinSequenceSet of x
b is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,x
width b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width b) is finite width b -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width b ) } is set
X is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of x
(A,n,K,x,b,X) is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,x
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[i,j] is set
{i,j} is non empty finite V37() set
{i} is non empty trivial finite V37() 1 -element set
{{i,j},{i}} is non empty finite V37() set
Indices b is set
dom b is finite Element of bool NAT
[:(dom b),(Seg (width b)):] is Relation-like finite set
(A,n,K,x,b,X) * (i,j) is Element of x
b * (i,j) is Element of x
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x is non empty set
x * is functional non empty FinSequence-membered FinSequenceSet of x
b is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,x
Col (b,A) is Relation-like NAT -defined x -valued Function-like finite len b -element FinSequence-like FinSubsequence-like Element of (len b) -tuples_on x
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len b) -tuples_on x is functional non empty FinSequence-membered FinSequenceSet of x
{ b₁ where b₁ is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like Element of x * : len b₁ = len b } is set
(A,n,K,x,b,(Col (b,A))) is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,x
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[j,N] is set
{j,N} is non empty finite V37() set
{j} is non empty trivial finite V37() 1 -element set
{{j,N},{j}} is non empty finite V37() set
Indices b is set
dom b is finite Element of bool NAT
width b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width b) is finite width b -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width b ) } is set
[:(dom b),(Seg (width b)):] is Relation-like finite set
len (Col (b,A)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(A,n,K,x,b,(Col (b,A))) * (j,N) is Element of x
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(Col (b,A)) . i is set
b * (j,N) is Element of x
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(A,n,K,x,b,(Col (b,A))) * (j,N) is Element of x
b * (j,N) is Element of x
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(A,n,K,x,b,(Col (b,A))) * (j,N) is Element of x
b * (j,N) is Element of x
(A,n,K,x,b,(Col (b,A))) * (j,N) is Element of x
b * (j,N) is Element of x
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x is non empty set
x * is functional non empty FinSequence-membered FinSequenceSet of x
b is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of x
X is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,x
X @ is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of x *
(A,n,K,x,X,b) is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,x
B is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,n,x
ReplaceLine (B,A,b) is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,n,x
(ReplaceLine (B,A,b)) @ is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of x *
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len (ReplaceLine (B,A,b)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width (ReplaceLine (B,A,b)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len ((ReplaceLine (B,A,b)) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len (A,n,K,x,X,b) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(X @) @ is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of x *
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width (ReplaceLine (B,A,b)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len ((ReplaceLine (B,A,b)) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len (ReplaceLine (B,A,b)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width ((ReplaceLine (B,A,b)) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Indices (A,n,K,x,X,b) is set
dom (A,n,K,x,X,b) is finite Element of bool NAT
width (A,n,K,x,X,b) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width (A,n,K,x,X,b)) is finite width (A,n,K,x,X,b) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (A,n,K,x,X,b) ) } is set
[:(dom (A,n,K,x,X,b)),(Seg (width (A,n,K,x,X,b))):] is Relation-like finite set
Indices X is set
dom X is finite Element of bool NAT
width X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width X) is finite width X -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width X ) } is set
[:(dom X),(Seg (width X)):] is Relation-like finite set
Indices (ReplaceLine (B,A,b)) is set
dom (ReplaceLine (B,A,b)) is finite Element of bool NAT
Seg (width (ReplaceLine (B,A,b))) is finite width (ReplaceLine (B,A,b)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (ReplaceLine (B,A,b)) ) } is set
[:(dom (ReplaceLine (B,A,b))),(Seg (width (ReplaceLine (B,A,b)))):] is Relation-like finite set
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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width B ) } is set
[:(dom B),(Seg (width B)):] is Relation-like finite set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[i,j] is set
{i,j} is non empty finite V37() set
{i} is non empty trivial finite V37() 1 -element set
{{i,j},{i}} is non empty finite V37() set
N is Relation-like NAT -defined x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,K,x
Indices N is set
dom N is finite Element of bool NAT
width N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width N) is finite width N -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width N ) } is set
[:(dom N),(Seg (width N)):] is Relation-like finite set
[j,i] is set
{j,i} is non empty finite V37() set
{j} is non empty trivial finite V37() 1 -element set
{{j,i},{j}} is non empty finite V37() set
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(A,n,K,x,X,b) * (i,j) is Element of x
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
b . COL is set
(ReplaceLine (B,A,b)) * (L,COL) is Element of x
N * (i,j) is Element of x
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(A,n,K,x,X,b) * (i,j) is Element of x
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
X * (COL,L) is Element of x
B * (j,i) is Element of x
(ReplaceLine (B,A,b)) * (L,COL) is Element of x
N * (i,j) is Element of x
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Permutations (K + 1) is non empty permutational set
len (Permutations (K + 1)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (K + 1))) is finite len (Permutations (K + 1)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (K + 1)) ) } is set
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg (K + 1) is non empty finite K + 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K + 1 ) } is set
Permutations K is non empty permutational set
len (Permutations K) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations K)) is finite len (Permutations K) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations K) ) } is set
Seg K is finite K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
A is Relation-like Seg (len (Permutations (K + 1))) -defined Seg (len (Permutations (K + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (K + 1)))) quasi_total onto bijective finite Element of Permutations (K + 1)
A . n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[:(Seg (K + 1)),(Seg (K + 1)):] is Relation-like finite set
bool [:(Seg (K + 1)),(Seg (K + 1)):] is cup-closed diff-closed preBoolean finite V37() set
j is Relation-like Seg (K + 1) -defined Seg (K + 1) -valued Function-like one-to-one non empty V14( Seg (K + 1)) quasi_total onto bijective finite Element of bool [:(Seg (K + 1)),(Seg (K + 1)):]
dom j is non empty finite Element of bool (Seg (K + 1))
bool (Seg (K + 1)) is cup-closed diff-closed preBoolean finite V37() set
rng j is non empty finite set
N is set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
A . i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
1 + {} is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A . (i + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . COL) - 1 is V108() V109() V110() set
COL + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A . (COL + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . (COL + 1)) - 1 is V108() V109() V110() set
j . i is set
j . n is set
(A . i) - 1 is V108() V109() V110() set
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . L) - 1 is V108() V109() V110() set
L + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . (L + 1)) - 1 is V108() V109() V110() set
COL + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . COL) - 1 is V108() V109() V110() set
COL + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A . (COL + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . (COL + 1)) - 1 is V108() V109() V110() set
j . (i + 1) is set
j . n is set
(A . (i + 1)) - 1 is V108() V109() V110() set
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . L) - 1 is V108() V109() V110() set
L + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . (L + 1)) - 1 is V108() V109() V110() set
COL + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
[:(Seg K),(Seg K):] is Relation-like finite set
bool [:(Seg K),(Seg K):] is cup-closed diff-closed preBoolean finite V37() set
N is Relation-like Seg K -defined Seg K -valued Function-like V14( Seg K) quasi_total finite Element of bool [:(Seg K),(Seg K):]
dom N is finite Element of bool (Seg K)
bool (Seg K) is cup-closed diff-closed preBoolean finite V37() set
i is set
j is set
N . i is set
N . j is set
COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
{} + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
COL + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
L + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A . COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
N . COL is set
j . COL is set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j . L is set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
N . L is set
(A . L) - 1 is V108() V109() V110() set
(A . COL) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j . L is set
j . n is set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j . (L + 1) is set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . (L + 1)) - 1 is V108() V109() V110() set
(A . COL) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j . (L + 1) is set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
N . COL is set
(A . COL) - 1 is V108() V109() V110() set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
N . L is set
j . L is set
(A . L) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . L) - 1 is V108() V109() V110() set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . (L + 1)) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j . COL is set
j . n is set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . (L + 1)) - 1 is V108() V109() V110() set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . (COL + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
N . COL is set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j . (COL + 1) is set
j . L is set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . L) - 1 is V108() V109() V110() set
(A . (COL + 1)) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j . L is set
j . n is set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
N . L is set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . (L + 1)) - 1 is V108() V109() V110() set
(A . (COL + 1)) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j . (L + 1) is set
j . n is set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . (COL + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
N . COL is set
(A . (COL + 1)) - 1 is V108() V109() V110() set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . L) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j . (COL + 1) is set
j . n is set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . L) - 1 is V108() V109() V110() set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . (L + 1)) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j . (COL + 1) is set
j . n is set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . (L + 1)) - 1 is V108() V109() V110() set
A . L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . (L + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . COL is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . (COL + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
finSeg i is finite i -element Element of Fin NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i ) } is set
card (finSeg i) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i is Relation-like Seg (len (Permutations K)) -defined Seg (len (Permutations K)) -valued Function-like one-to-one V14( Seg (len (Permutations K))) quasi_total onto bijective finite Element of Permutations K
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i . j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . j) - 1 is V108() V109() V110() set
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A . (j + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . (j + 1)) - 1 is V108() V109() V110() set
b is Relation-like Seg (len (Permutations K)) -defined Seg (len (Permutations K)) -valued Function-like one-to-one V14( Seg (len (Permutations K))) quasi_total onto bijective finite Element of Permutations K
X is Relation-like Seg (len (Permutations K)) -defined Seg (len (Permutations K)) -valued Function-like one-to-one V14( Seg (len (Permutations K))) quasi_total onto bijective finite Element of Permutations K
[:(Seg K),(Seg K):] is Relation-like finite set
bool [:(Seg K),(Seg K):] is cup-closed diff-closed preBoolean finite V37() set
dom b is finite Element of bool (Seg (len (Permutations K)))
bool (Seg (len (Permutations K))) is cup-closed diff-closed preBoolean finite V37() set
B is set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
A . i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A . (i + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b . i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
X . i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A . i) - 1 is V108() V109() V110() set
(A . (i + 1)) - 1 is V108() V109() V110() set
b . B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
X . B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
dom X is finite Element of bool (Seg (len (Permutations K)))
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
n + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Seg (n + 1) is non empty finite n + 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n + 1 ) } is set
Permutations (n + 1) is non empty permutational set
len (Permutations (n + 1)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (n + 1))) is finite len (Permutations (n + 1)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (n + 1)) ) } is set
Permutations n is non empty permutational set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{ b₁ where b₁ is Relation-like Seg (len (Permutations (n + 1))) -defined Seg (len (Permutations (n + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (n + 1)))) quasi_total onto bijective finite Element of Permutations (n + 1) : b₁ . K = A } is set
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
b + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Permutations (b + 1) is non empty permutational set
len (Permutations (b + 1)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (b + 1))) is finite len (Permutations (b + 1)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (b + 1)) ) } is set
B is set
[:B,(Permutations n):] is Relation-like set
bool [:B,(Permutations n):] is cup-closed diff-closed preBoolean set
i is set
j is Relation-like Seg (len (Permutations (b + 1))) -defined Seg (len (Permutations (b + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (b + 1)))) quasi_total onto bijective finite Element of Permutations (b + 1)
j . K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,j) is Relation-like Seg (len (Permutations b)) -defined Seg (len (Permutations b)) -valued Function-like one-to-one V14( Seg (len (Permutations b))) quasi_total onto bijective finite Element of Permutations b
Permutations b is non empty permutational set
len (Permutations b) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations b)) is finite len (Permutations b) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations b) ) } is set
N is Relation-like Seg (len (Permutations (b + 1))) -defined Seg (len (Permutations (b + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (b + 1)))) quasi_total onto bijective finite Element of Permutations (b + 1)
N . K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,N) is Relation-like Seg (len (Permutations b)) -defined Seg (len (Permutations b)) -valued Function-like one-to-one V14( Seg (len (Permutations b))) quasi_total onto bijective finite Element of Permutations b
i is Relation-like B -defined Permutations n -valued Function-like V14(B) quasi_total Element of bool [:B,(Permutations n):]
j is set
N is set
i . j is set
i . N is set
i is Relation-like Seg (len (Permutations (b + 1))) -defined Seg (len (Permutations (b + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (b + 1)))) quasi_total onto bijective finite Element of Permutations (b + 1)
i . K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,i) is Relation-like Seg (len (Permutations b)) -defined Seg (len (Permutations b)) -valued Function-like one-to-one V14( Seg (len (Permutations b))) quasi_total onto bijective finite Element of Permutations b
Permutations b is non empty permutational set
len (Permutations b) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations b)) is finite len (Permutations b) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations b) ) } is set
COL is Relation-like Seg (len (Permutations (b + 1))) -defined Seg (len (Permutations (b + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (b + 1)))) quasi_total onto bijective finite Element of Permutations (b + 1)
COL . K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,COL) is Relation-like Seg (len (Permutations b)) -defined Seg (len (Permutations b)) -valued Function-like one-to-one V14( Seg (len (Permutations b))) quasi_total onto bijective finite Element of Permutations b
[:(Seg (n + 1)),(Seg (n + 1)):] is Relation-like finite set
bool [:(Seg (n + 1)),(Seg (n + 1)):] is cup-closed diff-closed preBoolean finite V37() set
j9 is Relation-like Seg (n + 1) -defined Seg (n + 1) -valued Function-like one-to-one non empty V14( Seg (n + 1)) quasi_total onto bijective finite Element of bool [:(Seg (n + 1)),(Seg (n + 1)):]
dom j9 is non empty finite Element of bool (Seg (n + 1))
bool (Seg (n + 1)) is cup-closed diff-closed preBoolean finite V37() set
i9 is Relation-like Seg (n + 1) -defined Seg (n + 1) -valued Function-like one-to-one non empty V14( Seg (n + 1)) quasi_total onto bijective finite Element of bool [:(Seg (n + 1)),(Seg (n + 1)):]
dom i9 is non empty finite Element of bool (Seg (n + 1))
Laa is set
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
i . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,i) . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,COL) . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(i . x) - 1 is V108() V109() V110() set
(COL . x) - 1 is V108() V109() V110() set
i . Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,i) . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,COL) . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(COL . x) - 1 is V108() V109() V110() set
(i . x) - 1 is V108() V109() V110() set
(i . x) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(COL . x) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j9 . x is set
j9 . K is set
i9 . x is set
i9 . K is set
i . Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i . Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x - 1 is V108() V109() V110() set
y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
y + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
i . (y + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . (y + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,i) . y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,COL) . y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(i . x) - 1 is V108() V109() V110() set
(COL . x) - 1 is V108() V109() V110() set
i . Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i . (y + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . (y + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,i) . y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,b,COL) . y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(COL . x) - 1 is V108() V109() V110() set
(i . x) - 1 is V108() V109() V110() set
(i . x) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(COL . x) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
j9 . x is set
j9 . K is set
i9 . x is set
i9 . K is set
i . Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i . (y + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . (y + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Permutations b is non empty permutational set
len (Permutations b) is non empty V26() V27() V28() cardinal set
b ! is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len B is V26() V27() V28() cardinal set
N is finite set
card N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n ! is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i is finite set
card i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j is Relation-like Seg (len (Permutations (n + 1))) -defined Seg (len (Permutations (n + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (n + 1)))) quasi_total onto bijective finite Element of Permutations (n + 1)
j . K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i . j is set
(K,n,j) is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n
len (Permutations n) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations n) ) } is set
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
n + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Permutations (n + 1) is non empty permutational set
len (Permutations (n + 1)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (n + 1))) is finite len (Permutations (n + 1)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (n + 1)) ) } is set
Seg (n + 1) is non empty finite n + 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n + 1 ) } is set
K is Relation-like Seg (len (Permutations (n + 1))) -defined Seg (len (Permutations (n + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (n + 1)))) quasi_total onto bijective finite Element of Permutations (n + 1)
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() left_unital doubleLoopStr
the carrier of A is non empty non trivial set
power A is Relation-like [: the carrier of A,NAT:] -defined the carrier of A -valued Function-like V14([: the carrier of A,NAT:]) quasi_total Element of bool [:[: the carrier of A,NAT:], the carrier of A:]
[: the carrier of A,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of A,NAT:], the carrier of A:] is Relation-like non trivial non finite set
bool [:[: the carrier of A,NAT:], the carrier of A:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ A is Element of the carrier of A
1. A is V52(A) Element of the carrier of A
- (1_ A) is Element of the carrier of A
x is Element of the carrier of A
- (x,K) is Element of the carrier of A
X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K . X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(X,n,K) is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n
Permutations n is non empty permutational set
len (Permutations n) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations n) ) } is set
- (x,(X,n,K)) is Element of the carrier of A
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
X + B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A,(power A),(- (1_ A)),(X + B)) is Element of the carrier of A
(A,(power A),(- (1_ A)),(X + B)) * (- (x,(X,n,K))) is Element of the carrier of A
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like V14([: the carrier of A, the carrier of A:]) 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:] is Relation-like set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is cup-closed diff-closed preBoolean set
the multF of A . ((A,(power A),(- (1_ A)),(X + B)),(- (x,(X,n,K)))) is Element of the carrier of A
[:(Seg (n + 1)),(Seg (n + 1)):] is Relation-like finite set
bool [:(Seg (n + 1)),(Seg (n + 1)):] is cup-closed diff-closed preBoolean finite V37() set
rng K is finite set
dom K is finite Element of bool (Seg (len (Permutations (n + 1))))
bool (Seg (len (Permutations (n + 1)))) is cup-closed diff-closed preBoolean finite V37() set
1 + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
2 * 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
2 * 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
2 * 2 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Permutations 2 is non empty permutational set
len (Permutations 2) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Permutations 2 is non empty permutational set
len (Permutations 2) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
- x is Element of the carrier of A
(1_ A) * x is Element of the carrier of A
the multF of A . ((1_ A),x) is Element of the carrier of A
- ((1_ A) * x) is Element of the carrier of A
2 * 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(2 * 1) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
n - 2 is V108() V109() V110() set
(X + B) mod 2 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
2 * N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(2 * N) + {} is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(X + B) mod 2 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
2 * N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(2 * N) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(X + B) mod 2 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
- x is Element of the carrier of A
x * (1_ A) is Element of the carrier of A
the multF of A . (x,(1_ A)) is Element of the carrier of A
- (x * (1_ A)) is Element of the carrier of A
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i + 2 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Permutations (i + 2) is non empty permutational set
len (Permutations (i + 2)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (i + 2))) is finite len (Permutations (i + 2)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (i + 2)) ) } is set
Seg (i + 2) is non empty finite i + 2 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i + 2 ) } is set
2Set (Seg (i + 2)) is non empty finite set
Fin (2Set (Seg (i + 2))) is non empty cup-closed diff-closed preBoolean set
{ {b₁,X} where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : {b₁,X} in 2Set (Seg (i + 2)) } is set
i + 2 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Seg (i + 2) is non empty finite i + 2 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= i + 2 ) } is set
2Set (Seg (i + 2)) is non empty finite set
COL is Relation-like Seg (len (Permutations (i + 2))) -defined Seg (len (Permutations (i + 2))) -valued Function-like one-to-one V14( Seg (len (Permutations (i + 2)))) quasi_total onto bijective finite Element of Permutations (i + 2)
Part_sgn (COL,A) is Relation-like 2Set (Seg (i + 2)) -defined the carrier of A -valued Function-like non empty V14( 2Set (Seg (i + 2))) quasi_total finite Element of bool [:(2Set (Seg (i + 2))), the carrier of A:]
[:(2Set (Seg (i + 2))), the carrier of A:] is Relation-like set
bool [:(2Set (Seg (i + 2))), the carrier of A:] is cup-closed diff-closed preBoolean set
i9 is finite Element of Fin (2Set (Seg (i + 2)))
the multF of A $$ (i9,(Part_sgn (COL,A))) is Element of the carrier of A
j + 2 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Seg (j + 2) is non empty finite j + 2 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= j + 2 ) } is set
2Set (Seg (j + 2)) is non empty finite set
[:(Seg (i + 2)),(Seg (i + 2)):] is Relation-like finite set
bool [:(Seg (i + 2)),(Seg (i + 2)):] is cup-closed diff-closed preBoolean finite V37() set
Permutations (j + 2) is non empty permutational set
len (Permutations (j + 2)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (j + 2))) is finite len (Permutations (j + 2)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (j + 2)) ) } is set
[:(Seg (j + 2)),(Seg (j + 2)):] is Relation-like finite set
bool [:(Seg (j + 2)),(Seg (j + 2)):] is cup-closed diff-closed preBoolean finite V37() set
z is Relation-like Seg (len (Permutations (j + 2))) -defined Seg (len (Permutations (j + 2))) -valued Function-like one-to-one V14( Seg (len (Permutations (j + 2)))) quasi_total onto bijective finite Element of Permutations (j + 2)
Part_sgn (z,A) is Relation-like 2Set (Seg (j + 2)) -defined the carrier of A -valued Function-like non empty V14( 2Set (Seg (j + 2))) quasi_total finite Element of bool [:(2Set (Seg (j + 2))), the carrier of A:]
j + 2 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Seg (j + 2) is non empty finite j + 2 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= j + 2 ) } is set
2Set (Seg (j + 2)) is non empty finite set
[:(2Set (Seg (j + 2))), the carrier of A:] is Relation-like set
bool [:(2Set (Seg (j + 2))), the carrier of A:] is cup-closed diff-closed preBoolean set
FinOmega (2Set (Seg (i + 2))) is finite Element of Fin (2Set (Seg (i + 2)))
(2Set (Seg (i + 2))) \ i9 is finite Element of bool (2Set (Seg (i + 2)))
bool (2Set (Seg (i + 2))) is cup-closed diff-closed preBoolean finite V37() set
Q is finite Element of Fin (2Set (Seg (i + 2)))
i9 \/ Q is finite Element of Fin (2Set (Seg (i + 2)))
(2Set (Seg (i + 2))) \/ i9 is non empty finite set
[:(2Set (Seg (j + 2))),(2Set (Seg (i + 2))):] is Relation-like finite set
bool [:(2Set (Seg (j + 2))),(2Set (Seg (i + 2))):] is cup-closed diff-closed preBoolean finite V37() set
Q19 is Relation-like 2Set (Seg (j + 2)) -defined 2Set (Seg (i + 2)) -valued Function-like non empty V14( 2Set (Seg (j + 2))) quasi_total finite Element of bool [:(2Set (Seg (j + 2))),(2Set (Seg (i + 2))):]
rng Q19 is non empty finite set
(Part_sgn (COL,A)) * Q19 is Relation-like 2Set (Seg (j + 2)) -defined the carrier of A -valued Function-like non empty V14( 2Set (Seg (j + 2))) quasi_total finite Element of bool [:(2Set (Seg (j + 2))), the carrier of A:]
[:(2Set (Seg (j + 2))), the carrier of A:] is Relation-like set
bool [:(2Set (Seg (j + 2))), the carrier of A:] is cup-closed diff-closed preBoolean set
dom ((Part_sgn (COL,A)) * Q19) is non empty finite Element of bool (2Set (Seg (j + 2)))
bool (2Set (Seg (j + 2))) is cup-closed diff-closed preBoolean finite V37() set
x is Relation-like Seg (i + 2) -defined Seg (i + 2) -valued Function-like one-to-one non empty V14( Seg (i + 2)) quasi_total onto bijective finite Element of bool [:(Seg (i + 2)),(Seg (i + 2)):]
dom x is non empty finite Element of bool (Seg (i + 2))
bool (Seg (i + 2)) is cup-closed diff-closed preBoolean finite V37() set
domf is set
k is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
m is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{k,m} is non empty finite V37() set
ProjD is Relation-like Seg (j + 2) -defined Seg (j + 2) -valued Function-like one-to-one non empty V14( Seg (j + 2)) quasi_total onto bijective finite Element of bool [:(Seg (j + 2)),(Seg (j + 2)):]
dom ProjD is non empty finite Element of bool (Seg (j + 2))
bool (Seg (j + 2)) is cup-closed diff-closed preBoolean finite V37() set
k is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
ProjD . k is set
m is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
z . m is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
z . k is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
m + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
k + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(j + 2) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
((Part_sgn (COL,A)) * Q19) . domf is set
Q19 . domf is set
(Part_sgn (COL,A)) . (Q19 . domf) is set
COL . k is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . m is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(COL . k) - 1 is V108() V109() V110() set
(COL . m) - 1 is V108() V109() V110() set
(Part_sgn (COL,A)) . domf is set
(Part_sgn (z,A)) . domf is set
COL . k is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . m is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(COL . m) - 1 is V108() V109() V110() set
(z . m) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
x . m is set
(Part_sgn (COL,A)) . domf is set
(Part_sgn (z,A)) . domf is set
COL . k is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . m is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(COL . k) - 1 is V108() V109() V110() set
(z . k) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
x . k is set
(Part_sgn (COL,A)) . domf is set
(Part_sgn (z,A)) . domf is set
COL . k is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . m is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
((Part_sgn (COL,A)) * Q19) . domf is set
(Part_sgn (z,A)) . domf is set
((Part_sgn (COL,A)) * Q19) . domf is set
{k,m} is non empty finite V37() set
Q19 . {k,m} is set
(Part_sgn (COL,A)) . (Q19 . {k,m}) is set
{k,(m + 1)} is non empty finite V37() set
COL . k is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . (m + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(COL . k) - 1 is V108() V109() V110() set
(COL . (m + 1)) - 1 is V108() V109() V110() set
(Part_sgn (COL,A)) . {k,(m + 1)} is set
(Part_sgn (z,A)) . domf is set
COL . k is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . (m + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x . (m + 1) is set
(COL . (m + 1)) - 1 is V108() V109() V110() set
(z . m) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(Part_sgn (COL,A)) . {k,(m + 1)} is set
(Part_sgn (z,A)) . domf is set
COL . k is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . (m + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(COL . k) - 1 is V108() V109() V110() set
(z . k) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
x . k is set
(Part_sgn (COL,A)) . {k,(m + 1)} is set
(Part_sgn (z,A)) . domf is set
COL . k is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . (m + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
((Part_sgn (COL,A)) * Q19) . domf is set
{k,m} is non empty finite V37() set
Q19 . {k,m} is set
(Part_sgn (COL,A)) . (Q19 . {k,m}) is set
{(k + 1),(m + 1)} is non empty finite V37() set
COL . (k + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . (m + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(COL . (k + 1)) - 1 is V108() V109() V110() set
(COL . (m + 1)) - 1 is V108() V109() V110() set
(Part_sgn (COL,A)) . {(k + 1),(m + 1)} is set
(Part_sgn (z,A)) . domf is set
{(m + 1),(k + 1)} is non empty finite V37() set
(Part_sgn (COL,A)) . {(m + 1),(k + 1)} is set
COL . (k + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . (m + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x . (m + 1) is set
(COL . (m + 1)) - 1 is V108() V109() V110() set
(z . m) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(Part_sgn (COL,A)) . {(k + 1),(m + 1)} is set
(Part_sgn (z,A)) . domf is set
COL . (k + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . (m + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x . (k + 1) is set
(COL . (k + 1)) - 1 is V108() V109() V110() set
(z . k) + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(Part_sgn (COL,A)) . {(k + 1),(m + 1)} is set
(Part_sgn (z,A)) . domf is set
COL . (k + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL . (m + 1) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Fin (2Set (Seg (j + 2))) is non empty cup-closed diff-closed preBoolean set
dom Q19 is non empty finite Element of bool (2Set (Seg (j + 2)))
domf is finite Element of Fin (2Set (Seg (j + 2)))
Q19 .: domf is finite Element of Fin (2Set (Seg (i + 2)))
FinOmega (2Set (Seg (j + 2))) is finite Element of Fin (2Set (Seg (j + 2)))
dom (Part_sgn (z,A)) is non empty finite Element of bool (2Set (Seg (j + 2)))
bool (2Set (Seg (j + 2))) is cup-closed diff-closed preBoolean finite V37() set
the multF of A $$ (Q,(Part_sgn (COL,A))) is Element of the carrier of A
sgn (z,A) is Element of the carrier of A
FinOmega (2Set (Seg (j + 2))) is finite Element of Fin (2Set (Seg (j + 2)))
Fin (2Set (Seg (j + 2))) is non empty cup-closed diff-closed preBoolean set
the multF of A $$ ((FinOmega (2Set (Seg (j + 2)))),(Part_sgn (z,A))) is Element of the carrier of A
sgn (COL,A) is Element of the carrier of A
FinOmega (2Set (Seg (i + 2))) is finite Element of Fin (2Set (Seg (i + 2)))
Fin (2Set (Seg (i + 2))) is non empty cup-closed diff-closed preBoolean set
the multF of A $$ ((FinOmega (2Set (Seg (i + 2)))),(Part_sgn (COL,A))) is Element of the carrier of A
(A,(power A),(- (1_ A)),(X + B)) * (sgn (z,A)) is Element of the carrier of A
the multF of A . ((A,(power A),(- (1_ A)),(X + B)),(sgn (z,A))) is Element of the carrier of A
((A,(power A),(- (1_ A)),(X + B)) * (sgn (z,A))) * x is Element of the carrier of A
the multF of A . (((A,(power A),(- (1_ A)),(X + B)) * (sgn (z,A))),x) is Element of the carrier of A
(sgn (z,A)) * x is Element of the carrier of A
the multF of A . ((sgn (z,A)),x) is Element of the carrier of A
(A,(power A),(- (1_ A)),(X + B)) * ((sgn (z,A)) * x) is Element of the carrier of A
the multF of A . ((A,(power A),(- (1_ A)),(X + B)),((sgn (z,A)) * x)) is Element of the carrier of A
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
n + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Permutations (n + 1) is non empty permutational set
len (Permutations (n + 1)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (n + 1))) is finite len (Permutations (n + 1)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (n + 1)) ) } is set
Seg (n + 1) is non empty finite n + 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n + 1 ) } is set
Permutations n is non empty permutational set
K is Relation-like Seg (len (Permutations (n + 1))) -defined Seg (len (Permutations (n + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (n + 1)))) quasi_total onto bijective finite Element of Permutations (n + 1)
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
power A is Relation-like [: the carrier of A,NAT:] -defined the carrier of A -valued Function-like V14([: the carrier of A,NAT:]) quasi_total Element of bool [:[: the carrier of A,NAT:], the carrier of A:]
[: the carrier of A,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of A,NAT:], the carrier of A:] is Relation-like non trivial non finite set
bool [:[: the carrier of A,NAT:], the carrier of A:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ A is Element of the carrier of A
1. A is V52(A) Element of the carrier of A
- (1_ A) is Element of the carrier of A
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K . X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
X + B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A,(power A),(- (1_ A)),(X + B)) is Element of the carrier of A
(X,n,K) is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n
len (Permutations n) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations n) ) } is set
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like V14([: the carrier of A, the carrier of A:]) 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:] is Relation-like set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is cup-closed diff-closed preBoolean set
N is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n + 1,n + 1, the carrier of A
(X,B,(n + 1),A,N) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of (n + 1) -' 1,(n + 1) -' 1, the carrier of A
(n + 1) -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Path_product N is Relation-like Permutations (n + 1) -defined the carrier of A -valued Function-like non empty V14( Permutations (n + 1)) quasi_total Element of bool [:(Permutations (n + 1)), the carrier of A:]
[:(Permutations (n + 1)), the carrier of A:] is Relation-like set
bool [:(Permutations (n + 1)), the carrier of A:] is cup-closed diff-closed preBoolean set
(Path_product N) . K is Element of the carrier of A
N * (X,B) is Element of the carrier of A
(A,(power A),(- (1_ A)),(X + B)) * (N * (X,B)) is Element of the carrier of A
the multF of A . ((A,(power A),(- (1_ A)),(X + B)),(N * (X,B))) is Element of the carrier of A
i is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of A
Path_product i is Relation-like Permutations n -defined the carrier of A -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of A:]
[:(Permutations n), the carrier of A:] is Relation-like set
bool [:(Permutations n), the carrier of A:] is cup-closed diff-closed preBoolean set
(Path_product i) . (X,n,K) is Element of the carrier of A
((A,(power A),(- (1_ A)),(X + B)) * (N * (X,B))) * ((Path_product i) . (X,n,K)) is Element of the carrier of A
the multF of A . (((A,(power A),(- (1_ A)),(X + B)) * (N * (X,B))),((Path_product i) . (X,n,K))) is Element of the carrier of A
Path_matrix ((X,n,K),i) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
Path_matrix (K,N) is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
len (Path_matrix (K,N)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom (Path_matrix (K,N)) is finite Element of bool NAT
Seg (x + 1) is non empty finite x + 1 -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x + 1 ) } is set
(Path_matrix (K,N)) . X is set
(Path_matrix (K,N)) . 1 is set
<*(N * (X,B))*> is Relation-like NAT -defined the carrier of A -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of A
1 -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 = 1 } is set
[1,(N * (X,B))] is set
{1,(N * (X,B))} is non empty finite set
{{1,(N * (X,B))},{1}} is non empty finite V37() set
{[1,(N * (X,B))]} is Relation-like Function-like constant non empty trivial finite 1 -element set
the multF of A $$ (Path_matrix (K,N)) is Element of the carrier of A
len (Path_matrix ((X,n,K),i)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
<*> the carrier of A is Relation-like non-empty empty-yielding NAT -defined the carrier of A -valued Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V108() V109() V110() ext-real non positive non negative FinSequence of the carrier of A
the multF of A $$ (Path_matrix ((X,n,K),i)) is Element of the carrier of A
the_unity_wrt the multF of A is Element of the carrier of A
(N * (X,B)) * ( the multF of A $$ (Path_matrix ((X,n,K),i))) is Element of the carrier of A
the multF of A . ((N * (X,B)),( the multF of A $$ (Path_matrix ((X,n,K),i)))) is Element of the carrier of A
len (Path_matrix ((X,n,K),i)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
[:NAT, the carrier of A:] is Relation-like non trivial non finite set
bool [:NAT, the carrier of A:] is non trivial cup-closed diff-closed preBoolean non finite set
(Path_matrix ((X,n,K),i)) . 1 is set
the multF of A $$ (Path_matrix ((X,n,K),i)) is Element of the carrier of A
L is Relation-like NAT -defined the carrier of A -valued Function-like non empty V14( NAT ) quasi_total Element of bool [:NAT, the carrier of A:]
L . 1 is Element of the carrier of A
L . n is Element of the carrier of A
(Path_matrix (K,N)) . 1 is set
the multF of A $$ (Path_matrix (K,N)) is Element of the carrier of A
i9 is Relation-like NAT -defined the carrier of A -valued Function-like non empty V14( NAT ) quasi_total Element of bool [:NAT, the carrier of A:]
i9 . 1 is Element of the carrier of A
i9 . (x + 1) is Element of the carrier of A
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
dom (Path_matrix ((X,n,K),i)) is finite Element of bool NAT
[:(Seg (x + 1)),(Seg (x + 1)):] is Relation-like finite set
bool [:(Seg (x + 1)),(Seg (x + 1)):] is cup-closed diff-closed preBoolean finite V37() set
[:(Seg n),(Seg n):] is Relation-like finite set
bool [:(Seg n),(Seg n):] is cup-closed diff-closed preBoolean finite V37() set
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(Path_matrix ((X,n,K),i)) . x is set
(Path_matrix (K,N)) . x is set
x + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(Path_matrix (K,N)) . (x + 1) is set
y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j9 is Relation-like Seg (x + 1) -defined Seg (x + 1) -valued Function-like one-to-one non empty V14( Seg (x + 1)) quasi_total onto bijective finite Element of bool [:(Seg (x + 1)),(Seg (x + 1)):]
rng j9 is non empty finite set
dom j9 is non empty finite Element of bool (Seg (x + 1))
bool (Seg (x + 1)) is cup-closed diff-closed preBoolean finite V37() set
Laa is Relation-like Seg n -defined Seg n -valued Function-like one-to-one V14( Seg n) quasi_total onto bijective finite Element of bool [:(Seg n),(Seg n):]
rng Laa is finite set
dom Laa is finite Element of bool (Seg n)
bool (Seg n) is cup-closed diff-closed preBoolean finite V37() set
(X,n,K) . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
z is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(x + 1) -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(x + 1) - 1 is V108() V109() V110() set
j9 . x is set
j9 . X is set
K . x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K . x) - 1 is V108() V109() V110() set
z + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
i * (x,z) is Element of the carrier of A
N * (x,z) is Element of the carrier of A
N * (x,(z + 1)) is Element of the carrier of A
j9 . y is set
j9 . X is set
K . y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K . y) - 1 is V108() V109() V110() set
z + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
i * (x,z) is Element of the carrier of A
N * ((x + 1),z) is Element of the carrier of A
(Path_matrix (K,N)) . y is set
N * ((x + 1),(z + 1)) is Element of the carrier of A
N * (y,(z + 1)) is Element of the carrier of A
j9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
L . j9 is Element of the carrier of A
i9 . j9 is Element of the carrier of A
j9 + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
L . (j9 + 1) is Element of the carrier of A
i9 . (j9 + 1) is Element of the carrier of A
Laa is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Laa + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
Seg x is finite x -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= x ) } is set
i9 . (Laa + 1) is Element of the carrier of A
(Path_matrix (K,N)) . (Laa + 1) is set
the multF of A . ((i9 . j9),((Path_matrix (K,N)) . (Laa + 1))) is set
L . (Laa + 1) is Element of the carrier of A
(Path_matrix ((X,n,K),i)) . (Laa + 1) is set
the multF of A . ((L . j9),((Path_matrix ((X,n,K),i)) . (Laa + 1))) is set
L . {} is Element of the carrier of A
i9 . {} is Element of the carrier of A
j9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i9 . j9 is Element of the carrier of A
j9 - 1 is V108() V109() V110() set
L . (j9 - 1) is set
j9 + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
i9 . (j9 + 1) is Element of the carrier of A
(j9 + 1) - 1 is V108() V109() V110() set
L . ((j9 + 1) - 1) is set
x is Element of the carrier of A
(N * (X,B)) * x is Element of the carrier of A
the multF of A . ((N * (X,B)),x) is Element of the carrier of A
x is Element of the carrier of A
(N * (X,B)) * x is Element of the carrier of A
the multF of A . ((N * (X,B)),x) is Element of the carrier of A
dom (Path_matrix ((X,n,K),i)) is finite Element of bool NAT
(Path_matrix ((X,n,K),i)) . j9 is set
(X,n,K) . j9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
i * (j9,((X,n,K) . j9)) is Element of the carrier of A
(Path_matrix (K,N)) . (j9 + 1) is set
the multF of A . ((i9 . j9),((Path_matrix (K,N)) . (j9 + 1))) is set
x * (N * (X,B)) is Element of the carrier of A
the multF of A . (x,(N * (X,B))) is Element of the carrier of A
y is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
L . y is Element of the carrier of A
y + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
{} + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
z is Element of the carrier of A
(Path_matrix ((X,n,K),i)) . (y + 1) is set
the multF of A . (z,((Path_matrix ((X,n,K),i)) . (y + 1))) is set
(N * (X,B)) * z is Element of the carrier of A
the multF of A . ((N * (X,B)),z) is Element of the carrier of A
((N * (X,B)) * z) * (i * (j9,((X,n,K) . j9))) is Element of the carrier of A
the multF of A . (((N * (X,B)) * z),(i * (j9,((X,n,K) . j9)))) is Element of the carrier of A
z * (i * (j9,((X,n,K) . j9))) is Element of the carrier of A
the multF of A . (z,(i * (j9,((X,n,K) . j9)))) is Element of the carrier of A
(N * (X,B)) * (z * (i * (j9,((X,n,K) . j9)))) is Element of the carrier of A
the multF of A . ((N * (X,B)),(z * (i * (j9,((X,n,K) . j9))))) is Element of the carrier of A
{} - 1 is V108() V109() V110() set
L . ({} - 1) is set
j9 is Element of the carrier of A
(N * (X,B)) * j9 is Element of the carrier of A
the multF of A . ((N * (X,B)),j9) is Element of the carrier of A
(x + 1) - 1 is V108() V109() V110() set
{} + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(N * (X,B)) * ( the multF of A $$ (Path_matrix ((X,n,K),i))) is Element of the carrier of A
the multF of A . ((N * (X,B)),( the multF of A $$ (Path_matrix ((X,n,K),i)))) is Element of the carrier of A
the multF of A $$ (Path_matrix (K,N)) is Element of the carrier of A
the multF of A $$ (Path_matrix ((X,n,K),i)) is Element of the carrier of A
(N * (X,B)) * ( the multF of A $$ (Path_matrix ((X,n,K),i))) is Element of the carrier of A
the multF of A . ((N * (X,B)),( the multF of A $$ (Path_matrix ((X,n,K),i)))) is Element of the carrier of A
the multF of A $$ (Path_matrix (K,N)) is Element of the carrier of A
the multF of A $$ (Path_matrix ((X,n,K),i)) is Element of the carrier of A
(N * (X,B)) * ( the multF of A $$ (Path_matrix ((X,n,K),i))) is Element of the carrier of A
the multF of A . ((N * (X,B)),( the multF of A $$ (Path_matrix ((X,n,K),i)))) is Element of the carrier of A
- (( the multF of A $$ (Path_matrix (K,N))),K) is Element of the carrier of A
- (( the multF of A $$ (Path_matrix (K,N))),(X,n,K)) is Element of the carrier of A
(A,(power A),(- (1_ A)),(X + B)) * (- (( the multF of A $$ (Path_matrix (K,N))),(X,n,K))) is Element of the carrier of A
the multF of A . ((A,(power A),(- (1_ A)),(X + B)),(- (( the multF of A $$ (Path_matrix (K,N))),(X,n,K)))) is Element of the carrier of A
(A,(power A),(- (1_ A)),(X + B)) * ((N * (X,B)) * ( the multF of A $$ (Path_matrix ((X,n,K),i)))) is Element of the carrier of A
the multF of A . ((A,(power A),(- (1_ A)),(X + B)),((N * (X,B)) * ( the multF of A $$ (Path_matrix ((X,n,K),i))))) is Element of the carrier of A
((A,(power A),(- (1_ A)),(X + B)) * (N * (X,B))) * ( the multF of A $$ (Path_matrix ((X,n,K),i))) is Element of the carrier of A
the multF of A . (((A,(power A),(- (1_ A)),(X + B)) * (N * (X,B))),( the multF of A $$ (Path_matrix ((X,n,K),i)))) is Element of the carrier of A
- (( the multF of A $$ (Path_matrix ((X,n,K),i))),(X,n,K)) is Element of the carrier of A
((A,(power A),(- (1_ A)),(X + B)) * (N * (X,B))) * (- (( the multF of A $$ (Path_matrix ((X,n,K),i))),(X,n,K))) is Element of the carrier of A
the multF of A . (((A,(power A),(- (1_ A)),(X + B)) * (N * (X,B))),(- (( the multF of A $$ (Path_matrix ((X,n,K),i))),(X,n,K)))) is Element of the carrier of A
- (( the multF of A $$ (Path_matrix (K,N))),K) is Element of the carrier of A
- (( the multF of A $$ (Path_matrix (K,N))),(X,n,K)) is Element of the carrier of A
(A,(power A),(- (1_ A)),(X + B)) * (- (( the multF of A $$ (Path_matrix (K,N))),(X,n,K))) is Element of the carrier of A
the multF of A . ((A,(power A),(- (1_ A)),(X + B)),(- (( the multF of A $$ (Path_matrix (K,N))),(X,n,K)))) is Element of the carrier of A
- ((N * (X,B)) * ( the multF of A $$ (Path_matrix ((X,n,K),i)))) is Element of the carrier of A
(A,(power A),(- (1_ A)),(X + B)) * (- ((N * (X,B)) * ( the multF of A $$ (Path_matrix ((X,n,K),i))))) is Element of the carrier of A
the multF of A . ((A,(power A),(- (1_ A)),(X + B)),(- ((N * (X,B)) * ( the multF of A $$ (Path_matrix ((X,n,K),i)))))) is Element of the carrier of A
- ( the multF of A $$ (Path_matrix ((X,n,K),i))) is Element of the carrier of A
(N * (X,B)) * (- ( the multF of A $$ (Path_matrix ((X,n,K),i)))) is Element of the carrier of A
the multF of A . ((N * (X,B)),(- ( the multF of A $$ (Path_matrix ((X,n,K),i))))) is Element of the carrier of A
(A,(power A),(- (1_ A)),(X + B)) * ((N * (X,B)) * (- ( the multF of A $$ (Path_matrix ((X,n,K),i))))) is Element of the carrier of A
the multF of A . ((A,(power A),(- (1_ A)),(X + B)),((N * (X,B)) * (- ( the multF of A $$ (Path_matrix ((X,n,K),i)))))) is Element of the carrier of A
((A,(power A),(- (1_ A)),(X + B)) * (N * (X,B))) * (- ( the multF of A $$ (Path_matrix ((X,n,K),i)))) is Element of the carrier of A
the multF of A . (((A,(power A),(- (1_ A)),(X + B)) * (N * (X,B))),(- ( the multF of A $$ (Path_matrix ((X,n,K),i))))) is Element of the carrier of A
- (( the multF of A $$ (Path_matrix ((X,n,K),i))),(X,n,K)) is Element of the carrier of A
((A,(power A),(- (1_ A)),(X + B)) * (N * (X,B))) * (- (( the multF of A $$ (Path_matrix ((X,n,K),i))),(X,n,K))) is Element of the carrier of A
the multF of A . (((A,(power A),(- (1_ A)),(X + B)) * (N * (X,B))),(- (( the multF of A $$ (Path_matrix ((X,n,K),i))),(X,n,K)))) is Element of the carrier of A
x is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() left_unital doubleLoopStr
the carrier of x is non empty non trivial set
the carrier of x * is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
A -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of A,A, the carrier of x
(n,K,A,x,b) is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of A -' 1,A -' 1, the carrier of x
Det (n,K,A,x,b) is Element of the carrier of x
Permutations (A -' 1) is non empty permutational set
the addF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V14([: the carrier of x, the carrier of x:]) quasi_total commutative associative Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is Relation-like set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is cup-closed diff-closed preBoolean set
FinOmega (Permutations (A -' 1)) is finite Element of Fin (Permutations (A -' 1))
Fin (Permutations (A -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (n,K,A,x,b) is Relation-like Permutations (A -' 1) -defined the carrier of x -valued Function-like non empty V14( Permutations (A -' 1)) quasi_total Element of bool [:(Permutations (A -' 1)), the carrier of x:]
[:(Permutations (A -' 1)), the carrier of x:] is Relation-like set
bool [:(Permutations (A -' 1)), the carrier of x:] is cup-closed diff-closed preBoolean set
the addF of x $$ ((FinOmega (Permutations (A -' 1))),(Path_product (n,K,A,x,b))) is Element of the carrier of x
x is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() left_unital doubleLoopStr
the carrier of x is non empty non trivial set
the carrier of x * is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
power x is Relation-like [: the carrier of x,NAT:] -defined the carrier of x -valued Function-like V14([: the carrier of x,NAT:]) quasi_total Element of bool [:[: the carrier of x,NAT:], the carrier of x:]
[: the carrier of x,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of x,NAT:], the carrier of x:] is Relation-like non trivial non finite set
bool [:[: the carrier of x,NAT:], the carrier of x:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ x is Element of the carrier of x
1. x is V52(x) Element of the carrier of x
- (1_ x) is Element of the carrier of x
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
n + K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(x,(power x),(- (1_ x)),(n + K)) is Element of the carrier of x
b is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of A,A, the carrier of x
(n,K,A,x,b) is Element of the carrier of x
A -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(n,K,A,x,b) is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of A -' 1,A -' 1, the carrier of x
Det (n,K,A,x,b) is Element of the carrier of x
Permutations (A -' 1) is non empty permutational set
the addF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V14([: the carrier of x, the carrier of x:]) quasi_total commutative associative Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is Relation-like set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is cup-closed diff-closed preBoolean set
FinOmega (Permutations (A -' 1)) is finite Element of Fin (Permutations (A -' 1))
Fin (Permutations (A -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (n,K,A,x,b) is Relation-like Permutations (A -' 1) -defined the carrier of x -valued Function-like non empty V14( Permutations (A -' 1)) quasi_total Element of bool [:(Permutations (A -' 1)), the carrier of x:]
[:(Permutations (A -' 1)), the carrier of x:] is Relation-like set
bool [:(Permutations (A -' 1)), the carrier of x:] is cup-closed diff-closed preBoolean set
the addF of x $$ ((FinOmega (Permutations (A -' 1))),(Path_product (n,K,A,x,b))) is Element of the carrier of x
(x,(power x),(- (1_ x)),(n + K)) * (n,K,A,x,b) is Element of the carrier of x
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V14([: the carrier of x, the carrier of x:]) quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
the multF of x . ((x,(power x),(- (1_ x)),(n + K)),(n,K,A,x,b)) is Element of the carrier of x
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
Permutations n is non empty permutational set
Fin (Permutations n) is non empty cup-closed diff-closed preBoolean set
len (Permutations n) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations n) ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is cup-closed diff-closed preBoolean set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
{ b₁ where b₁ is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n : b₁ . A = x } is set
n - 1 is V108() V109() V110() set
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Permutations (n -' 1) is non empty permutational set
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
b + 1 is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
(b + 1) -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(b + 1) - 1 is V108() V109() V110() set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
N is finite Element of Fin (Permutations n)
[:N,(Permutations (n -' 1)):] is Relation-like set
bool [:N,(Permutations (n -' 1)):] is cup-closed diff-closed preBoolean set
Permutations (b + 1) is non empty permutational set
len (Permutations (b + 1)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (b + 1))) is finite len (Permutations (b + 1)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (b + 1)) ) } is set
i is Relation-like N -defined Permutations (n -' 1) -valued Function-like V14(N) quasi_total finite Element of bool [:N,(Permutations (n -' 1)):]
j 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 j is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ (N,(Path_product j)) is Element of the carrier of K
j * (A,x) is Element of the carrier of K
(A,x,n,K,j) is Element of the carrier of K
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like V14([: the carrier of K,NAT:]) quasi_total Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ K is Element of the carrier of K
1. K is V52(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
A + x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,(power K),(- (1_ K)),(A + x)) is Element of the carrier of K
(A,x,n,K,j) is Element of the carrier of K
(A,x,n,K,j) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det (A,x,n,K,j) is Element of the carrier of K
Path_product (A,x,n,K,j) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
[:(Permutations (n -' 1)), the carrier of K:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (A,x,n,K,j))) is Element of the carrier of K
(K,(power K),(- (1_ K)),(A + x)) * (A,x,n,K,j) is Element of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V14([: the carrier of K, the carrier of K:]) quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the multF of K . ((K,(power K),(- (1_ K)),(A + x)),(A,x,n,K,j)) is Element of the carrier of K
(j * (A,x)) * (A,x,n,K,j) is Element of the carrier of K
the multF of K . ((j * (A,x)),(A,x,n,K,j)) is Element of the carrier of K
(K,(power K),(- (1_ K)),(A + x)) * (j * (A,x)) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(A + x)),(j * (A,x))) is Element of the carrier of K
Laa is finite Element of Fin (Permutations n)
x is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n
{x} is functional non empty trivial finite V37() 1 -element set
Laa \/ {x} is non empty finite set
rng i is finite set
i .: Laa is finite set
z is finite Element of Fin (Permutations n)
i .: z is finite set
the addF of K $$ (z,(Path_product j)) is Element of the carrier of K
ProjD is finite Element of Fin (Permutations (n -' 1))
the addF of K $$ (ProjD,(Path_product (A,x,n,K,j))) is Element of the carrier of K
((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))) * ( the addF of K $$ (ProjD,(Path_product (A,x,n,K,j)))) is Element of the carrier of K
the multF of K . (((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))),( the addF of K $$ (ProjD,(Path_product (A,x,n,K,j))))) is Element of the carrier of K
Q1 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n
Q1 . A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
dom i is finite Element of bool N
bool N is cup-closed diff-closed preBoolean finite V37() set
Im (i,x) is set
i .: {x} is finite set
i . x is set
{(i . x)} is non empty trivial finite 1 -element set
len (Permutations (n -' 1)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations (n -' 1))) is finite len (Permutations (n -' 1)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations (n -' 1)) ) } is set
Q19 is Relation-like Seg (len (Permutations (b + 1))) -defined Seg (len (Permutations (b + 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (b + 1)))) quasi_total onto bijective finite Element of Permutations (b + 1)
(A,b,Q19) is Relation-like Seg (len (Permutations b)) -defined Seg (len (Permutations b)) -valued Function-like one-to-one V14( Seg (len (Permutations b))) quasi_total onto bijective finite Element of Permutations b
Permutations b is non empty permutational set
len (Permutations b) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations b)) is finite len (Permutations b) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= len (Permutations b) ) } is set
Q is Relation-like Seg (len (Permutations (n -' 1))) -defined Seg (len (Permutations (n -' 1))) -valued Function-like one-to-one V14( Seg (len (Permutations (n -' 1)))) quasi_total onto bijective finite Element of Permutations (n -' 1)
(Path_product j) . Q1 is Element of the carrier of K
(Path_product (A,x,n,K,j)) . Q is Element of the carrier of K
((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))) * ((Path_product (A,x,n,K,j)) . Q) is Element of the carrier of K
the multF of K . (((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))),((Path_product (A,x,n,K,j)) . Q)) is Element of the carrier of K
y is finite Element of Fin (Permutations (n -' 1))
Pf is set
i . Pf is set
(Path_product j) . x is Element of the carrier of K
(Path_product (A,x,n,K,j)) . (i . x) is set
{Q} is functional non empty trivial finite V37() 1 -element set
y \/ {Q} is non empty finite set
the addF of K $$ (y,(Path_product (A,x,n,K,j))) is Element of the carrier of K
( the addF of K $$ (y,(Path_product (A,x,n,K,j)))) + ((Path_product (A,x,n,K,j)) . Q) is Element of the carrier of K
the addF of K . (( the addF of K $$ (y,(Path_product (A,x,n,K,j)))),((Path_product (A,x,n,K,j)) . Q)) is Element of the carrier of K
((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))) * (( the addF of K $$ (y,(Path_product (A,x,n,K,j)))) + ((Path_product (A,x,n,K,j)) . Q)) is Element of the carrier of K
the multF of K . (((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))),(( the addF of K $$ (y,(Path_product (A,x,n,K,j)))) + ((Path_product (A,x,n,K,j)) . Q))) is Element of the carrier of K
((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))) * ( the addF of K $$ (y,(Path_product (A,x,n,K,j)))) is Element of the carrier of K
the multF of K . (((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))),( the addF of K $$ (y,(Path_product (A,x,n,K,j))))) is Element of the carrier of K
(((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))) * ( the addF of K $$ (y,(Path_product (A,x,n,K,j))))) + (((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))) * ((Path_product (A,x,n,K,j)) . Q)) is Element of the carrier of K
the addF of K . ((((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))) * ( the addF of K $$ (y,(Path_product (A,x,n,K,j))))),(((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))) * ((Path_product (A,x,n,K,j)) . Q))) is Element of the carrier of K
the addF of K $$ (Laa,(Path_product j)) is Element of the carrier of K
(Path_product j) . x is Element of the carrier of K
the addF of K . (( the addF of K $$ (Laa,(Path_product j))),((Path_product j) . x)) is Element of the carrier of K
{}. (Permutations n) is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V108() V109() V110() ext-real non positive non negative Element of Fin (Permutations n)
Laa is finite Element of Fin (Permutations n)
i .: Laa is finite set
the addF of K $$ (Laa,(Path_product j)) is Element of the carrier of K
x is finite Element of Fin (Permutations (n -' 1))
the addF of K $$ (x,(Path_product (A,x,n,K,j))) is Element of the carrier of K
((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))) * ( the addF of K $$ (x,(Path_product (A,x,n,K,j)))) is Element of the carrier of K
the multF of K . (((K,(power K),(- (1_ K)),(A + x)) * (j * (A,x))),( the addF of K $$ (x,(Path_product (A,x,n,K,j))))) is Element of the carrier of K
{}. (Permutations (n -' 1)) is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V108() V109() V110() ext-real non positive non negative Element of Fin (Permutations (n -' 1))
the_unity_wrt the addF of K is Element of the carrier of K
0. K is V52(K) Element of the carrier of K
dom i is finite Element of bool N
bool N is cup-closed diff-closed preBoolean finite V37() set
rng i is finite set
i .: N is finite set
(j * (A,x)) * (K,(power K),(- (1_ K)),(A + x)) is Element of the carrier of K
the multF of K . ((j * (A,x)),(K,(power K),(- (1_ K)),(A + x))) is Element of the carrier of K
((j * (A,x)) * (K,(power K),(- (1_ K)),(A + x))) * (Det (A,x,n,K,j)) is Element of the carrier of K
the multF of K . (((j * (A,x)) * (K,(power K),(- (1_ K)),(A + x))),(Det (A,x,n,K,j))) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
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
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(x,b,n,K,A) is Element of the carrier of K
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(x,b,n,K,A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det (x,b,n,K,A) is Element of the carrier of K
Permutations (n -' 1) 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is cup-closed diff-closed preBoolean set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (x,b,n,K,A) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
[:(Permutations (n -' 1)), the carrier of K:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (x,b,n,K,A))) is Element of the carrier of K
(b,x,n,K,(A @)) is Element of the carrier of K
(b,x,n,K,(A @)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det (b,x,n,K,(A @)) is Element of the carrier of K
Path_product (b,x,n,K,(A @)) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (b,x,n,K,(A @)))) is Element of the carrier of K
(x,b,n,K,A) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det ((x,b,n,K,A) @) is Element of the carrier of K
Path_product ((x,b,n,K,A) @) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product ((x,b,n,K,A) @))) is Element of the carrier of K
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
b is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of x,x, the carrier of K
Indices b is set
dom b is finite Element of bool NAT
width b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width b) is finite width b -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width b ) } is set
[:(dom b),(Seg (width b)):] is Relation-like finite set
x 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
Indices x is set
dom x is finite Element of bool NAT
width x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width x) is finite width x -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width x ) } is set
[:(dom x),(Seg (width x)):] is Relation-like finite set
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
Indices b is set
dom b is finite Element of bool NAT
width b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width b) is finite width b -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width b ) } is set
[:(dom b),(Seg (width b)):] is Relation-like finite set
X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[X,B] is set
{X,B} is non empty finite V37() set
{X} is non empty trivial finite V37() 1 -element set
{{X,B},{X}} is non empty finite V37() set
x * (X,B) is Element of the carrier of K
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(i,j,n,K,A) is Element of the carrier of K
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like V14([: the carrier of K,NAT:]) quasi_total Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ K is Element of the carrier of K
1. K is V52(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
i + j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,(power K),(- (1_ K)),(i + j)) is Element of the carrier of K
(i,j,n,K,A) is Element of the carrier of K
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(i,j,n,K,A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det (i,j,n,K,A) is Element of the carrier of K
Permutations (n -' 1) 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is cup-closed diff-closed preBoolean set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (i,j,n,K,A) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
[:(Permutations (n -' 1)), the carrier of K:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (i,j,n,K,A))) is Element of the carrier of K
(K,(power K),(- (1_ K)),(i + j)) * (i,j,n,K,A) 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 V14([: the carrier of K, the carrier of K:]) 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 . ((K,(power K),(- (1_ K)),(i + j)),(i,j,n,K,A)) is Element of the carrier of K
b * (X,B) is Element of the carrier of K
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of A
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom X is finite Element of bool NAT
rng X is finite set
B is set
i is set
X . i is set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x * (K,j) is Element of the carrier of A
(K,j,n,A,x) is Element of the carrier of A
power A is Relation-like [: the carrier of A,NAT:] -defined the carrier of A -valued Function-like V14([: the carrier of A,NAT:]) quasi_total Element of bool [:[: the carrier of A,NAT:], the carrier of A:]
[: the carrier of A,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of A,NAT:], the carrier of A:] is Relation-like non trivial non finite set
bool [:[: the carrier of A,NAT:], the carrier of A:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ A is Element of the carrier of A
1. A is V52(A) Element of the carrier of A
- (1_ A) is Element of the carrier of A
K + j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A,(power A),(- (1_ A)),(K + j)) is Element of the carrier of A
(K,j,n,A,x) is Element of the carrier of A
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,j,n,A,x) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of A
Det (K,j,n,A,x) is Element of the carrier of A
Permutations (n -' 1) 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 V14([: the carrier of A, the carrier of A:]) 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 set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is cup-closed diff-closed preBoolean set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (K,j,n,A,x) is Relation-like Permutations (n -' 1) -defined the carrier of A -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of A:]
[:(Permutations (n -' 1)), the carrier of A:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of A:] is cup-closed diff-closed preBoolean set
the addF of A $$ ((FinOmega (Permutations (n -' 1))),(Path_product (K,j,n,A,x))) is Element of the carrier of A
(A,(power A),(- (1_ A)),(K + j)) * (K,j,n,A,x) is Element of the carrier of A
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like V14([: the carrier of A, the carrier of A:]) quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
the multF of A . ((A,(power A),(- (1_ A)),(K + j)),(K,j,n,A,x)) is Element of the carrier of A
(x * (K,j)) * (K,j,n,A,x) is Element of the carrier of A
the multF of A . ((x * (K,j)),(K,j,n,A,x)) is Element of the carrier of A
X . j is set
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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom B is finite Element of bool NAT
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
B . i is set
x * (K,i) is Element of the carrier of A
(K,i,n,A,x) is Element of the carrier of A
power A is Relation-like [: the carrier of A,NAT:] -defined the carrier of A -valued Function-like V14([: the carrier of A,NAT:]) quasi_total Element of bool [:[: the carrier of A,NAT:], the carrier of A:]
[: the carrier of A,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of A,NAT:], the carrier of A:] is Relation-like non trivial non finite set
bool [:[: the carrier of A,NAT:], the carrier of A:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ A is Element of the carrier of A
1. A is V52(A) Element of the carrier of A
- (1_ A) is Element of the carrier of A
K + i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A,(power A),(- (1_ A)),(K + i)) is Element of the carrier of A
(K,i,n,A,x) is Element of the carrier of A
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,i,n,A,x) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of A
Det (K,i,n,A,x) is Element of the carrier of A
Permutations (n -' 1) 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 V14([: the carrier of A, the carrier of A:]) 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 set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is cup-closed diff-closed preBoolean set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (K,i,n,A,x) is Relation-like Permutations (n -' 1) -defined the carrier of A -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of A:]
[:(Permutations (n -' 1)), the carrier of A:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of A:] is cup-closed diff-closed preBoolean set
the addF of A $$ ((FinOmega (Permutations (n -' 1))),(Path_product (K,i,n,A,x))) is Element of the carrier of A
(A,(power A),(- (1_ A)),(K + i)) * (K,i,n,A,x) is Element of the carrier of A
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like V14([: the carrier of A, the carrier of A:]) quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
the multF of A . ((A,(power A),(- (1_ A)),(K + i)),(K,i,n,A,x)) is Element of the carrier of A
(x * (K,i)) * (K,i,n,A,x) is Element of the carrier of A
the multF of A . ((x * (K,i)),(K,i,n,A,x)) is Element 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom b is finite Element of bool NAT
X is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom X is finite Element of bool NAT
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b . B is set
x * (K,B) is Element of the carrier of A
(K,B,n,A,x) is Element of the carrier of A
power A is Relation-like [: the carrier of A,NAT:] -defined the carrier of A -valued Function-like V14([: the carrier of A,NAT:]) quasi_total Element of bool [:[: the carrier of A,NAT:], the carrier of A:]
[: the carrier of A,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of A,NAT:], the carrier of A:] is Relation-like non trivial non finite set
bool [:[: the carrier of A,NAT:], the carrier of A:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ A is Element of the carrier of A
1. A is V52(A) Element of the carrier of A
- (1_ A) is Element of the carrier of A
K + B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A,(power A),(- (1_ A)),(K + B)) is Element of the carrier of A
(K,B,n,A,x) is Element of the carrier of A
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,B,n,A,x) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of A
Det (K,B,n,A,x) is Element of the carrier of A
Permutations (n -' 1) 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 V14([: the carrier of A, the carrier of A:]) 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 set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is cup-closed diff-closed preBoolean set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (K,B,n,A,x) is Relation-like Permutations (n -' 1) -defined the carrier of A -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of A:]
[:(Permutations (n -' 1)), the carrier of A:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of A:] is cup-closed diff-closed preBoolean set
the addF of A $$ ((FinOmega (Permutations (n -' 1))),(Path_product (K,B,n,A,x))) is Element of the carrier of A
(A,(power A),(- (1_ A)),(K + B)) * (K,B,n,A,x) is Element of the carrier of A
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like V14([: the carrier of A, the carrier of A:]) quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
the multF of A . ((A,(power A),(- (1_ A)),(K + B)),(K,B,n,A,x)) is Element of the carrier of A
(x * (K,B)) * (K,B,n,A,x) is Element of the carrier of A
the multF of A . ((x * (K,B)),(K,B,n,A,x)) is Element of the carrier of A
X . B is set
A is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of A
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom X is finite Element of bool NAT
rng X is finite set
B is set
i is set
X . i is set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x * (j,K) is Element of the carrier of A
(j,K,n,A,x) is Element of the carrier of A
power A is Relation-like [: the carrier of A,NAT:] -defined the carrier of A -valued Function-like V14([: the carrier of A,NAT:]) quasi_total Element of bool [:[: the carrier of A,NAT:], the carrier of A:]
[: the carrier of A,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of A,NAT:], the carrier of A:] is Relation-like non trivial non finite set
bool [:[: the carrier of A,NAT:], the carrier of A:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ A is Element of the carrier of A
1. A is V52(A) Element of the carrier of A
- (1_ A) is Element of the carrier of A
j + K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A,(power A),(- (1_ A)),(j + K)) is Element of the carrier of A
(j,K,n,A,x) is Element of the carrier of A
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(j,K,n,A,x) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of A
Det (j,K,n,A,x) is Element of the carrier of A
Permutations (n -' 1) 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 V14([: the carrier of A, the carrier of A:]) 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 set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is cup-closed diff-closed preBoolean set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (j,K,n,A,x) is Relation-like Permutations (n -' 1) -defined the carrier of A -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of A:]
[:(Permutations (n -' 1)), the carrier of A:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of A:] is cup-closed diff-closed preBoolean set
the addF of A $$ ((FinOmega (Permutations (n -' 1))),(Path_product (j,K,n,A,x))) is Element of the carrier of A
(A,(power A),(- (1_ A)),(j + K)) * (j,K,n,A,x) is Element of the carrier of A
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like V14([: the carrier of A, the carrier of A:]) quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
the multF of A . ((A,(power A),(- (1_ A)),(j + K)),(j,K,n,A,x)) is Element of the carrier of A
(x * (j,K)) * (j,K,n,A,x) is Element of the carrier of A
the multF of A . ((x * (j,K)),(j,K,n,A,x)) is Element of the carrier of A
X . j is set
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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom B is finite Element of bool NAT
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
B . i is set
x * (i,K) is Element of the carrier of A
(i,K,n,A,x) is Element of the carrier of A
power A is Relation-like [: the carrier of A,NAT:] -defined the carrier of A -valued Function-like V14([: the carrier of A,NAT:]) quasi_total Element of bool [:[: the carrier of A,NAT:], the carrier of A:]
[: the carrier of A,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of A,NAT:], the carrier of A:] is Relation-like non trivial non finite set
bool [:[: the carrier of A,NAT:], the carrier of A:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ A is Element of the carrier of A
1. A is V52(A) Element of the carrier of A
- (1_ A) is Element of the carrier of A
i + K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A,(power A),(- (1_ A)),(i + K)) is Element of the carrier of A
(i,K,n,A,x) is Element of the carrier of A
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(i,K,n,A,x) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of A
Det (i,K,n,A,x) is Element of the carrier of A
Permutations (n -' 1) 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 V14([: the carrier of A, the carrier of A:]) 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 set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is cup-closed diff-closed preBoolean set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (i,K,n,A,x) is Relation-like Permutations (n -' 1) -defined the carrier of A -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of A:]
[:(Permutations (n -' 1)), the carrier of A:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of A:] is cup-closed diff-closed preBoolean set
the addF of A $$ ((FinOmega (Permutations (n -' 1))),(Path_product (i,K,n,A,x))) is Element of the carrier of A
(A,(power A),(- (1_ A)),(i + K)) * (i,K,n,A,x) is Element of the carrier of A
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like V14([: the carrier of A, the carrier of A:]) quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
the multF of A . ((A,(power A),(- (1_ A)),(i + K)),(i,K,n,A,x)) is Element of the carrier of A
(x * (i,K)) * (i,K,n,A,x) is Element of the carrier of A
the multF of A . ((x * (i,K)),(i,K,n,A,x)) is Element 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom b is finite Element of bool NAT
X is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom X is finite Element of bool NAT
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
b . B is set
x * (B,K) is Element of the carrier of A
(B,K,n,A,x) is Element of the carrier of A
power A is Relation-like [: the carrier of A,NAT:] -defined the carrier of A -valued Function-like V14([: the carrier of A,NAT:]) quasi_total Element of bool [:[: the carrier of A,NAT:], the carrier of A:]
[: the carrier of A,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of A,NAT:], the carrier of A:] is Relation-like non trivial non finite set
bool [:[: the carrier of A,NAT:], the carrier of A:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ A is Element of the carrier of A
1. A is V52(A) Element of the carrier of A
- (1_ A) is Element of the carrier of A
B + K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(A,(power A),(- (1_ A)),(B + K)) is Element of the carrier of A
(B,K,n,A,x) is Element of the carrier of A
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(B,K,n,A,x) is Relation-like NAT -defined the carrier of A * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of A
Det (B,K,n,A,x) is Element of the carrier of A
Permutations (n -' 1) 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 V14([: the carrier of A, the carrier of A:]) 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 set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is cup-closed diff-closed preBoolean set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (B,K,n,A,x) is Relation-like Permutations (n -' 1) -defined the carrier of A -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of A:]
[:(Permutations (n -' 1)), the carrier of A:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of A:] is cup-closed diff-closed preBoolean set
the addF of A $$ ((FinOmega (Permutations (n -' 1))),(Path_product (B,K,n,A,x))) is Element of the carrier of A
(A,(power A),(- (1_ A)),(B + K)) * (B,K,n,A,x) is Element of the carrier of A
the multF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like V14([: the carrier of A, the carrier of A:]) quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
the multF of A . ((A,(power A),(- (1_ A)),(B + K)),(B,K,n,A,x)) is Element of the carrier of A
(x * (B,K)) * (B,K,n,A,x) is Element of the carrier of A
the multF of A . ((x * (B,K)),(B,K,n,A,x)) is Element of the carrier of A
X . B is set
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is cup-closed diff-closed preBoolean set
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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
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 V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product i)) is Element of the carrier of K
(n,B,K,i) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (n,B,K,i) is Element of the carrier of K
the addF of K $$ (n,B,K,i) is Element of the carrier of K
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
finSeg A is finite A -element Element of Fin NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= A ) } is set
[:(Fin (Permutations n)), the carrier of K:] is Relation-like set
bool [:(Fin (Permutations n)), the carrier of K:] is cup-closed diff-closed preBoolean set
i is Relation-like Fin (Permutations n) -defined the carrier of K -valued Function-like non empty V14( Fin (Permutations n)) quasi_total Element of bool [:(Fin (Permutations n)), the carrier of K:]
j is finite Element of Fin (Permutations n)
COL is finite Element of Fin (Permutations n)
i . j is Element of the carrier of K
i . COL is Element of the carrier of K
the addF of K . ((i . j),(i . COL)) is Element of the carrier of K
j \/ COL is finite Element of Fin (Permutations n)
i . (j \/ COL) is Element of the carrier of K
the addF of K $$ (j,(Path_product i)) is Element of the carrier of K
the addF of K $$ (COL,(Path_product i)) is Element of the carrier of K
the addF of K $$ ((j \/ COL),(Path_product i)) is Element of the carrier of K
len (Permutations n) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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 V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n : b₁ . B = a₁ } is set
j is non empty set
j is Relation-like Function-like set
dom j is set
rng j is set
COL is set
L is set
j . L is set
{ b₁ where b₁ is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n : b₁ . B = L } is set
i9 is set
j9 is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n
j9 . B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[:j,(Fin (Permutations n)):] is Relation-like set
bool [:j,(Fin (Permutations n)):] is cup-closed diff-closed preBoolean set
i . (FinOmega (Permutations n)) is Element of the carrier of K
COL is Relation-like j -defined Fin (Permutations n) -valued Function-like non empty V14(j) quasi_total Element of bool [:j,(Fin (Permutations n)):]
i * COL is Relation-like j -defined the carrier of K -valued Function-like non empty V14(j) quasi_total Element of bool [:j, the carrier of K:]
[:j, the carrier of K:] is Relation-like set
bool [:j, the carrier of K:] is cup-closed diff-closed preBoolean set
dom (i * COL) is non empty Element of bool j
bool j is cup-closed diff-closed preBoolean set
id j is Relation-like j -defined j -valued Function-like one-to-one non empty V14(j) quasi_total onto bijective V112() V114() V115() V119() Element of bool [:j,j:]
[:j,j:] is Relation-like set
bool [:j,j:] is cup-closed diff-closed preBoolean set
(i * COL) * (id j) is Relation-like j -defined the carrier of K -valued Function-like non empty V14(j) quasi_total Element of bool [:j, the carrier of K:]
COL .: j is set
union (COL .: j) is set
i9 is set
j9 is Relation-like j -defined j -valued Function-like one-to-one non empty V14(j) quasi_total onto bijective Element of bool [:j,j:]
j9 . B is set
{ b₁ where b₁ is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n : b₁ . B = j9 . B } is set
rng j9 is non empty set
dom j9 is non empty Element of bool j
COL . (j9 . B) is set
len (n,B,K,i) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom (n,B,K,i) is finite Element of bool NAT
dom (id j) is non empty Element of bool j
Fin j is non empty cup-closed diff-closed preBoolean set
{}. (Fin (Permutations n)) is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V108() V109() V110() ext-real non positive non negative Element of Fin (Fin (Permutations n))
Fin (Fin (Permutations n)) is non empty cup-closed diff-closed preBoolean set
i . ({}. (Fin (Permutations n))) is set
{}. (Permutations n) is Relation-like non-empty empty-yielding NAT -defined Function-like one-to-one constant functional empty V26() V27() V28() V30() V31() V32() finite finite-yielding V37() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V108() V109() V110() ext-real non positive non negative Element of Fin (Permutations n)
the addF of K $$ (({}. (Permutations n)),(Path_product i)) is Element of the carrier of K
i . {} is set
the_unity_wrt the addF of K is Element of the carrier of K
Laa is set
j9 is finite Element of Fin j
x is set
COL . Laa is set
COL . x is set
y is set
{ b₁ where b₁ is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n : b₁ . B = x } is set
{ b₁ where b₁ is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n : b₁ . B = Laa } is set
z is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n
z . B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
ProjD is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n
ProjD . B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
rng COL is non empty set
Laa is set
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
COL . x is set
{ b₁ where b₁ is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n : b₁ . B = x } is set
y is finite Element of Fin (Permutations n)
(i * COL) . x is set
i . (COL . x) is set
the addF of K $$ (y,(Path_product i)) is Element of the carrier of K
i * (B,x) is Element of the carrier of K
(B,x,n,K,i) is Element of the carrier of K
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like V14([: the carrier of K,NAT:]) quasi_total Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ K is Element of the carrier of K
1. K is V52(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
B + x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,(power K),(- (1_ K)),(B + x)) is Element of the carrier of K
(B,x,n,K,i) is Element of the carrier of K
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(B,x,n,K,i) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det (B,x,n,K,i) is Element of the carrier of K
Permutations (n -' 1) is non empty permutational set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (B,x,n,K,i) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
[:(Permutations (n -' 1)), the carrier of K:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (B,x,n,K,i))) is Element of the carrier of K
(K,(power K),(- (1_ K)),(B + x)) * (B,x,n,K,i) 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 V14([: the carrier of K, the carrier of K:]) 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 . ((K,(power K),(- (1_ K)),(B + x)),(B,x,n,K,i)) is Element of the carrier of K
(i * (B,x)) * (B,x,n,K,i) is Element of the carrier of K
the multF of K . ((i * (B,x)),(B,x,n,K,i)) is Element of the carrier of K
(n,B,K,i) . Laa is set
(i * COL) . Laa is set
[#] ((n,B,K,i),(the_unity_wrt the addF of K)) is Relation-like NAT -defined the carrier of K -valued Function-like non empty V14( NAT ) quasi_total Element of bool [:NAT, the carrier of K:]
[:NAT, the carrier of K:] is Relation-like non trivial non finite set
bool [:NAT, the carrier of K:] is non trivial cup-closed diff-closed preBoolean non finite set
rng (id j) is non empty set
([#] ((n,B,K,i),(the_unity_wrt the addF of K))) | (dom (n,B,K,i)) is Relation-like NAT -defined dom (n,B,K,i) -defined NAT -defined the carrier of K -valued Function-like finite Element of bool [:NAT, the carrier of K:]
x is set
y is set
dom COL is non empty Element of bool j
z is set
COL . z is set
{ b₁ where b₁ is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n : b₁ . B = z } is set
ProjD is Relation-like Seg (len (Permutations n)) -defined Seg (len (Permutations n)) -valued Function-like one-to-one V14( Seg (len (Permutations n))) quasi_total onto bijective finite Element of Permutations n
ProjD . B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
COL .: j9 is finite Element of Fin (Fin (Permutations n))
the addF of K $$ ((COL .: j9),i) is Element of the carrier of K
the addF of K $$ (j9,(i * COL)) is Element of the carrier of K
findom (n,B,K,i) is finite Element of Fin NAT
the addF of K $$ ((findom (n,B,K,i)),([#] ((n,B,K,i),(the_unity_wrt the addF of K)))) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
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
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(n,x,K,A) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(n,x,K,(A @)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len (n,x,K,(A @)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len (n,x,K,A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
dom (n,x,K,A) is finite Element of bool NAT
(n,x,K,A) . i is set
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
A * (i,B) is Element of the carrier of K
(i,B,n,K,A) is Element of the carrier of K
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like V14([: the carrier of K,NAT:]) quasi_total Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ K is Element of the carrier of K
1. K is V52(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
i + B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,(power K),(- (1_ K)),(i + B)) is Element of the carrier of K
(i,B,n,K,A) is Element of the carrier of K
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(i,B,n,K,A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det (i,B,n,K,A) is Element of the carrier of K
Permutations (n -' 1) 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is cup-closed diff-closed preBoolean set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (i,B,n,K,A) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
[:(Permutations (n -' 1)), the carrier of K:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (i,B,n,K,A))) is Element of the carrier of K
(K,(power K),(- (1_ K)),(i + B)) * (i,B,n,K,A) 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 V14([: the carrier of K, the carrier of K:]) 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 . ((K,(power K),(- (1_ K)),(i + B)),(i,B,n,K,A)) is Element of the carrier of K
(A * (i,B)) * (i,B,n,K,A) is Element of the carrier of K
the multF of K . ((A * (i,B)),(i,B,n,K,A)) is Element of the carrier of K
Indices A is set
dom A is finite Element of bool NAT
width A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width A) is finite width A -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
[:(dom A),(Seg (width A)):] is Relation-like finite set
[:(Seg n),(Seg n):] is Relation-like finite set
[i,x] is set
{i,x} is non empty finite V37() set
{i} is non empty trivial finite V37() 1 -element set
{{i,x},{i}} is non empty finite V37() set
dom (n,x,K,(A @)) is finite Element of bool NAT
(n,x,K,(A @)) . i is set
(A @) * (B,i) is Element of the carrier of K
(B,i,n,K,(A @)) is Element of the carrier of K
B + i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,(power K),(- (1_ K)),(B + i)) is Element of the carrier of K
(B,i,n,K,(A @)) is Element of the carrier of K
(B,i,n,K,(A @)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det (B,i,n,K,(A @)) is Element of the carrier of K
Path_product (B,i,n,K,(A @)) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (B,i,n,K,(A @)))) is Element of the carrier of K
(K,(power K),(- (1_ K)),(B + i)) * (B,i,n,K,(A @)) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(B + i)),(B,i,n,K,(A @))) is Element of the carrier of K
((A @) * (B,i)) * (B,i,n,K,(A @)) is Element of the carrier of K
the multF of K . (((A @) * (B,i)),(B,i,n,K,(A @))) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x 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 x 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is 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 x is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product x)) is Element of the carrier of K
(n,A,K,x) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (n,A,K,x) is Element of the carrier of K
the addF of K $$ (n,A,K,x) is Element of the carrier of K
x @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (x @) is Element of the carrier of K
Path_product (x @) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (x @))) is Element of the carrier of K
(n,A,K,(x @)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (n,A,K,(x @)) is Element of the carrier of K
the addF of K $$ (n,A,K,(x @)) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Permutations n is non empty permutational set
len (Permutations n) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (len (Permutations n)) is finite len (Permutations n) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 FinSequence of the carrier of K
len A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x 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,K,x) 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
X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Line ((n,K,x),X) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (n,K,x) -element FinSequence-like FinSubsequence-like Element of (width (n,K,x)) -tuples_on the carrier of K
width (n,K,x) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width (n,K,x)) -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,K,x) } is set
mlt ((Line ((n,K,x),X)),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 is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is cup-closed diff-closed preBoolean set
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line ((n,K,x),X)),A) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ReplaceLine (x,X,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
(n,X,K,(ReplaceLine (x,X,A))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
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 = B } is set
i is Relation-like NAT -defined the carrier of K -valued Function-like finite B -element FinSequence-like FinSubsequence-like Element of B -tuples_on the carrier of K
N is Relation-like NAT -defined the carrier of K -valued Function-like finite B -element FinSequence-like FinSubsequence-like Element of B -tuples_on the carrier of K
mlt (i,N) is Relation-like NAT -defined the carrier of K -valued Function-like finite B -element FinSequence-like FinSubsequence-like Element of B -tuples_on the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,i,N) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len (n,X,K,(ReplaceLine (x,X,A))) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom (n,X,K,(ReplaceLine (x,X,A))) is finite Element of bool NAT
width x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(Line ((n,K,x),X)) . i9 is set
(n,K,x) * (X,i9) is Element of the carrier of K
Indices x is set
dom x is finite Element of bool NAT
Seg (width x) is finite width x -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width x ) } is set
[:(dom x),(Seg (width x)):] is Relation-like finite set
[:(Seg n),(Seg n):] is Relation-like finite set
[X,i9] is set
{X,i9} is non empty finite V37() set
{X} is non empty trivial finite V37() 1 -element set
{{X,i9},{X}} is non empty finite V37() set
(ReplaceLine (x,X,A)) * (X,i9) is Element of the carrier of K
A . i9 is set
Indices (n,K,x) is set
dom (n,K,x) is finite Element of bool NAT
Seg (width (n,K,x)) is finite width (n,K,x) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (n,K,x) ) } is set
[:(dom (n,K,x)),(Seg (width (n,K,x))):] is Relation-like finite set
(X,i9,n,K,x) is Element of the carrier of K
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like V14([: the carrier of K,NAT:]) quasi_total Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ K is Element of the carrier of K
1. K is V52(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
X + i9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,(power K),(- (1_ K)),(X + i9)) is Element of the carrier of K
(X,i9,n,K,x) is Element of the carrier of K
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(X,i9,n,K,x) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det (X,i9,n,K,x) is Element of the carrier of K
Permutations (n -' 1) 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 V14([: the carrier of K, the carrier of K:]) quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (X,i9,n,K,x) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
[:(Permutations (n -' 1)), the carrier of K:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (X,i9,n,K,x))) is Element of the carrier of K
(K,(power K),(- (1_ K)),(X + i9)) * (X,i9,n,K,x) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(X + i9)),(X,i9,n,K,x)) is Element of the carrier of K
(mlt (i,N)) . i9 is set
(X,i9,n,K,x) * ((ReplaceLine (x,X,A)) * (X,i9)) is Element of the carrier of K
the multF of K . ((X,i9,n,K,x),((ReplaceLine (x,X,A)) * (X,i9))) is Element of the carrier of K
(X,i9,n,K,(ReplaceLine (x,X,A))) is Element of the carrier of K
(X,i9,n,K,(ReplaceLine (x,X,A))) is Element of the carrier of K
(X,i9,n,K,(ReplaceLine (x,X,A))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det (X,i9,n,K,(ReplaceLine (x,X,A))) is Element of the carrier of K
Path_product (X,i9,n,K,(ReplaceLine (x,X,A))) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (X,i9,n,K,(ReplaceLine (x,X,A))))) is Element of the carrier of K
(K,(power K),(- (1_ K)),(X + i9)) * (X,i9,n,K,(ReplaceLine (x,X,A))) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(X + i9)),(X,i9,n,K,(ReplaceLine (x,X,A)))) is Element of the carrier of K
(n,X,K,(ReplaceLine (x,X,A))) . i9 is set
len (mlt (i,N)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg K is finite K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() left_unital doubleLoopStr
the carrier of x is non empty non trivial set
the carrier of x * is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
b is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of x
Line (b,A) is Relation-like NAT -defined the carrier of x -valued Function-like finite width b -element FinSequence-like FinSubsequence-like Element of (width b) -tuples_on the carrier of x
width b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width b) -tuples_on the carrier of x is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
{ b₁ where b₁ is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of x * : len b₁ = width b } is set
(K,x,b) is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of x
(K,x,b) @ is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of x
Col (((K,x,b) @),n) is Relation-like NAT -defined the carrier of x -valued Function-like finite len ((K,x,b) @) -element FinSequence-like FinSubsequence-like Element of (len ((K,x,b) @)) -tuples_on the carrier of x
len ((K,x,b) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len ((K,x,b) @)) -tuples_on the carrier of x is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
{ b₁ where b₁ is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of x * : len b₁ = len ((K,x,b) @) } is set
(Line (b,A)) "*" (Col (((K,x,b) @),n)) is Element of the carrier of x
mlt ((Line (b,A)),(Col (((K,x,b) @),n))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V14([: the carrier of x, the carrier of x:]) quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is Relation-like set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is cup-closed diff-closed preBoolean set
K300( the carrier of x, the carrier of x, the carrier of x, the multF of x,(Line (b,A)),(Col (((K,x,b) @),n))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
Sum (mlt ((Line (b,A)),(Col (((K,x,b) @),n)))) is Element of the carrier of x
the addF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V14([: the carrier of x, the carrier of x:]) quasi_total commutative associative Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
the addF of x $$ (mlt ((Line (b,A)),(Col (((K,x,b) @),n)))) is Element of the carrier of x
ReplaceLine (b,n,(Line (b,A))) is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of x
Det (ReplaceLine (b,n,(Line (b,A)))) is Element of the carrier of x
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 (ReplaceLine (b,n,(Line (b,A)))) is Relation-like Permutations K -defined the carrier of x -valued Function-like non empty V14( Permutations K) quasi_total Element of bool [:(Permutations K), the carrier of x:]
[:(Permutations K), the carrier of x:] is Relation-like set
bool [:(Permutations K), the carrier of x:] is cup-closed diff-closed preBoolean set
the addF of x $$ ((FinOmega (Permutations K)),(Path_product (ReplaceLine (b,n,(Line (b,A)))))) is Element of the carrier of x
len (K,x,b) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
dom (K,x,b) is finite Element of bool NAT
Line ((K,x,b),n) is Relation-like NAT -defined the carrier of x -valued Function-like finite width (K,x,b) -element FinSequence-like FinSubsequence-like Element of (width (K,x,b)) -tuples_on the carrier of x
width (K,x,b) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width (K,x,b)) -tuples_on the carrier of x is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
{ b₁ where b₁ is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of x * : len b₁ = width (K,x,b) } is set
len (Line (b,A)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,n,x,(ReplaceLine (b,n,(Line (b,A))))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
Sum (K,n,x,(ReplaceLine (b,n,(Line (b,A))))) is Element of the carrier of x
the addF of x $$ (K,n,x,(ReplaceLine (b,n,(Line (b,A))))) is Element of the carrier of x
mlt ((Col (((K,x,b) @),n)),(Line (b,A))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
K300( the carrier of x, the carrier of x, the carrier of x, the multF of x,(Col (((K,x,b) @),n)),(Line (b,A))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
Sum (mlt ((Col (((K,x,b) @),n)),(Line (b,A)))) is Element of the carrier of x
the addF of x $$ (mlt ((Col (((K,x,b) @),n)),(Line (b,A)))) is Element of the carrier of x
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 V52(K) Element of the carrier of K
1. (K,n) 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
Det A 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is 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 V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product A)) is Element of the carrier of K
(n,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
(n,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
((Det A) ") * ((n,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
A * (((Det A) ") * ((n,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
Indices (A * (((Det A) ") * ((n,K,A) @))) is set
dom (A * (((Det A) ") * ((n,K,A) @))) is finite Element of bool NAT
width (A * (((Det A) ") * ((n,K,A) @))) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width (A * (((Det A) ") * ((n,K,A) @)))) is finite width (A * (((Det A) ") * ((n,K,A) @))) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (A * (((Det A) ") * ((n,K,A) @))) ) } is set
[:(dom (A * (((Det A) ") * ((n,K,A) @)))),(Seg (width (A * (((Det A) ") * ((n,K,A) @))))):] is Relation-like finite set
Indices (1. (K,n)) is set
dom (1. (K,n)) is finite Element of bool NAT
width (1. (K,n)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width (1. (K,n))) is finite width (1. (K,n)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (K,n)) ) } is set
[:(dom (1. (K,n))),(Seg (width (1. (K,n)))):] is Relation-like finite set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[i,j] is set
{i,j} is non empty finite V37() set
{i} is non empty trivial finite V37() 1 -element set
{{i,j},{i}} is non empty finite V37() set
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
N -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 = N } is set
Col (((n,K,A) @),j) is Relation-like NAT -defined the carrier of K -valued Function-like finite len ((n,K,A) @) -element FinSequence-like FinSubsequence-like Element of (len ((n,K,A) @)) -tuples_on the carrier of K
len ((n,K,A) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len ((n,K,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 = len ((n,K,A) @) } is set
Line (A,i) 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 V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative 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
len (((Det A) ") * ((n,K,A) @)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
width ((n,K,A) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Col ((((Det A) ") * ((n,K,A) @)),j) is Relation-like NAT -defined the carrier of K -valued Function-like finite len (((Det A) ") * ((n,K,A) @)) -element FinSequence-like FinSubsequence-like Element of (len (((Det A) ") * ((n,K,A) @))) -tuples_on the carrier of K
(len (((Det A) ") * ((n,K,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 = len (((Det A) ") * ((n,K,A) @)) } is set
COL is Relation-like NAT -defined the carrier of K -valued Function-like finite N -element FinSequence-like FinSubsequence-like Element of N -tuples_on the carrier of K
((Det A) ") * COL is Relation-like NAT -defined the carrier of K -valued Function-like finite N -element FinSequence-like FinSubsequence-like Element of N -tuples_on the carrier of K
((Det A) ") multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty V14( the carrier of K) quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is 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 V14([: the carrier of K, the carrier of K:]) 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 Function-like one-to-one non empty V14( the carrier of K) quasi_total onto bijective V112() V114() V115() V119() Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,((Det A) "),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty V14( the carrier of K) quasi_total Element of bool [: the carrier of K, the carrier of K:]
K303( the carrier of K, the carrier of K,COL,(((Det A) ") multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
mlt ((Line (A,i)),(Col ((((Det A) ") * ((n,K,A) @)),j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line (A,i)),(Col ((((Det A) ") * ((n,K,A) @)),j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
L is Relation-like NAT -defined the carrier of K -valued Function-like finite N -element FinSequence-like FinSubsequence-like Element of N -tuples_on the carrier of K
mlt (L,COL) is Relation-like NAT -defined the carrier of K -valued Function-like finite N -element FinSequence-like FinSubsequence-like Element of N -tuples_on the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,L,COL) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((Det A) ") * (mlt (L,COL)) is Relation-like NAT -defined the carrier of K -valued Function-like finite N -element FinSequence-like FinSubsequence-like Element of N -tuples_on the carrier of K
K303( the carrier of K, the carrier of K,(mlt (L,COL)),(((Det A) ") multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(Line (A,i)) "*" (Col ((((Det A) ") * ((n,K,A) @)),j)) is Element of the carrier of K
Sum (mlt ((Line (A,i)),(Col ((((Det A) ") * ((n,K,A) @)),j)))) is Element of the carrier of K
the addF of K $$ (mlt ((Line (A,i)),(Col ((((Det A) ") * ((n,K,A) @)),j)))) is Element of the carrier of K
(Line (A,i)) "*" (Col (((n,K,A) @),j)) is Element of the carrier of K
mlt ((Line (A,i)),(Col (((n,K,A) @),j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line (A,i)),(Col (((n,K,A) @),j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line (A,i)),(Col (((n,K,A) @),j)))) is Element of the carrier of K
the addF of K $$ (mlt ((Line (A,i)),(Col (((n,K,A) @),j)))) is Element of the carrier of K
((Det A) ") * ((Line (A,i)) "*" (Col (((n,K,A) @),j))) is Element of the carrier of K
the multF of K . (((Det A) "),((Line (A,i)) "*" (Col (((n,K,A) @),j)))) is Element of the carrier of K
j9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
i9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Line (A,i9) 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
ReplaceLine (A,j9,(Line (A,i9))) 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 (ReplaceLine (A,j9,(Line (A,i9)))) is Element of the carrier of K
Path_product (ReplaceLine (A,j9,(Line (A,i9)))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (ReplaceLine (A,j9,(Line (A,i9)))))) is Element of the carrier of K
((Det A) ") * (Det (ReplaceLine (A,j9,(Line (A,i9))))) is Element of the carrier of K
the multF of K . (((Det A) "),(Det (ReplaceLine (A,j9,(Line (A,i9)))))) is Element of the carrier of K
(A * (((Det A) ") * ((n,K,A) @))) * (i,j) is Element of the carrier of K
ReplaceLine (A,j,(Line (A,i))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (ReplaceLine (A,j,(Line (A,i)))) is Element of the carrier of K
Path_product (ReplaceLine (A,j,(Line (A,i)))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (ReplaceLine (A,j,(Line (A,i)))))) is Element of the carrier of K
((Det A) ") * (Det (ReplaceLine (A,j,(Line (A,i))))) is Element of the carrier of K
the multF of K . (((Det A) "),(Det (ReplaceLine (A,j,(Line (A,i)))))) is Element of the carrier of K
(Det A) * ((Det A) ") is Element of the carrier of K
the multF of K . ((Det A),((Det A) ")) is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V52(K) Element of the carrier of K
(1. (K,n)) * (i,j) is Element of the carrier of K
(1. (K,n)) * (i,j) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
(n,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
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
x is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Col ((n,K,A),b) is Relation-like NAT -defined the carrier of K -valued Function-like finite len (n,K,A) -element FinSequence-like FinSubsequence-like Element of (len (n,K,A)) -tuples_on the carrier of K
len (n,K,A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len (n,K,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 = len (n,K,A) } is set
mlt ((Col ((n,K,A),b)),x) 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 is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is cup-closed diff-closed preBoolean set
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Col ((n,K,A),b)),x) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
ReplaceLine ((A @),b,x) 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,b,K,(ReplaceLine ((A @),b,x))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
X -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 = X } is set
N is Relation-like NAT -defined the carrier of K -valued Function-like finite X -element FinSequence-like FinSubsequence-like Element of X -tuples_on the carrier of K
j is Relation-like NAT -defined the carrier of K -valued Function-like finite X -element FinSequence-like FinSubsequence-like Element of X -tuples_on the carrier of K
mlt (N,j) is Relation-like NAT -defined the carrier of K -valued Function-like finite X -element FinSequence-like FinSubsequence-like Element of X -tuples_on the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,N,j) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len (n,b,K,(ReplaceLine ((A @),b,x))) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Indices (A @) is set
dom (A @) is finite Element of bool NAT
width (A @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width (A @)) is finite width (A @) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (A @) ) } is set
[:(dom (A @)),(Seg (width (A @))):] is Relation-like finite set
[:(Seg n),(Seg n):] is Relation-like finite set
dom (n,b,K,(ReplaceLine ((A @),b,x))) is finite Element of bool NAT
L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(b,L,n,K,(A @)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(L,b,n,K,A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
(L,b,n,K,A) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
(b,L,n,K,(A @)) is Element of the carrier of K
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like V14([: the carrier of K,NAT:]) quasi_total Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ K is Element of the carrier of K
1. K is V52(K) Element of the carrier of K
- (1_ K) is Element of the carrier of K
b + L is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,(power K),(- (1_ K)),(b + L)) is Element of the carrier of K
(b,L,n,K,(A @)) is Element of the carrier of K
Det (b,L,n,K,(A @)) is Element of the carrier of K
Permutations (n -' 1) 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 V14([: the carrier of K, the carrier of K:]) quasi_total commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (b,L,n,K,(A @)) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
[:(Permutations (n -' 1)), the carrier of K:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (b,L,n,K,(A @)))) is Element of the carrier of K
(K,(power K),(- (1_ K)),(b + L)) * (b,L,n,K,(A @)) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(b + L)),(b,L,n,K,(A @))) is Element of the carrier of K
(L,b,n,K,A) is Element of the carrier of K
L + b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,(power K),(- (1_ K)),(L + b)) is Element of the carrier of K
(L,b,n,K,A) is Element of the carrier of K
Det (L,b,n,K,A) is Element of the carrier of K
Path_product (L,b,n,K,A) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (L,b,n,K,A))) is Element of the carrier of K
(K,(power K),(- (1_ K)),(L + b)) * (L,b,n,K,A) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(L + b)),(L,b,n,K,A)) is Element of the carrier of K
Indices (n,K,A) is set
dom (n,K,A) is finite Element of bool NAT
width (n,K,A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width (n,K,A)) is finite width (n,K,A) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (n,K,A) ) } is set
[:(dom (n,K,A)),(Seg (width (n,K,A))):] is Relation-like finite set
[L,b] is set
{L,b} is non empty finite V37() set
{L} is non empty trivial finite V37() 1 -element set
{{L,b},{L}} is non empty finite V37() set
(n,K,A) * (L,b) is Element of the carrier of K
(Col ((n,K,A),b)) . L is set
Indices A is set
dom A is finite Element of bool NAT
width A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width A) is finite width A -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width A ) } is set
[:(dom A),(Seg (width A)):] is Relation-like finite set
[b,L] is set
{b,L} is non empty finite V37() set
{b} is non empty trivial finite V37() 1 -element set
{{b,L},{b}} is non empty finite V37() set
(ReplaceLine ((A @),b,x)) * (b,L) is Element of the carrier of K
x . L is set
(mlt (N,j)) . L is set
(L,b,n,K,A) * ((ReplaceLine ((A @),b,x)) * (b,L)) is Element of the carrier of K
the multF of K . ((L,b,n,K,A),((ReplaceLine ((A @),b,x)) * (b,L))) is Element of the carrier of K
(b,L,n,K,(ReplaceLine ((A @),b,x))) is Element of the carrier of K
(b,L,n,K,(ReplaceLine ((A @),b,x))) is Element of the carrier of K
(b,L,n,K,(ReplaceLine ((A @),b,x))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det (b,L,n,K,(ReplaceLine ((A @),b,x))) is Element of the carrier of K
Path_product (b,L,n,K,(ReplaceLine ((A @),b,x))) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (b,L,n,K,(ReplaceLine ((A @),b,x))))) is Element of the carrier of K
(K,(power K),(- (1_ K)),(b + L)) * (b,L,n,K,(ReplaceLine ((A @),b,x))) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(b + L)),(b,L,n,K,(ReplaceLine ((A @),b,x)))) is Element of the carrier of K
(n,b,K,(ReplaceLine ((A @),b,x))) . L is set
len (mlt (N,j)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg K is finite K -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= K ) } is set
A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() left_unital doubleLoopStr
the carrier of x is non empty non trivial set
the carrier of x * is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
b is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of x
(K,x,b) is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of x
(K,x,b) @ is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of x
Line (((K,x,b) @),n) is Relation-like NAT -defined the carrier of x -valued Function-like finite width ((K,x,b) @) -element FinSequence-like FinSubsequence-like Element of (width ((K,x,b) @)) -tuples_on the carrier of x
width ((K,x,b) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width ((K,x,b) @)) -tuples_on the carrier of x is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
{ b₁ where b₁ is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of x * : len b₁ = width ((K,x,b) @) } is set
Col (b,A) is Relation-like NAT -defined the carrier of x -valued Function-like finite len b -element FinSequence-like FinSubsequence-like Element of (len b) -tuples_on the carrier of x
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len b) -tuples_on the carrier of x is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
{ b₁ where b₁ is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of x * : len b₁ = len b } is set
(Line (((K,x,b) @),n)) "*" (Col (b,A)) is Element of the carrier of x
mlt ((Line (((K,x,b) @),n)),(Col (b,A))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
the multF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V14([: the carrier of x, the carrier of x:]) quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like set
[:[: the carrier of x, the carrier of x:], the carrier of x:] is Relation-like set
bool [:[: the carrier of x, the carrier of x:], the carrier of x:] is cup-closed diff-closed preBoolean set
K300( the carrier of x, the carrier of x, the carrier of x, the multF of x,(Line (((K,x,b) @),n)),(Col (b,A))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
Sum (mlt ((Line (((K,x,b) @),n)),(Col (b,A)))) is Element of the carrier of x
the addF of x is Relation-like [: the carrier of x, the carrier of x:] -defined the carrier of x -valued Function-like V14([: the carrier of x, the carrier of x:]) quasi_total commutative associative Element of bool [:[: the carrier of x, the carrier of x:], the carrier of x:]
the addF of x $$ (mlt ((Line (((K,x,b) @),n)),(Col (b,A)))) is Element of the carrier of x
b @ is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of x
Line ((b @),A) is Relation-like NAT -defined the carrier of x -valued Function-like finite width (b @) -element FinSequence-like FinSubsequence-like Element of (width (b @)) -tuples_on the carrier of x
width (b @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width (b @)) -tuples_on the carrier of x is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
{ b₁ where b₁ is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of x * : len b₁ = width (b @) } is set
ReplaceLine ((b @),n,(Line ((b @),A))) is Relation-like NAT -defined the carrier of x * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of K,K, the carrier of x
Det (ReplaceLine ((b @),n,(Line ((b @),A)))) is Element of the carrier of x
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 (ReplaceLine ((b @),n,(Line ((b @),A)))) is Relation-like Permutations K -defined the carrier of x -valued Function-like non empty V14( Permutations K) quasi_total Element of bool [:(Permutations K), the carrier of x:]
[:(Permutations K), the carrier of x:] is Relation-like set
bool [:(Permutations K), the carrier of x:] is cup-closed diff-closed preBoolean set
the addF of x $$ ((FinOmega (Permutations K)),(Path_product (ReplaceLine ((b @),n,(Line ((b @),A)))))) is Element of the carrier of x
width (K,x,b) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len (Line ((b @),A)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,n,x,(ReplaceLine ((b @),n,(Line ((b @),A))))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
Sum (K,n,x,(ReplaceLine ((b @),n,(Line ((b @),A))))) is Element of the carrier of x
the addF of x $$ (K,n,x,(ReplaceLine ((b @),n,(Line ((b @),A))))) is Element of the carrier of x
Col ((K,x,b),n) is Relation-like NAT -defined the carrier of x -valued Function-like finite len (K,x,b) -element FinSequence-like FinSubsequence-like Element of (len (K,x,b)) -tuples_on the carrier of x
len (K,x,b) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len (K,x,b)) -tuples_on the carrier of x is functional non empty FinSequence-membered FinSequenceSet of the carrier of x
{ b₁ where b₁ is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of x * : len b₁ = len (K,x,b) } is set
mlt ((Col ((K,x,b),n)),(Line ((b @),A))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
K300( the carrier of x, the carrier of x, the carrier of x, the multF of x,(Col ((K,x,b),n)),(Line ((b @),A))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
Sum (mlt ((Col ((K,x,b),n)),(Line ((b @),A)))) is Element of the carrier of x
the addF of x $$ (mlt ((Col ((K,x,b),n)),(Line ((b @),A)))) is Element of the carrier of x
(Line (((K,x,b) @),n)) "*" (Line ((b @),A)) is Element of the carrier of x
mlt ((Line (((K,x,b) @),n)),(Line ((b @),A))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
K300( the carrier of x, the carrier of x, the carrier of x, the multF of x,(Line (((K,x,b) @),n)),(Line ((b @),A))) is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of x
Sum (mlt ((Line (((K,x,b) @),n)),(Line ((b @),A)))) is Element of the carrier of x
the addF of x $$ (mlt ((Line (((K,x,b) @),n)),(Line ((b @),A)))) is Element of the carrier of x
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 V52(K) Element of the carrier of K
1. (K,n) 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
Det A 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is 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 V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product A)) is Element of the carrier of K
(n,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
(n,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
((Det A) ") * ((n,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) ") * ((n,K,A) @)) * 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
Indices ((((Det A) ") * ((n,K,A) @)) * A) is set
dom ((((Det A) ") * ((n,K,A) @)) * A) is finite Element of bool NAT
width ((((Det A) ") * ((n,K,A) @)) * A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width ((((Det A) ") * ((n,K,A) @)) * A)) is finite width ((((Det A) ") * ((n,K,A) @)) * A) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ((((Det A) ") * ((n,K,A) @)) * A) ) } is set
[:(dom ((((Det A) ") * ((n,K,A) @)) * A)),(Seg (width ((((Det A) ") * ((n,K,A) @)) * A))):] is Relation-like finite set
Indices (1. (K,n)) is set
dom (1. (K,n)) is finite Element of bool NAT
width (1. (K,n)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width (1. (K,n))) is finite width (1. (K,n)) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (1. (K,n)) ) } is set
[:(dom (1. (K,n))),(Seg (width (1. (K,n)))):] is Relation-like finite set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[i,j] is set
{i,j} is non empty finite V37() set
{i} is non empty trivial finite V37() 1 -element set
{{i,j},{i}} is non empty finite V37() set
N is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
N -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 = N } is set
Col (A,j) is Relation-like NAT -defined the carrier of K -valued Function-like finite len A -element FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of K
len A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len 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 = len A } is set
Line (((n,K,A) @),i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((n,K,A) @) -element FinSequence-like FinSubsequence-like Element of (width ((n,K,A) @)) -tuples_on the carrier of K
width ((n,K,A) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width ((n,K,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 ((n,K,A) @) } is set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg n),(Seg n):] is Relation-like finite set
len ((n,K,A) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Line ((((Det A) ") * ((n,K,A) @)),i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (((Det A) ") * ((n,K,A) @)) -element FinSequence-like FinSubsequence-like Element of (width (((Det A) ") * ((n,K,A) @))) -tuples_on the carrier of K
width (((Det A) ") * ((n,K,A) @)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width (((Det A) ") * ((n,K,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 (((Det A) ") * ((n,K,A) @)) } is set
L is Relation-like NAT -defined the carrier of K -valued Function-like finite N -element FinSequence-like FinSubsequence-like Element of N -tuples_on the carrier of K
((Det A) ") * L is Relation-like NAT -defined the carrier of K -valued Function-like finite N -element FinSequence-like FinSubsequence-like Element of N -tuples_on the carrier of K
((Det A) ") multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty V14( the carrier of K) quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is 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 V14([: the carrier of K, the carrier of K:]) 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 Function-like one-to-one non empty V14( the carrier of K) quasi_total onto bijective V112() V114() V115() V119() Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,((Det A) "),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty V14( the carrier of K) quasi_total Element of bool [: the carrier of K, the carrier of K:]
K303( the carrier of K, the carrier of K,L,(((Det A) ") multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
mlt ((Line ((((Det A) ") * ((n,K,A) @)),i)),(Col (A,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line ((((Det A) ") * ((n,K,A) @)),i)),(Col (A,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
COL is Relation-like NAT -defined the carrier of K -valued Function-like finite N -element FinSequence-like FinSubsequence-like Element of N -tuples_on the carrier of K
mlt (L,COL) is Relation-like NAT -defined the carrier of K -valued Function-like finite N -element FinSequence-like FinSubsequence-like Element of N -tuples_on the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,L,COL) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((Det A) ") * (mlt (L,COL)) is Relation-like NAT -defined the carrier of K -valued Function-like finite N -element FinSequence-like FinSubsequence-like Element of N -tuples_on the carrier of K
K303( the carrier of K, the carrier of K,(mlt (L,COL)),(((Det A) ") multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(Line ((((Det A) ") * ((n,K,A) @)),i)) "*" (Col (A,j)) is Element of the carrier of K
Sum (mlt ((Line ((((Det A) ") * ((n,K,A) @)),i)),(Col (A,j)))) is Element of the carrier of K
the addF of K $$ (mlt ((Line ((((Det A) ") * ((n,K,A) @)),i)),(Col (A,j)))) is Element of the carrier of K
(Line (((n,K,A) @),i)) "*" (Col (A,j)) is Element of the carrier of K
mlt ((Line (((n,K,A) @),i)),(Col (A,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line (((n,K,A) @),i)),(Col (A,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line (((n,K,A) @),i)),(Col (A,j)))) is Element of the carrier of K
the addF of K $$ (mlt ((Line (((n,K,A) @),i)),(Col (A,j)))) is Element of the carrier of K
((Det A) ") * ((Line (((n,K,A) @),i)) "*" (Col (A,j))) is Element of the carrier of K
the multF of K . (((Det A) "),((Line (((n,K,A) @),i)) "*" (Col (A,j)))) is Element of the carrier of K
i9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
A @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
j9 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Line ((A @),j9) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (A @) -element FinSequence-like FinSubsequence-like Element of (width (A @)) -tuples_on the carrier of K
width (A @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width (A @)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width (A @) } is set
ReplaceLine ((A @),i9,(Line ((A @),j9))) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (ReplaceLine ((A @),i9,(Line ((A @),j9)))) is Element of the carrier of K
Path_product (ReplaceLine ((A @),i9,(Line ((A @),j9)))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (ReplaceLine ((A @),i9,(Line ((A @),j9)))))) is Element of the carrier of K
((Det A) ") * (Det (ReplaceLine ((A @),i9,(Line ((A @),j9))))) is Element of the carrier of K
the multF of K . (((Det A) "),(Det (ReplaceLine ((A @),i9,(Line ((A @),j9)))))) is Element of the carrier of K
((((Det A) ") * ((n,K,A) @)) * A) * (i,j) is Element of the carrier of K
Line ((A @),j) 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
ReplaceLine ((A @),i,(Line ((A @),j))) 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 (ReplaceLine ((A @),i,(Line ((A @),j)))) is Element of the carrier of K
Path_product (ReplaceLine ((A @),i,(Line ((A @),j)))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (ReplaceLine ((A @),i,(Line ((A @),j)))))) is Element of the carrier of K
((Det A) ") * (Det (ReplaceLine ((A @),i,(Line ((A @),j))))) is Element of the carrier of K
the multF of K . (((Det A) "),(Det (ReplaceLine ((A @),i,(Line ((A @),j)))))) is Element of the carrier of K
Det (A @) is Element of the carrier of K
Path_product (A @) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (A @))) is Element of the carrier of K
((Det A) ") * (Det A) is Element of the carrier of K
the multF of K . (((Det A) "),(Det A)) is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V52(K) Element of the carrier of K
(1. (K,n)) * (i,j) is Element of the carrier of K
(1. (K,n)) * (i,j) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 V52(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
Det A 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is 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 V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product A)) is Element of the carrier of K
b is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
1_ K is Element of the carrier of K
1. K is V52(K) Element of the carrier of K
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
A * b is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
1. (K,n) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Det (1. (K,n)) is Element of the carrier of K
Path_product (1. (K,n)) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (1. (K,n)))) is Element of the carrier of K
1_ K is Element of the carrier of K
1. K is V52(K) 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 V14( Permutations n) 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 V14([: the carrier of K, the carrier of K:]) 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
x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(n,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
(n,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
((Det A) ") * ((n,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
A * (((Det A) ") * ((n,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
1. (K,n) 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) ") * ((n,K,A) @)) * 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
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 V52(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
Det A 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is 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 V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product A)) is 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
(n,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
(n,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
((Det A) ") * ((n,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
A * (((Det A) ") * ((n,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
1. (K,n) 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) ") * ((n,K,A) @)) * 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
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
power K is Relation-like [: the carrier of K,NAT:] -defined the carrier of K -valued Function-like V14([: the carrier of K,NAT:]) quasi_total Element of bool [:[: the carrier of K,NAT:], the carrier of K:]
[: the carrier of K,NAT:] is Relation-like non trivial non finite set
[:[: the carrier of K,NAT:], the carrier of K:] is Relation-like non trivial non finite set
bool [:[: the carrier of K,NAT:], the carrier of K:] is non trivial cup-closed diff-closed preBoolean non finite set
1_ K is Element of the carrier of K
1. K is V52(K) Element of the carrier of K
- (1_ K) is 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
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
Indices (A ~) is set
dom (A ~) is finite Element of bool NAT
width (A ~) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width (A ~)) is finite width (A ~) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (A ~) ) } is set
[:(dom (A ~)),(Seg (width (A ~))):] is Relation-like finite set
Det A 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is 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 V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product A)) is Element of the carrier of K
(Det A) " is Element of the carrier of K
0. K is V52(K) Element of the carrier of K
(n,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
X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[X,B] is set
{X,B} is non empty finite V37() set
{X} is non empty trivial finite V37() 1 -element set
{{X,B},{X}} is non empty finite V37() set
(A ~) * (X,B) is Element of the carrier of K
X + B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,(power K),(- (1_ K)),(X + B)) is Element of the carrier of K
((Det A) ") * (K,(power K),(- (1_ K)),(X + 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 V14([: the carrier of K, the carrier of K:]) 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) "),(K,(power K),(- (1_ K)),(X + B))) is Element of the carrier of K
(B,X,n,K,A) is Element of the carrier of K
n -' 1 is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(B,X,n,K,A) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n -' 1,n -' 1, the carrier of K
Det (B,X,n,K,A) is Element of the carrier of K
Permutations (n -' 1) is non empty permutational set
FinOmega (Permutations (n -' 1)) is finite Element of Fin (Permutations (n -' 1))
Fin (Permutations (n -' 1)) is non empty cup-closed diff-closed preBoolean set
Path_product (B,X,n,K,A) is Relation-like Permutations (n -' 1) -defined the carrier of K -valued Function-like non empty V14( Permutations (n -' 1)) quasi_total Element of bool [:(Permutations (n -' 1)), the carrier of K:]
[:(Permutations (n -' 1)), the carrier of K:] is Relation-like set
bool [:(Permutations (n -' 1)), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations (n -' 1))),(Path_product (B,X,n,K,A))) is Element of the carrier of K
(((Det A) ") * (K,(power K),(- (1_ K)),(X + B))) * (B,X,n,K,A) is Element of the carrier of K
the multF of K . ((((Det A) ") * (K,(power K),(- (1_ K)),(X + B))),(B,X,n,K,A)) is Element of the carrier of K
(n,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
Indices ((n,K,A) @) is set
dom ((n,K,A) @) is finite Element of bool NAT
width ((n,K,A) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width ((n,K,A) @)) is finite width ((n,K,A) @) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ((n,K,A) @) ) } is set
[:(dom ((n,K,A) @)),(Seg (width ((n,K,A) @))):] is Relation-like finite set
[B,X] is set
{B,X} is non empty finite V37() set
{B} is non empty trivial finite V37() 1 -element set
{{B,X},{B}} is non empty finite V37() set
Indices (n,K,A) is set
dom (n,K,A) is finite Element of bool NAT
width (n,K,A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width (n,K,A)) is finite width (n,K,A) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (n,K,A) ) } is set
[:(dom (n,K,A)),(Seg (width (n,K,A))):] is Relation-like finite set
((Det A) ") * ((n,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) ") * ((n,K,A) @)) * (X,B) is Element of the carrier of K
((n,K,A) @) * (X,B) is Element of the carrier of K
((Det A) ") * (((n,K,A) @) * (X,B)) is Element of the carrier of K
the multF of K . (((Det A) "),(((n,K,A) @) * (X,B))) is Element of the carrier of K
(n,K,A) * (B,X) is Element of the carrier of K
((Det A) ") * ((n,K,A) * (B,X)) is Element of the carrier of K
the multF of K . (((Det A) "),((n,K,A) * (B,X))) is Element of the carrier of K
(B,X,n,K,A) is Element of the carrier of K
B + X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
(K,(power K),(- (1_ K)),(B + X)) is Element of the carrier of K
(K,(power K),(- (1_ K)),(B + X)) * (B,X,n,K,A) is Element of the carrier of K
the multF of K . ((K,(power K),(- (1_ K)),(B + X)),(B,X,n,K,A)) is Element of the carrier of K
((Det A) ") * (B,X,n,K,A) is Element of the carrier of K
the multF of K . (((Det A) "),(B,X,n,K,A)) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 V52(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
Det A 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is 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 V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product A)) is 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
(Det A) " is Element of the carrier of K
(A ~) * 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
1. (K,n) 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,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
width (n,K,A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(n,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
len ((n,K,A) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width ((n,K,A) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width (A ~) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
len X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
A * X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
(A ~) * B is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
Indices X is set
dom X is finite Element of bool NAT
width X is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg (width X) is finite width X -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width X ) } is set
[:(dom X),(Seg (width X)):] is Relation-like finite set
len A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len B is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(1. (K,n)) * X is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[i,j] is set
{i,j} is non empty finite V37() set
{i} is non empty trivial finite V37() 1 -element set
{{i,j},{i}} is non empty finite V37() set
X * (i,j) is Element of the carrier of K
Col (B,j) is Relation-like NAT -defined the carrier of K -valued Function-like finite len B -element FinSequence-like FinSubsequence-like Element of (len B) -tuples_on the carrier of K
(len 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 = len B } is set
(i,n,n, the carrier of K,A,(Col (B,j))) 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,n,n, the carrier of K,A,(Col (B,j))) is Element of the carrier of K
Path_product (i,n,n, the carrier of K,A,(Col (B,j))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (i,n,n, the carrier of K,A,(Col (B,j))))) is Element of the carrier of K
((Det A) ") * (Det (i,n,n, the carrier of K,A,(Col (B,j)))) 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 V14([: the carrier of K, the carrier of K:]) 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 (i,n,n, the carrier of K,A,(Col (B,j))))) is Element of the carrier of K
len (Col (B,j)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
[:(Seg n),(Seg (width X)):] is Relation-like finite set
Line ((A ~),i) 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 ~)) -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
(Line ((A ~),i)) "*" (Col (B,j)) is Element of the carrier of K
mlt ((Line ((A ~),i)),(Col (B,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line ((A ~),i)),(Col (B,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line ((A ~),i)),(Col (B,j)))) is Element of the carrier of K
the addF of K $$ (mlt ((Line ((A ~),i)),(Col (B,j)))) is Element of the carrier of K
((Det A) ") * ((n,K,A) @) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of K
Line ((((Det A) ") * ((n,K,A) @)),i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (((Det A) ") * ((n,K,A) @)) -element FinSequence-like FinSubsequence-like Element of (width (((Det A) ") * ((n,K,A) @))) -tuples_on the carrier of K
width (((Det A) ") * ((n,K,A) @)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width (((Det A) ") * ((n,K,A) @))) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width (((Det A) ") * ((n,K,A) @)) } is set
(Line ((((Det A) ") * ((n,K,A) @)),i)) "*" (Col (B,j)) is Element of the carrier of K
mlt ((Line ((((Det A) ") * ((n,K,A) @)),i)),(Col (B,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line ((((Det A) ") * ((n,K,A) @)),i)),(Col (B,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line ((((Det A) ") * ((n,K,A) @)),i)),(Col (B,j)))) is Element of the carrier of K
the addF of K $$ (mlt ((Line ((((Det A) ") * ((n,K,A) @)),i)),(Col (B,j)))) is Element of the carrier of K
Line (((n,K,A) @),i) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((n,K,A) @) -element FinSequence-like FinSubsequence-like Element of (width ((n,K,A) @)) -tuples_on the carrier of K
(width ((n,K,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 ((n,K,A) @) } is set
((Det A) ") * (Line (((n,K,A) @),i)) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ((n,K,A) @) -element FinSequence-like FinSubsequence-like Element of (width ((n,K,A) @)) -tuples_on the carrier of K
((Det A) ") multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty V14( the carrier of K) quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is cup-closed diff-closed preBoolean set
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued Function-like one-to-one non empty V14( the carrier of K) quasi_total onto bijective V112() V114() V115() V119() Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,((Det A) "),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty V14( the carrier of K) quasi_total Element of bool [: the carrier of K, the carrier of K:]
K303( the carrier of K, the carrier of K,(Line (((n,K,A) @),i)),(((Det A) ") multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
(((Det A) ") * (Line (((n,K,A) @),i))) "*" (Col (B,j)) is Element of the carrier of K
mlt ((((Det A) ") * (Line (((n,K,A) @),i))),(Col (B,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(((Det A) ") * (Line (((n,K,A) @),i))),(Col (B,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((((Det A) ") * (Line (((n,K,A) @),i))),(Col (B,j)))) is Element of the carrier of K
the addF of K $$ (mlt ((((Det A) ") * (Line (((n,K,A) @),i))),(Col (B,j)))) is Element of the carrier of K
mlt ((Line (((n,K,A) @),i)),(Col (B,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line (((n,K,A) @),i)),(Col (B,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((Det A) ") * (mlt ((Line (((n,K,A) @),i)),(Col (B,j)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K303( the carrier of K, the carrier of K,(mlt ((Line (((n,K,A) @),i)),(Col (B,j)))),(((Det A) ") multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (((Det A) ") * (mlt ((Line (((n,K,A) @),i)),(Col (B,j))))) is Element of the carrier of K
the addF of K $$ (((Det A) ") * (mlt ((Line (((n,K,A) @),i)),(Col (B,j))))) is Element of the carrier of K
(Line (((n,K,A) @),i)) "*" (Col (B,j)) is Element of the carrier of K
Sum (mlt ((Line (((n,K,A) @),i)),(Col (B,j)))) is Element of the carrier of K
the addF of K $$ (mlt ((Line (((n,K,A) @),i)),(Col (B,j)))) is Element of the carrier of K
((Det A) ") * ((Line (((n,K,A) @),i)) "*" (Col (B,j))) is Element of the carrier of K
the multF of K . (((Det A) "),((Line (((n,K,A) @),i)) "*" (Col (B,j)))) is Element of the carrier of K
Col ((n,K,A),i) is Relation-like NAT -defined the carrier of K -valued Function-like finite len (n,K,A) -element FinSequence-like FinSubsequence-like Element of (len (n,K,A)) -tuples_on the carrier of K
len (n,K,A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len (n,K,A)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len (n,K,A) } is set
(Col ((n,K,A),i)) "*" (Col (B,j)) is Element of the carrier of K
mlt ((Col ((n,K,A),i)),(Col (B,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Col ((n,K,A),i)),(Col (B,j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Col ((n,K,A),i)),(Col (B,j)))) is Element of the carrier of K
the addF of K $$ (mlt ((Col ((n,K,A),i)),(Col (B,j)))) is Element of the carrier of K
((Det A) ") * ((Col ((n,K,A),i)) "*" (Col (B,j))) is Element of the carrier of K
the multF of K . (((Det A) "),((Col ((n,K,A),i)) "*" (Col (B,j)))) is 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
ReplaceLine ((A @),i,(Col (B,j))) 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,i,K,(ReplaceLine ((A @),i,(Col (B,j))))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (n,i,K,(ReplaceLine ((A @),i,(Col (B,j))))) is Element of the carrier of K
the addF of K $$ (n,i,K,(ReplaceLine ((A @),i,(Col (B,j))))) is Element of the carrier of K
((Det A) ") * (Sum (n,i,K,(ReplaceLine ((A @),i,(Col (B,j)))))) is Element of the carrier of K
the multF of K . (((Det A) "),(Sum (n,i,K,(ReplaceLine ((A @),i,(Col (B,j))))))) is Element of the carrier of K
Det (ReplaceLine ((A @),i,(Col (B,j)))) is Element of the carrier of K
Path_product (ReplaceLine ((A @),i,(Col (B,j)))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (ReplaceLine ((A @),i,(Col (B,j)))))) is Element of the carrier of K
((Det A) ") * (Det (ReplaceLine ((A @),i,(Col (B,j))))) is Element of the carrier of K
the multF of K . (((Det A) "),(Det (ReplaceLine ((A @),i,(Col (B,j)))))) is Element of the carrier of K
(ReplaceLine ((A @),i,(Col (B,j)))) @ 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 ((ReplaceLine ((A @),i,(Col (B,j)))) @) is Element of the carrier of K
Path_product ((ReplaceLine ((A @),i,(Col (B,j)))) @) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product ((ReplaceLine ((A @),i,(Col (B,j)))) @))) is Element of the carrier of K
((Det A) ") * (Det ((ReplaceLine ((A @),i,(Col (B,j)))) @)) is Element of the carrier of K
the multF of K . (((Det A) "),(Det ((ReplaceLine ((A @),i,(Col (B,j)))) @))) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 V52(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
Det A 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is 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 V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product A)) is 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
(Det A) " is Element of the carrier of K
A * (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
1. (K,n) 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 A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
width x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
b is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
b * (A ~) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
Indices x is set
dom x is finite Element of bool NAT
Seg (width x) is finite width x -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width x ) } is set
[:(dom x),(Seg (width x)):] is Relation-like finite set
len A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(n,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
(n,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
len ((n,K,A) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
width ((n,K,A) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len (n,K,A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
dom (n,K,A) is finite Element of bool NAT
len (A ~) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
x * (1. (K,n)) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Line (b,i) 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
j is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
[i,j] is set
{i,j} is non empty finite V37() set
{i} is non empty trivial finite V37() 1 -element set
{{i,j},{i}} is non empty finite V37() set
x * (i,j) is Element of the carrier of K
ReplaceLine (A,j,(Line (b,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 (ReplaceLine (A,j,(Line (b,i)))) is Element of the carrier of K
Path_product (ReplaceLine (A,j,(Line (b,i)))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (ReplaceLine (A,j,(Line (b,i)))))) is Element of the carrier of K
((Det A) ") * (Det (ReplaceLine (A,j,(Line (b,i))))) is Element of the carrier of K
the multF of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V14([: the carrier of K, the carrier of K:]) quasi_total having_a_unity commutative associative Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the multF of K . (((Det A) "),(Det (ReplaceLine (A,j,(Line (b,i)))))) is Element of the carrier of K
len (Line (b,i)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Col ((A ~),j) is Relation-like NAT -defined the carrier of K -valued Function-like finite len (A ~) -element FinSequence-like FinSubsequence-like Element of (len (A ~)) -tuples_on the carrier of K
(len (A ~)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = len (A ~) } is set
(Line (b,i)) "*" (Col ((A ~),j)) is Element of the carrier of K
mlt ((Line (b,i)),(Col ((A ~),j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line (b,i)),(Col ((A ~),j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line (b,i)),(Col ((A ~),j)))) is Element of the carrier of K
the addF of K $$ (mlt ((Line (b,i)),(Col ((A ~),j)))) is Element of the carrier of K
((Det A) ") * ((n,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
Col ((((Det A) ") * ((n,K,A) @)),j) is Relation-like NAT -defined the carrier of K -valued Function-like finite len (((Det A) ") * ((n,K,A) @)) -element FinSequence-like FinSubsequence-like Element of (len (((Det A) ") * ((n,K,A) @))) -tuples_on the carrier of K
len (((Det A) ") * ((n,K,A) @)) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len (((Det A) ") * ((n,K,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 = len (((Det A) ") * ((n,K,A) @)) } is set
(Line (b,i)) "*" (Col ((((Det A) ") * ((n,K,A) @)),j)) is Element of the carrier of K
mlt ((Line (b,i)),(Col ((((Det A) ") * ((n,K,A) @)),j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line (b,i)),(Col ((((Det A) ") * ((n,K,A) @)),j))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line (b,i)),(Col ((((Det A) ") * ((n,K,A) @)),j)))) is Element of the carrier of K
the addF of K $$ (mlt ((Line (b,i)),(Col ((((Det A) ") * ((n,K,A) @)),j)))) is Element of the carrier of K
Col (((n,K,A) @),j) is Relation-like NAT -defined the carrier of K -valued Function-like finite len ((n,K,A) @) -element FinSequence-like FinSubsequence-like Element of (len ((n,K,A) @)) -tuples_on the carrier of K
(len ((n,K,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 = len ((n,K,A) @) } is set
((Det A) ") * (Col (((n,K,A) @),j)) is Relation-like NAT -defined the carrier of K -valued Function-like finite len ((n,K,A) @) -element FinSequence-like FinSubsequence-like Element of (len ((n,K,A) @)) -tuples_on the carrier of K
((Det A) ") multfield is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty V14( the carrier of K) quasi_total Element of bool [: the carrier of K, the carrier of K:]
bool [: the carrier of K, the carrier of K:] is cup-closed diff-closed preBoolean set
id the carrier of K is Relation-like the carrier of K -defined the carrier of K -valued Function-like one-to-one non empty V14( the carrier of K) quasi_total onto bijective V112() V114() V115() V119() Element of bool [: the carrier of K, the carrier of K:]
K224( the carrier of K, the carrier of K, the multF of K,((Det A) "),(id the carrier of K)) is Relation-like the carrier of K -defined the carrier of K -valued Function-like non empty V14( the carrier of K) quasi_total Element of bool [: the carrier of K, the carrier of K:]
K303( the carrier of K, the carrier of K,(Col (((n,K,A) @),j)),(((Det A) ") multfield)) 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,i)) "*" (((Det A) ") * (Col (((n,K,A) @),j))) is Element of the carrier of K
mlt ((Line (b,i)),(((Det A) ") * (Col (((n,K,A) @),j)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line (b,i)),(((Det A) ") * (Col (((n,K,A) @),j)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line (b,i)),(((Det A) ") * (Col (((n,K,A) @),j))))) is Element of the carrier of K
the addF of K $$ (mlt ((Line (b,i)),(((Det A) ") * (Col (((n,K,A) @),j))))) is Element of the carrier of K
(((Det A) ") * (Col (((n,K,A) @),j))) "*" (Line (b,i)) is Element of the carrier of K
mlt ((((Det A) ") * (Col (((n,K,A) @),j))),(Line (b,i))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(((Det A) ") * (Col (((n,K,A) @),j))),(Line (b,i))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((((Det A) ") * (Col (((n,K,A) @),j))),(Line (b,i)))) is Element of the carrier of K
the addF of K $$ (mlt ((((Det A) ") * (Col (((n,K,A) @),j))),(Line (b,i)))) is Element of the carrier of K
mlt ((Col (((n,K,A) @),j)),(Line (b,i))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Col (((n,K,A) @),j)),(Line (b,i))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
((Det A) ") * (mlt ((Col (((n,K,A) @),j)),(Line (b,i)))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K303( the carrier of K, the carrier of K,(mlt ((Col (((n,K,A) @),j)),(Line (b,i)))),(((Det A) ") multfield)) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (((Det A) ") * (mlt ((Col (((n,K,A) @),j)),(Line (b,i))))) is Element of the carrier of K
the addF of K $$ (((Det A) ") * (mlt ((Col (((n,K,A) @),j)),(Line (b,i))))) is Element of the carrier of K
(Col (((n,K,A) @),j)) "*" (Line (b,i)) is Element of the carrier of K
Sum (mlt ((Col (((n,K,A) @),j)),(Line (b,i)))) is Element of the carrier of K
the addF of K $$ (mlt ((Col (((n,K,A) @),j)),(Line (b,i)))) is Element of the carrier of K
((Det A) ") * ((Col (((n,K,A) @),j)) "*" (Line (b,i))) is Element of the carrier of K
the multF of K . (((Det A) "),((Col (((n,K,A) @),j)) "*" (Line (b,i)))) is Element of the carrier of K
Line ((n,K,A),j) is Relation-like NAT -defined the carrier of K -valued Function-like finite width (n,K,A) -element FinSequence-like FinSubsequence-like Element of (width (n,K,A)) -tuples_on the carrier of K
width (n,K,A) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width (n,K,A)) -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
{ b₁ where b₁ is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of K * : len b₁ = width (n,K,A) } is set
(Line ((n,K,A),j)) "*" (Line (b,i)) is Element of the carrier of K
mlt ((Line ((n,K,A),j)),(Line (b,i))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
K300( the carrier of K, the carrier of K, the carrier of K, the multF of K,(Line ((n,K,A),j)),(Line (b,i))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (mlt ((Line ((n,K,A),j)),(Line (b,i)))) is Element of the carrier of K
the addF of K $$ (mlt ((Line ((n,K,A),j)),(Line (b,i)))) is Element of the carrier of K
((Det A) ") * ((Line ((n,K,A),j)) "*" (Line (b,i))) is Element of the carrier of K
the multF of K . (((Det A) "),((Line ((n,K,A),j)) "*" (Line (b,i)))) is Element of the carrier of K
(n,j,K,(ReplaceLine (A,j,(Line (b,i))))) is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
Sum (n,j,K,(ReplaceLine (A,j,(Line (b,i))))) is Element of the carrier of K
the addF of K $$ (n,j,K,(ReplaceLine (A,j,(Line (b,i))))) is Element of the carrier of K
((Det A) ") * (Sum (n,j,K,(ReplaceLine (A,j,(Line (b,i)))))) is Element of the carrier of K
the multF of K . (((Det A) "),(Sum (n,j,K,(ReplaceLine (A,j,(Line (b,i))))))) is Element of the carrier of K
n is non empty set
K is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of n
<*K*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,K] is set
{1,K} is non empty finite V37() set
{{1,K},{1}} is non empty finite V37() set
{[1,K]} is Relation-like Function-like constant non empty trivial finite 1 -element set
n * is functional non empty FinSequence-membered FinSequenceSet of n
len K is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
n is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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
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 *
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 carrier of n,A) is Relation-like NAT -defined the carrier of n * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular Matrix of 1, len A, the carrier of n
len A is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
[1,A] is set
{1,A} is non empty finite V37() set
{{1,A},{1}} is non empty finite V37() set
{[1,A]} is Relation-like Function-like constant non empty trivial finite 1 -element set
( the carrier of n,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 *
K * (( the carrier of n,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 *
( the carrier of n,A) * 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 *
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 V52(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
Det A 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is 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 V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product A)) is 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
(Det A) " is Element of the carrier of K
x is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,A,x) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
( the carrier of K,x) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular Matrix of 1, len x, the carrier of K
[1,x] is set
{1,x} is non empty finite V37() set
{{1,x},{1}} is non empty finite V37() set
{[1,x]} is Relation-like Function-like constant non empty trivial finite 1 -element set
( the carrier of K,x) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
A * (( the carrier of K,x) @) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
b is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
( the carrier of K,b) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular Matrix of 1, len b, the carrier of K
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
[1,b] is set
{1,b} is non empty finite V37() set
{{1,b},{1}} is non empty finite V37() set
{[1,b]} is Relation-like Function-like constant non empty trivial finite 1 -element set
( the carrier of K,b) @ is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
(K,(A ~),b) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
(A ~) * (( the carrier of K,b) @) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
len ( the carrier of K,x) is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
width ( the carrier of K,x) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
len (( the carrier of K,x) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Line (( the carrier of K,x),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,x) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,x)) -tuples_on the carrier of K
(width ( the carrier of K,x)) -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 ( the carrier of K,x) } is set
( the carrier of K,x) . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len ( the carrier of K,b) is non empty V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real positive non negative Element of NAT
dom ( the carrier of K,b) is non empty trivial finite 1 -element Element of bool NAT
Line (( the carrier of K,b),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,b) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,b)) -tuples_on the carrier of K
width ( the carrier of K,b) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width ( the carrier of K,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 ( the carrier of K,b) } is set
( the carrier of K,b) . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x . i is set
(i,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 (i,n,n, the carrier of K,A,b) is Element of the carrier of K
Path_product (i,n,n, the carrier of K,A,b) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (i,n,n, the carrier of K,A,b))) is Element of the carrier of K
((Det A) ") * (Det (i,n,n, the carrier of K,A,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 V14([: the carrier of K, the carrier of K:]) 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 (i,n,n, the carrier of K,A,b))) is Element of the carrier of K
width (( the carrier of K,x) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
Indices (( the carrier of K,x) @) is set
dom (( the carrier of K,x) @) is finite Element of bool NAT
Seg (width (( the carrier of K,x) @)) is finite width (( the carrier of K,x) @) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width (( the carrier of K,x) @) ) } is set
[:(dom (( the carrier of K,x) @)),(Seg (width (( the carrier of K,x) @))):] is Relation-like finite set
[:(Seg n),(Seg 1):] is Relation-like finite set
[i,1] is set
{i,1} is non empty finite V37() set
{i} is non empty trivial finite V37() 1 -element set
{{i,1},{i}} is non empty finite V37() set
[1,i] is set
{1,i} is non empty finite V37() set
{{1,i},{1}} is non empty finite V37() set
Indices ( the carrier of K,x) is set
dom ( the carrier of K,x) is non empty trivial finite 1 -element Element of bool NAT
Seg (width ( the carrier of K,x)) is finite width ( the carrier of K,x) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,x) ) } is set
[:(dom ( the carrier of K,x)),(Seg (width ( the carrier of K,x))):] is Relation-like finite set
(( the carrier of K,x) @) * (i,1) is Element of the carrier of K
( the carrier of K,x) * (1,i) is Element of the carrier of K
(Line (( the carrier of K,x),1)) . i is set
Col ((( the carrier of K,b) @),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite len (( the carrier of K,b) @) -element FinSequence-like FinSubsequence-like Element of (len (( the carrier of K,b) @)) -tuples_on the carrier of K
len (( the carrier of K,b) @) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(len (( the carrier of K,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 = len (( the carrier of K,b) @) } is set
(i,n,n, the carrier of K,A,(Col ((( the carrier of K,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 (i,n,n, the carrier of K,A,(Col ((( the carrier of K,b) @),1))) is Element of the carrier of K
Path_product (i,n,n, the carrier of K,A,(Col ((( the carrier of K,b) @),1))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (i,n,n, the carrier of K,A,(Col ((( the carrier of K,b) @),1))))) is Element of the carrier of K
((Det A) ") * (Det (i,n,n, the carrier of K,A,(Col ((( the carrier of K,b) @),1)))) is Element of the carrier of K
the multF of K . (((Det A) "),(Det (i,n,n, the carrier of K,A,(Col ((( the carrier of K,b) @),1))))) is Element of the carrier of K
(i,n,n, the carrier of K,A,(Line (( the carrier of K,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 (i,n,n, the carrier of K,A,(Line (( the carrier of K,b),1))) is Element of the carrier of K
Path_product (i,n,n, the carrier of K,A,(Line (( the carrier of K,b),1))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (i,n,n, the carrier of K,A,(Line (( the carrier of K,b),1))))) is Element of the carrier of K
((Det A) ") * (Det (i,n,n, the carrier of K,A,(Line (( the carrier of K,b),1)))) is Element of the carrier of K
the multF of K . (((Det A) "),(Det (i,n,n, the carrier of K,A,(Line (( the carrier of K,b),1))))) is Element of the carrier of K
n is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
Seg n is finite n -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
K is non empty non degenerated non trivial right_complementable almost_left_invertible V95() unital associative commutative V132() V133() V134() right-distributive left-distributive right_unital well-unital V146() 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 V52(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
Det A 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 V14([: the carrier of K, the carrier of K:]) 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 set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is 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 V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
[:(Permutations n), the carrier of K:] is Relation-like set
bool [:(Permutations n), the carrier of K:] is cup-closed diff-closed preBoolean set
the addF of K $$ ((FinOmega (Permutations n)),(Path_product A)) is 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
(Det A) " is Element of the carrier of K
x is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
len x is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(K,A,x) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
( the carrier of K,x) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular Matrix of 1, len x, the carrier of K
[1,x] is set
{1,x} is non empty finite V37() set
{{1,x},{1}} is non empty finite V37() set
{[1,x]} is Relation-like Function-like constant non empty trivial finite 1 -element set
( the carrier of K,x) * A is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
b is Relation-like NAT -defined the carrier of K -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of K
( the carrier of K,b) is Relation-like NAT -defined the carrier of K * -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like tabular Matrix of 1, len b, the carrier of K
len b is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
[1,b] is set
{1,b} is non empty finite V37() set
{{1,b},{1}} is non empty finite V37() set
{[1,b]} is Relation-like Function-like constant non empty trivial finite 1 -element set
(K,(A ~),b) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
( the carrier of K,b) * (A ~) is Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like tabular FinSequence of the carrier of K *
width ( the carrier of K,x) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
[:(Seg 1),(Seg n):] is Relation-like finite set
Indices ( the carrier of K,x) is set
dom ( the carrier of K,x) is non empty trivial finite 1 -element Element of bool NAT
Seg (width ( the carrier of K,x)) is finite width ( the carrier of K,x) -element Element of bool NAT
{ b₁ where b₁ is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= width ( the carrier of K,x) ) } is set
[:(dom ( the carrier of K,x)),(Seg (width ( the carrier of K,x))):] is Relation-like finite set
Line (( the carrier of K,x),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,x) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,x)) -tuples_on the carrier of K
(width ( the carrier of K,x)) -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 ( the carrier of K,x) } is set
( the carrier of K,x) . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
i is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative set
x . i is set
ReplaceLine (A,i,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 (ReplaceLine (A,i,b)) is Element of the carrier of K
Path_product (ReplaceLine (A,i,b)) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (ReplaceLine (A,i,b)))) is Element of the carrier of K
((Det A) ") * (Det (ReplaceLine (A,i,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 V14([: the carrier of K, the carrier of K:]) 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 (ReplaceLine (A,i,b)))) is Element of the carrier of K
[1,i] is set
{1,i} is non empty finite V37() set
{{1,i},{1}} is non empty finite V37() set
Line (( the carrier of K,b),1) is Relation-like NAT -defined the carrier of K -valued Function-like finite width ( the carrier of K,b) -element FinSequence-like FinSubsequence-like Element of (width ( the carrier of K,b)) -tuples_on the carrier of K
width ( the carrier of K,b) is V26() V27() V28() V32() finite cardinal V108() V109() V110() ext-real non negative Element of NAT
(width ( the carrier of K,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 ( the carrier of K,b) } is set
( the carrier of K,b) . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
( the carrier of K,x) * (1,i) is Element of the carrier of K
(Line (( the carrier of K,x),1)) . i is set
ReplaceLine (A,i,(Line (( the carrier of K,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 (ReplaceLine (A,i,(Line (( the carrier of K,b),1)))) is Element of the carrier of K
Path_product (ReplaceLine (A,i,(Line (( the carrier of K,b),1)))) is Relation-like Permutations n -defined the carrier of K -valued Function-like non empty V14( Permutations n) quasi_total Element of bool [:(Permutations n), the carrier of K:]
the addF of K $$ ((FinOmega (Permutations n)),(Path_product (ReplaceLine (A,i,(Line (( the carrier of K,b),1)))))) is Element of the carrier of K
((Det A) ") * (Det (ReplaceLine (A,i,(Line (( the carrier of K,b),1))))) is Element of the carrier of K
the multF of K . (((Det A) "),(Det (ReplaceLine (A,i,(Line (( the carrier of K,b),1)))))) is Element of the carrier of K